From 4f20679dd350207c12479e5097289cdc611515b7 Mon Sep 17 00:00:00 2001 From: Joe Thorley Date: Mon, 21 Oct 2024 07:45:05 -0700 Subject: [PATCH 01/66] dev/ version --- README.Rmd | 2 +- README.md | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/README.Rmd b/README.Rmd index 349018f0..20730b82 100644 --- a/README.Rmd +++ b/README.Rmd @@ -58,7 +58,7 @@ pak::pak("bcgov/ssdtools") #### Website -The website for the development version is at . +The website for the development version is at . ## Introduction diff --git a/README.md b/README.md index edead287..7f94a7b5 100644 --- a/README.md +++ b/README.md @@ -62,7 +62,7 @@ pak::pak("bcgov/ssdtools") #### Website The website for the development version is at -. +. ## Introduction From 89fca04d4b1f6aeaf5b74c0998ca1e298821db88 Mon Sep 17 00:00:00 2001 From: Duncan Kennedy Date: Mon, 2 Dec 2024 09:47:27 -0700 Subject: [PATCH 02/66] upadated docs --- Meta/vignette.rds | Bin 0 -> 431 bytes doc/additional-technical-details.R | 12 + doc/additional-technical-details.Rmd | 221 +++++++ doc/customising-plots.R | 70 +++ doc/customising-plots.Rmd | 137 +++++ doc/distributions.R | 315 ++++++++++ doc/distributions.Rmd | 837 +++++++++++++++++++++++++++ doc/model-averaging.R | 173 ++++++ doc/model-averaging.Rmd | 538 +++++++++++++++++ doc/small-sample-bias.R | 1 + doc/small-sample-bias.pdf | Bin 0 -> 329051 bytes doc/small-sample-bias.pdf.asis | 6 + doc/ssdtools.R | 94 +++ doc/ssdtools.Rmd | 267 +++++++++ 14 files changed, 2671 insertions(+) create mode 100644 Meta/vignette.rds create mode 100644 doc/additional-technical-details.R create mode 100644 doc/additional-technical-details.Rmd create mode 100644 doc/customising-plots.R create mode 100644 doc/customising-plots.Rmd create mode 100644 doc/distributions.R create mode 100644 doc/distributions.Rmd create mode 100644 doc/model-averaging.R create mode 100644 doc/model-averaging.Rmd create mode 100644 doc/small-sample-bias.R create mode 100644 doc/small-sample-bias.pdf create mode 100644 doc/small-sample-bias.pdf.asis create mode 100644 doc/ssdtools.R create mode 100644 doc/ssdtools.Rmd diff --git a/Meta/vignette.rds b/Meta/vignette.rds new file mode 100644 index 0000000000000000000000000000000000000000..d048d773130666ceddad857736f78debf6d6b996 GIT binary patch literal 431 zcmV;g0Z{%QiwFP!000001D#UaP69CyU0_up8ZV%UMoImE-G3l%)I>=L@ZjsXtbP#jsn(QPYpG*jeJ9J?bsFj>Kq*I_ z#2`Fm;UwmvvM58~Li&S<>b&!CDlv(; zx`fBXaM20vUQU~I4Q(nJjG4-+Dz~D;doe^M zUAXsLn7X|Y$F*bis+eDK5U`ZR)F`n2#6IRpru^+jjVzh_rrDADs-NmuEEZoyMxg*{ Z7@1&`0)qdRse0vo{{XcH0taFQ008op)5HJ( literal 0 HcmV?d00001 diff --git a/doc/additional-technical-details.R b/doc/additional-technical-details.R new file mode 100644 index 00000000..78b30508 --- /dev/null +++ b/doc/additional-technical-details.R @@ -0,0 +1,12 @@ +## ----include = FALSE---------------------------------------------------------- +knitr::opts_chunk$set( + collapse = TRUE, + comment = "#>", + fig.width = 6, + fig.height = 4, + echo = TRUE +) + +## ----results = "asis", echo = FALSE------------------------------------------- +cat(ssdtools::ssd_licensing_md()) + diff --git a/doc/additional-technical-details.Rmd b/doc/additional-technical-details.Rmd new file mode 100644 index 00000000..6031b314 --- /dev/null +++ b/doc/additional-technical-details.Rmd @@ -0,0 +1,221 @@ +--- +title: "Additional Technical Details" +author: "ssdtools Team" +date: '`r format(Sys.time(), "%Y-%m-%d", tz = "UTC")`' +bibliography: references.bib +mathfont: Courier +latex_engine: MathJax +output: rmarkdown::html_vignette +vignette: > + %\VignetteIndexEntry{Additional Technical Details} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteEncoding{UTF-8} +--- + + + + + + +```{r, include = FALSE} +knitr::opts_chunk$set( + collapse = TRUE, + comment = "#>", + fig.width = 6, + fig.height = 4, + echo = TRUE +) +``` + +## Small sample bias + +The ssdtools package uses the method of Maximum Likelihood (ML) to estimate parameters for each distribution that is fit to the data. +Statistical theory says that maximum likelihood estimators are asymptotically unbiased, but does not guarantee performance in small samples. +A detailed account of the issue of small sample bias in estimates can be found in the following [pdf](https://github.com/bcgov/ssdtools/blob/master/vignettes/small-sample-bias.pdf). + +## Investigations into setting minimum sample sizes for uni-modal and bi-modal distributions in ssdtools + +Most jurisdictions require a minimum sample size for fitting a valid SSD. +The current Australian and New Zealand minimum is 5, in order to fit the two-parameter log-normal distribution [@warne2018]. +In ssdtools the default minimum sample size is 6 [@thorleysschwartz2018joss], which is consistent with the current methodology for British Columbia [@ccme]. + +Here we report on a series of simulation studies designed to inform a final decision on the default minimum sample size to adopt for both the uni-modal 2 parameter distributions, as well as the bi-modal 5 parameter distributions. + +### Bias and CI coverage and interval width + +#### Simulations based on ssddata + +We used the example datasets in the [`ssddata`](https://github.com/open-AIMS/ssddata) package in R [@ssddata] to undertake a simulation study to examine bias, coverage and confidence interval (CI) widths using the recommended default set of six distributions (lognormal, log-Gumbel, log-logistic, gamma, Weibull, and the lognormal-log-normal mixture), with model averaged estimates obtained using the multi-method, and confidence intervals estimated using the recommended weighted sample bootstrap method (see @fox_methodologies_2024). +A total of 20 unique datasets were extracted from ssddata and used to define the parameters for the simulation study as follows: + + 1. Each dataset was extracted from `ssddata` and fit using the default distribution set as recommended in [@fox_methodologies_2022] and [@fox_methodologies_2024]. + 2. Of the six default distributions, the parameters for the distribution having the highest weight for each dataset was used to generate new random datasets of varying values of N, including (but not limited to): 5 (current ANZG minimum), 6 (current BC minimum), and 8 (current ANZG preferred). + 3. For each randomly generated dataset, `ssdtools` was used to re-fit the data, and model averaged estimates were obtained using the multi-method, with upper and lower confidence limits (CLs) produced using the recommended weighted sample method (see [Confidence Intervals for Hazard Concentrations](https://bcgov.github.io/ssdtools/articles/confidence-intervals.html) vignette). + +The individual ssdtools fits are shown below for each of the 20 simulation datasets from ssddata, for the six recommended default distributions, as well as the model averaged CDF (black line): + +![](images/fitted_dists.png) + +This simulation process was repeated a minimum of 1,000 times for each dataset, and the results collated across all iterations. +For each simulated dataset the true HCx values were obtained directly from the parameter estimates of from data generating distribution. +From these, relative bias was calculated as the scaled-difference between the estimated HCx values and the true HCx value, i.e $$\frac{\widehat {HC}x-HCx}{HCx}$$ where $\widehat {HC}x$ is the estimate of the true value, $HCx$; coverage was calculated as the proportion of simulations where the true $HCx$ value fell within the lower and upper 95% confidence limits; and the scaled confidence interval width was calculated as $$\frac{UL-LL}{HCx}$$ where $UL$ and $LL$ are the upper and lower limits respectively. + +Bias, confidence interval width and coverage as a function of sample size across ~1000 simulations of 20 datasets using the multi model averaging method and the weighted sample method for estimating confidence intervals via ssdtools are shown below: + +![](images/ssdata_sims_collated.png) + +The simulation results showed significant gains in terms of reduced bias from N=5 to N=6, as well as in coverage, which improved substantially between N=5 and N=6. +There is also a small additional gain in coverage at N=7, where the median of the simulations reaches the expected 95% but this is only the case for HC1. + +#### Simulations based on EnviroTox + +In addition to the analysis based on the 20 ssddata example datasets, we also ran an expanded simulation study based on the EnviroTox dataset analysed by Yanagihara et al. (2024). +Combined with the ssddata examples, this includes a total of 353 example datasets to use as case studies. +Using this larger dataset as a basis for simulations, we followed the same procedure as described above to examine relative bias, as well as changes in the AICc weights (see below) for various sample sizes. +Estimates of coverage and confidence interval widths were not obtained for this larger dataset due to the computationally intensive bootstrap method of obtaining confidence intervals. + +Bias of sample size across ~1000 simulations of 353 datasets using the multi model averaging method via ssdtools, for HC5, 10 and 20 (0.05, 0.1, and 0.2) are shown below: + +![](images/all_sims_bias.png) + +Note that simulation results are shown separately for those derived from each of the six distributions as the underlying source data generating distribution. + +The bias results for this larger combined dataset did not show the same level of improvement from N=5 to N=6 as that based on the smaller ssddata simulation. +However, there as a gradual improvement in bias with increasing N. +There is no strong evidence for preferring N=6 over N=7 in the context of bias from either of the simulation studies. +We note that the bias was highest at these small sample sizes for data generated using a gamma distribution, likely reflecting the extreme left-tailed nature of this distribution. + +### AICc based model weights + +Aside from considerations of bias, coverage and confidence interval width, it is also prudent to examine how the weights of the different distributions changed with sample size, for data generated using the six different default distributions, to more fully investigate sample size issues associated with the use of the mixture distributions. +This was examined using the simulation study across the larger combined ssddata and EnviroTox datasets (353 datasets) to ensure a wide range of potential representations of each of the six default distributions was considered. + +Below is a plot of the mean AICc weights as a function of sample size (N) obtained for data simulated using the best fit distribution to 353 datasets. +Results are shown separately for the six different source distributions, with the upper plot (A) showing the AICc weight of the source (data generating) distributions, when fit using the default set of six distributions using ssdtools; and the lower plot (B) showing the AICc weight of the lognormal-lognormal mixture distribution for each of the source (data generating) distributions. + +![](images/weights_collated.png) + +We found that the AICc weights for the five unimodal distributions were relatively similar (~0.2) for very low N. +This is because for small N it is difficult to discern differences between the distributions in the candidate list. +The weights increase to above 0.5 as N increases (i.e. their converge to the true underlying generating distribution at high N, upper plot). +For very low sample sizes (N=5, 6 or 7) the source (data-generating) distribution is not preferentially weighted by the AICc (upper plot, A). + +For the lnorm_lnorm mixture, AICc weights can be >0.5 at relatively N (>8, lower plot, B). +We also looked specifically at the AICc weight of the lognormal-lognormal mixture as one of six distributions in the default candidate set, across simulation based on all six source generating distributions. +This was done to examine the potential for erroneously highly weighting the lognormal-lognormal mixture distribution by chance, when data are generated instead using one of the five unimodal distributions. +For all the unimodal source distributions the lognormal-lognormal mixture never has high AICc weight, even at very high sample sizes (N=256, lower plot, B). +The AICc weights are particularly low for the lnorm_lnorm mixture at low sample sizes for all the uni-modal source distributions (a desirable property) (lower plot, B). + +### Conclusions + +Overall, the results suggest that (relative) bias as a function of N behaves as expected. +The N=6 recommendation appears to be well-supported as coverage is particularly low for N=5, but acceptable for N=6. +The lognormal-lognormal mixture AICc weights suggest that this distribution will only be preferred (i.e. have a high AICc weight) when (i) there is clear evidence of bimodality in the source data; and (ii) large N. + +Based on these results, our recommendation is that only a single minimum sample size of N=6 be adopted (for both unimodal and bimodal), since our results suggest any gains in increasing this to 7 are minimal. + +## The inverse Pareto and inverse Weibull as limiting distributions of the Burr Type-III distribution + +### Burr III distribution + +The probability density function, ${f_X}(x;b,c,k)$ and cumulative distribution function, ${F_X}(x;b,c,k)$ for the Burr III distribution (also known as the *Dagum* distribution) as used in `ssdtools` are + +
+Burr III distribution + +\[\begin{array}{*{20}{c}} +{{f_X}(x;b,c,k) = \frac{{b\,k\,c}}{{{x^2}}}\frac{{{{\left( {\frac{b}{x}} \right)}^{c - 1}}}}{{{{\left[ {1 + {{\left( {\frac{b}{x}} \right)}^c}} \right]}^{k + 1}}}}{\rm{ }}}&{b,c,k,x > 0} +\end{array}\]
\[\begin{array}{*{20}{c}} +{{F_X}(x;b,c,k) = \frac{1}{{{{\left[ {1 + {{\left( {\frac{b}{x}} \right)}^c}} \right]}^k}}}{\rm{ }}}&{b,c,k,x > 0} +\end{array}\] + +
+ +### Inverse Pareto distribution + +Let $X \sim Burr(b,c,k)$ have the *pdf* given in the box above. +It is well known that the distribution of $Y = \frac{1}{X}$ is the *inverse Burr* distribution (also known as the *SinghMaddala* distribution) for which + +\[\begin{array}{*{20}{c}} +{{f_Y}(y;b,c,k) = \frac{{c{\kern 1pt} {\kern 1pt} k{{\left( {\frac{y}{b}} \right)}^c}}}{{y{\kern 1pt} {{\left[ {1 + {{\left( {\frac{y}{b}} \right)}^c}} \right]}^{k + 1}}}}}&{b,c,k,y > 0} +\end{array}\] + +\[\begin{array}{*{20}{c}} +{{F_Y}(y;b,c,k) = 1 - \frac{1}{{{{\left[ {1 + {{\left( {\frac{y}{b}} \right)}^c}} \right]}^k}}}}&{b,c,k,y > 0} +\end{array}\] + +We now consider the limiting distribution when $c \to \infty$ and $k \to 0$ in such a way that the product $ck$ remains constant, i.e. $ck = \lambda$. + +Now, +\[\begin{array}{l} +\mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } \left\{ {{F_Y}(y;b,c,k)} \right\} = 1 - \mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } \frac{1}{{{{\left[ {1 + {{\left( {\frac{y}{b}} \right)}^c}} \right]}^k}}}\\ +\\ +and\\ +\\ +\mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } {\left[ {1 + {{\left( {\frac{y}{b}} \right)}^c}} \right]^k} = \mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } \left\{ {{{\left( {\frac{y}{b}} \right)}^{ck}}{{\left[ {1 + {{\left( {\frac{b}{y}} \right)}^c}} \right]}^k}} \right\}\\ +\\ +and\\ +\\ +\mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } \left\{ {{{\left( {\frac{y}{b}} \right)}^{ck}}{{\left[ {1 + {{\left( {\frac{b}{y}} \right)}^c}} \right]}^k}} \right\} = \mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } \left\{ {{{\left( {\frac{y}{b}} \right)}^{ck}}} \right\}\mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } \left\{ {{{\left[ {1 + {{\left( {\frac{b}{y}} \right)}^c}} \right]}^k}} \right\}\\ + = \mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } \left\{ {{{\left( {\frac{y}{b}} \right)}^{ck}}} \right\}\; \cdot \,1\\ + = {\left( {\frac{y}{b}} \right)^\lambda } +\end{array}\] + +Therefore, \[\begin{array}{*{20}{c}} +{\mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } \left\{ {{F_Y}(y;b,c,k)} \right\} = 1 - {{\left( {\frac{b}{y}} \right)}^\lambda }}&{y \ge b} +\end{array}\] + +which we recognise as the (American) Pareto distribution. So, if the limiting distribution of $Y = \frac{1}{X}$ is a Pareto distribution, then the limiting distribution of $X = \frac{1}{Y}$ is the (American) *inverse Pareto* distribution + +\[\begin{array}{l} +{f_X}\left( {x;\alpha ,\beta } \right) = \lambda {b^\lambda }{x^{\lambda - 1}};{\rm{ }}0 \le x \le {\textstyle{1 \over b}};{\rm{ }}\lambda {\rm{,}}b > 0\\ +{F_X}\left( {x;\alpha ,\beta } \right) = {\left( {xb} \right)^\lambda };{\rm{ }}0 \le x \le {\textstyle{1 \over b}};{\rm{ }}\lambda {\rm{,}}b > 0 +\end{array}\] + +For completeness, the MLEs of this distribution have closed-form expressions and are given by + +\[\begin{array}{l} +\hat \lambda = {\left[ {\ln \left( {\frac{{{g_X}}}{{\hat b}}} \right)} \right]^{ - 1}}\\ +\hat b = \frac{1}{{\max \left\{ {{X_i}} \right\}}}{\rm{ }} +\end{array}\] + +and $\rm{ }{g_X}$ is the *geometric mean* of the data. + +### Inverse Weibull distribution + +Let $X \sim Burr(b,c,k)$ have the *pdf* given in the box above. We make the transformation $Y = \frac{{b{\kern 1pt} {k^{\tfrac{1}{c}}}{\kern 1pt} \theta }}{X}$ + +where $\theta$ is a parameter (constant). The distribution of $Y$ is also a Burr distribution and has *cdf* ${G_Y}\left( y \right) = 1 - \frac{1}{{{{\left[ {1 + {{\left( {\frac{y}{{{k^{\tfrac{1}{c}}}{\kern 1pt} \theta }}} \right)}^c}} \right]}^k}}}$. + +We are interested in the limiting behaviour of this Burr distribution as $k \to \infty$. + +Now, $\mathop {\lim }\limits_{k \to \infty } {G_Y}\left( y \right) = 1 - \mathop {\lim }\limits_{k \to \infty } {\left[ {1 + {{\left( {\frac{y}{{{k^{\tfrac{1}{c}}}{\kern 1pt} \theta }}} \right)}^c}} \right]^{ - k}}$ + +${ = 1 - \mathop {\lim }\limits_{k \to \infty } {{\left[ {1 + \frac{{{{\left( {\frac{y}{\theta }} \right)}^c}}}{{k{\kern 1pt} }}} \right]}^{ - k}}}$ + +$\begin{matrix} + =1-\exp \left[ -{{\left( \frac{y}{\theta } \right)}^{c}} \right] \\ + \left\{ \text{using the fact that }\underset{n\to \infty }{\mathop{\lim }}\,{{\left( 1+{}^{z}\!\!\diagup\!\!{}_{n}\; \right)}^{-n}}={{e}^{-z}} \right\} \\ +\end{matrix}$ + +We recognise the last expression as the *cdf* of a Weibull distribution with parameters $c$ and $\theta$. + +## References + +
+ +```{r, results = "asis", echo = FALSE} +cat(ssdtools::ssd_licensing_md()) +``` diff --git a/doc/customising-plots.R b/doc/customising-plots.R new file mode 100644 index 00000000..5d69f82b --- /dev/null +++ b/doc/customising-plots.R @@ -0,0 +1,70 @@ +## ----include = FALSE---------------------------------------------------------- +knitr::opts_chunk$set( + collapse = TRUE, + comment = "#>", + fig.width = 6, + fig.height = 4 +) + +## ----warning=FALSE, message=FALSE, fig.alt="A plot of the CCME boron data with the six default distributions."---- +library(ssdtools) +library(ggplot2) + +fits <- ssd_fit_dists(ssddata::ccme_boron) +ssd_plot_cdf(fits) + +## ----fig.alt="A plot of the CCME boron data with the model average distribution and the six default distributions."---- +ssd_plot_cdf(fits, average = NA, big.mark = " ") + + scale_color_manual(name = "Distribution", breaks = c("average", names(fits)), values = 1:7) + + theme_bw() + +## ----fig.alt = "A plot of the CCME boron data."------------------------------- +ggplot(ssddata::ccme_boron) + + geom_ssdpoint(aes(x = Conc)) + + ylab("Probability density") + + xlab("Concentration") + +## ----fig.alt = "A plot of CCME boron data with the ranges of the censored data indicated by horizontal lines."---- +ggplot(ssddata::ccme_boron) + + geom_ssdsegment(aes(x = Conc, xend = Conc * 4)) + + ylab("Probability density") + + xlab("Concenration") + +## ----fig.alt="A plot of the confidence intervals for the CCME boron data."---- +ggplot(boron_pred) + + geom_xribbon(aes(xmin = lcl, xmax = ucl, y = proportion)) + + ylab("Probability density") + + xlab("Concenration") + +## ----fig.alt="A plot of hazard concentrations as dotted lines."--------------- +ggplot() + + geom_hcintersect(xintercept = c(1, 2, 3), yintercept = c(0.05, 0.1, 0.2)) + + ylab("Probability density") + + xlab("Concenration") + +## ----fig.alt="A plot of censored CCME boron data with confidence intervals"---- +gp <- ggplot(boron_pred, aes(x = est)) + + geom_xribbon(aes(xmin = lcl, xmax = ucl, y = proportion), alpha = 0.2) + + geom_line(aes(y = proportion)) + + geom_ssdsegment(data = ssddata::ccme_boron, aes(x = Conc / 2, xend = Conc * 2)) + + geom_ssdpoint(data = ssddata::ccme_boron, aes(x = Conc / 2)) + + geom_ssdpoint(data = ssddata::ccme_boron, aes(x = Conc * 2)) + + scale_y_continuous("Species Affected (%)", labels = scales::percent) + + xlab("Concentration (mg/L)") + + expand_limits(y = c(0, 1)) + +gp + +## ----fig.alt="A plot of censored CCME boron data on log scale with confidence intervals, mathematical and the 5% hazard concentration."---- +gp + + scale_x_log10( + latex2exp::TeX("Boron $(\\mu g$/L)$") + ) + + geom_hcintersect(xintercept = ssd_hc(fits)$est, yintercept = 0.05) + +## ----eval = FALSE------------------------------------------------------------- +# ggsave("file_name.png", dpi = 300) + +## ----results = "asis", echo = FALSE------------------------------------------- +cat(ssdtools::ssd_licensing_md()) + diff --git a/doc/customising-plots.Rmd b/doc/customising-plots.Rmd new file mode 100644 index 00000000..b5b3d415 --- /dev/null +++ b/doc/customising-plots.Rmd @@ -0,0 +1,137 @@ +--- +title: "Customising Plots" +author: "ssdtools Team" +date: '`r format(Sys.time(), "%Y-%m-%d", tz = "UTC")`' +bibliography: references.bib +mathfont: Courier +latex_engine: MathJax +output: rmarkdown::html_vignette +vignette: > + %\VignetteIndexEntry{Customising Plots} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteEncoding{UTF-8} +--- + +```{r, include = FALSE} +knitr::opts_chunk$set( + collapse = TRUE, + comment = "#>", + fig.width = 6, + fig.height = 4 +) +``` + +## Plotting the cumulative distributions + +The `ssdtools` package plots the cumulative distribution functions using `ssd_plot_cdf()`. + +For example, consider the CCME boron data from the [`ssddata`](https://github.com/open-AIMS/ssddata) package. +We can fit, and then plot the cdfs as follows. + +```{r warning=FALSE, message=FALSE, fig.alt="A plot of the CCME boron data with the six default distributions."} +library(ssdtools) +library(ggplot2) + +fits <- ssd_fit_dists(ssddata::ccme_boron) +ssd_plot_cdf(fits) +``` + +This graphic is a [ggplot](https://ggplot2.tidyverse.org) object and so can be customized in the usual way. + +For example, we can add the model-averaged cdf by setting `average = NA`, change the string used to separate thousands using `big.mark`, customize the color scale with `scale_color_manual()` and change the theme. + +```{r, fig.alt="A plot of the CCME boron data with the model average distribution and the six default distributions."} +ssd_plot_cdf(fits, average = NA, big.mark = " ") + + scale_color_manual(name = "Distribution", breaks = c("average", names(fits)), values = 1:7) + + theme_bw() +``` + +## ggplot Geoms + +The `ssdtools` package provides four ggplot geoms to allow you construct your own plots. + +### `geom_ssdpoint()` + +The first is `geom_ssdpoint()` which plots species sensitivity data + +```{r, fig.alt = "A plot of the CCME boron data."} +ggplot(ssddata::ccme_boron) + + geom_ssdpoint(aes(x = Conc)) + + ylab("Probability density") + + xlab("Concentration") +``` + +### `geom_ssdsegments()` + +The second is `geom_ssdsegments()` which plots the ranges of censored species sensitivity data + +```{r, fig.alt = "A plot of CCME boron data with the ranges of the censored data indicated by horizontal lines."} +ggplot(ssddata::ccme_boron) + + geom_ssdsegment(aes(x = Conc, xend = Conc * 4)) + + ylab("Probability density") + + xlab("Concenration") +``` + +### `geom_xribbon()` + +The third is `geom_xribbon()` which plots species sensitivity confidence intervals + +```{r, fig.alt="A plot of the confidence intervals for the CCME boron data."} +ggplot(boron_pred) + + geom_xribbon(aes(xmin = lcl, xmax = ucl, y = proportion)) + + ylab("Probability density") + + xlab("Concenration") +``` + +### `geom_hcintersect()` + +And the fourth is `geom_hcintersect()` which plots hazard concentrations + +```{r, fig.alt="A plot of hazard concentrations as dotted lines."} +ggplot() + + geom_hcintersect(xintercept = c(1, 2, 3), yintercept = c(0.05, 0.1, 0.2)) + + ylab("Probability density") + + xlab("Concenration") +``` + +### Putting it together + +Geoms can be combined as follows + +```{r, fig.alt="A plot of censored CCME boron data with confidence intervals"} +gp <- ggplot(boron_pred, aes(x = est)) + + geom_xribbon(aes(xmin = lcl, xmax = ucl, y = proportion), alpha = 0.2) + + geom_line(aes(y = proportion)) + + geom_ssdsegment(data = ssddata::ccme_boron, aes(x = Conc / 2, xend = Conc * 2)) + + geom_ssdpoint(data = ssddata::ccme_boron, aes(x = Conc / 2)) + + geom_ssdpoint(data = ssddata::ccme_boron, aes(x = Conc * 2)) + + scale_y_continuous("Species Affected (%)", labels = scales::percent) + + xlab("Concentration (mg/L)") + + expand_limits(y = c(0, 1)) + +gp +``` + +To log the x-axis and include mathematical notation and add the HC5 value use the following code. + +```{r, fig.alt="A plot of censored CCME boron data on log scale with confidence intervals, mathematical and the 5% hazard concentration."} +gp + + scale_x_log10( + latex2exp::TeX("Boron $(\\mu g$/L)$") + ) + + geom_hcintersect(xintercept = ssd_hc(fits)$est, yintercept = 0.05) +``` + +## Saving plots + +The most recent plot can be saved as a file using `ggsave()`, which also allows the user to set the resolution. + +```{r, eval = FALSE} +ggsave("file_name.png", dpi = 300) +``` + +
+ +```{r, results = "asis", echo = FALSE} +cat(ssdtools::ssd_licensing_md()) +``` diff --git a/doc/distributions.R b/doc/distributions.R new file mode 100644 index 00000000..e079c254 --- /dev/null +++ b/doc/distributions.R @@ -0,0 +1,315 @@ +## ----include = FALSE---------------------------------------------------------- +knitr::opts_chunk$set( + collapse = TRUE, + comment = "#>", + fig.width = 6, + fig.height = 4 +) + +## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5, fig.cap="Sample Burr probability density (A) and cumulative probability (B) functions."---- +par(mfrow = c(1, 2)) + +f <- function(x, b, c, k) { + z1 <- (b / x)^(c - 1) + z2 <- (b / x)^c + y <- (b * c * k / x^2) * z1 / (1 + z2)^(k + 1) + return(y) +} + +conc <- seq(0, 10, by = 0.005) +plot(conc, f(conc, 1, 3, 5), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733") +lines(conc, f(conc, 1, 1, 2), col = "#F2A61C") +lines(conc, f(conc, 1, 2, 2), col = "#1CADF2") +lines(conc, f(conc, 1, 2, 5), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +F <- function(x, b, c, k) { + z2 <- (b / x)^c + y <- 1 / (1 + z2)^k + return(y) +} +conc <- seq(0, 10, by = 0.005) +plot(conc, F(conc, 1, 3, 5), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733") +lines(conc, F(conc, 1, 1, 2), col = "#F2A61C") +lines(conc, F(conc, 1, 2, 2), col = "#1CADF2") +lines(conc, F(conc, 1, 2, 5), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") + +## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5, fig.cap="Sample lognormal lognormal mixture probability density (A) and cumulative probability (B) functions."---- +par(mfrow = c(1, 2)) + +f <- function(x, m1, s1, m2, s2, p) { + y <- p * dlnorm(x, m1, s1) + (1 - p) * dlnorm(x, m2, s2) + return(y) +} + +conc <- seq(0, 5, by = 0.0025) +m1 <- 1 +s1 <- .2 +m2 <- 1.8 +s2 <- 1.5 +p <- 0.25 +plot(conc, f(conc, m1, s1, m2, s2, p), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, f(conc, 0.09, 0.5, 1, 0.08, 0.9), col = "#F2A61C") +lines(conc, f(conc, 0.5, 0.5, 1, 0.08, 0.9), col = "#1CADF2") +lines(conc, f(conc, 0.7, 1.5, 0.5, 0.1, .7), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +F <- function(x, m1, s1, m2, s2, p) { + y <- p * plnorm(x, m1, s1) + (1 - p) * plnorm(x, m2, s2) + return(y) +} + +conc <- seq(0, 10, by = 0.0025) +m1 <- 1 +s1 <- .2 +m2 <- 1.8 +s2 <- 1.5 +p <- 0.25 +plot(conc, F(conc, m1, s1, m2, s2, p), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, F(conc, 0.09, 0.5, 1, 0.08, 0.9), col = "#F2A61C") +lines(conc, F(conc, 0.5, 0.5, 1, 0.08, 0.9), col = "#1CADF2") +lines(conc, F(conc, 0.7, 1.5, 0.5, 0.1, .7), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") + +## ----message=FALSE, echo=FALSE, fig.width=7,fig.height=5, fig.cap="Currently recommended default distributions."---- +library(ssdtools) +library(ggplot2) + +# theme_set(theme_bw()) + +set.seed(7) + +ssd_plot_cdf(ssd_match_moments(meanlog = 2, sdlog = 2)) + + scale_color_ssd() + +## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample lognormal probability density (A) and cumulative probability (B) functions."---- +par(mfrow = c(1, 2)) + +conc <- seq(0, 10, by = 0.005) +m1 <- 1 +s1 <- .2 +m2 <- 1.8 +s2 <- 1.5 +p <- 0.25 +plot(conc, dlnorm(conc, m1, s1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, dlnorm(conc, 0.4, 2), col = "#F2A61C") +lines(conc, dlnorm(conc, m1 * 2, s1), col = "#1CADF2") +lines(conc, dlnorm(conc, 0.9, 1.5), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +conc <- seq(0, 10, by = 0.005) +m1 <- 1 +s1 <- .2 +m2 <- 1.8 +s2 <- 1.5 +p <- 0.25 +plot(conc, plnorm(conc, m1, s1), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, plnorm(conc, 0.4, 2), col = "#F2A61C") +lines(conc, plnorm(conc, m1 * 2, s1), col = "#1CADF2") +lines(conc, plnorm(conc, 0.9, 1.5), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") + +## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Log logistic probability density (A) and cumulative probability (B) functions."---- +par(mfrow = c(1, 2)) + +f <- function(x, a, b) { + z1 <- (x / a)^(b - 1) + z2 <- (x / a)^b + y <- (b / a) * z1 / (1 + z2)^2 + return(y) +} + +conc <- seq(0, 10, by = 0.005) +plot(conc, f(conc, 3.2, 3.5), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) +lines(conc, f(conc, 1.5, 1.5), col = "#F2A61C") +lines(conc, f(conc, 1, 1), col = "#1CADF2") +lines(conc, f(conc, 1, 4), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +F <- function(x, a, b) { + z2 <- (x / a)^(-b) + y <- 1 / (1 + z2) + return(y) +} +conc <- seq(0, 10, by = 0.005) +plot(conc, F(conc, 3.2, 3.5), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) +lines(conc, F(conc, 1.5, 1.5), col = "#F2A61C") +lines(conc, F(conc, 1, 1), col = "#1CADF2") +lines(conc, F(conc, 1, 4), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") + +## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample gamma probability density (A) and cumulative probability (B) functions."---- +par(mfrow = c(1, 2)) + +conc <- seq(0, 10, by = 0.005) +plot(conc, dgamma(conc, 5, 5), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) +lines(conc, dgamma(conc, 4, 1), col = "#F2A61C") +lines(conc, dgamma(conc, 0.9, 1), col = "#1CADF2") +lines(conc, dgamma(conc, 2, 1.), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +conc <- seq(0, 10, by = 0.005) +plot(conc, pgamma(conc, 5, 5), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) +lines(conc, pgamma(conc, 4, 1), col = "#F2A61C") +lines(conc, pgamma(conc, 0.9, 1), col = "#1CADF2") +lines(conc, pgamma(conc, 2, 1.), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") + +## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Log-Gumbel probability density (A) and cumulative probability (B) functions."---- +par(mfrow = c(1, 2)) + +f <- function(x, a, b) { + y <- b * exp(-(a * x)^(-b)) / (a^b * x^(b + 1)) + return(y) +} + +conc <- seq(0, 10, by = 0.005) +plot(conc, f(conc, 0.2, 5), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) +lines(conc, f(conc, 0.5, 1.5), col = "#F2A61C") +lines(conc, f(conc, 1, 2), col = "#1CADF2") +lines(conc, f(conc, 10, .5), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +F <- function(x, a, b) { + y <- exp(-(a * x)^(-b)) + return(y) +} + +conc <- seq(0, 10, by = 0.005) +plot(conc, F(conc, 0.2, 5), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) +lines(conc, F(conc, 0.5, 1.5), col = "#F2A61C") +lines(conc, F(conc, 1, 2), col = "#1CADF2") +lines(conc, F(conc, 10, .5), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") + +## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Gompertz probability density (A) and cumulative probability (B) functions."---- +par(mfrow = c(1, 2)) + +f <- function(x, n, b) { + y <- n * b * exp(n + b * x - n * exp(b * x)) + return(y) +} + +conc <- seq(0, 10, by = 0.005) +plot(conc, f(conc, 0.089, 1.25), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 2)) +lines(conc, f(conc, 0.001, 3.5), col = "#F2A61C") +lines(conc, f(conc, 0.0005, 1.1), col = "#1CADF2") +lines(conc, f(conc, 0.01, 5), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +F <- function(x, n, b) { + y <- 1 - exp(-n * exp(b * x - 1)) + return(y) +} + +conc <- seq(0, 10, by = 0.005) +plot(conc, F(conc, 0.089, 1.25), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, F(conc, 0.001, 3.5), col = "#F2A61C") +lines(conc, F(conc, 0.0005, 1.1), col = "#1CADF2") +lines(conc, F(conc, 0.01, 5), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") + +## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Weibull probability density (A) and cumulative probability (B) functions."---- +par(mfrow = c(1, 2)) + +conc <- seq(0, 10, by = 0.005) +plot(conc, dweibull(conc, 4.321, 4.949), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, dweibull(conc, 0.838, 0.911), col = "#F2A61C") +lines(conc, dweibull(conc, 1, 1.546), col = "#1CADF2") +lines(conc, dweibull(conc, 17.267, 7.219), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +conc <- seq(0, 10, by = 0.005) +plot(conc, pweibull(conc, 4.321, 4.949), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, pweibull(conc, 0.838, 0.911), col = "#F2A61C") +lines(conc, pweibull(conc, 1, 1.546), col = "#1CADF2") +lines(conc, pweibull(conc, 17.267, 7.219), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") + +## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample North American Pareto probability density (A) and cumulative probability (B) functions."---- +par(mfrow = c(1, 2)) + +conc <- seq(0, 10, by = 0.005) +plot(conc, extraDistr::dpareto(conc, 3, 2), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.5)) +lines(conc, extraDistr::dpareto(conc, 0.838, 0.911), col = "#F2A61C") +lines(conc, extraDistr::dpareto(conc, 4, 4), col = "#1CADF2") +lines(conc, extraDistr::dpareto(conc, 10, 7), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +conc <- seq(0, 10, by = 0.005) +plot(conc, extraDistr::ppareto(conc, 3, 2), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, extraDistr::ppareto(conc, 0.838, 0.911), col = "#F2A61C") +lines(conc, extraDistr::ppareto(conc, 4, 4), col = "#1CADF2") +lines(conc, extraDistr::ppareto(conc, 10, 7), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") + +conc <- seq(0, 10, by = 0.005) +plot(conc, extraDistr::ppareto(conc, 3, 2), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, extraDistr::ppareto(conc, 0.838, 0.911), col = "#F2A61C") +lines(conc, extraDistr::ppareto(conc, 4, 4), col = "#1CADF2") +lines(conc, extraDistr::ppareto(conc, 10, 7), col = "#1F1CF2") + +## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Sample North American inverse Pareto probability density (A) and cumulative probability (B) functions."---- +par(mfrow = c(1, 2)) + +f <- function(x, a, b) { + y <- a * (b^a) * x^(a - 1) + return(y) +} + +conc <- seq(0, 10, by = 0.001) +plot(conc, f(conc, 5, 0.1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 0.5)) +lines(conc, f(conc, 3, 0.1), col = "#F2A61C") +lines(conc, f(conc, 0.5, 0.1), col = "#1CADF2") +lines(conc, f(conc, 0.1, 0.1), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +F <- function(x, a, b) { + y <- (b * x)^a + return(y) +} + +conc <- seq(0, 10, by = 0.001) +plot(conc, F(conc, 5, 0.1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, F(conc, 3, 0.1), col = "#F2A61C") +lines(conc, F(conc, 0.5, 0.1), col = "#1CADF2") +lines(conc, F(conc, 0.1, 0.1), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") + +## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample European Pareto probability density (A) and cumulative probability (B) functions."---- +par(mfrow = c(1, 2)) + +conc <- seq(0, 10, by = 0.005) +plot(conc, actuar::dpareto(conc, 1, 1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, actuar::dpareto(conc, 2, 3), col = "#F2A61C") +lines(conc, actuar::dpareto(conc, 0.5, 1), col = "#1CADF2") +lines(conc, actuar::dpareto(conc, 10.5, 6.5), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +conc <- seq(0, 10, by = 0.005) +plot(conc, actuar::ppareto(conc, 1, 1), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, actuar::ppareto(conc, 2, 3), col = "#F2A61C") +lines(conc, actuar::ppareto(conc, 0.5, 1), col = "#1CADF2") +lines(conc, actuar::ppareto(conc, 10.5, 6.5), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") + +## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample European inverse Pareto probability density (A) and cumulative probability (B) functions."---- +par(mfrow = c(1, 2)) + +conc <- seq(0, 10, by = 0.001) +plot(conc, actuar::dinvpareto(conc, 1, 1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 0.8)) +lines(conc, actuar::dinvpareto(conc, 1, 3), col = "#F2A61C") +lines(conc, actuar::dinvpareto(conc, 10, 0.1), col = "#1CADF2") +lines(conc, actuar::dinvpareto(conc, 2.045, 0.98), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +conc <- seq(0, 10, by = 0.001) +plot(conc, actuar::pinvpareto(conc, 1, 1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, actuar::pinvpareto(conc, 1, 3), col = "#F2A61C") +lines(conc, actuar::pinvpareto(conc, 10, 0.1), col = "#1CADF2") +lines(conc, actuar::pinvpareto(conc, 2.045, 0.98), col = "#1F1CF2") + +## ----results = "asis", echo = FALSE------------------------------------------- +cat(ssdtools::ssd_licensing_md()) + diff --git a/doc/distributions.Rmd b/doc/distributions.Rmd new file mode 100644 index 00000000..8ab423d4 --- /dev/null +++ b/doc/distributions.Rmd @@ -0,0 +1,837 @@ +--- +title: "Distributions in ssdtools" +author: "ssdtools Team" +date: '`r format(Sys.time(), "%Y-%m-%d", tz = "UTC")`' +bibliography: references.bib +mathfont: Courier +latex_engine: MathJax +output: rmarkdown::html_vignette +vignette: > + %\VignetteIndexEntry{Distributions in ssdtools} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteEncoding{UTF-8} +--- + +```{r, include = FALSE} +knitr::opts_chunk$set( + collapse = TRUE, + comment = "#>", + fig.width = 6, + fig.height = 4 +) +``` + + + + + +## Distributions for SSD modelling + +Many authors have noted that there is no guiding theory in ecotoxicology to justify a particular distributional form for the SSD other than that its domain be restricted to the positive real line [@newman_2000; @Zajdlik_2005; @fox_2016]. +Distributions selected to use in model averaging of SSDs must be bounded by zero given that effect concentrations cannot be negative. +They must also be continuous, and generally unbounded on the right. +Furthermore, the selected distributions within the candidate model set should provide a variety of shapes to capture the diversity of shapes in empirical species sensitivity distributions. + +To date `r length(ssdtools::ssd_dists())` distributions have been implemented in `ssdtools`, although only `r length(ssdtools::ssd_dists_bcanz())` appear in the default set. +Here we provide a detailed account of all `r length(ssdtools::ssd_dists())` of the distributions available in `ssdtools`, and guidance on their use. + +### Original `ssdtools` distributions + +The log-normal, log-logistic and Gamma distributions have been widely used in SSD modelling, and were part of the original distribution set for early releases of `ssdtools` as developed by @thorley_ssdtools_2018. +They were adopted as the default set of three distributions in early updates of `ssdtools` and the associated ShinyApp [@dalgarno_shinyssdtools_2021]. +All three distributions show good convergence properties and are retained as part of the default model set in version 2.0 of `ssdtools`. + +In addition to the log-normal, log-logistic and Gamma distributions, the original version of `ssdtools` as developed by [Thorley and Schwarz (2018)](https://joss.theoj.org/papers/10.21105/joss.01082.pdf) also included three additional distributions in the candidate model set, including the log-gumbel, Gompertz and Weibull distributions. +Of these, the log-Gumbel (otherwise known as the inverse Weibull, see below) shows relatively good convergence [see Figure 32, @fox_methodologies_2022], and is also one of the limiting distributions of the Burr Type 3 distribution implemented in `ssdtools`, and has been retained in the default model set. +The Gompertz and Weibull distributions, however can exhibit unstable behaviour, sometimes showing poor convergence, and therefore been excluded from the default set [see Figure 32, @fox_methodologies_2022] + +### Burr III distribution + +A history of Burrlioz and the primary distributions it used were recently summmarized by @fox_recent_2021. + +> In 2000, Australia and New Zealand (Australian and New Zealand Environment and Conservation Council/Agriculture and Resource Management Council of Australia and New Zealand 2000) adopted an SSD‐based method for deriving WQBs, following a critical review of multiple WQB derivation methods (Warne 1998). +A distinct feature of the method was the use of a 3‐parameter Burr distribution to model the empirical SSD, which was implemented in the Burrlioz software tool (Campbell et al. 2000). +This represented a generalization of the methods previously employed by Aldenberg and Slob (1993) because the log–logistic distribution +was shown to be a specific case of the Burr family (Tadikamalla 1980). +Recent revision of the derivation method recognized that using the 3‐parameter Burr distributions for small sample sizes (<8 species) created additional uncertainty by estimating more parameters than could be justified, essentially overfitting the data (Batley et al. 2018). +Consequently, the method, and the updated software (Burrlioz Ver 2.0), now uses a 2‐parameter log–logistic distribution for these small +data sets, whereas the Burr type III distribution is used for data sets of 8 species or more (Batley et al. 2018; Australian and +New Zealand Guidelines 2018). + +[@fox_recent_2021] + +The probability density function, ${f_X}(x;b,c,k)$ and cumulative distribution function, ${F_X}(x;b,c,k)$ for the Burr III distribution are: + + + + + + + + + +
+ Burr III Distribution + + $$f_X(x;b,c,k) = \frac{{b\,k\,c}}{{x^2}} \frac{{\left( \frac{b}{x} \right)}^{c - 1}}{{\left[ 1 + \left( \frac{b}{x} \right)^c \right]}^{k + 1}}, \quad b, c, k, x > 0$$ + +
+ + $$F_X(x;b,c,k) = \frac{1}{{\left[ 1 + \left( \frac{b}{x} \right)^c \right]}^k}, \quad b, c, k, x > 0$$ + +
+ +
+Burr III distribution + +$${{f_X}(x;b,c,k) = \frac{{b\,k\,c}}{{{x^2}}}\frac{{{{\left( {\frac{b}{x}} \right)}^{c - 1}}}}{{{{\left[ {1 + {{\left( {\frac{b}{x}} \right)}^c}} \right]}^{k + 1}}}}{\rm{ }}} {b,c,k,x > 0}$$ +
+ + +$${{F_X}(x;b,c,k) = \frac{1}{{{{\left[ {1 + {{\left( {\frac{b}{x}} \right)}^c}} \right]}^k}}}{\rm{ }}} {b,c,k,x > 0}$$ +
+ +```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5, fig.cap="Sample Burr probability density (A) and cumulative probability (B) functions."} +par(mfrow = c(1, 2)) + +f <- function(x, b, c, k) { + z1 <- (b / x)^(c - 1) + z2 <- (b / x)^c + y <- (b * c * k / x^2) * z1 / (1 + z2)^(k + 1) + return(y) +} + +conc <- seq(0, 10, by = 0.005) +plot(conc, f(conc, 1, 3, 5), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733") +lines(conc, f(conc, 1, 1, 2), col = "#F2A61C") +lines(conc, f(conc, 1, 2, 2), col = "#1CADF2") +lines(conc, f(conc, 1, 2, 5), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +F <- function(x, b, c, k) { + z2 <- (b / x)^c + y <- 1 / (1 + z2)^k + return(y) +} +conc <- seq(0, 10, by = 0.005) +plot(conc, F(conc, 1, 3, 5), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733") +lines(conc, F(conc, 1, 1, 2), col = "#F2A61C") +lines(conc, F(conc, 1, 2, 2), col = "#1CADF2") +lines(conc, F(conc, 1, 2, 5), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") +``` + +While the Burr type III distribution was adopted as the default distribution in Burrlioz, it is well known (e.g., Tadikamalla (1980)) that the Burr III distribution is related to several other theoretical distributions, some of which only exist as limiting cases of the Burr III, i.e., as one or more of the Burr III parameters approaches either zero or infinity. +The Burrlioz software incorporates logic that aims to identify situations where parameter estimates are tending towards either very large or very small values. +In such cases, fitting a Burr III distribution is abandoned and one of the limiting +distributions is fitted instead. + +Specifically: + + - As c tends to infinity the Burr III distribution tends to the inverse (North American) Pareto distribution (see [technical details](https://bcgov.github.io/ssdtools/articles/E_additional-technical-details.html)) + + - As k tends to infinity the Burr III distribution tends to the inverse Weibull (log-Gumbel) distribution (see [technical details](https://bcgov.github.io/ssdtools/articles/E_additional-technical-details.html)) + +In practical terms, if the Burr III distribution is fitted and k is estimated to be greater than 100, the estimation procedure is carried out again using an inverse Weibull distribution. +Similarly, if c is greater than 80 an (American) Pareto distribution is fitted. This is necessary to +ensure numerical stability. + +Since the Burr type III, inverse Pareto and inverse Weibull (log Gumbel) distributions are used by the Burrlioz software, these have been implemented in `ssdtools`. +However, we have found there are stability issues with both the Burr type III, as well as the inverse Pareto distributions, which currently precludes their inclusion in the default model set (see @fox_methodologies_2022, and below for more details). + +### Bimodal distributions + +The use of statistical mixture-models was promoted by Fox as a convenient and more realistic way of modelling bimodal toxicity data [@fisher2019]. +Although parameter heavy, statistical mixture models provide a better conceptual match to the inherent underlying data generating process since they directly model bimodality as a mixture of 2 underlying univariate distributions that represent, for example, different modes of action [@fox_recent_2021]. +It has been postulated that a mixture-model would only be selected in a model-averaging context when the fit afforded by the mixture is demonstrably better than the fit afforded by any single distribution. +This is a consequence of the high penalty in AICc associated with the increased number of parameters (p in Equation 7 of [@fox_recent_2021]) and will be most pronounced for relatively small sample sizes. + +The TMB version of `ssdtools` now includes the option of fitting two mixture distributions, individually or as part of a model average set. +These can be fitted using `ssdtools` by supplying the strings "llogis_llogis" and/or "lnorm_lnorm" to the *dists* argument in the *ssd_fit_dists* call. + +The underlying code for these mixtures has three components: the likelihood function required for TMB; exported R functions to allow the usual methods for a distribution to be called (p, q and r); and a set of supporting R functions (see @fox_methodologies_2022 Appendix D for more details). +Both mixtures have five parameters - two parameters for each of the component distributions and a mixing parameter (pmix) that defines the weighting of the two distributions in the ‘mixture.’ + +```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5, fig.cap="Sample lognormal lognormal mixture probability density (A) and cumulative probability (B) functions."} +par(mfrow = c(1, 2)) + +f <- function(x, m1, s1, m2, s2, p) { + y <- p * dlnorm(x, m1, s1) + (1 - p) * dlnorm(x, m2, s2) + return(y) +} + +conc <- seq(0, 5, by = 0.0025) +m1 <- 1 +s1 <- .2 +m2 <- 1.8 +s2 <- 1.5 +p <- 0.25 +plot(conc, f(conc, m1, s1, m2, s2, p), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, f(conc, 0.09, 0.5, 1, 0.08, 0.9), col = "#F2A61C") +lines(conc, f(conc, 0.5, 0.5, 1, 0.08, 0.9), col = "#1CADF2") +lines(conc, f(conc, 0.7, 1.5, 0.5, 0.1, .7), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +F <- function(x, m1, s1, m2, s2, p) { + y <- p * plnorm(x, m1, s1) + (1 - p) * plnorm(x, m2, s2) + return(y) +} + +conc <- seq(0, 10, by = 0.0025) +m1 <- 1 +s1 <- .2 +m2 <- 1.8 +s2 <- 1.5 +p <- 0.25 +plot(conc, F(conc, m1, s1, m2, s2, p), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, F(conc, 0.09, 0.5, 1, 0.08, 0.9), col = "#F2A61C") +lines(conc, F(conc, 0.5, 0.5, 1, 0.08, 0.9), col = "#1CADF2") +lines(conc, F(conc, 0.7, 1.5, 0.5, 0.1, .7), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") +``` + +
As can be see from the plot above, the mixture distributions provide a highly flexible means of modelling *bimodality* in an emprical SSD. +This happens, for example, when the toxicity data for some toxicant include both animal and plant species, or there are different modes of action operating. +Unfortunately, this increased flexibility comes with a high penalty in the model-averaging process. +The combination of small sample sizes and a high parameter count (typically 5 or more) means that mixture distributions will be down-weighted - even when they do a good job at describing the data. +For this reason, when attempting to model bimodal data, we suggest looking at the fit using the default set of distributions and then examining the fit with just one of either the log-normal mixture or the log-logistic mixture. +Keep in mind that this should only be done if the sample size is not pathologically small. +As a guide, Prof. David Fox recommends as an *absolute minimum* $n \ge 3k + 1$ but preferably $n \ge 5k + 1$ where $k$ is the number of model parameters. + +## Default Distributions + +While there is a variety of distributions available in `ssdtools`, the inclusion of all of them for estimating a model-averaged SSD is not recommended. + +By default, `ssdtools` uses the (corrected) Akaike Information Criterion for small sample size (AICc) as a measure of relative quality of fit for different distributions and as the basis for calculating the model-averaged weights. +However, the choice of distributions used to fit a model-averaged SSD can have a profound effect on the estimated *HCx* values. + +Deciding on a final default set of distributions to adopt using the model averaging approach is non-trivial, and we acknowledge that there is probably no definitive ‘solution’ to this issue. +However, the default set should be underpinned by a guiding principle of parsimony, i.e., the set should be as large as is necessary to cover a wide variety of distributional shapes and contingencies but no bigger. +Further, the default set should result in model-averaged estimates of *HCx* values that: 1) minimise bias; 2) have actual coverages of confidence intervals that are close to the nominal level of confidence; 3) estimated *HCx* and confidence intervals of *HCx* are robust to small changes in the data; and 4) represent a positively continuous distribution that has both right and left tails. + +The `ssdtools` development team has undertaken extensive simulation studies, as well as some detailed technical examinations of the various candidate distributions to examine issues of bias, coverage and numerical stability. +A detailed account of our findings can be found in our report [@fox_methodologies_2022] and are not repeated in detail here, although some of the issues associated with individual distributions are outlined below. + +### Currently recommended default distributions + +The default list of candidate distributions in `ssdtools` is comprised of the following: log-normal; log-logistic; gamma; inverse Weibull (log-Gumbel); Weibull; mixture of two log-normal distributions + +The default distributions are plotted below with a mean of 2 and standard deviation of 2 on the (natural) log concentration scale or around 7.4 on the concentration scale. + +```{r, message=FALSE, echo=FALSE, fig.width=7,fig.height=5, fig.cap="Currently recommended default distributions."} +library(ssdtools) +library(ggplot2) + +# theme_set(theme_bw()) + +set.seed(7) + +ssd_plot_cdf(ssd_match_moments(meanlog = 2, sdlog = 2)) + + scale_color_ssd() +``` + + +## Distributions currently implemented in `ssdtools` + +#### Burr Type III distribution + +The Burr Type 3 is a flexible three parameter distribution can be fitted using `ssdtools` by supplying the string `burrIII3` to the `dists` argument in the `ssd_fit_dists` call. + +
+Burr III distribution + +

See above for details + +
+ +
+The Burr family of distributions has been central to the derivation of guideline values in Australia and New Zealand for over 20 yr [@fox_recent_2021]. While offering a high degree of flexibility, experience with these distributions during that time has repeatedly highlighted numerical stability and convergence issues when parameters are estimated using maximum likelihood [@fox_recent_2021]. +This is thought to be due to the high degree of collinearity between parameter estimates and/or relatively flat likelihood profiles [@fox_recent_2021], and is one of the motivations behind the logic coded into Burrlioz to revert to either of the two limiting distributions. +Burr Type 3 distribution is not currently one of the recommended distributionsin the default model set. +This is because of 1) the convergence issues associated with the Burr Type 3 distribution, 2) the fact that reverting to a limiting two parameter distribution does not fit easily within a model averaging framework, and 3) that one of the two limiting distributions (the inverse Pareto, see below) also has estimation and convergence issues. + +#### Log-normal + +The [log-normal](https://en.wikipedia.org/wiki/Log-normal_distribution) distribution is a commonly used distribution in the natural sciences - particularly as a probability model to describe right (positive)-skewed phenomena such as *concentration* data. + +A random variable, $X$ is lognormally distributed if the *logarithm* of $X$ is normally distributed. The *pdf* of $X$ is given by + +
+Lognormal distribution + +$${f_X}\left( {x;\mu ,\sigma } \right) = \frac{1}{{\sqrt {2\pi } {\kern 1pt} \sigma x}}\exp \left[ { - \frac{{{{\left( {\ln x - \mu } \right)}^2}}}{{2{\sigma ^2}}}} \right]{\rm{ ; }}x,\sigma > 0,\; - \infty < \mu < \infty $$ + +
+ +The lognormal distribution was selected as the starting distribution given the data are for effect concentrations. +The log-normal distribution can be fitted using `ssdtools` supplying the string `lnorm` to the `dists` argument in the `ssd_fit_dists` call. + +```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample lognormal probability density (A) and cumulative probability (B) functions."} +par(mfrow = c(1, 2)) + +conc <- seq(0, 10, by = 0.005) +m1 <- 1 +s1 <- .2 +m2 <- 1.8 +s2 <- 1.5 +p <- 0.25 +plot(conc, dlnorm(conc, m1, s1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, dlnorm(conc, 0.4, 2), col = "#F2A61C") +lines(conc, dlnorm(conc, m1 * 2, s1), col = "#1CADF2") +lines(conc, dlnorm(conc, 0.9, 1.5), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +conc <- seq(0, 10, by = 0.005) +m1 <- 1 +s1 <- .2 +m2 <- 1.8 +s2 <- 1.5 +p <- 0.25 +plot(conc, plnorm(conc, m1, s1), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, plnorm(conc, 0.4, 2), col = "#F2A61C") +lines(conc, plnorm(conc, m1 * 2, s1), col = "#1CADF2") +lines(conc, plnorm(conc, 0.9, 1.5), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") +``` + +#### Log-logistic distribution + +Like the *lognormal* distribution, the [log-logistic](https://en.wikipedia.org/wiki/Log-logistic_distribution) is similarly defined, that is: if $X$ has a log-logistic distribution, then $Y = \ln (X)$ has a *logistic* distribution. + +
+ + + + + + + + + + + + + + + + + + + + + +Log-logistic distribution + +$${{f_Y}\left( {y;\alpha ,\beta } \right) = \frac{{\left( {{\raise0.5ex\hbox{$\scriptstyle \beta $} +\kern-0.1em/\kern-0.15em +\lower0.25ex\hbox{$\scriptstyle \alpha $}}} \right){{\left( {{\raise0.5ex\hbox{$\scriptstyle y$} +\kern-0.1em/\kern-0.15em +\lower0.25ex\hbox{$\scriptstyle \alpha $}}} \right)}^{\beta - 1}}}}{{{{\left[ {1 + {{\left( {{\raise0.5ex\hbox{$\scriptstyle y$} +\kern-0.1em/\kern-0.15em +\lower0.25ex\hbox{$\scriptstyle \alpha $}}} \right)}^\beta }} \right]}^2}}};} {y,}$$ + +$${{F_Y}\left( {y;\alpha ,\beta } \right) = \frac{{{{\left( {{\raise0.5ex\hbox{$\scriptstyle y$} +\kern-0.1em/\kern-0.15em +\lower0.25ex\hbox{$\scriptstyle \alpha $}}} \right)}^\beta }}}{{1 + {{\left( {{\raise0.5ex\hbox{$\scriptstyle y$} +\kern-0.1em/\kern-0.15em +\lower0.25ex\hbox{$\scriptstyle \alpha $}}} \right)}^\beta }}};} {y,}$$ + +$$\alpha ,\beta > 0$$ + +
+ +
letting $\mu = \ln \left( \alpha \right)$ and $s = \frac{1}{\beta }$ we have: + +
+Logistic distribution + +$${{f_X}\left( {x;\mu ,s} \right) = \frac{{{e^{ - x}}}}{{{{\left( {1 + {e^{ - x}}} \right)}^2}}};} {x,s > 0,\; - \infty < \mu < \infty }$$ + +$${{F_X}\left( {x;\mu ,s} \right) = \frac{1}{{1 + {e^{ - \frac{{\left( {x - \mu } \right)}}{s}}}}};} {x,s > 0,\; - \infty < \mu < \infty }$$ +
+ + +
+The log-logistic distribution is often used as a candidate SSD primarily because of its analytic tractability [@aldenberg_confidence_1993]. +We included it because it has wider tails than the log-normal and because it is a specific case of the more general Burr family of distributions [@shao_estimation_2000, @burr_cumulative_1942]. +The log-logistic distribution can be fitted using `ssdtools` by supplying the string `lnorm` to the `dists` argument in the `ssd_fit_dists` call. + +```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Log logistic probability density (A) and cumulative probability (B) functions."} +par(mfrow = c(1, 2)) + +f <- function(x, a, b) { + z1 <- (x / a)^(b - 1) + z2 <- (x / a)^b + y <- (b / a) * z1 / (1 + z2)^2 + return(y) +} + +conc <- seq(0, 10, by = 0.005) +plot(conc, f(conc, 3.2, 3.5), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) +lines(conc, f(conc, 1.5, 1.5), col = "#F2A61C") +lines(conc, f(conc, 1, 1), col = "#1CADF2") +lines(conc, f(conc, 1, 4), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +F <- function(x, a, b) { + z2 <- (x / a)^(-b) + y <- 1 / (1 + z2) + return(y) +} +conc <- seq(0, 10, by = 0.005) +plot(conc, F(conc, 3.2, 3.5), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) +lines(conc, F(conc, 1.5, 1.5), col = "#F2A61C") +lines(conc, F(conc, 1, 1), col = "#1CADF2") +lines(conc, F(conc, 1, 4), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") +``` + +#### Gamma distribution +The two-parameter [gamma](https://en.wikipedia.org/wiki/Gamma_distribution) distribution has the following *pdf* and *cdf*. + +
+Gamma distribution + +$${{f_X}\left( {x;b,c} \right) = \frac{{{x^{c - 1}}{e^{ - \left( {\frac{x}{b}} \right)}}}}{{{b^c}\,\Gamma \left( c \right)}}} {0 \le x}$$ + +$${{F_X}\left( {x;b,c} \right) = \frac{1}{{\,\Gamma \left( c \right)}}\,\gamma \left( {c,\frac{x}{b}} \right)} {0 \le x}$$ +$$ < \infty ;b,c > 0$$ + +
+ +
where $\Gamma \left( \cdot \right)$ is the *gamma* function (in `R` this is simply `gamma(x)`) and $\gamma \left( \cdot \right)$ is the (lower) *incomplete gamma* function \[\gamma \left( {x,a} \right) = \int\limits_0^x {{t^{a - 1}}} \,{e^{ - t}}\,dt\] (this can be computed using the `gammainc` function from the `pracma` package in `R`). + +For use in modeling species sensitivity data, the gamma distribution has two key features that provide additional flexibility relative to the log-normal distribution: 1) it is asymmetrical on the logarithmic scale; and 2) it has wider tails. + +The gamma distribution can be fitted using `ssdtools` by supplying the string "gamma" to the *dists* argument in the *ssd_fit_dists* call. + +```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample gamma probability density (A) and cumulative probability (B) functions."} +par(mfrow = c(1, 2)) + +conc <- seq(0, 10, by = 0.005) +plot(conc, dgamma(conc, 5, 5), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) +lines(conc, dgamma(conc, 4, 1), col = "#F2A61C") +lines(conc, dgamma(conc, 0.9, 1), col = "#1CADF2") +lines(conc, dgamma(conc, 2, 1.), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +conc <- seq(0, 10, by = 0.005) +plot(conc, pgamma(conc, 5, 5), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) +lines(conc, pgamma(conc, 4, 1), col = "#F2A61C") +lines(conc, pgamma(conc, 0.9, 1), col = "#1CADF2") +lines(conc, pgamma(conc, 2, 1.), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") +``` + +#### Log-gumbel (inverse Weibull) distribution + +The log-gumbel distribution is a two-parameter distribution commonly used to model extreme values. +The log-gumbel distribution can be fitted using `ssdtools` by supplying the string `lgumbel` to the `dists` argument in the `ssd_fit_dists` call. +The two-parameter log-gumbel distribution has the following *pdf* and *cdf*: + +
+Log-Gumbel distribution + +$${{f_X}\left( {x;\alpha ,\beta } \right) = \frac{{\beta \,{e^{ - {{(\alpha x)}^{ - \beta }}}}}}{{{\alpha ^\beta }\,{x^{\beta + 1}}}}} ;\quad x,\alpha ,\beta > 0\\$$ + +$${{F_X}\left( {x;\alpha ,\beta } \right) = {e^{ - {{(\alpha x)}^{ - \beta }}}}} ;\quad x,\alpha ,\beta > 0$$ + +
+ + + + + + + + + + + +```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Log-Gumbel probability density (A) and cumulative probability (B) functions."} +par(mfrow = c(1, 2)) + +f <- function(x, a, b) { + y <- b * exp(-(a * x)^(-b)) / (a^b * x^(b + 1)) + return(y) +} + +conc <- seq(0, 10, by = 0.005) +plot(conc, f(conc, 0.2, 5), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) +lines(conc, f(conc, 0.5, 1.5), col = "#F2A61C") +lines(conc, f(conc, 1, 2), col = "#1CADF2") +lines(conc, f(conc, 10, .5), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +F <- function(x, a, b) { + y <- exp(-(a * x)^(-b)) + return(y) +} + +conc <- seq(0, 10, by = 0.005) +plot(conc, F(conc, 0.2, 5), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) +lines(conc, F(conc, 0.5, 1.5), col = "#F2A61C") +lines(conc, F(conc, 1, 2), col = "#1CADF2") +lines(conc, F(conc, 10, .5), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") +``` + + +#### Gompertz distribution + +The Gompertz distribution is a flexible distribution that exhibits both positive and negative skewness. +The Gompertz distribution can be fitted using `ssdtools` by supplying the string `gompertz` to the `dists` argument in the `ssd_fit_dists` call. +We consider two parameterisations of the Gompertz distribution. +The first, as given in [Wikipedia](https://en.wikipedia.org/wiki/Gompertz_distribution) and also used in `ssdtools` [Gompertz] has the following *pdf* and *cdf*: + +
+Gompertz distribution: Parameterisation I + + + + + + + + + + +$${{f_X}\left( {x;\eta ,b} \right) = b{\kern 1pt} \eta \exp \left( {\eta + bx - \eta {e^{bx}}} \right)} ; \quad 0 \le x < \infty ,\eta ,b > 0\\$$ + + +$${{F_X}\left( {x;\eta ,b} \right) = 1 - \exp \left[ { - \eta \left( {{e^{bx}} - 1} \right)} \right]} ; \quad 0 \le x < \infty ,\eta ,b > 0$$ + + +
+ +The second parameterisation in which the *product* $b\eta$ in the formulae above is replaced by the parameter $a$ giving: + +
+Gompertz distribution: Parameterisation II + + + + + + + + + + +$${{f_X}(x;a,b) = \frac{a}{b}\;{e^{\frac{x}{b}}}\;\exp \left[ { - a\left( {{e^{\frac{x}{b}}}\; - 1} \right)} \right]} {;\quad 0 \le x < \infty,a,b > 0}$$ + +$${{F_X}(x;a,b) = 1 - \;\exp \left[ { - a\left( {{e^{\frac{x}{b}}}\; - 1} \right)} \right]} {;\quad 0 \le x < \infty,a,b > 0}$$ + +
+ +```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Gompertz probability density (A) and cumulative probability (B) functions."} +par(mfrow = c(1, 2)) + +f <- function(x, n, b) { + y <- n * b * exp(n + b * x - n * exp(b * x)) + return(y) +} + +conc <- seq(0, 10, by = 0.005) +plot(conc, f(conc, 0.089, 1.25), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 2)) +lines(conc, f(conc, 0.001, 3.5), col = "#F2A61C") +lines(conc, f(conc, 0.0005, 1.1), col = "#1CADF2") +lines(conc, f(conc, 0.01, 5), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +F <- function(x, n, b) { + y <- 1 - exp(-n * exp(b * x - 1)) + return(y) +} + +conc <- seq(0, 10, by = 0.005) +plot(conc, F(conc, 0.089, 1.25), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, F(conc, 0.001, 3.5), col = "#F2A61C") +lines(conc, F(conc, 0.0005, 1.1), col = "#1CADF2") +lines(conc, F(conc, 0.01, 5), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") +``` + +The Gompertz distribution is available in `ssdtools`, however parameter estimation can be somewhat unstable [@fox_methodologies_2022], and for this reason it is not currently included in the default set. + +#### Weibull distribution + +The inclusion of the Weibull distribution and inverse Pareto distribution (see next) in `ssdtools` was primarily necessitated by the need to maintain consistency with the calculations undertaken in `Burrlioz`. +As mentioned earlier, both the Weibull and inverse Pareto distributions arise as *limiting distributions* when the Burr parameters $c$ and $k$ tend to either zero and/or infinity in specific ways. + +The two-parameter [Weibull](https://en.wikipedia.org/wiki/Weibull_distribution) distribution has the following *pdf* and *cdf*: + +
+Weibull distribution + + + + + + + + + + +$${{f_X}(x;a,b) = \frac{c}{\theta }\,{{\left( {\frac{x}{\theta }} \right)}^{c - 1}}{e^{ - {{\left( {\frac{x}{\theta }} \right)}^c}}}} {;\quad 0 \le x < \infty ,c,\theta > 0}$$ + + +$${{F_X}(x;a,b) = 1 - \;{e^{ - {{\left( {\frac{x}{\theta }} \right)}^c}}}} {;\quad 0 \le x < \infty ,c,\theta > 0}$$ + +
+ +The Weibull distribution can be fitted in `ssdtools` by supplying the string `weibull` to the `dists` argument in the `ssd_fit_dists` call. + +```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Weibull probability density (A) and cumulative probability (B) functions."} +par(mfrow = c(1, 2)) + +conc <- seq(0, 10, by = 0.005) +plot(conc, dweibull(conc, 4.321, 4.949), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, dweibull(conc, 0.838, 0.911), col = "#F2A61C") +lines(conc, dweibull(conc, 1, 1.546), col = "#1CADF2") +lines(conc, dweibull(conc, 17.267, 7.219), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +conc <- seq(0, 10, by = 0.005) +plot(conc, pweibull(conc, 4.321, 4.949), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, pweibull(conc, 0.838, 0.911), col = "#F2A61C") +lines(conc, pweibull(conc, 1, 1.546), col = "#1CADF2") +lines(conc, pweibull(conc, 17.267, 7.219), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") +``` + + +#### Inverse Pareto distribution + +The inverse Pareto distribution can be fitted using `ssdtools` by supplying the string `invpareto` to the `dists` argument in the `ssd_fit_dists` call. + +While the inverse Pareto distribution is implemented in the `Burrlioz 2.0` software, it is important to understand that it is done so only as one of the limiting Burr distributions (see [technical details](https://bcgov.github.io/ssdtools/articles/E_additional-technical-details.html)). +The inverse Pareto is not offered as a stand-alone option in the `Burrlioz 2.0` software. We have spent considerable time and effort exploring the properties of the inverse Pareto distribution, including deriving bias correction equations and alternative methods for deriving confidence intervals [@fox_methodologies_2022]. +This work has substantial value for improving the current `Burrlioz 2.0` method, and our bias corrections should be adopted when deriving *HCx* estimates from the inverse Pareto where parameters have been estimated using maximum likelihood. + +As is the case with the `Burrlioz 2.0` software, we have decided not to include the inverse Pareto distribution in the default candidate set in `ssdtools` although it is offered ass a user-selectable distribution to use in the model-fitting process. + +As with many statistical distributions, different 'variants' exist. These 'variants' are not so much different distributions as they are simple re-parameterisations. +For example, many distributions have a *scale* parameter, $\beta$ and some authors and texts will use $\beta$ while others use $\frac{1}{\beta }$. +An example of this re-parameterisation was given above for the Gompertz distribution. +While the choice of mathematical representation may be purely preferential, it is sometimes done for mathematical convenience. +For example, Parameterisation I of the Gompertz distribution above was obtained by letting $a = b\eta$ in Parameterisation II. +This re-expression involving parameters $b$ and $\eta$ would be particularly useful when trying to fit a distribution for which one of $\left\{ {b,\,\eta } \right\}$ was very small and the other was very large. + +It has already been noted that the particular parameterisation of the (Inverse)Pareto distribution used in both `Burrlioz 2.0` and `ssdtools` was **not** a matter of preference, but rather was dictated by mathematical considerations which demonstrated convergence of the Burr distribution to one specific version of the (Inverse)Pareto distribution. +While the mathematics provides an elegant solution to an otherwise problematic situation, this version of the (Inverse)Pareto distribution is not particularly use as a stand-alone distribution for fitting an SSD (other than as a special, limiting case of the Burr distribution). + +The two versions of the (Inverse)Pareto distribution are known as the *European* and *North American* versions. Their *pdfs* and *cdfs* are given below. + +
+ +
+                    (Inverse)Pareto distribution - North American version

Pareto distribution + + + + + + + + + + + + + + + + + + + + +$${{f_X}\left( {x;\alpha ,\beta } \right) = \alpha {\kern 1pt} {\beta ^\alpha }{\kern 1pt} {x^{ - \left( {\alpha + 1} \right)}}} ; \quad x > \beta ;\;\alpha ,\beta > 0\\$$ + + +$${{F_X}\left( {x;\alpha ,\beta } \right) = 1 - {{\left( {\frac{\beta }{x}} \right)}^\alpha }}; \quad x > \beta ;\;\alpha ,\beta > 0$$ + + +Now, if X has the Pareto distribution above, then \[Y = \frac{1}{X}\] has an inverse Pareto distribution.

inverse Pareto distribution + + +$${{g_Y}\left( {y;\alpha ,\beta } \right) = \alpha {\kern 1pt} {\beta ^\alpha }{\kern 1pt} {y^{\alpha - 1}}} ;\quad y \le \frac{1}{\beta };\;\;\alpha ,\beta > 0\\$$ + +$${{G_Y}\left( {y;\alpha ,\beta } \right) = {{\left( {\beta \,y} \right)}^\alpha }}; \quad 0 < y \le \frac{1}{\beta };\;\;\alpha ,\beta > 0$$ + +
+ +

Importantly, we see that the *North American* versions of these distributions are bounded with the Pareto distribution bounded **below** by $\beta$ and the inverse Pareto distribution bounded *above* by $\frac{1}{\beta }$.
As an aside, the *mle* of $\beta$ in the Pareto distribution is \[\hat \beta = \min \left\{ {{X_1}, \ldots ,{X_n}} \right\}\] and the *mle* of $\frac{1}{\beta }$ in the inverse Pareto is \[\begin{array}{*{20}{l}} +{\tilde \beta = \max \left\{ {{Y_1}, \ldots ,{Y_n}} \right\}}\\ +{\quad = \max \left\{ {\frac{1}{{{X_1}}}, \ldots ,\frac{1}{{{X_n}}}} \right\} = \frac{1}{{\min \left\{ {{X_1}, \ldots ,{X_n}} \right\}}}}\\ +{\quad = \frac{1}{{\hat \beta }}} +\end{array}\]. + +and the *mle* of $\alpha$ is: \[\hat \alpha = {\left[ {\ln \left( {\frac{g}{{\hat \beta }}} \right)} \right]^{ - 1}}\] where $g$ is the *geometric mean*: \[g = {\left[ {\prod\limits_{i = 1}^n {{X_i}} } \right]^{\frac{1}{n}}}\] + +Thus, it doesn't matter whether you're fitting a Pareto or inverse Pareto distribution to your data - the parameter estimates are the same. + +Because it is *bounded*, the North American version of the (Inverse)Pareto distribution is not useful as a stand-alone SSD - more so for the *inverse Pareto* distribution since it is bounded from *above*. + +```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample North American Pareto probability density (A) and cumulative probability (B) functions."} +par(mfrow = c(1, 2)) + +conc <- seq(0, 10, by = 0.005) +plot(conc, extraDistr::dpareto(conc, 3, 2), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.5)) +lines(conc, extraDistr::dpareto(conc, 0.838, 0.911), col = "#F2A61C") +lines(conc, extraDistr::dpareto(conc, 4, 4), col = "#1CADF2") +lines(conc, extraDistr::dpareto(conc, 10, 7), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +conc <- seq(0, 10, by = 0.005) +plot(conc, extraDistr::ppareto(conc, 3, 2), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, extraDistr::ppareto(conc, 0.838, 0.911), col = "#F2A61C") +lines(conc, extraDistr::ppareto(conc, 4, 4), col = "#1CADF2") +lines(conc, extraDistr::ppareto(conc, 10, 7), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") + +conc <- seq(0, 10, by = 0.005) +plot(conc, extraDistr::ppareto(conc, 3, 2), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, extraDistr::ppareto(conc, 0.838, 0.911), col = "#F2A61C") +lines(conc, extraDistr::ppareto(conc, 4, 4), col = "#1CADF2") +lines(conc, extraDistr::ppareto(conc, 10, 7), col = "#1F1CF2") +``` + +```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Sample North American inverse Pareto probability density (A) and cumulative probability (B) functions."} +par(mfrow = c(1, 2)) + +f <- function(x, a, b) { + y <- a * (b^a) * x^(a - 1) + return(y) +} + +conc <- seq(0, 10, by = 0.001) +plot(conc, f(conc, 5, 0.1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 0.5)) +lines(conc, f(conc, 3, 0.1), col = "#F2A61C") +lines(conc, f(conc, 0.5, 0.1), col = "#1CADF2") +lines(conc, f(conc, 0.1, 0.1), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +F <- function(x, a, b) { + y <- (b * x)^a + return(y) +} + +conc <- seq(0, 10, by = 0.001) +plot(conc, F(conc, 5, 0.1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, F(conc, 3, 0.1), col = "#F2A61C") +lines(conc, F(conc, 0.5, 0.1), col = "#1CADF2") +lines(conc, F(conc, 0.1, 0.1), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") +``` + +We see from the *pdf* plots that the alternative, *European* version of the inverse Pareto distribution is a more realistic candidate. + +
+ +
+                    (Inverse)Pareto distribution - European version

Pareto distribution + + +$${{f_X}\left( {x;\alpha ,\beta } \right) = \frac{{\alpha \,\beta \,{x^{\alpha - 1}}}}{{{{\left( {x + \beta } \right)}^{\alpha + 1}}}}} ; \quad x,\;\alpha ,\beta > 0\\$$ + + +$${{F_X}\left( {x;\alpha ,\beta } \right) = 1 - {{\left( {\frac{\beta }{{x + \beta }}} \right)}^\alpha }}; +\quad x,\;\alpha ,\beta > 0$$ + + +Now, if X has the Pareto distribution above, then \[Y = \frac{1}{X}\] has an inverse Pareto distribution.

inverse Pareto distribution + +$${{g_Y}\left( {y;\alpha ,\beta } \right) = \frac{{\alpha {\kern 1pt} {\beta ^\alpha }{\kern 1pt} {y^{\alpha - 1}}}}{{{{\left( {1 + \beta y} \right)}^{\alpha + 1}}}}}; +\quad y,\;\alpha ,\beta > 0\\$$ + + +$${{G_Y}\left( {y;\alpha ,\beta } \right) = {{\left( {\frac{{\beta y}}{{1 + \beta y}}} \right)}^\alpha }}; +\quad y,\;\alpha ,\beta > 0$$ + +
+ +We note in passing that *both* versions of these Pareto and inverse Pareto distrbutions are available in `R`. +For example, the `R`package `extraDistr` has North American versions, while the `actuar` package has European versions. + +```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample European Pareto probability density (A) and cumulative probability (B) functions."} +par(mfrow = c(1, 2)) + +conc <- seq(0, 10, by = 0.005) +plot(conc, actuar::dpareto(conc, 1, 1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, actuar::dpareto(conc, 2, 3), col = "#F2A61C") +lines(conc, actuar::dpareto(conc, 0.5, 1), col = "#1CADF2") +lines(conc, actuar::dpareto(conc, 10.5, 6.5), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +conc <- seq(0, 10, by = 0.005) +plot(conc, actuar::ppareto(conc, 1, 1), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, actuar::ppareto(conc, 2, 3), col = "#F2A61C") +lines(conc, actuar::ppareto(conc, 0.5, 1), col = "#1CADF2") +lines(conc, actuar::ppareto(conc, 10.5, 6.5), col = "#1F1CF2") +legend("topleft", "(B)", bty = "n") +``` + +```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample European inverse Pareto probability density (A) and cumulative probability (B) functions."} +par(mfrow = c(1, 2)) + +conc <- seq(0, 10, by = 0.001) +plot(conc, actuar::dinvpareto(conc, 1, 1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 0.8)) +lines(conc, actuar::dinvpareto(conc, 1, 3), col = "#F2A61C") +lines(conc, actuar::dinvpareto(conc, 10, 0.1), col = "#1CADF2") +lines(conc, actuar::dinvpareto(conc, 2.045, 0.98), col = "#1F1CF2") +legend("topleft", "(A)", bty = "n") + +conc <- seq(0, 10, by = 0.001) +plot(conc, actuar::pinvpareto(conc, 1, 1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) +lines(conc, actuar::pinvpareto(conc, 1, 3), col = "#F2A61C") +lines(conc, actuar::pinvpareto(conc, 10, 0.1), col = "#1CADF2") +lines(conc, actuar::pinvpareto(conc, 2.045, 0.98), col = "#1F1CF2") +``` + +#### Inverse Weibull distribution (see log-Gumbel, above) + +The inverse Weibull is mathematically equivalent to the log-Gumbel distribution described above. +While which is also a limiting distribution of the Burr Type 3, this distribution does not show the same instability issues, and is unbounded to the right. +It therefore represents a valid SSD distribution and is included in the default model set as a distribution in its own right. + +The inverse Weibull (log-Gumbel) distribution can be fitted in `ssdtools` by supplying the string `lgumbel` to the `dists` argument in the `ssd_fit_dists` call. + +## Relationships among distributions in `ssdtools` + +##### NOTES +1. In the diagram below, $X$ denotes the random variable in the box at the *beginning* of the arrow and the expression beside the arrow indicates the mathematical transformation of $X$ such that the resultant *transformed* data has the distribution identified in the box at the *end* of the arrow. + +2. *Reciprocal* transformations ($\frac{1}{X}$) are **bi-directional** ($\leftrightarrow$). + +3. Although the *negative exponential* distribution is **not** explicitly included in `ssdtools`, it is a special case of the *gamma* distribution with $c=1$. It is included in this figure as it is related to other distributions that *are* included in `ssdtools`. + +4. The *European* versions of the Pareto and inverse Pareto distributions are **unbounded**; the *North American* versions are **bounded**. + + +
+ +![](images/relationships.png){width=100%} + +## References + +
+ +```{r, results = "asis", echo = FALSE} +cat(ssdtools::ssd_licensing_md()) +``` diff --git a/doc/model-averaging.R b/doc/model-averaging.R new file mode 100644 index 00000000..27ed1e72 --- /dev/null +++ b/doc/model-averaging.R @@ -0,0 +1,173 @@ +## ----include = FALSE---------------------------------------------------------- +knitr::opts_chunk$set( + collapse = TRUE, + comment = "#>", + fig.width = 6, + fig.height = 4 +) +library(tinytex) + +## ----include=FALSE------------------------------------------------------------ +# this just loads the package and suppresses the load package message +library(ssdtools) +require(ggplot2) + +## ----echo=FALSE,warning=FALSE, message=FALSE,class.output="scroll-100"-------- +samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +print(samp) +# knitr::kable(samp,caption="Some toxicity data (concentrations)") + +## ----echo=FALSE,fig.cap="Emprirical cdf (black); Model 1(green); and Model 2 (blue)", fig.width=7,fig.height=4.5---- +samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +samp <- sort(samp) +plot(ecdf(samp), main = "Empirical and fitted SSDs", xlab = "Concentration", ylab = "Probability") +xx <- seq(0.01, 3, by = 0.01) +lines(xx, plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10]))), col = "#77d408") +lines(xx, plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))), col = "#08afd4") +# lines(xx,(0.4419*plnorm(xx,meanlog=mean(log(samp[5:15])),sd=sd(log(samp[5:15])))+ +# 0.5581*plnorm(xx,meanlog=mean(log(samp[1:10])),sd=sd(log(samp[1:10])))) ,col="#d40830") + +## ----echo=FALSE,fig.cap="Empirical cdf (black); Model 1(green); Model 2 (blue); and averaged Model (red)",fig.width=7,fig.height=4.5---- +samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +samp <- sort(samp) +plot(ecdf(samp), main = "Empirical and fitted SSDs", xlab = "Concentration", ylab = "Probability") +xx <- seq(0.01, 3, by = 0.01) +lines(xx, plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10]))), col = "#77d408") +lines(xx, plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))), col = "#08afd4") +lines(xx, (0.4419 * plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))) + + 0.5581 * plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10])))), col = "#d40830") + +## ----echo=TRUE---------------------------------------------------------------- +# Model 1 HC20 +cat("Model 1 HC20 =", qlnorm(0.2, -1.067, 0.414)) + +# Model 2 HC20 +cat("Model 2 HC20 =", qlnorm(0.2, -0.387, 0.617)) + +## ----echo=FALSE,fig.width=7,fig.height=5-------------------------------------- +samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +samp <- sort(samp) +xx <- seq(0.01, 3, by = 0.01) + + +plot(xx, (0.4419 * plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))) + + 0.5581 * plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10])))), +col = "#d40830", type = "l", xlab = "Concentration", ylab = "Probability" +) +segments(0.33, -1, 0.33, 0.292, col = "blue", lty = 21) +segments(-1, 0.292, 0.33, 0.292, col = "blue", lty = 21) +mtext("0.3", side = 2, at = 0.3, cex = 0.8, col = "blue") +mtext("0.33", side = 1, at = 0.33, cex = 0.8, col = "blue") + +## ----echo=FALSE, fig.width=8,fig.height=6------------------------------------- +t <- seq(0.01, 0.99, by = 0.001) + + +F <- 0.4419 * qlnorm(t, -1.067, 0.414) + 0.5581 * qlnorm(t, -0.387, 0.617) + +plot(xx, (0.4419 * plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))) + + 0.5581 * plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10])))), +col = "#d40830", type = "l", xlab = "Concentration", ylab = "Probability" +) + +lines(F, t, col = "#51c157", lwd = 1.75) + +segments(-1, 0.2, 0.34, 0.2, col = "black", lty = 21, lwd = 2) +segments(0.28, 0.2, 0.28, -1, col = "red", lty = 21, lwd = 2) +segments(0.34, 0.2, 0.34, -1, col = "#51c157", lty = 21, lwd = 2) +segments(1.12, -1, 1.12, 0.9, col = "grey", lty = 21, lwd = 1.7) + +text(0.25, 0.6, "Correct MA-SSD", col = "red", cex = 0.75) +text(0.75, 0.4, "Erroneous MA-SSD", col = "#51c157", cex = 0.75) +mtext("1.12", side = 1, at = 1.12, cex = 0.8, col = "grey") + +## ----echo=FALSE,warning=FALSE, results="markup",message=FALSE,class.output="scroll-100"---- +samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +print(samp) +# knitr::kable(samp,caption="Some toxicity data (concentrations)") + +## ----echo=TRUE,results='hide',warning=FALSE,message=FALSE--------------------- +dat <- data.frame(Conc = c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59)) +library(goftest) +library(EnvStats) # this is required for the Pareto cdf (ppareto) + +# Examine the fit for the gamma distribution (NB: parameters estimated from the data) +cvm.test(dat$Conc, null = "pgamma", shape = 2.0591977, scale = 0.3231032, estimated = TRUE) + +# Examine the fit for the lognormal distribution (NB: parameters estimated from the data) +cvm.test(dat$Conc, null = "plnorm", meanlog = -0.6695120, sd = 0.7199573, estimated = TRUE) + +# Examine the fit for the Pareto distribution (NB: parameters estimated from the data) +cvm.test(dat$Conc, null = "ppareto", location = 0.1800000, shape = 0.9566756, estimated = TRUE) + +## ----echo=FALSE,fig.cap="Emprirical cdf (black); lognormal (green); gamma (blue); and Pareeto (red)", fig.width=7,fig.height=4.5---- +library(EnvStats) +samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +samp <- sort(samp) +plot(ecdf(samp), main = "Empirical and fitted SSDs", xlab = "Concentration", ylab = "Probability") +xx <- seq(0.01, 3, by = 0.01) +lines(xx, plnorm(xx, meanlog = -0.6695120, sd = 0.7199573), col = "#77d408") +lines(xx, pgamma(xx, shape = 2.0591977, scale = 0.3231032), col = "#08afd4") +lines(xx, ppareto(xx, location = 0.1800000, shape = 0.9566756), col = "red") +# lines(xx,(0.4419*plnorm(xx,meanlog=mean(log(samp[5:15])),sd=sd(log(samp[5:15])))+ +# 0.5581*plnorm(xx,meanlog=mean(log(samp[1:10])),sd=sd(log(samp[1:10])))) ,col="#d40830") + +## ----echo=TRUE---------------------------------------------------------------- +sum(log(dgamma(dat$Conc, shape = 2.0591977, scale = 0.3231032))) +sum(log(dlnorm(dat$Conc, meanlog = -0.6695120, sdlog = 0.7199573))) +sum(log(EnvStats::dpareto(dat$Conc, location = 0.1800000, shape = 0.9566756))) + +## ----echo=FALSE,results='markup'---------------------------------------------- +dat <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +k <- 2 # number of parameters for each of the distributions +# Gamma distribution +aic1 <- 2 * k - 2 * sum(log(dgamma(dat, shape = 2.0591977, scale = 0.3231032))) +cat("AIC for gamma distribution =", aic1, "\n") + +# lognormal distribution +aic2 <- 2 * k - 2 * sum(log(dlnorm(dat, meanlog = -0.6695120, sdlog = 0.7199573))) +cat("AIC for lognormal distribution =", aic2, "\n") + +# Pareto distribution +aic3 <- 2 * k - 2 * sum(log(EnvStats::dpareto(dat, location = 0.1800000, shape = 0.9566756))) +cat("AIC for Pareto distribution =", aic3, "\n") + +## ----echo=TRUE,results='hide'------------------------------------------------- +dat <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +aic <- NULL +k <- 2 # number of parameters for each of the distributions + + +aic[1] <- 2 * k - 2 * sum(log(dgamma(dat, shape = 2.0591977, scale = 0.3231032))) # Gamma distribution + +aic[2] <- 2 * k - 2 * sum(log(dlnorm(dat, meanlog = -0.6695120, sdlog = 0.7199573))) # lognormal distribution + +aic[3] <- 2 * k - 2 * sum(log(EnvStats::dpareto(dat, location = 0.1800000, shape = 0.9566756))) # Pareto distribution + +delta <- aic - min(aic) # compute the delta values + +aic.w <- exp(-0.5 * delta) +aic.w <- round(aic.w / sum(aic.w), 4) + +cat( + " AIC weight for gamma distribution =", aic.w[1], "\n", + "AIC weight for lognormal distribution =", aic.w[2], "\n", + "AIC weight for pareto distribution =", aic.w[3], "\n" +) + +## ----echo=FALSE,fig.cap="Empirical cdf (black) and model-averaged fit (magenta)",fig.width=8,fig.height=5---- +samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +samp <- sort(samp) +plot(ecdf(samp), main = "Empirical and fitted SSDs", xlab = "Concentration", ylab = "Probability") +xx <- seq(0.01, 3, by = 0.005) + +lines(xx, plnorm(xx, meanlog = -0.6695120, sd = 0.7199573), col = "#959495", lty = 2) +lines(xx, pgamma(xx, shape = 2.0591977, scale = 0.3231032), col = "#959495", lty = 3) +lines(xx, ppareto(xx, location = 0.1800000, shape = 0.9566756), col = "#959495", lty = 4) +lines(xx, 0.1191 * pgamma(xx, shape = 2.0591977, scale = 0.3231032) + + 0.3985 * plnorm(xx, meanlog = -0.6695120, sd = 0.7199573) + + 0.4824 * ppareto(xx, location = 0.1800000, shape = 0.9566756), col = "#FF33D5", lwd = 1.5) + +## ----results = "asis", echo = FALSE------------------------------------------- +cat(ssdtools::ssd_licensing_md()) + diff --git a/doc/model-averaging.Rmd b/doc/model-averaging.Rmd new file mode 100644 index 00000000..a00e73b7 --- /dev/null +++ b/doc/model-averaging.Rmd @@ -0,0 +1,538 @@ +--- +title: "Model Averaging SSDs" +author: "ssdtools Team" +date: '`r format(Sys.time(), "%Y-%m-%d", tz = "UTC")`' +bibliography: references.bib +mathfont: Courier +latex_engine: MathJax +output: rmarkdown::html_vignette +vignette: > + %\VignetteIndexEntry{Model Averaging SSDs} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteEncoding{UTF-8} +--- + +```{r, include = FALSE} +knitr::opts_chunk$set( + collapse = TRUE, + comment = "#>", + fig.width = 6, + fig.height = 4 +) +library(tinytex) +``` + + +```{css, echo=FALSE} +pre { + max-height: 300px; + overflow-y: auto; +} + +pre[class] { + max-height: 200px; +} +``` + +```{css, echo=FALSE} +.scroll-200 { + max-height: 200px; + overflow-y: auto; + background-color: inherit; +} +``` + + + + + +```{r include=FALSE} +# this just loads the package and suppresses the load package message +library(ssdtools) +require(ggplot2) +``` + +## Background + +> Many authors have noted that there is no guiding theory in ecotoxicology to justify any particular distributional form for the SSD other than that its domain be restricted to the positive real line [@newman_2000], [@Zajdlik_2005], [@chapman_2007], [@fox_2016]. +Indeed, [@chapman_2007] described the identification of a suitable probability model as one of the most important and difficult choices in the use of SSDs. +Compounding this lack of clarity about the functional form of the SSD is the omnipresent, and equally vexatious issue of small sample size, meaning that any plausible candidate model is unlikely to be rejected [@fox_recent_2021]. +The ssdtools R package uses a model averaging procedure to avoid the need to a-priori select a candidate distribution and instead uses a measure of ‘fit’ for each model to compute weights to be applied to an initial set of candidate distributions. +The method, as applied in the SSD context is described in detail in [@fox_recent_2021], and potentially provides a level of flexibility and parsimony that is difficult to achieve with a single SSD distribution. + +[@fox_methodologies_2022] + +## Preliminaries + +Before we jump into model averaging and in particular, SSD Model Averaging, let's backup a little and consider why we average and the advantages and disadvantages of averaging. + +#### The pros and cons of averaging + +We're all familiar with the process of averaging. +Indeed, *averages* are pervasive in everyday life - we talk of average income; mean sea level; average global temperature; average height, weight, age etc. etc. So what's the obsession with *averaging*? +It's simple really - it's what statisticians call data reduction which is just a fancy name to describe the process of summarising a lot of *raw data* using a small number of (hopefully) representative summary statistics such as the mean and the standard deviation. +Clearly, it's a lot easier to work with just a single mean than all the individual data values. +That's the upside. +The downside is that the process of data reduction decimates your original data - you lose information in the process. +Nevertheless, the benefits tend to outweigh this information loss. +Indeed, much of 'conventional' statistical theory and practice is focused on the mean. +Examples include T-tests, ANOVA, regression, and clustering. +When we talk of an 'average' we are usually referring to the simple, *arithmetic mean*:\[\bar{X}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{X}_{i}}}\] although we recognize there are other types of mean including the geometric mean, the harmonic mean and the weighted mean. +The last of these is particularly pertinent to model averaging. + +### Weighted Averages + +For the simple arithmetic mean, all of the individual values receive the same weighting - they each contribute $\frac{1}{n}$ to the summation. +While this is appropriate in many cases, it's not useful when the components contribute to varying degrees. +An example familiar to ecotoxicologists is that of a *time-varying* concentration as shown in the figure below.
+ +![](images/Figure1.jpg){width=90% align="center"} + +From the figure we see there are 5 concentrations going from left to to right: $\left\{ 0.25,0.95,0.25,0.12,0.5 \right\}$. +If we were to take the simple arithmetic mean of these concentrations we get $\bar{X}=0.414$. +But this ignores the different *durations* of these 5 concentrations. +Of the 170 hours, 63 were at concentration 0.25, 25 at concentration 0.95, 23 at concentration 0.25, 23 at concentration 0.12, and 36 at concentration 0.50. +So if we were to *weight* these concentrations by time have:
\[{{{\bar{X}}}_{TW}}=\frac{\left( 63\cdot 0.25+25\cdot 0.95+23\cdot 0.25+23\cdot 0.12+36\cdot 0.50 \right)}{\left( 63+25+23+23+36 \right)}=\frac{56.01}{170}=0.33\]
So, our formula for a *weighted average* is:\[\bar{X}=\sum\limits_{i=1}^{n}{{{w}_{i}}{{X}_{i}}}\] with $0\le {{w}_{i}}\le 1$ and $\sum\limits_{i=1}^{n}{{{w}_{i}}=1}$. +
Note, the simple arithmetic mean is just a special case of the weighted mean with $\sum\limits_{i=1}^{n}{{{w}_{i}}=\frac{1}{n}}$ ; $\forall i=1,\ldots ,n$ + +## Model Averaging +The *weighted average* acknowledges that the elements in the computation are not of equal 'importance'. +In the example above, this importance was based on the *proportion of time* that the concentration was at a particular level. +Bayesians are well-versed in this concept - the elicitation of *prior distributions* for model parameters provides a mechanism for weighting the degree to which the analysis is informed by existing knowledge versus using a purely data-driven approach. +Model averaging is usually used in the context of estimating model parameters or quantities derived from a fitted model - for example an EC50 derived from a C-R model. +Let's motivate the discussion using the following small dataset of toxicity estimates for some chemical. + + +```{r echo=FALSE,warning=FALSE, message=FALSE,class.output="scroll-100"} +samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +print(samp) +# knitr::kable(samp,caption="Some toxicity data (concentrations)") +``` + +Now, suppose we have only two possibilities for fitting an SSD - both lognormal distributions. +Model 1 is the LN(-1.067,0.414) distribution while Model 2 is the LN(-0.387,0.617) distribution. +A plot of the empirical *cdf* and Models 1 and 2 is shown below. + +```{r echo=FALSE,fig.cap="Emprirical cdf (black); Model 1(green); and Model 2 (blue)", fig.width=7,fig.height=4.5} +samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +samp <- sort(samp) +plot(ecdf(samp), main = "Empirical and fitted SSDs", xlab = "Concentration", ylab = "Probability") +xx <- seq(0.01, 3, by = 0.01) +lines(xx, plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10]))), col = "#77d408") +lines(xx, plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))), col = "#08afd4") +# lines(xx,(0.4419*plnorm(xx,meanlog=mean(log(samp[5:15])),sd=sd(log(samp[5:15])))+ +# 0.5581*plnorm(xx,meanlog=mean(log(samp[1:10])),sd=sd(log(samp[1:10])))) ,col="#d40830") +``` + +
We see that Model 1 fits well in the lower, left region and poorly in the upper region, while the reverse is true for Model 2. +So using *either* Model 1 **or** Model 2 is going to result in a poor fit overall. +However, the obvious thing to do is to **combine** both models. +We could just try using 50% of Model 1 and 50% of Model 2, but that may be sub-optimal. +It turns out that the best fit is obtained by using 44% of Model 1 and 56% of Model 2. Redrawing the plot and adding the *weighted average* of Models 1 and 2 is shown below. + +```{r echo=FALSE,fig.cap="Empirical cdf (black); Model 1(green); Model 2 (blue); and averaged Model (red)",fig.width=7,fig.height=4.5} +samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +samp <- sort(samp) +plot(ecdf(samp), main = "Empirical and fitted SSDs", xlab = "Concentration", ylab = "Probability") +xx <- seq(0.01, 3, by = 0.01) +lines(xx, plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10]))), col = "#77d408") +lines(xx, plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))), col = "#08afd4") +lines(xx, (0.4419 * plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))) + + 0.5581 * plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10])))), col = "#d40830") +``` + +
Clearly the strategy has worked - we now have an excellent fitting SSD. +What about estimation of an *HC20*? +It's a simple matter to work out the *individual* *HC20* values for Models 1&2 using the appropriate `qlnorm()` function in `R`. +Thus we have: + +```{r, echo=TRUE} +# Model 1 HC20 +cat("Model 1 HC20 =", qlnorm(0.2, -1.067, 0.414)) + +# Model 2 HC20 +cat("Model 2 HC20 =", qlnorm(0.2, -0.387, 0.617)) +``` +What about the averaged distribution? +An intuitively appealing approach would be to apply the same weights to the individual *HC20* values as was applied to the respective models. That is `0.44*0.2428209 + 0.56*0.4040243 = 0.33`. + +So our model-averaged *HC20* estimate is 0.33. +As a check, we can determine the *fraction affected* at concentration = 0.33 - it should of course be 20%. Let's take a look at the plot. + +```{r echo=FALSE,fig.width=7,fig.height=5} +samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +samp <- sort(samp) +xx <- seq(0.01, 3, by = 0.01) + + +plot(xx, (0.4419 * plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))) + + 0.5581 * plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10])))), +col = "#d40830", type = "l", xlab = "Concentration", ylab = "Probability" +) +segments(0.33, -1, 0.33, 0.292, col = "blue", lty = 21) +segments(-1, 0.292, 0.33, 0.292, col = "blue", lty = 21) +mtext("0.3", side = 2, at = 0.3, cex = 0.8, col = "blue") +mtext("0.33", side = 1, at = 0.33, cex = 0.8, col = "blue") +``` + +Something's wrong - the fraction affected at concentration 0.33 is 30% - **not the required 20%**. +This issue is taken up in the next section + +## Model Averaged SSDs +As we've just seen, applying the model weights to component *HCx* values and summing does **not** produce the correct result. +The reason for this can be explained mathematically as follows (*if your not interested in the mathematical explanation - skip ahead to the next section*). + +

The fallacy of weighting individual *HCx* values

+ +The correct expression for a model-averaged SSD is: $$G\left( x \right) = \sum\limits_{i = 1}^k {{w_i}} {F_i}\left( x \right)$$ where ${F_i}\left( \cdot \right)$ is the *i^th^* component SSD (i.e. *cdf*) and *w~i~* is the weight assigned to ${F_i}\left( \cdot \right)$. +
Notice that the function $G\left( x \right)$ is a proper *cumulative distribution function* (*cdf*) which means for a given quantile, *x*, $G\left( x \right)$ returns the *cumulative probability*: $$P\left[ {X \leqslant x} \right]$$ + +
Now, the *incorrect* approach takes a weighted sum of the component *inverse cdf's*, that is: + +$$H\left( p \right) = \sum\limits_{i = 1}^k {{w_i}} {F_i}^{ - 1}\left( p \right)$$ +where ${F_i}^{ - 1}\left( \cdot \right)$ is the *i^th^* *inverse cdf*. +Notice that ${G_i}\left( \cdot \right)$ is a function of a *quantile* and returns a **probability** while ${H_i}\left( \cdot \right)$ is a function of a *probability* and returns an **quantile**. +

Now, the correct method of determining the *HCx* is to work with the proper model-averaged *cdf* $G\left( x \right)$. +This means finding the **inverse** function ${G^{ - 1}}\left( p \right)$. +We'll address how we do this in a moment.

+The reason why $H\left( p \right)$ does **not** return the correct result is because of the implicit assumption that the inverse of $G\left( x \right)$ is equivalent to $H\left( p \right)$. +This is akin to stating the inverse of a *sum* is equal to the sum of the inverses i.e. +$$\sum\limits_{i = 1}^n {\frac{1}{{{X_i}}}} = \frac{1}{{\sum\limits_{i = 1}^n {{X_i}} }}{\text{ ???}}$$ +
+For the mathematical nerds: +There are some very special cases where the above identity does in fact hold, but for that you need to use **complex numbers**. +

For example, consider two complex numbers $${\text{a = }}\frac{{\left( {5 - i} \right)}}{2}{\text{ and }}b = - 1.683 - 1.915i$$ +It can be shown that $$\frac{1}{{a + b}} = \frac{1}{a} + \frac{1}{b} = 0.126 + 0.372i$$ +

+ +

Back to the issue at hand, and since we're not dealing with complex numbers, it's safe to say:$${G^{ - 1}}\left( p \right) \ne H\left( p \right)$$ +

+If you need a visual demonstration, we can plot $G\left( x \right)$ and the *inverse* of $H\left( p \right)$ both as functions of *x* (a quantile) for our two-component lognormal distribution above. + + +```{r echo=FALSE, fig.width=8,fig.height=6} +t <- seq(0.01, 0.99, by = 0.001) + + +F <- 0.4419 * qlnorm(t, -1.067, 0.414) + 0.5581 * qlnorm(t, -0.387, 0.617) + +plot(xx, (0.4419 * plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))) + + 0.5581 * plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10])))), +col = "#d40830", type = "l", xlab = "Concentration", ylab = "Probability" +) + +lines(F, t, col = "#51c157", lwd = 1.75) + +segments(-1, 0.2, 0.34, 0.2, col = "black", lty = 21, lwd = 2) +segments(0.28, 0.2, 0.28, -1, col = "red", lty = 21, lwd = 2) +segments(0.34, 0.2, 0.34, -1, col = "#51c157", lty = 21, lwd = 2) +segments(1.12, -1, 1.12, 0.9, col = "grey", lty = 21, lwd = 1.7) + +text(0.25, 0.6, "Correct MA-SSD", col = "red", cex = 0.75) +text(0.75, 0.4, "Erroneous MA-SSD", col = "#51c157", cex = 0.75) +mtext("1.12", side = 1, at = 1.12, cex = 0.8, col = "grey") +``` + +Clearly, the two functions are **not** the same and thus *HCx* values derived from each will nearly always be different (as indicated by the positions of the vertical red and green dashed lines in the Figure above corresponding to the 2 values of the *HC20*). +(Note: The two curves do cross over at a concentration of about 1.12 corresponding to the 90^th^ percentile, but in the region of ecotoxicological interest, there is no such cross-over and so the two approaches will **always** yield different *HCx* values with this difference → 0 as x → 0). +

We next discuss the use of a model-averaged SSD to obtain the *correct* model-averaged *HCx*. + +## Computing a model-averaged *HCx* +A proper *HCx* needs to satisfy what David Fox refers to as **the inversion principle**. +

More formally, the inversion principle states that an *HCx* (denoted as ${\varphi _x}$) must satisfy the following:

$$df\left( {{\varphi _x}} \right) = x\quad \quad and\quad \quad qf\left( x \right) = {\varphi _x}$$

+where $df\left( \cdot \right)$ is a model-averaged *distribution function* (i.e. SSD) and $qf\left( \cdot \right)$ is a model-averaged *quantile function*. +For this equality to hold, it is necessary that $qf\left( p \right) = d{f^{ - 1}}\left( p \right)$.

+
So, in our example above, the green curve was taken to be $qf\left( x \right)$ and this was used to derive ${\varphi _x}$ but the *fraction affected* $\left\{ { = df\left( {{\varphi _x}} \right)} \right\}$ at ${\varphi _x}$ is computed using the red curve. +

In `ssdtools` the following is a check that the inversion principle holds:

+
+ +``` +# Obtain a model-averaged HCx using the ssd_hc() function +hcp<-ssd_hc(x, p = p) +# Check that the inversion principle holds +ssd_hp(x, hcp, multi_est = TRUE) == p # this should result in logical `TRUE` +``` +*Note: if the `multi_est` argument is set to `FALSE` the test will fail*. +
+ +

The *inversion principle* ensures that we only use a **single** distribution function to compute both the *HCx* *and* the fraction affected. +Referring to the figure below, the *HCx* is obtained from the MA-SSD (red curve) by following the → arrows while the fraction affected is obtained by following the ← arrows. + +![](images/Figure2.jpg){width=100% align="center"} + +

Finally, we'll briefly discuss how the *HCx* is computed in `R` using the same method as has been implemented in `ssdtools`.

+ +### Computing the *HCx* in `R`/`ssdtools` +Recall, our MA-SSD was given as +$$G\left( x \right) = \sum\limits_{i = 1}^k {{w_i}} {F_i}\left( x \right)$$ +and an *HCx* is obtained from the MA-SSD by essentially working 'in reverse' by starting at a value of $x$ on the **vertical** scale in the Figure above and following the → arrows and reading off the corresponding value on the horizontal scale. +

Obviously, we need to be able to 'codify' this process in `R` (or any other computer language).
Mathematically this is equivalent to seeking a solution to the following equation:$${x:G\left( x \right) = p}$$ +or, equivalently:$$x:G\left( x \right) - p = 0$$ +for some fraction affected, $p$. +

+

Finding the solution to this last equation is referred to as *finding the root(s)* of the function $G\left( x \right)$ or, as is made clear in the figure below, *finding the zero-crossing* of the function $G\left( x \right)$ for the case $p=0.2$. + +![](images/uniroot.jpg){width=100% align="center"} + +
+

In `R` finding the roots of $x:G\left( x \right) - p = 0$ is achieved using the `uniroot()` function.

+ Help on the `uniroot` function can be found [here](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/uniroot.html) + +## Where do the model-averaged weights come from? +This is a little more complex, although we'll try to provide a non-mathematical explanation. +For those interested in going deeper, a more comprehensive treatment can be found in [@model_averaging] and [@fletcher]. + +

This time, we'll look at fitting a gamma, lognormal, and pareto distribution to our sample data: + +```{r echo=FALSE,warning=FALSE, results="markup",message=FALSE,class.output="scroll-100"} +samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +print(samp) +# knitr::kable(samp,caption="Some toxicity data (concentrations)") +``` +
The adequacy (or otherwise) of a fitted model can be assessed using a variety of numerical measures known as **goodness-of-fit** or GoF statistics. +These are invariably based on a measure of discrepancy between the emprical data and the hypothesized model. +Common GoF statistics used to test whether the hypothesis of some specified theoretical probability distribution is plausible for a given data set include: *Kolmogorov-Smirnov test; Anderson-Darling test; Shapiro-Wilk test;and Cramer-von Mises test*. + [The Cramer-von Mises](https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion) test is a good choice and is readily performed using the `cvm.test()` function in the `goftest` package in `R` as follows: + +```{r, echo=TRUE,results='hide',warning=FALSE,message=FALSE} +dat <- data.frame(Conc = c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59)) +library(goftest) +library(EnvStats) # this is required for the Pareto cdf (ppareto) + +# Examine the fit for the gamma distribution (NB: parameters estimated from the data) +cvm.test(dat$Conc, null = "pgamma", shape = 2.0591977, scale = 0.3231032, estimated = TRUE) + +# Examine the fit for the lognormal distribution (NB: parameters estimated from the data) +cvm.test(dat$Conc, null = "plnorm", meanlog = -0.6695120, sd = 0.7199573, estimated = TRUE) + +# Examine the fit for the Pareto distribution (NB: parameters estimated from the data) +cvm.test(dat$Conc, null = "ppareto", location = 0.1800000, shape = 0.9566756, estimated = TRUE) +``` + +``` + Cramer-von Mises test of goodness-of-fit + Braun's adjustment using 4 groups + Null hypothesis: Gamma distribution + with parameters shape = 2.0591977, scale = 0.3231032 + Parameters assumed to have been estimated from data + +data: dat$Conc +omega2max = 0.34389, p-value = 0.3404 + + + Cramer-von Mises test of goodness-of-fit + Braun's adjustment using 4 groups + Null hypothesis: log-normal distribution + with parameter meanlog = -0.669512 + Parameters assumed to have been estimated from data + +data: dat$Conc +omega2max = 0.32845, p-value = 0.3719 + + + Cramer-von Mises test of goodness-of-fit + Braun's adjustment using 4 groups + Null hypothesis: distribution ‘ppareto’ + with parameters location = 0.18, shape = 0.9566756 + Parameters assumed to have been estimated from data + +data: dat$Conc +omega2max = 0.31391, p-value = 0.4015 +``` + +From this output and using a level of significance of $p = 0.05$, we see that none of the distributions is implausible. +However, if *forced* to choose just one distribution, we would choose the *Pareto* distribution (smaller values of the `omega2max` statistic are better). +However, this does not mean that the gamma and lognormal distributions are of no value in describing the data. +We can see from the plot below, that in fact both the gamma and lognormal distributions do a reasonable job over the range of toxicity values. +The use of the Pareto may be a questionable choice given it is truncated at 0.18 (which is the minimum value of our toxicity data). +
+ +```{r echo=FALSE,fig.cap="Emprirical cdf (black); lognormal (green); gamma (blue); and Pareeto (red)", fig.width=7,fig.height=4.5} +library(EnvStats) +samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +samp <- sort(samp) +plot(ecdf(samp), main = "Empirical and fitted SSDs", xlab = "Concentration", ylab = "Probability") +xx <- seq(0.01, 3, by = 0.01) +lines(xx, plnorm(xx, meanlog = -0.6695120, sd = 0.7199573), col = "#77d408") +lines(xx, pgamma(xx, shape = 2.0591977, scale = 0.3231032), col = "#08afd4") +lines(xx, ppareto(xx, location = 0.1800000, shape = 0.9566756), col = "red") +# lines(xx,(0.4419*plnorm(xx,meanlog=mean(log(samp[5:15])),sd=sd(log(samp[5:15])))+ +# 0.5581*plnorm(xx,meanlog=mean(log(samp[1:10])),sd=sd(log(samp[1:10])))) ,col="#d40830") +``` +

+ +


As in the earlier example, we might expect to find a better fitting distribution by combining *all three distributions* using a *weighted SSD*. +The issue we face now is *how do we choose the weights* to reflect the relative fits of the three distributions? +Like all tests of statistical significance, a *p-value* is computed from the value of the relevant *test statistic* - in this case, the value of the `omega2max` test statistic. +For this particular test, it's a case of the smaller the better. +From the output above we see that the `omega2max` values are $0.344$ for the gamma distribution, $0.328$ for the lognormal distribution, and $0.314$ for the Pareto distribution.

+ +

We might somewhat naively compute the relative weights as:
${w_1} = \frac{{{{0.344}^{ - 1}}}}{{\left( {{{0.344}^{ - 1}} + {{0.328}^{ - 1}} + {{0.314}^{ - 1}}} \right)}} = 0.318$         ${w_2} = \frac{{{{0.328}^{ - 1}}}}{{\left( {{{0.344}^{ - 1}} + {{0.328}^{ - 1}} + {{0.314}^{ - 1}}} \right)}} = 0.333$     and ${w_3} = \frac{{{{0.314}^{ - 1}}}}{{\left( {{{0.344}^{ - 1}} + {{0.328}^{ - 1}} + {{0.314}^{ - 1}}} \right)}} = 0.349$
   (we use *reciprocals* since smaller values of `omega2max` represent better fits). +As will be seen shortly - these are incorrect. +

However, being based on a simplistic measure of discrepancy between the *observed* and *hypothesized* distributions, the `omega2max` statistic is a fairly 'blunt instrument' and has no grounding in information theory which *is* the basis for determining the weights that we seek. +

+ +
A discussion of *information theoretic* methods for assessing goodness-of-fit is beyond the scope of this vignette. Interested readers should consult [@model_averaging]. +A commonly used metric to determine the model-average weights is the **Akaike Information Criterion** or [AIC](https://en.wikipedia.org/wiki/Akaike_information_criterion). +The formula for the $AIC$ is: +$$AIC = 2k - 2\ln \left( \ell \right)$$ +where $k$ is the number of model parameters and $\ell$ is the *likelihood* for that model. +Again, a full discussion of statistical likelihood is beyond the present scope. +A relatively gentle introduction can be found [here](https://ep-news.web.cern.ch/what-likelihood-function-and-how-it-used-particle-physics). +

The likelihood for our three distributions can be computed in `R` as follows: + + +```{r, echo=TRUE} +sum(log(dgamma(dat$Conc, shape = 2.0591977, scale = 0.3231032))) +sum(log(dlnorm(dat$Conc, meanlog = -0.6695120, sdlog = 0.7199573))) +sum(log(EnvStats::dpareto(dat$Conc, location = 0.1800000, shape = 0.9566756))) +``` + +From which the *AIC* values readily follow: + +```{r echo=FALSE,results='markup'} +dat <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +k <- 2 # number of parameters for each of the distributions +# Gamma distribution +aic1 <- 2 * k - 2 * sum(log(dgamma(dat, shape = 2.0591977, scale = 0.3231032))) +cat("AIC for gamma distribution =", aic1, "\n") + +# lognormal distribution +aic2 <- 2 * k - 2 * sum(log(dlnorm(dat, meanlog = -0.6695120, sdlog = 0.7199573))) +cat("AIC for lognormal distribution =", aic2, "\n") + +# Pareto distribution +aic3 <- 2 * k - 2 * sum(log(EnvStats::dpareto(dat, location = 0.1800000, shape = 0.9566756))) +cat("AIC for Pareto distribution =", aic3, "\n") +``` +

+ +

As with the `omega2max` statistic, **smaller** values of *AIC* are better. Thus, a comparison of the AIC values above gives the ranking of distributional fits (best to worst) as: *Pareto > lognormal > gamma* + +

+ +### Computing model weights from the `AIC` +We will simply provide a formula for computing the model weights from the `AIC` values. +More detailed information can be found [here](https://training.visionanalytix.com/ssd-model-averaging/). + +The *AIC* for the *i^th^* distribution fitted to the data is +$$AI{C_i} = 2{k_i} - 2\ln \left( {{L_i}} \right)$$ +where ${L_i}$ is the *i^th^ likelihood* and ${k_i}$ is the *number of parameters* for the *i^th^ distribution*. +Next, we form the differences: +$${\Delta _i} = AI{C_i} - AI{C_0}$$ +where $AI{C_0}$ is the *AIC* for the **best-fitting** model (i.e.$AI{C_0} = \mathop {\min }\limits_i \left\{ {AI{C_i}} \right\}$ ). +The *model-averaged weights* ${w_i}$ are then computed as + +
+AIC Model averaging weights + +$${w}_{i}=\frac{\exp \left\{ -\frac{1}{2}{{\Delta }_{i}} \right\}}{\sum{\exp \left\{ -\frac{1}{2}{{\Delta }_{i}} \right\}}}$$ + +
+
+ +

The model-averaged weights for the gamma, lognormal, and Pareto distributions used in the previous example can be computed 'manually' in `R` as follows: + + +```{r echo=TRUE,results='hide'} +dat <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +aic <- NULL +k <- 2 # number of parameters for each of the distributions + + +aic[1] <- 2 * k - 2 * sum(log(dgamma(dat, shape = 2.0591977, scale = 0.3231032))) # Gamma distribution + +aic[2] <- 2 * k - 2 * sum(log(dlnorm(dat, meanlog = -0.6695120, sdlog = 0.7199573))) # lognormal distribution + +aic[3] <- 2 * k - 2 * sum(log(EnvStats::dpareto(dat, location = 0.1800000, shape = 0.9566756))) # Pareto distribution + +delta <- aic - min(aic) # compute the delta values + +aic.w <- exp(-0.5 * delta) +aic.w <- round(aic.w / sum(aic.w), 4) + +cat( + " AIC weight for gamma distribution =", aic.w[1], "\n", + "AIC weight for lognormal distribution =", aic.w[2], "\n", + "AIC weight for pareto distribution =", aic.w[3], "\n" +) +``` +``` + AIC weight for gamma distribution = 0.1191 + AIC weight for lognormal distribution = 0.3985 + AIC weight for pareto distribution = 0.4824 +``` +

+ +Finally, let's look at the fitted *model-averaged SSD*: + +```{r echo=FALSE,fig.cap="Empirical cdf (black) and model-averaged fit (magenta)",fig.width=8,fig.height=5} +samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) +samp <- sort(samp) +plot(ecdf(samp), main = "Empirical and fitted SSDs", xlab = "Concentration", ylab = "Probability") +xx <- seq(0.01, 3, by = 0.005) + +lines(xx, plnorm(xx, meanlog = -0.6695120, sd = 0.7199573), col = "#959495", lty = 2) +lines(xx, pgamma(xx, shape = 2.0591977, scale = 0.3231032), col = "#959495", lty = 3) +lines(xx, ppareto(xx, location = 0.1800000, shape = 0.9566756), col = "#959495", lty = 4) +lines(xx, 0.1191 * pgamma(xx, shape = 2.0591977, scale = 0.3231032) + + 0.3985 * plnorm(xx, meanlog = -0.6695120, sd = 0.7199573) + + 0.4824 * ppareto(xx, location = 0.1800000, shape = 0.9566756), col = "#FF33D5", lwd = 1.5) +``` + +As can be seen from the figure above, the model-averaged fit provides a very good fit to the empirical data. + +### Correcting for distributions having differing numbers of parameters + +In deriving the AIC, Akaike had to make certain, strong assumptions. +In addition, the bias factor (the $2k$ term) was derived from theoretical considerations (such as mathematical *expectation*) that relate to *infinite* sample sizes. +For small sample sizes, the AIC is likely to select models having too many parameters (i.e models which *over-fit*). +In 1978, Sugiura proposed a modification to the AIC to address this problem, although it too relied on a number of assumptions. +This 'correction' to the AIC for small samples (referred to as $AI{{C}_{c}}$) is + +
+Corrected Akaike Information Criterion (AICc) + +$$AI{{C}_{c}}=AIC+\frac{2{{k}^{2}}+2k}{n-k-1}$$ + +where n is the sample size and k is the number of parameters. + +
+ +
It is clear from the formula for $AI{{C}_{c}}$ that for   $n\gg k$,    $AI{{C}_{c}}\simeq AIC$. The issue of sample size is ubiquitous in statistics, but even more so in ecotoxicology where logistical and practical limitations invariably mean we are dealing with (pathologically) small sample sizes. There are no hard and fast rules as to what constitutes an *appropriate* sample size for SSD modelling. However, Professor David Fox's personal rule of thumb which works quite well is: + + +
Sample size rule-of-thumb for SSD modelling $$n\ge 5k+1$$ + +where n is the sample size and k is the number of parameters. +
+ +
Since most of the common SSD models are 2-parameter, we should be aiming to have a sample size of at least 11. +For 3-parameter models (like the Burr III), the minimum sample size is 16 and if we wanted to fit a mixture of two, 2-parameter models (eg. *logNormal-logNormal* or *logLogistic-logLogistic*) the sample size should be *at least* 26. +Sadly, this is rarely the case in practice! Jurisdictional guidance material may specify minimum sample size requirements, and should be adhered to where available and relevant. + +### Model-Averaging in `ssdtools` + +Please see the [Getting started with ssdtools](https://bcgov.github.io/ssdtools/articles/ssdtools.html) vignette for examples of obtaining model-averaged *HCx* values and predictions using `ssdtools`. + +## References + +
+ +```{r, results = "asis", echo = FALSE} +cat(ssdtools::ssd_licensing_md()) +``` diff --git a/doc/small-sample-bias.R b/doc/small-sample-bias.R new file mode 100644 index 00000000..49425038 --- /dev/null +++ b/doc/small-sample-bias.R @@ -0,0 +1 @@ +### This is an R script tangled from 'small-sample-bias.pdf.asis' diff --git a/doc/small-sample-bias.pdf b/doc/small-sample-bias.pdf new file mode 100644 index 0000000000000000000000000000000000000000..be1d75477187b03371b3a1efe973552b1cea40c2 GIT binary patch literal 329051 zcmdqIV~}Ls)-_t_vTeJ|wr$&$W!vgPmu=g&tuEWPZGZioIB(n=ao&jU{J6jFj>wG6 zoolQ;*N(O48f(mvWC|i;w2XAD&}1_ULo3j%gbajsMwZY#JkazqCbnkI=7fx#%xr}J zI-u#rEUcYP90}>gtPPz_L`(p7#wO5we9%tLjwXgS(C%xKm{u`3>{st!;euWQBj%MS zjMIb|xh=ADs2%2>s{=j~5d!$*PlWgO$8pB*=hETnnlS(`_6|;+o=8%7Q6*B*xPz^= z{jG6k-|?54iiQz-m9wx^NhI*C#j%9U27zKdR5F=DAT3Y3Pxd0Z@Kqg$)0XxP^xsDj zIxQS1zIsT5jkuDG;qRUg$0RagBcb6WQtNj>#nPd4Q-L1?E*>+ zZRunJ;wch+(i+Gk_#bAH*mLNkCJY$fQ((7{`?~l}qa^5LNOfTr;}w?0LEEgoP{SrU 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z*Y`FA?%Gg=xBlb`Y(*J&#_r$Z=fw5`s+C&DIMrk?HLZO=DYf2`&q@tN*L=QFwW1Dl zE4Rufqmq^Qa;N_S+O6ki)_L}Wafv0xf&XF|%vA^PEDFrZxBi=N%V(d>K}@nC%B4{r zoSE*R@;tB`hdj?(>{2;C&(QqVKX2XqI*BJ$j@pDfb&`_r)>192BQQ_#wKm?Vqed3! zYqlEmDFvQb>S;j|<^82mlB*65wn#4oy(om?&o<4y#&bWEOTVa6$aJ-b6a-H*SkKL$ zwGMvA7=DC2e!7Bw%klR{BFrY1hUwHMwAL?8C{2P5Oen3Vgie!8!!G-M&kyApksY& literal 0 HcmV?d00001 diff --git a/doc/small-sample-bias.pdf.asis b/doc/small-sample-bias.pdf.asis new file mode 100644 index 00000000..16a48f21 --- /dev/null +++ b/doc/small-sample-bias.pdf.asis @@ -0,0 +1,6 @@ +%\VignetteIndexEntry{Small sample bias in estimates} +%\VignetteEngine{R.rsp::asis} +%\VignetteKeyword{PDF} +%\VignetteKeyword{HTML} +%\VignetteKeyword{vignette} +%\VignetteKeyword{package} diff --git a/doc/ssdtools.R b/doc/ssdtools.R new file mode 100644 index 00000000..06f80ff1 --- /dev/null +++ b/doc/ssdtools.R @@ -0,0 +1,94 @@ +## ----include = FALSE---------------------------------------------------------- +knitr::opts_chunk$set( + collapse = TRUE, + comment = "#>", + fig.width = 6, + fig.height = 4 +) + +## ----eval = FALSE------------------------------------------------------------- +# install.packages(c("ssdtools", "tidyverse")) + +## ----message = FALSE---------------------------------------------------------- +library(ssdtools) +library(ggplot2) + +## ----eval = FALSE------------------------------------------------------------- +# data <- read_csv(file = "path/to/file.csv") + +## ----------------------------------------------------------------------------- +ssddata::ccme_boron + +## ----------------------------------------------------------------------------- +ssd_dists_all() + +## ----------------------------------------------------------------------------- +fits <- ssd_fit_dists(ssddata::ccme_boron, dists = c("llogis", "lnorm", "gamma")) + +## ----------------------------------------------------------------------------- +tidy(fits) + +## ----fig.alt="A plot of the CCME boron dataset with the gamma, log-logistic and log-normal distributions with a simple black and white background color scheme."---- +theme_set(theme_bw()) # set plot theme + +autoplot(fits) + + ggtitle("Species Sensitivity Distributions for Boron") + + scale_colour_ssd() + +## ----------------------------------------------------------------------------- +ssd_gof(fits) + +## ----eval = FALSE------------------------------------------------------------- +# set.seed(99) +# boron_pred <- predict(fits, ci = TRUE) + +## ----------------------------------------------------------------------------- +boron_pred + +## ----fig.alt="A plot of the CCME boron dataset species colored by group and the model average species sensitivity distribution with a simple black and white background color scheme."---- +ssd_plot(ssddata::ccme_boron, boron_pred, + color = "Group", label = "Species", + xlab = "Concentration (mg/L)", ribbon = TRUE +) + + expand_limits(x = 5000) + # to ensure the species labels fit + ggtitle("Species Sensitivity for Boron") + + scale_colour_ssd() + +## ----------------------------------------------------------------------------- +set.seed(99) +boron_hc5 <- ssd_hc(fits, proportion = 0.05, ci = TRUE) +print(boron_hc5) +boron_pc <- ssd_hp(fits, conc = boron_hc5$est, ci = TRUE) +print(boron_pc) + +## ----------------------------------------------------------------------------- +boron_censored <- ssddata::ccme_boron |> + dplyr::mutate(left = Conc, right = Conc) + +boron_censored$left[c(3, 6, 8)] <- NA + +## ----------------------------------------------------------------------------- +dists <- ssd_fit_dists(boron_censored, + dists = ssd_dists_bcanz(n = 2), + left = "left", right = "right" +) + +## ----------------------------------------------------------------------------- +ssd_gof(dists) + +## ----------------------------------------------------------------------------- +ssd_hc(dists, average = FALSE) +ssd_hc(dists) + +## ----fig.alt="A plot of the left censored CCME boron dataset with the model average species sensitivity distribution and arrows indicating the censoring."---- +set.seed(99) +pred <- predict(dists, ci = TRUE, parametric = FALSE) + +ssd_plot(boron_censored, pred, + left = "left", right = "right", + xlab = "Concentration (mg/L)" +) + +## ----results = "asis", echo = FALSE------------------------------------------- +cat(ssdtools::ssd_licensing_md()) + diff --git a/doc/ssdtools.Rmd b/doc/ssdtools.Rmd new file mode 100644 index 00000000..a553fa05 --- /dev/null +++ b/doc/ssdtools.Rmd @@ -0,0 +1,267 @@ +--- +title: "Getting Started with ssdtools" +author: "ssdtools Team" +date: '`r format(Sys.time(), "%Y-%m-%d", tz = "UTC")`' +bibliography: references.bib +mathfont: Courier +latex_engine: MathJax +output: rmarkdown::html_vignette +vignette: > + %\VignetteIndexEntry{Getting Started with ssdtools} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteEncoding{UTF-8} +--- + +```{r, include = FALSE} +knitr::opts_chunk$set( + collapse = TRUE, + comment = "#>", + fig.width = 6, + fig.height = 4 +) +``` + +## Introduction + +`ssdtools` is an R package to fit Species Sensitivity Distributions (SSDs) using Maximum Likelihood and model averaging. + +SSDs are cumulative probability distributions that are used to estimate the percent of species that are affected and/or protected by a given concentration of a chemical. +The concentration that affects 5% of the species is referred to as the 5% Hazard Concentration (*HC~5~*). This is equivalent to a 95% protection value (*PC~95~*). +For more information on SSDs the reader is referred to @posthuma_species_2001. + +In order to use `ssdtools` you need to install R (see below) or use the Shiny [app](https://bcgov-env.shinyapps.io/ssdtools/). +The shiny app includes a user guide. +This vignette is a user manual for the R package. + +## Philosophy + +`ssdtools` provides the key functionality required to fit SSDs using Maximum Likelihood and [model averaging](https://bcgov.github.io/ssdtools/articles/A_model_averaging.html) in R. +It is intended to be used in conjunction with [tidyverse](https://www.tidyverse.org) packages such as `readr` to input data, `tidyr` and `dplyr` to group and manipulate data and `ggplot2` [@ggplot2] to plot data. +As such it endeavors to fulfill the tidyverse [manifesto](https://tidyverse.tidyverse.org/articles/manifesto.html). + +## Installing + +In order to install R [@r] the appropriate binary for the users operating system should be downloaded from [CRAN](https://cran.r-project.org) and then installed. + +Once R is installed, the `ssdtools` package can be installed (together with the tidyverse) by executing the following code at the R console + +```{r, eval = FALSE} +install.packages(c("ssdtools", "tidyverse")) +``` + +The `ssdtools` package (and ggplot2 package) can then be loaded into the current session using + +```{r, message = FALSE} +library(ssdtools) +library(ggplot2) +``` + +## Getting Help + +To get additional information on a particular function just type `?` followed by the name of the function at the R console. +For example `?ssd_gof` brings up the R documentation for the `ssdtools` goodness of fit function. + +For more information on using R the reader is referred to [R for Data Science](https://r4ds.had.co.nz) [@wickham_r_2016]. + +If you discover a bug in `ssdtools` please file an issue with a [reprex](https://reprex.tidyverse.org/articles/reprex-dos-and-donts.html) (repeatable example) at . + +## Inputting Data + +Once the `ssdtools` package has been loaded the next task is to input some data. +An easy way to do this is to save the concentration data for a *single* chemical as a column called `Conc` in a comma separated file (`.csv`). +Each row should be the sensitivity concentration for a separate species. +If species and/or group information is available then this can be saved as `Species` and `Group` columns. +The `.csv` file can then be read into R using the following + +```{r, eval = FALSE} +data <- read_csv(file = "path/to/file.csv") +``` + +For the purposes of this manual we use the CCME dataset for boron from the [`ssddata`](https://github.com/open-AIMS/ssddata) package. + +```{r} +ssddata::ccme_boron +``` + +## Fitting Distributions + +The function `ssd_fit_dists()` inputs a data frame and fits one or more distributions. +The user can specify a subset of the following `r length(ssd_dists_all())` distributions. Please see the [distributions](https://bcgov.github.io/ssdtools/articles/B_distributions.html) and [model averaging](https://bcgov.github.io/ssdtools/articles/A_model_averaging.html) vignettes for more information regarding appropriate use of distributions and the use of model-averaged SSDs. + +```{r} +ssd_dists_all() +``` + +using the `dists` argument. + +```{r} +fits <- ssd_fit_dists(ssddata::ccme_boron, dists = c("llogis", "lnorm", "gamma")) +``` + +## Coefficients + +The estimates for the various terms can be extracted using the tidyverse generic `tidy` function (or the base R generic `coef` function). + +```{r} +tidy(fits) +``` + +## Plots + +It is generally more informative to plot the fits using the `autoplot` generic function (a wrapper on `ssd_plot_cdf()`). +As `autoplot` returns a `ggplot` object it can be modified prior to plotting. +For more information see the [customising plots](https://bcgov.github.io/ssdtools/articles/A_model_averaging.html) vignette. + +```{r, fig.alt="A plot of the CCME boron dataset with the gamma, log-logistic and log-normal distributions with a simple black and white background color scheme."} +theme_set(theme_bw()) # set plot theme + +autoplot(fits) + + ggtitle("Species Sensitivity Distributions for Boron") + + scale_colour_ssd() +``` + +## Selecting One Distribution + +Given multiple distributions the user is faced with choosing the "best" distribution (or as discussed below averaging the results weighted by the fit). + +```{r} +ssd_gof(fits) +``` + +The `ssd_gof()` function returns three test statistics that can be used to evaluate the fit of the various distributions to the data. + +- [Anderson-Darling](https://en.wikipedia.org/wiki/Anderson–Darling_test) (`ad`) statistic, +- [Kolmogorov-Smirnov](https://en.wikipedia.org/wiki/Kolmogorov–Smirnov_test) (`ks`) statistic and +- [Cramer-von Mises](https://en.wikipedia.org/wiki/Cramér–von_Mises_criterion) (`cvm`) statistic + +and three information criteria + +- Akaike's Information Criterion (`AIC`), +- Akaike's Information Criterion corrected for sample size (`AICc`) and +- Bayesian Information Criterion (`BIC`) + +Note if `ssd_gof()` is called with `pvalue = TRUE` then the p-values rather than the statistics are returned for the ad, ks and cvm tests. + +Following @burnham_model_2002 we recommend the `AICc` for model selection. +The best predictive model is that with the lowest `AICc` (indicated by the model with a `delta` value of 0 in the goodness of fit table). +In the current example the best predictive model is the gamma distribution but both the lnorm and llogis distributions have some support. + +For further information on the advantages of an information theoretic approach in the context of selecting SSDs the reader is referred to @fox_recent_2021. + +## Averaging Multiple Distributions + +Often other distributions will fit the data almost as well as the best distribution as evidenced by `delta` values < 2 [@burnham_model_2002]. +In general, the recommended approach is to estimate the average fit based on the relative weights of the distributions [@burnham_model_2002]. +The `AICc` based weights are indicated by the `weight` column in the goodness of fit table. +A detailed introduction to model averaging can be found in the [Model averaging](https://bcgov.github.io/ssdtools/articles/A_model_averaging.html) vignette. +A discussion on the recommended set of default distributions can be found in the [Distributions](https://bcgov.github.io/ssdtools/articles/B_distributions.html) vignette. + +## Estimating the Fit + +The `predict` function can be used to generate model-averaged estimates (or if `average = FALSE` estimates for each distribution individual) by bootstrapping. +Model averaging is based on `AICc` unless the data censored is which case `AICc` is undefined. +In this situation model averaging is only possible if the distributions have the same number of parameters (so that `AIC` can be used to compare the models). + +```{r, eval = FALSE} +set.seed(99) +boron_pred <- predict(fits, ci = TRUE) +``` + +The resultant object is a data frame of the estimated concentration (`est`) with standard error (`se`) and lower (`lcl`) and upper (`ucl`) 95% confidence limits (CLs) by percent of species affected (`percent`). +The object includes the number of bootstraps (`nboot`) data sets generated as well as the proportion of the data sets that successfully fitted (`pboot`). + +```{r} +boron_pred +``` + +The data frame of the estimates can then be plotted together with the original data using the `ssd_plot()` function to summarize an analysis. +Once again the returned object is a `ggplot` object which can be customized prior to plotting. + +```{r, fig.alt="A plot of the CCME boron dataset species colored by group and the model average species sensitivity distribution with a simple black and white background color scheme."} +ssd_plot(ssddata::ccme_boron, boron_pred, + color = "Group", label = "Species", + xlab = "Concentration (mg/L)", ribbon = TRUE +) + + expand_limits(x = 5000) + # to ensure the species labels fit + ggtitle("Species Sensitivity for Boron") + + scale_colour_ssd() +``` + +In the above plot the model-averaged 95% confidence interval is indicated by the shaded band and the model-averaged 5%/95% Hazard/Protection Concentration (*HC5*/ *PC~95~*) by the dotted line. +Hazard/Protection concentrations are discussed below. + +## Hazard/Protection Concentrations + +The 5% hazard concentration (*HC5*) is the concentration that affects 5% of the species tested. This is equivalent to the 95% protection concentration which protects 95% of species (*PC~95~*). The hazard and protection concentrations are directly interchangeable, and terminology depends simply on user preference. + +The hazard/protection concentrations can be obtained using the `ssd_hc()` function, which can be used to obtain any desired percentage value. The fitted SSD can also be used to determine the percentage of species protected at a given concentration using `ssd_hp()`. + +```{r} +set.seed(99) +boron_hc5 <- ssd_hc(fits, proportion = 0.05, ci = TRUE) +print(boron_hc5) +boron_pc <- ssd_hp(fits, conc = boron_hc5$est, ci = TRUE) +print(boron_pc) +``` + +### Censored Data + +Censored data is that for which only a lower and/or upper limit is known for a particular species. +If the `right` argument in `ssd_fit_dists()` is different to the `left` argument then the data are considered to be censored. + +It is important to note that `ssdtools` doesn't currently support right censored data. + +Let's produce some left censored data. + +```{r} +boron_censored <- ssddata::ccme_boron |> + dplyr::mutate(left = Conc, right = Conc) + +boron_censored$left[c(3, 6, 8)] <- NA +``` + +As the sample size `n` is undefined for censored data, `AICc` cannot be calculated. +However, if all the models have the same number of parameters, the `AIC` `delta` values are identical to those for `AICc`. +For this reason, `ssdtools` only permits model averaging of censored data for distributions with the same number of parameters. +We can call only the default two parameter models using `ssd_dists_bcanz(n = 2)`. + +```{r} +dists <- ssd_fit_dists(boron_censored, + dists = ssd_dists_bcanz(n = 2), + left = "left", right = "right" +) +``` + +There are less goodness-of-fit statistics available for fits to censored data (currently just `AIC` and `BIC`). + +```{r} +ssd_gof(dists) +``` + + +The model-averaged predictions are calculated using `AIC` +```{r} +ssd_hc(dists, average = FALSE) +ssd_hc(dists) +``` + +The confidence intervals can currently only be generated for censored data using non-parametric bootstrapping. +The horizontal lines in the plot indicate the censoring (range of possible values). + +```{r, fig.alt="A plot of the left censored CCME boron dataset with the model average species sensitivity distribution and arrows indicating the censoring."} +set.seed(99) +pred <- predict(dists, ci = TRUE, parametric = FALSE) + +ssd_plot(boron_censored, pred, + left = "left", right = "right", + xlab = "Concentration (mg/L)" +) +``` + +## References + +
+ +```{r, results = "asis", echo = FALSE} +cat(ssdtools::ssd_licensing_md()) +``` From d9b3aa03a638875bf271c09eb19e79967ed3778a Mon Sep 17 00:00:00 2001 From: Joe Thorley Date: Tue, 3 Dec 2024 07:43:06 -0800 Subject: [PATCH 03/66] add small-sample-bias back in --- .github/workflows/R-CMD-check.yaml | 2 +- .github/workflows/paper.yaml | 4 +- .github/workflows/test-coverage.yaml | 2 +- Meta/vignette.rds | Bin 431 -> 0 bytes doc/additional-technical-details.R | 12 - doc/additional-technical-details.Rmd | 221 ----- doc/customising-plots.R | 70 -- doc/customising-plots.Rmd | 137 --- doc/distributions.R | 315 ------- doc/distributions.Rmd | 837 ------------------ doc/model-averaging.R | 173 ---- doc/model-averaging.Rmd | 538 ----------- doc/small-sample-bias.R | 1 - doc/ssdtools.R | 94 -- doc/ssdtools.Rmd | 267 ------ {doc => vignettes}/small-sample-bias.pdf | Bin {doc => vignettes}/small-sample-bias.pdf.asis | 0 17 files changed, 4 insertions(+), 2669 deletions(-) delete mode 100644 Meta/vignette.rds delete mode 100644 doc/additional-technical-details.R delete mode 100644 doc/additional-technical-details.Rmd delete mode 100644 doc/customising-plots.R delete mode 100644 doc/customising-plots.Rmd delete mode 100644 doc/distributions.R delete mode 100644 doc/distributions.Rmd delete mode 100644 doc/model-averaging.R delete mode 100644 doc/model-averaging.Rmd delete mode 100644 doc/small-sample-bias.R delete mode 100644 doc/ssdtools.R delete mode 100644 doc/ssdtools.Rmd rename {doc => vignettes}/small-sample-bias.pdf (100%) rename {doc => vignettes}/small-sample-bias.pdf.asis (100%) diff --git a/.github/workflows/R-CMD-check.yaml b/.github/workflows/R-CMD-check.yaml index 58aedd38..571e5a4e 100644 --- a/.github/workflows/R-CMD-check.yaml +++ b/.github/workflows/R-CMD-check.yaml @@ -5,7 +5,7 @@ on: push: branches: [main, master] pull_request: - branches: [main, master, dev] + branches: [main, master] name: R-CMD-check diff --git a/.github/workflows/paper.yaml b/.github/workflows/paper.yaml index 17e2b7dd..b65787ec 100644 --- a/.github/workflows/paper.yaml +++ b/.github/workflows/paper.yaml @@ -2,9 +2,9 @@ on: push: - branches: [main] + branches: [main, joss-paper] pull_request: - branches: [main, dev] + branches: [main, joss-paper] name: paper diff --git a/.github/workflows/test-coverage.yaml b/.github/workflows/test-coverage.yaml index 9211f4b5..eb30c23a 100644 --- a/.github/workflows/test-coverage.yaml +++ b/.github/workflows/test-coverage.yaml @@ -4,7 +4,7 @@ on: push: branches: [main, master] pull_request: - branches: [main, master, dev] + branches: [main, master] name: test-coverage.yaml diff --git a/Meta/vignette.rds b/Meta/vignette.rds deleted file mode 100644 index d048d773130666ceddad857736f78debf6d6b996..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 431 zcmV;g0Z{%QiwFP!000001D#UaP69CyU0_up8ZV%UMoImE-G3l%)I>=L@ZjsXtbP#jsn(QPYpG*jeJ9J?bsFj>Kq*I_ z#2`Fm;UwmvvM58~Li&S<>b&!CDlv(; zx`fBXaM20vUQU~I4Q(nJjG4-+Dz~D;doe^M zUAXsLn7X|Y$F*bis+eDK5U`ZR)F`n2#6IRpru^+jjVzh_rrDADs-NmuEEZoyMxg*{ Z7@1&`0)qdRse0vo{{XcH0taFQ008op)5HJ( diff --git a/doc/additional-technical-details.R b/doc/additional-technical-details.R deleted file mode 100644 index 78b30508..00000000 --- a/doc/additional-technical-details.R +++ /dev/null @@ -1,12 +0,0 @@ -## ----include = FALSE---------------------------------------------------------- -knitr::opts_chunk$set( - collapse = TRUE, - comment = "#>", - fig.width = 6, - fig.height = 4, - echo = TRUE -) - -## ----results = "asis", echo = FALSE------------------------------------------- -cat(ssdtools::ssd_licensing_md()) - diff --git a/doc/additional-technical-details.Rmd b/doc/additional-technical-details.Rmd deleted file mode 100644 index 6031b314..00000000 --- a/doc/additional-technical-details.Rmd +++ /dev/null @@ -1,221 +0,0 @@ ---- -title: "Additional Technical Details" -author: "ssdtools Team" -date: '`r format(Sys.time(), "%Y-%m-%d", tz = "UTC")`' -bibliography: references.bib -mathfont: Courier -latex_engine: MathJax -output: rmarkdown::html_vignette -vignette: > - %\VignetteIndexEntry{Additional Technical Details} - %\VignetteEngine{knitr::rmarkdown} - %\VignetteEncoding{UTF-8} ---- - - - - - - -```{r, include = FALSE} -knitr::opts_chunk$set( - collapse = TRUE, - comment = "#>", - fig.width = 6, - fig.height = 4, - echo = TRUE -) -``` - -## Small sample bias - -The ssdtools package uses the method of Maximum Likelihood (ML) to estimate parameters for each distribution that is fit to the data. -Statistical theory says that maximum likelihood estimators are asymptotically unbiased, but does not guarantee performance in small samples. -A detailed account of the issue of small sample bias in estimates can be found in the following [pdf](https://github.com/bcgov/ssdtools/blob/master/vignettes/small-sample-bias.pdf). - -## Investigations into setting minimum sample sizes for uni-modal and bi-modal distributions in ssdtools - -Most jurisdictions require a minimum sample size for fitting a valid SSD. -The current Australian and New Zealand minimum is 5, in order to fit the two-parameter log-normal distribution [@warne2018]. -In ssdtools the default minimum sample size is 6 [@thorleysschwartz2018joss], which is consistent with the current methodology for British Columbia [@ccme]. - -Here we report on a series of simulation studies designed to inform a final decision on the default minimum sample size to adopt for both the uni-modal 2 parameter distributions, as well as the bi-modal 5 parameter distributions. - -### Bias and CI coverage and interval width - -#### Simulations based on ssddata - -We used the example datasets in the [`ssddata`](https://github.com/open-AIMS/ssddata) package in R [@ssddata] to undertake a simulation study to examine bias, coverage and confidence interval (CI) widths using the recommended default set of six distributions (lognormal, log-Gumbel, log-logistic, gamma, Weibull, and the lognormal-log-normal mixture), with model averaged estimates obtained using the multi-method, and confidence intervals estimated using the recommended weighted sample bootstrap method (see @fox_methodologies_2024). -A total of 20 unique datasets were extracted from ssddata and used to define the parameters for the simulation study as follows: - - 1. Each dataset was extracted from `ssddata` and fit using the default distribution set as recommended in [@fox_methodologies_2022] and [@fox_methodologies_2024]. - 2. Of the six default distributions, the parameters for the distribution having the highest weight for each dataset was used to generate new random datasets of varying values of N, including (but not limited to): 5 (current ANZG minimum), 6 (current BC minimum), and 8 (current ANZG preferred). - 3. For each randomly generated dataset, `ssdtools` was used to re-fit the data, and model averaged estimates were obtained using the multi-method, with upper and lower confidence limits (CLs) produced using the recommended weighted sample method (see [Confidence Intervals for Hazard Concentrations](https://bcgov.github.io/ssdtools/articles/confidence-intervals.html) vignette). - -The individual ssdtools fits are shown below for each of the 20 simulation datasets from ssddata, for the six recommended default distributions, as well as the model averaged CDF (black line): - -![](images/fitted_dists.png) - -This simulation process was repeated a minimum of 1,000 times for each dataset, and the results collated across all iterations. -For each simulated dataset the true HCx values were obtained directly from the parameter estimates of from data generating distribution. -From these, relative bias was calculated as the scaled-difference between the estimated HCx values and the true HCx value, i.e $$\frac{\widehat {HC}x-HCx}{HCx}$$ where $\widehat {HC}x$ is the estimate of the true value, $HCx$; coverage was calculated as the proportion of simulations where the true $HCx$ value fell within the lower and upper 95% confidence limits; and the scaled confidence interval width was calculated as $$\frac{UL-LL}{HCx}$$ where $UL$ and $LL$ are the upper and lower limits respectively. - -Bias, confidence interval width and coverage as a function of sample size across ~1000 simulations of 20 datasets using the multi model averaging method and the weighted sample method for estimating confidence intervals via ssdtools are shown below: - -![](images/ssdata_sims_collated.png) - -The simulation results showed significant gains in terms of reduced bias from N=5 to N=6, as well as in coverage, which improved substantially between N=5 and N=6. -There is also a small additional gain in coverage at N=7, where the median of the simulations reaches the expected 95% but this is only the case for HC1. - -#### Simulations based on EnviroTox - -In addition to the analysis based on the 20 ssddata example datasets, we also ran an expanded simulation study based on the EnviroTox dataset analysed by Yanagihara et al. (2024). -Combined with the ssddata examples, this includes a total of 353 example datasets to use as case studies. -Using this larger dataset as a basis for simulations, we followed the same procedure as described above to examine relative bias, as well as changes in the AICc weights (see below) for various sample sizes. -Estimates of coverage and confidence interval widths were not obtained for this larger dataset due to the computationally intensive bootstrap method of obtaining confidence intervals. - -Bias of sample size across ~1000 simulations of 353 datasets using the multi model averaging method via ssdtools, for HC5, 10 and 20 (0.05, 0.1, and 0.2) are shown below: - -![](images/all_sims_bias.png) - -Note that simulation results are shown separately for those derived from each of the six distributions as the underlying source data generating distribution. - -The bias results for this larger combined dataset did not show the same level of improvement from N=5 to N=6 as that based on the smaller ssddata simulation. -However, there as a gradual improvement in bias with increasing N. -There is no strong evidence for preferring N=6 over N=7 in the context of bias from either of the simulation studies. -We note that the bias was highest at these small sample sizes for data generated using a gamma distribution, likely reflecting the extreme left-tailed nature of this distribution. - -### AICc based model weights - -Aside from considerations of bias, coverage and confidence interval width, it is also prudent to examine how the weights of the different distributions changed with sample size, for data generated using the six different default distributions, to more fully investigate sample size issues associated with the use of the mixture distributions. -This was examined using the simulation study across the larger combined ssddata and EnviroTox datasets (353 datasets) to ensure a wide range of potential representations of each of the six default distributions was considered. - -Below is a plot of the mean AICc weights as a function of sample size (N) obtained for data simulated using the best fit distribution to 353 datasets. -Results are shown separately for the six different source distributions, with the upper plot (A) showing the AICc weight of the source (data generating) distributions, when fit using the default set of six distributions using ssdtools; and the lower plot (B) showing the AICc weight of the lognormal-lognormal mixture distribution for each of the source (data generating) distributions. - -![](images/weights_collated.png) - -We found that the AICc weights for the five unimodal distributions were relatively similar (~0.2) for very low N. -This is because for small N it is difficult to discern differences between the distributions in the candidate list. -The weights increase to above 0.5 as N increases (i.e. their converge to the true underlying generating distribution at high N, upper plot). -For very low sample sizes (N=5, 6 or 7) the source (data-generating) distribution is not preferentially weighted by the AICc (upper plot, A). - -For the lnorm_lnorm mixture, AICc weights can be >0.5 at relatively N (>8, lower plot, B). -We also looked specifically at the AICc weight of the lognormal-lognormal mixture as one of six distributions in the default candidate set, across simulation based on all six source generating distributions. -This was done to examine the potential for erroneously highly weighting the lognormal-lognormal mixture distribution by chance, when data are generated instead using one of the five unimodal distributions. -For all the unimodal source distributions the lognormal-lognormal mixture never has high AICc weight, even at very high sample sizes (N=256, lower plot, B). -The AICc weights are particularly low for the lnorm_lnorm mixture at low sample sizes for all the uni-modal source distributions (a desirable property) (lower plot, B). - -### Conclusions - -Overall, the results suggest that (relative) bias as a function of N behaves as expected. -The N=6 recommendation appears to be well-supported as coverage is particularly low for N=5, but acceptable for N=6. -The lognormal-lognormal mixture AICc weights suggest that this distribution will only be preferred (i.e. have a high AICc weight) when (i) there is clear evidence of bimodality in the source data; and (ii) large N. - -Based on these results, our recommendation is that only a single minimum sample size of N=6 be adopted (for both unimodal and bimodal), since our results suggest any gains in increasing this to 7 are minimal. - -## The inverse Pareto and inverse Weibull as limiting distributions of the Burr Type-III distribution - -### Burr III distribution - -The probability density function, ${f_X}(x;b,c,k)$ and cumulative distribution function, ${F_X}(x;b,c,k)$ for the Burr III distribution (also known as the *Dagum* distribution) as used in `ssdtools` are - -
-Burr III distribution - -\[\begin{array}{*{20}{c}} -{{f_X}(x;b,c,k) = \frac{{b\,k\,c}}{{{x^2}}}\frac{{{{\left( {\frac{b}{x}} \right)}^{c - 1}}}}{{{{\left[ {1 + {{\left( {\frac{b}{x}} \right)}^c}} \right]}^{k + 1}}}}{\rm{ }}}&{b,c,k,x > 0} -\end{array}\]
\[\begin{array}{*{20}{c}} -{{F_X}(x;b,c,k) = \frac{1}{{{{\left[ {1 + {{\left( {\frac{b}{x}} \right)}^c}} \right]}^k}}}{\rm{ }}}&{b,c,k,x > 0} -\end{array}\] - -
- -### Inverse Pareto distribution - -Let $X \sim Burr(b,c,k)$ have the *pdf* given in the box above. -It is well known that the distribution of $Y = \frac{1}{X}$ is the *inverse Burr* distribution (also known as the *SinghMaddala* distribution) for which - -\[\begin{array}{*{20}{c}} -{{f_Y}(y;b,c,k) = \frac{{c{\kern 1pt} {\kern 1pt} k{{\left( {\frac{y}{b}} \right)}^c}}}{{y{\kern 1pt} {{\left[ {1 + {{\left( {\frac{y}{b}} \right)}^c}} \right]}^{k + 1}}}}}&{b,c,k,y > 0} -\end{array}\] - -\[\begin{array}{*{20}{c}} -{{F_Y}(y;b,c,k) = 1 - \frac{1}{{{{\left[ {1 + {{\left( {\frac{y}{b}} \right)}^c}} \right]}^k}}}}&{b,c,k,y > 0} -\end{array}\] - -We now consider the limiting distribution when $c \to \infty$ and $k \to 0$ in such a way that the product $ck$ remains constant, i.e. $ck = \lambda$. - -Now, -\[\begin{array}{l} -\mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } \left\{ {{F_Y}(y;b,c,k)} \right\} = 1 - \mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } \frac{1}{{{{\left[ {1 + {{\left( {\frac{y}{b}} \right)}^c}} \right]}^k}}}\\ -\\ -and\\ -\\ -\mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } {\left[ {1 + {{\left( {\frac{y}{b}} \right)}^c}} \right]^k} = \mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } \left\{ {{{\left( {\frac{y}{b}} \right)}^{ck}}{{\left[ {1 + {{\left( {\frac{b}{y}} \right)}^c}} \right]}^k}} \right\}\\ -\\ -and\\ -\\ -\mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } \left\{ {{{\left( {\frac{y}{b}} \right)}^{ck}}{{\left[ {1 + {{\left( {\frac{b}{y}} \right)}^c}} \right]}^k}} \right\} = \mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } \left\{ {{{\left( {\frac{y}{b}} \right)}^{ck}}} \right\}\mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } \left\{ {{{\left[ {1 + {{\left( {\frac{b}{y}} \right)}^c}} \right]}^k}} \right\}\\ - = \mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } \left\{ {{{\left( {\frac{y}{b}} \right)}^{ck}}} \right\}\; \cdot \,1\\ - = {\left( {\frac{y}{b}} \right)^\lambda } -\end{array}\] - -Therefore, \[\begin{array}{*{20}{c}} -{\mathop {\mathop {\lim }\limits_{(c,k) \to (\infty ,0)} }\limits_{ck = \lambda } \left\{ {{F_Y}(y;b,c,k)} \right\} = 1 - {{\left( {\frac{b}{y}} \right)}^\lambda }}&{y \ge b} -\end{array}\] - -which we recognise as the (American) Pareto distribution. So, if the limiting distribution of $Y = \frac{1}{X}$ is a Pareto distribution, then the limiting distribution of $X = \frac{1}{Y}$ is the (American) *inverse Pareto* distribution - -\[\begin{array}{l} -{f_X}\left( {x;\alpha ,\beta } \right) = \lambda {b^\lambda }{x^{\lambda - 1}};{\rm{ }}0 \le x \le {\textstyle{1 \over b}};{\rm{ }}\lambda {\rm{,}}b > 0\\ -{F_X}\left( {x;\alpha ,\beta } \right) = {\left( {xb} \right)^\lambda };{\rm{ }}0 \le x \le {\textstyle{1 \over b}};{\rm{ }}\lambda {\rm{,}}b > 0 -\end{array}\] - -For completeness, the MLEs of this distribution have closed-form expressions and are given by - -\[\begin{array}{l} -\hat \lambda = {\left[ {\ln \left( {\frac{{{g_X}}}{{\hat b}}} \right)} \right]^{ - 1}}\\ -\hat b = \frac{1}{{\max \left\{ {{X_i}} \right\}}}{\rm{ }} -\end{array}\] - -and $\rm{ }{g_X}$ is the *geometric mean* of the data. - -### Inverse Weibull distribution - -Let $X \sim Burr(b,c,k)$ have the *pdf* given in the box above. We make the transformation $Y = \frac{{b{\kern 1pt} {k^{\tfrac{1}{c}}}{\kern 1pt} \theta }}{X}$ - -where $\theta$ is a parameter (constant). The distribution of $Y$ is also a Burr distribution and has *cdf* ${G_Y}\left( y \right) = 1 - \frac{1}{{{{\left[ {1 + {{\left( {\frac{y}{{{k^{\tfrac{1}{c}}}{\kern 1pt} \theta }}} \right)}^c}} \right]}^k}}}$. - -We are interested in the limiting behaviour of this Burr distribution as $k \to \infty$. - -Now, $\mathop {\lim }\limits_{k \to \infty } {G_Y}\left( y \right) = 1 - \mathop {\lim }\limits_{k \to \infty } {\left[ {1 + {{\left( {\frac{y}{{{k^{\tfrac{1}{c}}}{\kern 1pt} \theta }}} \right)}^c}} \right]^{ - k}}$ - -${ = 1 - \mathop {\lim }\limits_{k \to \infty } {{\left[ {1 + \frac{{{{\left( {\frac{y}{\theta }} \right)}^c}}}{{k{\kern 1pt} }}} \right]}^{ - k}}}$ - -$\begin{matrix} - =1-\exp \left[ -{{\left( \frac{y}{\theta } \right)}^{c}} \right] \\ - \left\{ \text{using the fact that }\underset{n\to \infty }{\mathop{\lim }}\,{{\left( 1+{}^{z}\!\!\diagup\!\!{}_{n}\; \right)}^{-n}}={{e}^{-z}} \right\} \\ -\end{matrix}$ - -We recognise the last expression as the *cdf* of a Weibull distribution with parameters $c$ and $\theta$. - -## References - -
- -```{r, results = "asis", echo = FALSE} -cat(ssdtools::ssd_licensing_md()) -``` diff --git a/doc/customising-plots.R b/doc/customising-plots.R deleted file mode 100644 index 5d69f82b..00000000 --- a/doc/customising-plots.R +++ /dev/null @@ -1,70 +0,0 @@ -## ----include = FALSE---------------------------------------------------------- -knitr::opts_chunk$set( - collapse = TRUE, - comment = "#>", - fig.width = 6, - fig.height = 4 -) - -## ----warning=FALSE, message=FALSE, fig.alt="A plot of the CCME boron data with the six default distributions."---- -library(ssdtools) -library(ggplot2) - -fits <- ssd_fit_dists(ssddata::ccme_boron) -ssd_plot_cdf(fits) - -## ----fig.alt="A plot of the CCME boron data with the model average distribution and the six default distributions."---- -ssd_plot_cdf(fits, average = NA, big.mark = " ") + - scale_color_manual(name = "Distribution", breaks = c("average", names(fits)), values = 1:7) + - theme_bw() - -## ----fig.alt = "A plot of the CCME boron data."------------------------------- -ggplot(ssddata::ccme_boron) + - geom_ssdpoint(aes(x = Conc)) + - ylab("Probability density") + - xlab("Concentration") - -## ----fig.alt = "A plot of CCME boron data with the ranges of the censored data indicated by horizontal lines."---- -ggplot(ssddata::ccme_boron) + - geom_ssdsegment(aes(x = Conc, xend = Conc * 4)) + - ylab("Probability density") + - xlab("Concenration") - -## ----fig.alt="A plot of the confidence intervals for the CCME boron data."---- -ggplot(boron_pred) + - geom_xribbon(aes(xmin = lcl, xmax = ucl, y = proportion)) + - ylab("Probability density") + - xlab("Concenration") - -## ----fig.alt="A plot of hazard concentrations as dotted lines."--------------- -ggplot() + - geom_hcintersect(xintercept = c(1, 2, 3), yintercept = c(0.05, 0.1, 0.2)) + - ylab("Probability density") + - xlab("Concenration") - -## ----fig.alt="A plot of censored CCME boron data with confidence intervals"---- -gp <- ggplot(boron_pred, aes(x = est)) + - geom_xribbon(aes(xmin = lcl, xmax = ucl, y = proportion), alpha = 0.2) + - geom_line(aes(y = proportion)) + - geom_ssdsegment(data = ssddata::ccme_boron, aes(x = Conc / 2, xend = Conc * 2)) + - geom_ssdpoint(data = ssddata::ccme_boron, aes(x = Conc / 2)) + - geom_ssdpoint(data = ssddata::ccme_boron, aes(x = Conc * 2)) + - scale_y_continuous("Species Affected (%)", labels = scales::percent) + - xlab("Concentration (mg/L)") + - expand_limits(y = c(0, 1)) - -gp - -## ----fig.alt="A plot of censored CCME boron data on log scale with confidence intervals, mathematical and the 5% hazard concentration."---- -gp + - scale_x_log10( - latex2exp::TeX("Boron $(\\mu g$/L)$") - ) + - geom_hcintersect(xintercept = ssd_hc(fits)$est, yintercept = 0.05) - -## ----eval = FALSE------------------------------------------------------------- -# ggsave("file_name.png", dpi = 300) - -## ----results = "asis", echo = FALSE------------------------------------------- -cat(ssdtools::ssd_licensing_md()) - diff --git a/doc/customising-plots.Rmd b/doc/customising-plots.Rmd deleted file mode 100644 index b5b3d415..00000000 --- a/doc/customising-plots.Rmd +++ /dev/null @@ -1,137 +0,0 @@ ---- -title: "Customising Plots" -author: "ssdtools Team" -date: '`r format(Sys.time(), "%Y-%m-%d", tz = "UTC")`' -bibliography: references.bib -mathfont: Courier -latex_engine: MathJax -output: rmarkdown::html_vignette -vignette: > - %\VignetteIndexEntry{Customising Plots} - %\VignetteEngine{knitr::rmarkdown} - %\VignetteEncoding{UTF-8} ---- - -```{r, include = FALSE} -knitr::opts_chunk$set( - collapse = TRUE, - comment = "#>", - fig.width = 6, - fig.height = 4 -) -``` - -## Plotting the cumulative distributions - -The `ssdtools` package plots the cumulative distribution functions using `ssd_plot_cdf()`. - -For example, consider the CCME boron data from the [`ssddata`](https://github.com/open-AIMS/ssddata) package. -We can fit, and then plot the cdfs as follows. - -```{r warning=FALSE, message=FALSE, fig.alt="A plot of the CCME boron data with the six default distributions."} -library(ssdtools) -library(ggplot2) - -fits <- ssd_fit_dists(ssddata::ccme_boron) -ssd_plot_cdf(fits) -``` - -This graphic is a [ggplot](https://ggplot2.tidyverse.org) object and so can be customized in the usual way. - -For example, we can add the model-averaged cdf by setting `average = NA`, change the string used to separate thousands using `big.mark`, customize the color scale with `scale_color_manual()` and change the theme. - -```{r, fig.alt="A plot of the CCME boron data with the model average distribution and the six default distributions."} -ssd_plot_cdf(fits, average = NA, big.mark = " ") + - scale_color_manual(name = "Distribution", breaks = c("average", names(fits)), values = 1:7) + - theme_bw() -``` - -## ggplot Geoms - -The `ssdtools` package provides four ggplot geoms to allow you construct your own plots. - -### `geom_ssdpoint()` - -The first is `geom_ssdpoint()` which plots species sensitivity data - -```{r, fig.alt = "A plot of the CCME boron data."} -ggplot(ssddata::ccme_boron) + - geom_ssdpoint(aes(x = Conc)) + - ylab("Probability density") + - xlab("Concentration") -``` - -### `geom_ssdsegments()` - -The second is `geom_ssdsegments()` which plots the ranges of censored species sensitivity data - -```{r, fig.alt = "A plot of CCME boron data with the ranges of the censored data indicated by horizontal lines."} -ggplot(ssddata::ccme_boron) + - geom_ssdsegment(aes(x = Conc, xend = Conc * 4)) + - ylab("Probability density") + - xlab("Concenration") -``` - -### `geom_xribbon()` - -The third is `geom_xribbon()` which plots species sensitivity confidence intervals - -```{r, fig.alt="A plot of the confidence intervals for the CCME boron data."} -ggplot(boron_pred) + - geom_xribbon(aes(xmin = lcl, xmax = ucl, y = proportion)) + - ylab("Probability density") + - xlab("Concenration") -``` - -### `geom_hcintersect()` - -And the fourth is `geom_hcintersect()` which plots hazard concentrations - -```{r, fig.alt="A plot of hazard concentrations as dotted lines."} -ggplot() + - geom_hcintersect(xintercept = c(1, 2, 3), yintercept = c(0.05, 0.1, 0.2)) + - ylab("Probability density") + - xlab("Concenration") -``` - -### Putting it together - -Geoms can be combined as follows - -```{r, fig.alt="A plot of censored CCME boron data with confidence intervals"} -gp <- ggplot(boron_pred, aes(x = est)) + - geom_xribbon(aes(xmin = lcl, xmax = ucl, y = proportion), alpha = 0.2) + - geom_line(aes(y = proportion)) + - geom_ssdsegment(data = ssddata::ccme_boron, aes(x = Conc / 2, xend = Conc * 2)) + - geom_ssdpoint(data = ssddata::ccme_boron, aes(x = Conc / 2)) + - geom_ssdpoint(data = ssddata::ccme_boron, aes(x = Conc * 2)) + - scale_y_continuous("Species Affected (%)", labels = scales::percent) + - xlab("Concentration (mg/L)") + - expand_limits(y = c(0, 1)) - -gp -``` - -To log the x-axis and include mathematical notation and add the HC5 value use the following code. - -```{r, fig.alt="A plot of censored CCME boron data on log scale with confidence intervals, mathematical and the 5% hazard concentration."} -gp + - scale_x_log10( - latex2exp::TeX("Boron $(\\mu g$/L)$") - ) + - geom_hcintersect(xintercept = ssd_hc(fits)$est, yintercept = 0.05) -``` - -## Saving plots - -The most recent plot can be saved as a file using `ggsave()`, which also allows the user to set the resolution. - -```{r, eval = FALSE} -ggsave("file_name.png", dpi = 300) -``` - -
- -```{r, results = "asis", echo = FALSE} -cat(ssdtools::ssd_licensing_md()) -``` diff --git a/doc/distributions.R b/doc/distributions.R deleted file mode 100644 index e079c254..00000000 --- a/doc/distributions.R +++ /dev/null @@ -1,315 +0,0 @@ -## ----include = FALSE---------------------------------------------------------- -knitr::opts_chunk$set( - collapse = TRUE, - comment = "#>", - fig.width = 6, - fig.height = 4 -) - -## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5, fig.cap="Sample Burr probability density (A) and cumulative probability (B) functions."---- -par(mfrow = c(1, 2)) - -f <- function(x, b, c, k) { - z1 <- (b / x)^(c - 1) - z2 <- (b / x)^c - y <- (b * c * k / x^2) * z1 / (1 + z2)^(k + 1) - return(y) -} - -conc <- seq(0, 10, by = 0.005) -plot(conc, f(conc, 1, 3, 5), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733") -lines(conc, f(conc, 1, 1, 2), col = "#F2A61C") -lines(conc, f(conc, 1, 2, 2), col = "#1CADF2") -lines(conc, f(conc, 1, 2, 5), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -F <- function(x, b, c, k) { - z2 <- (b / x)^c - y <- 1 / (1 + z2)^k - return(y) -} -conc <- seq(0, 10, by = 0.005) -plot(conc, F(conc, 1, 3, 5), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733") -lines(conc, F(conc, 1, 1, 2), col = "#F2A61C") -lines(conc, F(conc, 1, 2, 2), col = "#1CADF2") -lines(conc, F(conc, 1, 2, 5), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") - -## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5, fig.cap="Sample lognormal lognormal mixture probability density (A) and cumulative probability (B) functions."---- -par(mfrow = c(1, 2)) - -f <- function(x, m1, s1, m2, s2, p) { - y <- p * dlnorm(x, m1, s1) + (1 - p) * dlnorm(x, m2, s2) - return(y) -} - -conc <- seq(0, 5, by = 0.0025) -m1 <- 1 -s1 <- .2 -m2 <- 1.8 -s2 <- 1.5 -p <- 0.25 -plot(conc, f(conc, m1, s1, m2, s2, p), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, f(conc, 0.09, 0.5, 1, 0.08, 0.9), col = "#F2A61C") -lines(conc, f(conc, 0.5, 0.5, 1, 0.08, 0.9), col = "#1CADF2") -lines(conc, f(conc, 0.7, 1.5, 0.5, 0.1, .7), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -F <- function(x, m1, s1, m2, s2, p) { - y <- p * plnorm(x, m1, s1) + (1 - p) * plnorm(x, m2, s2) - return(y) -} - -conc <- seq(0, 10, by = 0.0025) -m1 <- 1 -s1 <- .2 -m2 <- 1.8 -s2 <- 1.5 -p <- 0.25 -plot(conc, F(conc, m1, s1, m2, s2, p), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, F(conc, 0.09, 0.5, 1, 0.08, 0.9), col = "#F2A61C") -lines(conc, F(conc, 0.5, 0.5, 1, 0.08, 0.9), col = "#1CADF2") -lines(conc, F(conc, 0.7, 1.5, 0.5, 0.1, .7), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") - -## ----message=FALSE, echo=FALSE, fig.width=7,fig.height=5, fig.cap="Currently recommended default distributions."---- -library(ssdtools) -library(ggplot2) - -# theme_set(theme_bw()) - -set.seed(7) - -ssd_plot_cdf(ssd_match_moments(meanlog = 2, sdlog = 2)) + - scale_color_ssd() - -## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample lognormal probability density (A) and cumulative probability (B) functions."---- -par(mfrow = c(1, 2)) - -conc <- seq(0, 10, by = 0.005) -m1 <- 1 -s1 <- .2 -m2 <- 1.8 -s2 <- 1.5 -p <- 0.25 -plot(conc, dlnorm(conc, m1, s1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, dlnorm(conc, 0.4, 2), col = "#F2A61C") -lines(conc, dlnorm(conc, m1 * 2, s1), col = "#1CADF2") -lines(conc, dlnorm(conc, 0.9, 1.5), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -conc <- seq(0, 10, by = 0.005) -m1 <- 1 -s1 <- .2 -m2 <- 1.8 -s2 <- 1.5 -p <- 0.25 -plot(conc, plnorm(conc, m1, s1), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, plnorm(conc, 0.4, 2), col = "#F2A61C") -lines(conc, plnorm(conc, m1 * 2, s1), col = "#1CADF2") -lines(conc, plnorm(conc, 0.9, 1.5), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") - -## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Log logistic probability density (A) and cumulative probability (B) functions."---- -par(mfrow = c(1, 2)) - -f <- function(x, a, b) { - z1 <- (x / a)^(b - 1) - z2 <- (x / a)^b - y <- (b / a) * z1 / (1 + z2)^2 - return(y) -} - -conc <- seq(0, 10, by = 0.005) -plot(conc, f(conc, 3.2, 3.5), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) -lines(conc, f(conc, 1.5, 1.5), col = "#F2A61C") -lines(conc, f(conc, 1, 1), col = "#1CADF2") -lines(conc, f(conc, 1, 4), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -F <- function(x, a, b) { - z2 <- (x / a)^(-b) - y <- 1 / (1 + z2) - return(y) -} -conc <- seq(0, 10, by = 0.005) -plot(conc, F(conc, 3.2, 3.5), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) -lines(conc, F(conc, 1.5, 1.5), col = "#F2A61C") -lines(conc, F(conc, 1, 1), col = "#1CADF2") -lines(conc, F(conc, 1, 4), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") - -## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample gamma probability density (A) and cumulative probability (B) functions."---- -par(mfrow = c(1, 2)) - -conc <- seq(0, 10, by = 0.005) -plot(conc, dgamma(conc, 5, 5), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) -lines(conc, dgamma(conc, 4, 1), col = "#F2A61C") -lines(conc, dgamma(conc, 0.9, 1), col = "#1CADF2") -lines(conc, dgamma(conc, 2, 1.), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -conc <- seq(0, 10, by = 0.005) -plot(conc, pgamma(conc, 5, 5), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) -lines(conc, pgamma(conc, 4, 1), col = "#F2A61C") -lines(conc, pgamma(conc, 0.9, 1), col = "#1CADF2") -lines(conc, pgamma(conc, 2, 1.), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") - -## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Log-Gumbel probability density (A) and cumulative probability (B) functions."---- -par(mfrow = c(1, 2)) - -f <- function(x, a, b) { - y <- b * exp(-(a * x)^(-b)) / (a^b * x^(b + 1)) - return(y) -} - -conc <- seq(0, 10, by = 0.005) -plot(conc, f(conc, 0.2, 5), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) -lines(conc, f(conc, 0.5, 1.5), col = "#F2A61C") -lines(conc, f(conc, 1, 2), col = "#1CADF2") -lines(conc, f(conc, 10, .5), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -F <- function(x, a, b) { - y <- exp(-(a * x)^(-b)) - return(y) -} - -conc <- seq(0, 10, by = 0.005) -plot(conc, F(conc, 0.2, 5), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) -lines(conc, F(conc, 0.5, 1.5), col = "#F2A61C") -lines(conc, F(conc, 1, 2), col = "#1CADF2") -lines(conc, F(conc, 10, .5), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") - -## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Gompertz probability density (A) and cumulative probability (B) functions."---- -par(mfrow = c(1, 2)) - -f <- function(x, n, b) { - y <- n * b * exp(n + b * x - n * exp(b * x)) - return(y) -} - -conc <- seq(0, 10, by = 0.005) -plot(conc, f(conc, 0.089, 1.25), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 2)) -lines(conc, f(conc, 0.001, 3.5), col = "#F2A61C") -lines(conc, f(conc, 0.0005, 1.1), col = "#1CADF2") -lines(conc, f(conc, 0.01, 5), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -F <- function(x, n, b) { - y <- 1 - exp(-n * exp(b * x - 1)) - return(y) -} - -conc <- seq(0, 10, by = 0.005) -plot(conc, F(conc, 0.089, 1.25), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, F(conc, 0.001, 3.5), col = "#F2A61C") -lines(conc, F(conc, 0.0005, 1.1), col = "#1CADF2") -lines(conc, F(conc, 0.01, 5), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") - -## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Weibull probability density (A) and cumulative probability (B) functions."---- -par(mfrow = c(1, 2)) - -conc <- seq(0, 10, by = 0.005) -plot(conc, dweibull(conc, 4.321, 4.949), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, dweibull(conc, 0.838, 0.911), col = "#F2A61C") -lines(conc, dweibull(conc, 1, 1.546), col = "#1CADF2") -lines(conc, dweibull(conc, 17.267, 7.219), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -conc <- seq(0, 10, by = 0.005) -plot(conc, pweibull(conc, 4.321, 4.949), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, pweibull(conc, 0.838, 0.911), col = "#F2A61C") -lines(conc, pweibull(conc, 1, 1.546), col = "#1CADF2") -lines(conc, pweibull(conc, 17.267, 7.219), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") - -## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample North American Pareto probability density (A) and cumulative probability (B) functions."---- -par(mfrow = c(1, 2)) - -conc <- seq(0, 10, by = 0.005) -plot(conc, extraDistr::dpareto(conc, 3, 2), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.5)) -lines(conc, extraDistr::dpareto(conc, 0.838, 0.911), col = "#F2A61C") -lines(conc, extraDistr::dpareto(conc, 4, 4), col = "#1CADF2") -lines(conc, extraDistr::dpareto(conc, 10, 7), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -conc <- seq(0, 10, by = 0.005) -plot(conc, extraDistr::ppareto(conc, 3, 2), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, extraDistr::ppareto(conc, 0.838, 0.911), col = "#F2A61C") -lines(conc, extraDistr::ppareto(conc, 4, 4), col = "#1CADF2") -lines(conc, extraDistr::ppareto(conc, 10, 7), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") - -conc <- seq(0, 10, by = 0.005) -plot(conc, extraDistr::ppareto(conc, 3, 2), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, extraDistr::ppareto(conc, 0.838, 0.911), col = "#F2A61C") -lines(conc, extraDistr::ppareto(conc, 4, 4), col = "#1CADF2") -lines(conc, extraDistr::ppareto(conc, 10, 7), col = "#1F1CF2") - -## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Sample North American inverse Pareto probability density (A) and cumulative probability (B) functions."---- -par(mfrow = c(1, 2)) - -f <- function(x, a, b) { - y <- a * (b^a) * x^(a - 1) - return(y) -} - -conc <- seq(0, 10, by = 0.001) -plot(conc, f(conc, 5, 0.1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 0.5)) -lines(conc, f(conc, 3, 0.1), col = "#F2A61C") -lines(conc, f(conc, 0.5, 0.1), col = "#1CADF2") -lines(conc, f(conc, 0.1, 0.1), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -F <- function(x, a, b) { - y <- (b * x)^a - return(y) -} - -conc <- seq(0, 10, by = 0.001) -plot(conc, F(conc, 5, 0.1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, F(conc, 3, 0.1), col = "#F2A61C") -lines(conc, F(conc, 0.5, 0.1), col = "#1CADF2") -lines(conc, F(conc, 0.1, 0.1), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") - -## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample European Pareto probability density (A) and cumulative probability (B) functions."---- -par(mfrow = c(1, 2)) - -conc <- seq(0, 10, by = 0.005) -plot(conc, actuar::dpareto(conc, 1, 1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, actuar::dpareto(conc, 2, 3), col = "#F2A61C") -lines(conc, actuar::dpareto(conc, 0.5, 1), col = "#1CADF2") -lines(conc, actuar::dpareto(conc, 10.5, 6.5), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -conc <- seq(0, 10, by = 0.005) -plot(conc, actuar::ppareto(conc, 1, 1), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, actuar::ppareto(conc, 2, 3), col = "#F2A61C") -lines(conc, actuar::ppareto(conc, 0.5, 1), col = "#1CADF2") -lines(conc, actuar::ppareto(conc, 10.5, 6.5), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") - -## ----echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample European inverse Pareto probability density (A) and cumulative probability (B) functions."---- -par(mfrow = c(1, 2)) - -conc <- seq(0, 10, by = 0.001) -plot(conc, actuar::dinvpareto(conc, 1, 1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 0.8)) -lines(conc, actuar::dinvpareto(conc, 1, 3), col = "#F2A61C") -lines(conc, actuar::dinvpareto(conc, 10, 0.1), col = "#1CADF2") -lines(conc, actuar::dinvpareto(conc, 2.045, 0.98), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -conc <- seq(0, 10, by = 0.001) -plot(conc, actuar::pinvpareto(conc, 1, 1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, actuar::pinvpareto(conc, 1, 3), col = "#F2A61C") -lines(conc, actuar::pinvpareto(conc, 10, 0.1), col = "#1CADF2") -lines(conc, actuar::pinvpareto(conc, 2.045, 0.98), col = "#1F1CF2") - -## ----results = "asis", echo = FALSE------------------------------------------- -cat(ssdtools::ssd_licensing_md()) - diff --git a/doc/distributions.Rmd b/doc/distributions.Rmd deleted file mode 100644 index 8ab423d4..00000000 --- a/doc/distributions.Rmd +++ /dev/null @@ -1,837 +0,0 @@ ---- -title: "Distributions in ssdtools" -author: "ssdtools Team" -date: '`r format(Sys.time(), "%Y-%m-%d", tz = "UTC")`' -bibliography: references.bib -mathfont: Courier -latex_engine: MathJax -output: rmarkdown::html_vignette -vignette: > - %\VignetteIndexEntry{Distributions in ssdtools} - %\VignetteEngine{knitr::rmarkdown} - %\VignetteEncoding{UTF-8} ---- - -```{r, include = FALSE} -knitr::opts_chunk$set( - collapse = TRUE, - comment = "#>", - fig.width = 6, - fig.height = 4 -) -``` - - - - - -## Distributions for SSD modelling - -Many authors have noted that there is no guiding theory in ecotoxicology to justify a particular distributional form for the SSD other than that its domain be restricted to the positive real line [@newman_2000; @Zajdlik_2005; @fox_2016]. -Distributions selected to use in model averaging of SSDs must be bounded by zero given that effect concentrations cannot be negative. -They must also be continuous, and generally unbounded on the right. -Furthermore, the selected distributions within the candidate model set should provide a variety of shapes to capture the diversity of shapes in empirical species sensitivity distributions. - -To date `r length(ssdtools::ssd_dists())` distributions have been implemented in `ssdtools`, although only `r length(ssdtools::ssd_dists_bcanz())` appear in the default set. -Here we provide a detailed account of all `r length(ssdtools::ssd_dists())` of the distributions available in `ssdtools`, and guidance on their use. - -### Original `ssdtools` distributions - -The log-normal, log-logistic and Gamma distributions have been widely used in SSD modelling, and were part of the original distribution set for early releases of `ssdtools` as developed by @thorley_ssdtools_2018. -They were adopted as the default set of three distributions in early updates of `ssdtools` and the associated ShinyApp [@dalgarno_shinyssdtools_2021]. -All three distributions show good convergence properties and are retained as part of the default model set in version 2.0 of `ssdtools`. - -In addition to the log-normal, log-logistic and Gamma distributions, the original version of `ssdtools` as developed by [Thorley and Schwarz (2018)](https://joss.theoj.org/papers/10.21105/joss.01082.pdf) also included three additional distributions in the candidate model set, including the log-gumbel, Gompertz and Weibull distributions. -Of these, the log-Gumbel (otherwise known as the inverse Weibull, see below) shows relatively good convergence [see Figure 32, @fox_methodologies_2022], and is also one of the limiting distributions of the Burr Type 3 distribution implemented in `ssdtools`, and has been retained in the default model set. -The Gompertz and Weibull distributions, however can exhibit unstable behaviour, sometimes showing poor convergence, and therefore been excluded from the default set [see Figure 32, @fox_methodologies_2022] - -### Burr III distribution - -A history of Burrlioz and the primary distributions it used were recently summmarized by @fox_recent_2021. - -> In 2000, Australia and New Zealand (Australian and New Zealand Environment and Conservation Council/Agriculture and Resource Management Council of Australia and New Zealand 2000) adopted an SSD‐based method for deriving WQBs, following a critical review of multiple WQB derivation methods (Warne 1998). -A distinct feature of the method was the use of a 3‐parameter Burr distribution to model the empirical SSD, which was implemented in the Burrlioz software tool (Campbell et al. 2000). -This represented a generalization of the methods previously employed by Aldenberg and Slob (1993) because the log–logistic distribution -was shown to be a specific case of the Burr family (Tadikamalla 1980). -Recent revision of the derivation method recognized that using the 3‐parameter Burr distributions for small sample sizes (<8 species) created additional uncertainty by estimating more parameters than could be justified, essentially overfitting the data (Batley et al. 2018). -Consequently, the method, and the updated software (Burrlioz Ver 2.0), now uses a 2‐parameter log–logistic distribution for these small -data sets, whereas the Burr type III distribution is used for data sets of 8 species or more (Batley et al. 2018; Australian and -New Zealand Guidelines 2018). - -[@fox_recent_2021] - -The probability density function, ${f_X}(x;b,c,k)$ and cumulative distribution function, ${F_X}(x;b,c,k)$ for the Burr III distribution are: - - - - - - - - - -
- Burr III Distribution - - $$f_X(x;b,c,k) = \frac{{b\,k\,c}}{{x^2}} \frac{{\left( \frac{b}{x} \right)}^{c - 1}}{{\left[ 1 + \left( \frac{b}{x} \right)^c \right]}^{k + 1}}, \quad b, c, k, x > 0$$ - -
- - $$F_X(x;b,c,k) = \frac{1}{{\left[ 1 + \left( \frac{b}{x} \right)^c \right]}^k}, \quad b, c, k, x > 0$$ - -
- -
-Burr III distribution - -$${{f_X}(x;b,c,k) = \frac{{b\,k\,c}}{{{x^2}}}\frac{{{{\left( {\frac{b}{x}} \right)}^{c - 1}}}}{{{{\left[ {1 + {{\left( {\frac{b}{x}} \right)}^c}} \right]}^{k + 1}}}}{\rm{ }}} {b,c,k,x > 0}$$ -
- - -$${{F_X}(x;b,c,k) = \frac{1}{{{{\left[ {1 + {{\left( {\frac{b}{x}} \right)}^c}} \right]}^k}}}{\rm{ }}} {b,c,k,x > 0}$$ -
- -```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5, fig.cap="Sample Burr probability density (A) and cumulative probability (B) functions."} -par(mfrow = c(1, 2)) - -f <- function(x, b, c, k) { - z1 <- (b / x)^(c - 1) - z2 <- (b / x)^c - y <- (b * c * k / x^2) * z1 / (1 + z2)^(k + 1) - return(y) -} - -conc <- seq(0, 10, by = 0.005) -plot(conc, f(conc, 1, 3, 5), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733") -lines(conc, f(conc, 1, 1, 2), col = "#F2A61C") -lines(conc, f(conc, 1, 2, 2), col = "#1CADF2") -lines(conc, f(conc, 1, 2, 5), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -F <- function(x, b, c, k) { - z2 <- (b / x)^c - y <- 1 / (1 + z2)^k - return(y) -} -conc <- seq(0, 10, by = 0.005) -plot(conc, F(conc, 1, 3, 5), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733") -lines(conc, F(conc, 1, 1, 2), col = "#F2A61C") -lines(conc, F(conc, 1, 2, 2), col = "#1CADF2") -lines(conc, F(conc, 1, 2, 5), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") -``` - -While the Burr type III distribution was adopted as the default distribution in Burrlioz, it is well known (e.g., Tadikamalla (1980)) that the Burr III distribution is related to several other theoretical distributions, some of which only exist as limiting cases of the Burr III, i.e., as one or more of the Burr III parameters approaches either zero or infinity. -The Burrlioz software incorporates logic that aims to identify situations where parameter estimates are tending towards either very large or very small values. -In such cases, fitting a Burr III distribution is abandoned and one of the limiting -distributions is fitted instead. - -Specifically: - - - As c tends to infinity the Burr III distribution tends to the inverse (North American) Pareto distribution (see [technical details](https://bcgov.github.io/ssdtools/articles/E_additional-technical-details.html)) - - - As k tends to infinity the Burr III distribution tends to the inverse Weibull (log-Gumbel) distribution (see [technical details](https://bcgov.github.io/ssdtools/articles/E_additional-technical-details.html)) - -In practical terms, if the Burr III distribution is fitted and k is estimated to be greater than 100, the estimation procedure is carried out again using an inverse Weibull distribution. -Similarly, if c is greater than 80 an (American) Pareto distribution is fitted. This is necessary to -ensure numerical stability. - -Since the Burr type III, inverse Pareto and inverse Weibull (log Gumbel) distributions are used by the Burrlioz software, these have been implemented in `ssdtools`. -However, we have found there are stability issues with both the Burr type III, as well as the inverse Pareto distributions, which currently precludes their inclusion in the default model set (see @fox_methodologies_2022, and below for more details). - -### Bimodal distributions - -The use of statistical mixture-models was promoted by Fox as a convenient and more realistic way of modelling bimodal toxicity data [@fisher2019]. -Although parameter heavy, statistical mixture models provide a better conceptual match to the inherent underlying data generating process since they directly model bimodality as a mixture of 2 underlying univariate distributions that represent, for example, different modes of action [@fox_recent_2021]. -It has been postulated that a mixture-model would only be selected in a model-averaging context when the fit afforded by the mixture is demonstrably better than the fit afforded by any single distribution. -This is a consequence of the high penalty in AICc associated with the increased number of parameters (p in Equation 7 of [@fox_recent_2021]) and will be most pronounced for relatively small sample sizes. - -The TMB version of `ssdtools` now includes the option of fitting two mixture distributions, individually or as part of a model average set. -These can be fitted using `ssdtools` by supplying the strings "llogis_llogis" and/or "lnorm_lnorm" to the *dists* argument in the *ssd_fit_dists* call. - -The underlying code for these mixtures has three components: the likelihood function required for TMB; exported R functions to allow the usual methods for a distribution to be called (p, q and r); and a set of supporting R functions (see @fox_methodologies_2022 Appendix D for more details). -Both mixtures have five parameters - two parameters for each of the component distributions and a mixing parameter (pmix) that defines the weighting of the two distributions in the ‘mixture.’ - -```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5, fig.cap="Sample lognormal lognormal mixture probability density (A) and cumulative probability (B) functions."} -par(mfrow = c(1, 2)) - -f <- function(x, m1, s1, m2, s2, p) { - y <- p * dlnorm(x, m1, s1) + (1 - p) * dlnorm(x, m2, s2) - return(y) -} - -conc <- seq(0, 5, by = 0.0025) -m1 <- 1 -s1 <- .2 -m2 <- 1.8 -s2 <- 1.5 -p <- 0.25 -plot(conc, f(conc, m1, s1, m2, s2, p), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, f(conc, 0.09, 0.5, 1, 0.08, 0.9), col = "#F2A61C") -lines(conc, f(conc, 0.5, 0.5, 1, 0.08, 0.9), col = "#1CADF2") -lines(conc, f(conc, 0.7, 1.5, 0.5, 0.1, .7), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -F <- function(x, m1, s1, m2, s2, p) { - y <- p * plnorm(x, m1, s1) + (1 - p) * plnorm(x, m2, s2) - return(y) -} - -conc <- seq(0, 10, by = 0.0025) -m1 <- 1 -s1 <- .2 -m2 <- 1.8 -s2 <- 1.5 -p <- 0.25 -plot(conc, F(conc, m1, s1, m2, s2, p), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, F(conc, 0.09, 0.5, 1, 0.08, 0.9), col = "#F2A61C") -lines(conc, F(conc, 0.5, 0.5, 1, 0.08, 0.9), col = "#1CADF2") -lines(conc, F(conc, 0.7, 1.5, 0.5, 0.1, .7), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") -``` - -
As can be see from the plot above, the mixture distributions provide a highly flexible means of modelling *bimodality* in an emprical SSD. -This happens, for example, when the toxicity data for some toxicant include both animal and plant species, or there are different modes of action operating. -Unfortunately, this increased flexibility comes with a high penalty in the model-averaging process. -The combination of small sample sizes and a high parameter count (typically 5 or more) means that mixture distributions will be down-weighted - even when they do a good job at describing the data. -For this reason, when attempting to model bimodal data, we suggest looking at the fit using the default set of distributions and then examining the fit with just one of either the log-normal mixture or the log-logistic mixture. -Keep in mind that this should only be done if the sample size is not pathologically small. -As a guide, Prof. David Fox recommends as an *absolute minimum* $n \ge 3k + 1$ but preferably $n \ge 5k + 1$ where $k$ is the number of model parameters. - -## Default Distributions - -While there is a variety of distributions available in `ssdtools`, the inclusion of all of them for estimating a model-averaged SSD is not recommended. - -By default, `ssdtools` uses the (corrected) Akaike Information Criterion for small sample size (AICc) as a measure of relative quality of fit for different distributions and as the basis for calculating the model-averaged weights. -However, the choice of distributions used to fit a model-averaged SSD can have a profound effect on the estimated *HCx* values. - -Deciding on a final default set of distributions to adopt using the model averaging approach is non-trivial, and we acknowledge that there is probably no definitive ‘solution’ to this issue. -However, the default set should be underpinned by a guiding principle of parsimony, i.e., the set should be as large as is necessary to cover a wide variety of distributional shapes and contingencies but no bigger. -Further, the default set should result in model-averaged estimates of *HCx* values that: 1) minimise bias; 2) have actual coverages of confidence intervals that are close to the nominal level of confidence; 3) estimated *HCx* and confidence intervals of *HCx* are robust to small changes in the data; and 4) represent a positively continuous distribution that has both right and left tails. - -The `ssdtools` development team has undertaken extensive simulation studies, as well as some detailed technical examinations of the various candidate distributions to examine issues of bias, coverage and numerical stability. -A detailed account of our findings can be found in our report [@fox_methodologies_2022] and are not repeated in detail here, although some of the issues associated with individual distributions are outlined below. - -### Currently recommended default distributions - -The default list of candidate distributions in `ssdtools` is comprised of the following: log-normal; log-logistic; gamma; inverse Weibull (log-Gumbel); Weibull; mixture of two log-normal distributions - -The default distributions are plotted below with a mean of 2 and standard deviation of 2 on the (natural) log concentration scale or around 7.4 on the concentration scale. - -```{r, message=FALSE, echo=FALSE, fig.width=7,fig.height=5, fig.cap="Currently recommended default distributions."} -library(ssdtools) -library(ggplot2) - -# theme_set(theme_bw()) - -set.seed(7) - -ssd_plot_cdf(ssd_match_moments(meanlog = 2, sdlog = 2)) + - scale_color_ssd() -``` - - -## Distributions currently implemented in `ssdtools` - -#### Burr Type III distribution - -The Burr Type 3 is a flexible three parameter distribution can be fitted using `ssdtools` by supplying the string `burrIII3` to the `dists` argument in the `ssd_fit_dists` call. - -
-Burr III distribution - -

See above for details - -
- -
-The Burr family of distributions has been central to the derivation of guideline values in Australia and New Zealand for over 20 yr [@fox_recent_2021]. While offering a high degree of flexibility, experience with these distributions during that time has repeatedly highlighted numerical stability and convergence issues when parameters are estimated using maximum likelihood [@fox_recent_2021]. -This is thought to be due to the high degree of collinearity between parameter estimates and/or relatively flat likelihood profiles [@fox_recent_2021], and is one of the motivations behind the logic coded into Burrlioz to revert to either of the two limiting distributions. -Burr Type 3 distribution is not currently one of the recommended distributionsin the default model set. -This is because of 1) the convergence issues associated with the Burr Type 3 distribution, 2) the fact that reverting to a limiting two parameter distribution does not fit easily within a model averaging framework, and 3) that one of the two limiting distributions (the inverse Pareto, see below) also has estimation and convergence issues. - -#### Log-normal - -The [log-normal](https://en.wikipedia.org/wiki/Log-normal_distribution) distribution is a commonly used distribution in the natural sciences - particularly as a probability model to describe right (positive)-skewed phenomena such as *concentration* data. - -A random variable, $X$ is lognormally distributed if the *logarithm* of $X$ is normally distributed. The *pdf* of $X$ is given by - -
-Lognormal distribution - -$${f_X}\left( {x;\mu ,\sigma } \right) = \frac{1}{{\sqrt {2\pi } {\kern 1pt} \sigma x}}\exp \left[ { - \frac{{{{\left( {\ln x - \mu } \right)}^2}}}{{2{\sigma ^2}}}} \right]{\rm{ ; }}x,\sigma > 0,\; - \infty < \mu < \infty $$ - -
- -The lognormal distribution was selected as the starting distribution given the data are for effect concentrations. -The log-normal distribution can be fitted using `ssdtools` supplying the string `lnorm` to the `dists` argument in the `ssd_fit_dists` call. - -```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample lognormal probability density (A) and cumulative probability (B) functions."} -par(mfrow = c(1, 2)) - -conc <- seq(0, 10, by = 0.005) -m1 <- 1 -s1 <- .2 -m2 <- 1.8 -s2 <- 1.5 -p <- 0.25 -plot(conc, dlnorm(conc, m1, s1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, dlnorm(conc, 0.4, 2), col = "#F2A61C") -lines(conc, dlnorm(conc, m1 * 2, s1), col = "#1CADF2") -lines(conc, dlnorm(conc, 0.9, 1.5), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -conc <- seq(0, 10, by = 0.005) -m1 <- 1 -s1 <- .2 -m2 <- 1.8 -s2 <- 1.5 -p <- 0.25 -plot(conc, plnorm(conc, m1, s1), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, plnorm(conc, 0.4, 2), col = "#F2A61C") -lines(conc, plnorm(conc, m1 * 2, s1), col = "#1CADF2") -lines(conc, plnorm(conc, 0.9, 1.5), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") -``` - -#### Log-logistic distribution - -Like the *lognormal* distribution, the [log-logistic](https://en.wikipedia.org/wiki/Log-logistic_distribution) is similarly defined, that is: if $X$ has a log-logistic distribution, then $Y = \ln (X)$ has a *logistic* distribution. - -
- - - - - - - - - - - - - - - - - - - - - -Log-logistic distribution - -$${{f_Y}\left( {y;\alpha ,\beta } \right) = \frac{{\left( {{\raise0.5ex\hbox{$\scriptstyle \beta $} -\kern-0.1em/\kern-0.15em -\lower0.25ex\hbox{$\scriptstyle \alpha $}}} \right){{\left( {{\raise0.5ex\hbox{$\scriptstyle y$} -\kern-0.1em/\kern-0.15em -\lower0.25ex\hbox{$\scriptstyle \alpha $}}} \right)}^{\beta - 1}}}}{{{{\left[ {1 + {{\left( {{\raise0.5ex\hbox{$\scriptstyle y$} -\kern-0.1em/\kern-0.15em -\lower0.25ex\hbox{$\scriptstyle \alpha $}}} \right)}^\beta }} \right]}^2}}};} {y,}$$ - -$${{F_Y}\left( {y;\alpha ,\beta } \right) = \frac{{{{\left( {{\raise0.5ex\hbox{$\scriptstyle y$} -\kern-0.1em/\kern-0.15em -\lower0.25ex\hbox{$\scriptstyle \alpha $}}} \right)}^\beta }}}{{1 + {{\left( {{\raise0.5ex\hbox{$\scriptstyle y$} -\kern-0.1em/\kern-0.15em -\lower0.25ex\hbox{$\scriptstyle \alpha $}}} \right)}^\beta }}};} {y,}$$ - -$$\alpha ,\beta > 0$$ - -
- -
letting $\mu = \ln \left( \alpha \right)$ and $s = \frac{1}{\beta }$ we have: - -
-Logistic distribution - -$${{f_X}\left( {x;\mu ,s} \right) = \frac{{{e^{ - x}}}}{{{{\left( {1 + {e^{ - x}}} \right)}^2}}};} {x,s > 0,\; - \infty < \mu < \infty }$$ - -$${{F_X}\left( {x;\mu ,s} \right) = \frac{1}{{1 + {e^{ - \frac{{\left( {x - \mu } \right)}}{s}}}}};} {x,s > 0,\; - \infty < \mu < \infty }$$ -
- - -
-The log-logistic distribution is often used as a candidate SSD primarily because of its analytic tractability [@aldenberg_confidence_1993]. -We included it because it has wider tails than the log-normal and because it is a specific case of the more general Burr family of distributions [@shao_estimation_2000, @burr_cumulative_1942]. -The log-logistic distribution can be fitted using `ssdtools` by supplying the string `lnorm` to the `dists` argument in the `ssd_fit_dists` call. - -```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Log logistic probability density (A) and cumulative probability (B) functions."} -par(mfrow = c(1, 2)) - -f <- function(x, a, b) { - z1 <- (x / a)^(b - 1) - z2 <- (x / a)^b - y <- (b / a) * z1 / (1 + z2)^2 - return(y) -} - -conc <- seq(0, 10, by = 0.005) -plot(conc, f(conc, 3.2, 3.5), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) -lines(conc, f(conc, 1.5, 1.5), col = "#F2A61C") -lines(conc, f(conc, 1, 1), col = "#1CADF2") -lines(conc, f(conc, 1, 4), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -F <- function(x, a, b) { - z2 <- (x / a)^(-b) - y <- 1 / (1 + z2) - return(y) -} -conc <- seq(0, 10, by = 0.005) -plot(conc, F(conc, 3.2, 3.5), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) -lines(conc, F(conc, 1.5, 1.5), col = "#F2A61C") -lines(conc, F(conc, 1, 1), col = "#1CADF2") -lines(conc, F(conc, 1, 4), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") -``` - -#### Gamma distribution -The two-parameter [gamma](https://en.wikipedia.org/wiki/Gamma_distribution) distribution has the following *pdf* and *cdf*. - -
-Gamma distribution - -$${{f_X}\left( {x;b,c} \right) = \frac{{{x^{c - 1}}{e^{ - \left( {\frac{x}{b}} \right)}}}}{{{b^c}\,\Gamma \left( c \right)}}} {0 \le x}$$ - -$${{F_X}\left( {x;b,c} \right) = \frac{1}{{\,\Gamma \left( c \right)}}\,\gamma \left( {c,\frac{x}{b}} \right)} {0 \le x}$$ -$$ < \infty ;b,c > 0$$ - -
- -
where $\Gamma \left( \cdot \right)$ is the *gamma* function (in `R` this is simply `gamma(x)`) and $\gamma \left( \cdot \right)$ is the (lower) *incomplete gamma* function \[\gamma \left( {x,a} \right) = \int\limits_0^x {{t^{a - 1}}} \,{e^{ - t}}\,dt\] (this can be computed using the `gammainc` function from the `pracma` package in `R`). - -For use in modeling species sensitivity data, the gamma distribution has two key features that provide additional flexibility relative to the log-normal distribution: 1) it is asymmetrical on the logarithmic scale; and 2) it has wider tails. - -The gamma distribution can be fitted using `ssdtools` by supplying the string "gamma" to the *dists* argument in the *ssd_fit_dists* call. - -```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample gamma probability density (A) and cumulative probability (B) functions."} -par(mfrow = c(1, 2)) - -conc <- seq(0, 10, by = 0.005) -plot(conc, dgamma(conc, 5, 5), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) -lines(conc, dgamma(conc, 4, 1), col = "#F2A61C") -lines(conc, dgamma(conc, 0.9, 1), col = "#1CADF2") -lines(conc, dgamma(conc, 2, 1.), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -conc <- seq(0, 10, by = 0.005) -plot(conc, pgamma(conc, 5, 5), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) -lines(conc, pgamma(conc, 4, 1), col = "#F2A61C") -lines(conc, pgamma(conc, 0.9, 1), col = "#1CADF2") -lines(conc, pgamma(conc, 2, 1.), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") -``` - -#### Log-gumbel (inverse Weibull) distribution - -The log-gumbel distribution is a two-parameter distribution commonly used to model extreme values. -The log-gumbel distribution can be fitted using `ssdtools` by supplying the string `lgumbel` to the `dists` argument in the `ssd_fit_dists` call. -The two-parameter log-gumbel distribution has the following *pdf* and *cdf*: - -
-Log-Gumbel distribution - -$${{f_X}\left( {x;\alpha ,\beta } \right) = \frac{{\beta \,{e^{ - {{(\alpha x)}^{ - \beta }}}}}}{{{\alpha ^\beta }\,{x^{\beta + 1}}}}} ;\quad x,\alpha ,\beta > 0\\$$ - -$${{F_X}\left( {x;\alpha ,\beta } \right) = {e^{ - {{(\alpha x)}^{ - \beta }}}}} ;\quad x,\alpha ,\beta > 0$$ - -
- - - - - - - - - - - -```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Log-Gumbel probability density (A) and cumulative probability (B) functions."} -par(mfrow = c(1, 2)) - -f <- function(x, a, b) { - y <- b * exp(-(a * x)^(-b)) / (a^b * x^(b + 1)) - return(y) -} - -conc <- seq(0, 10, by = 0.005) -plot(conc, f(conc, 0.2, 5), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) -lines(conc, f(conc, 0.5, 1.5), col = "#F2A61C") -lines(conc, f(conc, 1, 2), col = "#1CADF2") -lines(conc, f(conc, 10, .5), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -F <- function(x, a, b) { - y <- exp(-(a * x)^(-b)) - return(y) -} - -conc <- seq(0, 10, by = 0.005) -plot(conc, F(conc, 0.2, 5), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.1)) -lines(conc, F(conc, 0.5, 1.5), col = "#F2A61C") -lines(conc, F(conc, 1, 2), col = "#1CADF2") -lines(conc, F(conc, 10, .5), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") -``` - - -#### Gompertz distribution - -The Gompertz distribution is a flexible distribution that exhibits both positive and negative skewness. -The Gompertz distribution can be fitted using `ssdtools` by supplying the string `gompertz` to the `dists` argument in the `ssd_fit_dists` call. -We consider two parameterisations of the Gompertz distribution. -The first, as given in [Wikipedia](https://en.wikipedia.org/wiki/Gompertz_distribution) and also used in `ssdtools` [Gompertz] has the following *pdf* and *cdf*: - -
-Gompertz distribution: Parameterisation I - - - - - - - - - - -$${{f_X}\left( {x;\eta ,b} \right) = b{\kern 1pt} \eta \exp \left( {\eta + bx - \eta {e^{bx}}} \right)} ; \quad 0 \le x < \infty ,\eta ,b > 0\\$$ - - -$${{F_X}\left( {x;\eta ,b} \right) = 1 - \exp \left[ { - \eta \left( {{e^{bx}} - 1} \right)} \right]} ; \quad 0 \le x < \infty ,\eta ,b > 0$$ - - -
- -The second parameterisation in which the *product* $b\eta$ in the formulae above is replaced by the parameter $a$ giving: - -
-Gompertz distribution: Parameterisation II - - - - - - - - - - -$${{f_X}(x;a,b) = \frac{a}{b}\;{e^{\frac{x}{b}}}\;\exp \left[ { - a\left( {{e^{\frac{x}{b}}}\; - 1} \right)} \right]} {;\quad 0 \le x < \infty,a,b > 0}$$ - -$${{F_X}(x;a,b) = 1 - \;\exp \left[ { - a\left( {{e^{\frac{x}{b}}}\; - 1} \right)} \right]} {;\quad 0 \le x < \infty,a,b > 0}$$ - -
- -```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Gompertz probability density (A) and cumulative probability (B) functions."} -par(mfrow = c(1, 2)) - -f <- function(x, n, b) { - y <- n * b * exp(n + b * x - n * exp(b * x)) - return(y) -} - -conc <- seq(0, 10, by = 0.005) -plot(conc, f(conc, 0.089, 1.25), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 2)) -lines(conc, f(conc, 0.001, 3.5), col = "#F2A61C") -lines(conc, f(conc, 0.0005, 1.1), col = "#1CADF2") -lines(conc, f(conc, 0.01, 5), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -F <- function(x, n, b) { - y <- 1 - exp(-n * exp(b * x - 1)) - return(y) -} - -conc <- seq(0, 10, by = 0.005) -plot(conc, F(conc, 0.089, 1.25), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, F(conc, 0.001, 3.5), col = "#F2A61C") -lines(conc, F(conc, 0.0005, 1.1), col = "#1CADF2") -lines(conc, F(conc, 0.01, 5), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") -``` - -The Gompertz distribution is available in `ssdtools`, however parameter estimation can be somewhat unstable [@fox_methodologies_2022], and for this reason it is not currently included in the default set. - -#### Weibull distribution - -The inclusion of the Weibull distribution and inverse Pareto distribution (see next) in `ssdtools` was primarily necessitated by the need to maintain consistency with the calculations undertaken in `Burrlioz`. -As mentioned earlier, both the Weibull and inverse Pareto distributions arise as *limiting distributions* when the Burr parameters $c$ and $k$ tend to either zero and/or infinity in specific ways. - -The two-parameter [Weibull](https://en.wikipedia.org/wiki/Weibull_distribution) distribution has the following *pdf* and *cdf*: - -
-Weibull distribution - - - - - - - - - - -$${{f_X}(x;a,b) = \frac{c}{\theta }\,{{\left( {\frac{x}{\theta }} \right)}^{c - 1}}{e^{ - {{\left( {\frac{x}{\theta }} \right)}^c}}}} {;\quad 0 \le x < \infty ,c,\theta > 0}$$ - - -$${{F_X}(x;a,b) = 1 - \;{e^{ - {{\left( {\frac{x}{\theta }} \right)}^c}}}} {;\quad 0 \le x < \infty ,c,\theta > 0}$$ - -
- -The Weibull distribution can be fitted in `ssdtools` by supplying the string `weibull` to the `dists` argument in the `ssd_fit_dists` call. - -```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Weibull probability density (A) and cumulative probability (B) functions."} -par(mfrow = c(1, 2)) - -conc <- seq(0, 10, by = 0.005) -plot(conc, dweibull(conc, 4.321, 4.949), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, dweibull(conc, 0.838, 0.911), col = "#F2A61C") -lines(conc, dweibull(conc, 1, 1.546), col = "#1CADF2") -lines(conc, dweibull(conc, 17.267, 7.219), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -conc <- seq(0, 10, by = 0.005) -plot(conc, pweibull(conc, 4.321, 4.949), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, pweibull(conc, 0.838, 0.911), col = "#F2A61C") -lines(conc, pweibull(conc, 1, 1.546), col = "#1CADF2") -lines(conc, pweibull(conc, 17.267, 7.219), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") -``` - - -#### Inverse Pareto distribution - -The inverse Pareto distribution can be fitted using `ssdtools` by supplying the string `invpareto` to the `dists` argument in the `ssd_fit_dists` call. - -While the inverse Pareto distribution is implemented in the `Burrlioz 2.0` software, it is important to understand that it is done so only as one of the limiting Burr distributions (see [technical details](https://bcgov.github.io/ssdtools/articles/E_additional-technical-details.html)). -The inverse Pareto is not offered as a stand-alone option in the `Burrlioz 2.0` software. We have spent considerable time and effort exploring the properties of the inverse Pareto distribution, including deriving bias correction equations and alternative methods for deriving confidence intervals [@fox_methodologies_2022]. -This work has substantial value for improving the current `Burrlioz 2.0` method, and our bias corrections should be adopted when deriving *HCx* estimates from the inverse Pareto where parameters have been estimated using maximum likelihood. - -As is the case with the `Burrlioz 2.0` software, we have decided not to include the inverse Pareto distribution in the default candidate set in `ssdtools` although it is offered ass a user-selectable distribution to use in the model-fitting process. - -As with many statistical distributions, different 'variants' exist. These 'variants' are not so much different distributions as they are simple re-parameterisations. -For example, many distributions have a *scale* parameter, $\beta$ and some authors and texts will use $\beta$ while others use $\frac{1}{\beta }$. -An example of this re-parameterisation was given above for the Gompertz distribution. -While the choice of mathematical representation may be purely preferential, it is sometimes done for mathematical convenience. -For example, Parameterisation I of the Gompertz distribution above was obtained by letting $a = b\eta$ in Parameterisation II. -This re-expression involving parameters $b$ and $\eta$ would be particularly useful when trying to fit a distribution for which one of $\left\{ {b,\,\eta } \right\}$ was very small and the other was very large. - -It has already been noted that the particular parameterisation of the (Inverse)Pareto distribution used in both `Burrlioz 2.0` and `ssdtools` was **not** a matter of preference, but rather was dictated by mathematical considerations which demonstrated convergence of the Burr distribution to one specific version of the (Inverse)Pareto distribution. -While the mathematics provides an elegant solution to an otherwise problematic situation, this version of the (Inverse)Pareto distribution is not particularly use as a stand-alone distribution for fitting an SSD (other than as a special, limiting case of the Burr distribution). - -The two versions of the (Inverse)Pareto distribution are known as the *European* and *North American* versions. Their *pdfs* and *cdfs* are given below. - -
- -
-                    (Inverse)Pareto distribution - North American version

Pareto distribution - - - - - - - - - - - - - - - - - - - - -$${{f_X}\left( {x;\alpha ,\beta } \right) = \alpha {\kern 1pt} {\beta ^\alpha }{\kern 1pt} {x^{ - \left( {\alpha + 1} \right)}}} ; \quad x > \beta ;\;\alpha ,\beta > 0\\$$ - - -$${{F_X}\left( {x;\alpha ,\beta } \right) = 1 - {{\left( {\frac{\beta }{x}} \right)}^\alpha }}; \quad x > \beta ;\;\alpha ,\beta > 0$$ - - -Now, if X has the Pareto distribution above, then \[Y = \frac{1}{X}\] has an inverse Pareto distribution.

inverse Pareto distribution - - -$${{g_Y}\left( {y;\alpha ,\beta } \right) = \alpha {\kern 1pt} {\beta ^\alpha }{\kern 1pt} {y^{\alpha - 1}}} ;\quad y \le \frac{1}{\beta };\;\;\alpha ,\beta > 0\\$$ - -$${{G_Y}\left( {y;\alpha ,\beta } \right) = {{\left( {\beta \,y} \right)}^\alpha }}; \quad 0 < y \le \frac{1}{\beta };\;\;\alpha ,\beta > 0$$ - -
- -

Importantly, we see that the *North American* versions of these distributions are bounded with the Pareto distribution bounded **below** by $\beta$ and the inverse Pareto distribution bounded *above* by $\frac{1}{\beta }$.
As an aside, the *mle* of $\beta$ in the Pareto distribution is \[\hat \beta = \min \left\{ {{X_1}, \ldots ,{X_n}} \right\}\] and the *mle* of $\frac{1}{\beta }$ in the inverse Pareto is \[\begin{array}{*{20}{l}} -{\tilde \beta = \max \left\{ {{Y_1}, \ldots ,{Y_n}} \right\}}\\ -{\quad = \max \left\{ {\frac{1}{{{X_1}}}, \ldots ,\frac{1}{{{X_n}}}} \right\} = \frac{1}{{\min \left\{ {{X_1}, \ldots ,{X_n}} \right\}}}}\\ -{\quad = \frac{1}{{\hat \beta }}} -\end{array}\]. - -and the *mle* of $\alpha$ is: \[\hat \alpha = {\left[ {\ln \left( {\frac{g}{{\hat \beta }}} \right)} \right]^{ - 1}}\] where $g$ is the *geometric mean*: \[g = {\left[ {\prod\limits_{i = 1}^n {{X_i}} } \right]^{\frac{1}{n}}}\] - -Thus, it doesn't matter whether you're fitting a Pareto or inverse Pareto distribution to your data - the parameter estimates are the same. - -Because it is *bounded*, the North American version of the (Inverse)Pareto distribution is not useful as a stand-alone SSD - more so for the *inverse Pareto* distribution since it is bounded from *above*. - -```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample North American Pareto probability density (A) and cumulative probability (B) functions."} -par(mfrow = c(1, 2)) - -conc <- seq(0, 10, by = 0.005) -plot(conc, extraDistr::dpareto(conc, 3, 2), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1.5)) -lines(conc, extraDistr::dpareto(conc, 0.838, 0.911), col = "#F2A61C") -lines(conc, extraDistr::dpareto(conc, 4, 4), col = "#1CADF2") -lines(conc, extraDistr::dpareto(conc, 10, 7), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -conc <- seq(0, 10, by = 0.005) -plot(conc, extraDistr::ppareto(conc, 3, 2), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, extraDistr::ppareto(conc, 0.838, 0.911), col = "#F2A61C") -lines(conc, extraDistr::ppareto(conc, 4, 4), col = "#1CADF2") -lines(conc, extraDistr::ppareto(conc, 10, 7), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") - -conc <- seq(0, 10, by = 0.005) -plot(conc, extraDistr::ppareto(conc, 3, 2), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, extraDistr::ppareto(conc, 0.838, 0.911), col = "#F2A61C") -lines(conc, extraDistr::ppareto(conc, 4, 4), col = "#1CADF2") -lines(conc, extraDistr::ppareto(conc, 10, 7), col = "#1F1CF2") -``` - -```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample Sample North American inverse Pareto probability density (A) and cumulative probability (B) functions."} -par(mfrow = c(1, 2)) - -f <- function(x, a, b) { - y <- a * (b^a) * x^(a - 1) - return(y) -} - -conc <- seq(0, 10, by = 0.001) -plot(conc, f(conc, 5, 0.1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 0.5)) -lines(conc, f(conc, 3, 0.1), col = "#F2A61C") -lines(conc, f(conc, 0.5, 0.1), col = "#1CADF2") -lines(conc, f(conc, 0.1, 0.1), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -F <- function(x, a, b) { - y <- (b * x)^a - return(y) -} - -conc <- seq(0, 10, by = 0.001) -plot(conc, F(conc, 5, 0.1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, F(conc, 3, 0.1), col = "#F2A61C") -lines(conc, F(conc, 0.5, 0.1), col = "#1CADF2") -lines(conc, F(conc, 0.1, 0.1), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") -``` - -We see from the *pdf* plots that the alternative, *European* version of the inverse Pareto distribution is a more realistic candidate. - -
- -
-                    (Inverse)Pareto distribution - European version

Pareto distribution - - -$${{f_X}\left( {x;\alpha ,\beta } \right) = \frac{{\alpha \,\beta \,{x^{\alpha - 1}}}}{{{{\left( {x + \beta } \right)}^{\alpha + 1}}}}} ; \quad x,\;\alpha ,\beta > 0\\$$ - - -$${{F_X}\left( {x;\alpha ,\beta } \right) = 1 - {{\left( {\frac{\beta }{{x + \beta }}} \right)}^\alpha }}; -\quad x,\;\alpha ,\beta > 0$$ - - -Now, if X has the Pareto distribution above, then \[Y = \frac{1}{X}\] has an inverse Pareto distribution.

inverse Pareto distribution - -$${{g_Y}\left( {y;\alpha ,\beta } \right) = \frac{{\alpha {\kern 1pt} {\beta ^\alpha }{\kern 1pt} {y^{\alpha - 1}}}}{{{{\left( {1 + \beta y} \right)}^{\alpha + 1}}}}}; -\quad y,\;\alpha ,\beta > 0\\$$ - - -$${{G_Y}\left( {y;\alpha ,\beta } \right) = {{\left( {\frac{{\beta y}}{{1 + \beta y}}} \right)}^\alpha }}; -\quad y,\;\alpha ,\beta > 0$$ - -
- -We note in passing that *both* versions of these Pareto and inverse Pareto distrbutions are available in `R`. -For example, the `R`package `extraDistr` has North American versions, while the `actuar` package has European versions. - -```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample European Pareto probability density (A) and cumulative probability (B) functions."} -par(mfrow = c(1, 2)) - -conc <- seq(0, 10, by = 0.005) -plot(conc, actuar::dpareto(conc, 1, 1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, actuar::dpareto(conc, 2, 3), col = "#F2A61C") -lines(conc, actuar::dpareto(conc, 0.5, 1), col = "#1CADF2") -lines(conc, actuar::dpareto(conc, 10.5, 6.5), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -conc <- seq(0, 10, by = 0.005) -plot(conc, actuar::ppareto(conc, 1, 1), type = "l", ylab = "Cumulative probability", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, actuar::ppareto(conc, 2, 3), col = "#F2A61C") -lines(conc, actuar::ppareto(conc, 0.5, 1), col = "#1CADF2") -lines(conc, actuar::ppareto(conc, 10.5, 6.5), col = "#1F1CF2") -legend("topleft", "(B)", bty = "n") -``` - -```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample European inverse Pareto probability density (A) and cumulative probability (B) functions."} -par(mfrow = c(1, 2)) - -conc <- seq(0, 10, by = 0.001) -plot(conc, actuar::dinvpareto(conc, 1, 1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 0.8)) -lines(conc, actuar::dinvpareto(conc, 1, 3), col = "#F2A61C") -lines(conc, actuar::dinvpareto(conc, 10, 0.1), col = "#1CADF2") -lines(conc, actuar::dinvpareto(conc, 2.045, 0.98), col = "#1F1CF2") -legend("topleft", "(A)", bty = "n") - -conc <- seq(0, 10, by = 0.001) -plot(conc, actuar::pinvpareto(conc, 1, 1), type = "l", ylab = "Probability density", xlab = "Concentration", col = "#FF5733", ylim = c(0, 1)) -lines(conc, actuar::pinvpareto(conc, 1, 3), col = "#F2A61C") -lines(conc, actuar::pinvpareto(conc, 10, 0.1), col = "#1CADF2") -lines(conc, actuar::pinvpareto(conc, 2.045, 0.98), col = "#1F1CF2") -``` - -#### Inverse Weibull distribution (see log-Gumbel, above) - -The inverse Weibull is mathematically equivalent to the log-Gumbel distribution described above. -While which is also a limiting distribution of the Burr Type 3, this distribution does not show the same instability issues, and is unbounded to the right. -It therefore represents a valid SSD distribution and is included in the default model set as a distribution in its own right. - -The inverse Weibull (log-Gumbel) distribution can be fitted in `ssdtools` by supplying the string `lgumbel` to the `dists` argument in the `ssd_fit_dists` call. - -## Relationships among distributions in `ssdtools` - -##### NOTES -1. In the diagram below, $X$ denotes the random variable in the box at the *beginning* of the arrow and the expression beside the arrow indicates the mathematical transformation of $X$ such that the resultant *transformed* data has the distribution identified in the box at the *end* of the arrow. - -2. *Reciprocal* transformations ($\frac{1}{X}$) are **bi-directional** ($\leftrightarrow$). - -3. Although the *negative exponential* distribution is **not** explicitly included in `ssdtools`, it is a special case of the *gamma* distribution with $c=1$. It is included in this figure as it is related to other distributions that *are* included in `ssdtools`. - -4. The *European* versions of the Pareto and inverse Pareto distributions are **unbounded**; the *North American* versions are **bounded**. - - -
- -![](images/relationships.png){width=100%} - -## References - -
- -```{r, results = "asis", echo = FALSE} -cat(ssdtools::ssd_licensing_md()) -``` diff --git a/doc/model-averaging.R b/doc/model-averaging.R deleted file mode 100644 index 27ed1e72..00000000 --- a/doc/model-averaging.R +++ /dev/null @@ -1,173 +0,0 @@ -## ----include = FALSE---------------------------------------------------------- -knitr::opts_chunk$set( - collapse = TRUE, - comment = "#>", - fig.width = 6, - fig.height = 4 -) -library(tinytex) - -## ----include=FALSE------------------------------------------------------------ -# this just loads the package and suppresses the load package message -library(ssdtools) -require(ggplot2) - -## ----echo=FALSE,warning=FALSE, message=FALSE,class.output="scroll-100"-------- -samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -print(samp) -# knitr::kable(samp,caption="Some toxicity data (concentrations)") - -## ----echo=FALSE,fig.cap="Emprirical cdf (black); Model 1(green); and Model 2 (blue)", fig.width=7,fig.height=4.5---- -samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -samp <- sort(samp) -plot(ecdf(samp), main = "Empirical and fitted SSDs", xlab = "Concentration", ylab = "Probability") -xx <- seq(0.01, 3, by = 0.01) -lines(xx, plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10]))), col = "#77d408") -lines(xx, plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))), col = "#08afd4") -# lines(xx,(0.4419*plnorm(xx,meanlog=mean(log(samp[5:15])),sd=sd(log(samp[5:15])))+ -# 0.5581*plnorm(xx,meanlog=mean(log(samp[1:10])),sd=sd(log(samp[1:10])))) ,col="#d40830") - -## ----echo=FALSE,fig.cap="Empirical cdf (black); Model 1(green); Model 2 (blue); and averaged Model (red)",fig.width=7,fig.height=4.5---- -samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -samp <- sort(samp) -plot(ecdf(samp), main = "Empirical and fitted SSDs", xlab = "Concentration", ylab = "Probability") -xx <- seq(0.01, 3, by = 0.01) -lines(xx, plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10]))), col = "#77d408") -lines(xx, plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))), col = "#08afd4") -lines(xx, (0.4419 * plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))) + - 0.5581 * plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10])))), col = "#d40830") - -## ----echo=TRUE---------------------------------------------------------------- -# Model 1 HC20 -cat("Model 1 HC20 =", qlnorm(0.2, -1.067, 0.414)) - -# Model 2 HC20 -cat("Model 2 HC20 =", qlnorm(0.2, -0.387, 0.617)) - -## ----echo=FALSE,fig.width=7,fig.height=5-------------------------------------- -samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -samp <- sort(samp) -xx <- seq(0.01, 3, by = 0.01) - - -plot(xx, (0.4419 * plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))) + - 0.5581 * plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10])))), -col = "#d40830", type = "l", xlab = "Concentration", ylab = "Probability" -) -segments(0.33, -1, 0.33, 0.292, col = "blue", lty = 21) -segments(-1, 0.292, 0.33, 0.292, col = "blue", lty = 21) -mtext("0.3", side = 2, at = 0.3, cex = 0.8, col = "blue") -mtext("0.33", side = 1, at = 0.33, cex = 0.8, col = "blue") - -## ----echo=FALSE, fig.width=8,fig.height=6------------------------------------- -t <- seq(0.01, 0.99, by = 0.001) - - -F <- 0.4419 * qlnorm(t, -1.067, 0.414) + 0.5581 * qlnorm(t, -0.387, 0.617) - -plot(xx, (0.4419 * plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))) + - 0.5581 * plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10])))), -col = "#d40830", type = "l", xlab = "Concentration", ylab = "Probability" -) - -lines(F, t, col = "#51c157", lwd = 1.75) - -segments(-1, 0.2, 0.34, 0.2, col = "black", lty = 21, lwd = 2) -segments(0.28, 0.2, 0.28, -1, col = "red", lty = 21, lwd = 2) -segments(0.34, 0.2, 0.34, -1, col = "#51c157", lty = 21, lwd = 2) -segments(1.12, -1, 1.12, 0.9, col = "grey", lty = 21, lwd = 1.7) - -text(0.25, 0.6, "Correct MA-SSD", col = "red", cex = 0.75) -text(0.75, 0.4, "Erroneous MA-SSD", col = "#51c157", cex = 0.75) -mtext("1.12", side = 1, at = 1.12, cex = 0.8, col = "grey") - -## ----echo=FALSE,warning=FALSE, results="markup",message=FALSE,class.output="scroll-100"---- -samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -print(samp) -# knitr::kable(samp,caption="Some toxicity data (concentrations)") - -## ----echo=TRUE,results='hide',warning=FALSE,message=FALSE--------------------- -dat <- data.frame(Conc = c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59)) -library(goftest) -library(EnvStats) # this is required for the Pareto cdf (ppareto) - -# Examine the fit for the gamma distribution (NB: parameters estimated from the data) -cvm.test(dat$Conc, null = "pgamma", shape = 2.0591977, scale = 0.3231032, estimated = TRUE) - -# Examine the fit for the lognormal distribution (NB: parameters estimated from the data) -cvm.test(dat$Conc, null = "plnorm", meanlog = -0.6695120, sd = 0.7199573, estimated = TRUE) - -# Examine the fit for the Pareto distribution (NB: parameters estimated from the data) -cvm.test(dat$Conc, null = "ppareto", location = 0.1800000, shape = 0.9566756, estimated = TRUE) - -## ----echo=FALSE,fig.cap="Emprirical cdf (black); lognormal (green); gamma (blue); and Pareeto (red)", fig.width=7,fig.height=4.5---- -library(EnvStats) -samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -samp <- sort(samp) -plot(ecdf(samp), main = "Empirical and fitted SSDs", xlab = "Concentration", ylab = "Probability") -xx <- seq(0.01, 3, by = 0.01) -lines(xx, plnorm(xx, meanlog = -0.6695120, sd = 0.7199573), col = "#77d408") -lines(xx, pgamma(xx, shape = 2.0591977, scale = 0.3231032), col = "#08afd4") -lines(xx, ppareto(xx, location = 0.1800000, shape = 0.9566756), col = "red") -# lines(xx,(0.4419*plnorm(xx,meanlog=mean(log(samp[5:15])),sd=sd(log(samp[5:15])))+ -# 0.5581*plnorm(xx,meanlog=mean(log(samp[1:10])),sd=sd(log(samp[1:10])))) ,col="#d40830") - -## ----echo=TRUE---------------------------------------------------------------- -sum(log(dgamma(dat$Conc, shape = 2.0591977, scale = 0.3231032))) -sum(log(dlnorm(dat$Conc, meanlog = -0.6695120, sdlog = 0.7199573))) -sum(log(EnvStats::dpareto(dat$Conc, location = 0.1800000, shape = 0.9566756))) - -## ----echo=FALSE,results='markup'---------------------------------------------- -dat <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -k <- 2 # number of parameters for each of the distributions -# Gamma distribution -aic1 <- 2 * k - 2 * sum(log(dgamma(dat, shape = 2.0591977, scale = 0.3231032))) -cat("AIC for gamma distribution =", aic1, "\n") - -# lognormal distribution -aic2 <- 2 * k - 2 * sum(log(dlnorm(dat, meanlog = -0.6695120, sdlog = 0.7199573))) -cat("AIC for lognormal distribution =", aic2, "\n") - -# Pareto distribution -aic3 <- 2 * k - 2 * sum(log(EnvStats::dpareto(dat, location = 0.1800000, shape = 0.9566756))) -cat("AIC for Pareto distribution =", aic3, "\n") - -## ----echo=TRUE,results='hide'------------------------------------------------- -dat <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -aic <- NULL -k <- 2 # number of parameters for each of the distributions - - -aic[1] <- 2 * k - 2 * sum(log(dgamma(dat, shape = 2.0591977, scale = 0.3231032))) # Gamma distribution - -aic[2] <- 2 * k - 2 * sum(log(dlnorm(dat, meanlog = -0.6695120, sdlog = 0.7199573))) # lognormal distribution - -aic[3] <- 2 * k - 2 * sum(log(EnvStats::dpareto(dat, location = 0.1800000, shape = 0.9566756))) # Pareto distribution - -delta <- aic - min(aic) # compute the delta values - -aic.w <- exp(-0.5 * delta) -aic.w <- round(aic.w / sum(aic.w), 4) - -cat( - " AIC weight for gamma distribution =", aic.w[1], "\n", - "AIC weight for lognormal distribution =", aic.w[2], "\n", - "AIC weight for pareto distribution =", aic.w[3], "\n" -) - -## ----echo=FALSE,fig.cap="Empirical cdf (black) and model-averaged fit (magenta)",fig.width=8,fig.height=5---- -samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -samp <- sort(samp) -plot(ecdf(samp), main = "Empirical and fitted SSDs", xlab = "Concentration", ylab = "Probability") -xx <- seq(0.01, 3, by = 0.005) - -lines(xx, plnorm(xx, meanlog = -0.6695120, sd = 0.7199573), col = "#959495", lty = 2) -lines(xx, pgamma(xx, shape = 2.0591977, scale = 0.3231032), col = "#959495", lty = 3) -lines(xx, ppareto(xx, location = 0.1800000, shape = 0.9566756), col = "#959495", lty = 4) -lines(xx, 0.1191 * pgamma(xx, shape = 2.0591977, scale = 0.3231032) + - 0.3985 * plnorm(xx, meanlog = -0.6695120, sd = 0.7199573) + - 0.4824 * ppareto(xx, location = 0.1800000, shape = 0.9566756), col = "#FF33D5", lwd = 1.5) - -## ----results = "asis", echo = FALSE------------------------------------------- -cat(ssdtools::ssd_licensing_md()) - diff --git a/doc/model-averaging.Rmd b/doc/model-averaging.Rmd deleted file mode 100644 index a00e73b7..00000000 --- a/doc/model-averaging.Rmd +++ /dev/null @@ -1,538 +0,0 @@ ---- -title: "Model Averaging SSDs" -author: "ssdtools Team" -date: '`r format(Sys.time(), "%Y-%m-%d", tz = "UTC")`' -bibliography: references.bib -mathfont: Courier -latex_engine: MathJax -output: rmarkdown::html_vignette -vignette: > - %\VignetteIndexEntry{Model Averaging SSDs} - %\VignetteEngine{knitr::rmarkdown} - %\VignetteEncoding{UTF-8} ---- - -```{r, include = FALSE} -knitr::opts_chunk$set( - collapse = TRUE, - comment = "#>", - fig.width = 6, - fig.height = 4 -) -library(tinytex) -``` - - -```{css, echo=FALSE} -pre { - max-height: 300px; - overflow-y: auto; -} - -pre[class] { - max-height: 200px; -} -``` - -```{css, echo=FALSE} -.scroll-200 { - max-height: 200px; - overflow-y: auto; - background-color: inherit; -} -``` - - - - - -```{r include=FALSE} -# this just loads the package and suppresses the load package message -library(ssdtools) -require(ggplot2) -``` - -## Background - -> Many authors have noted that there is no guiding theory in ecotoxicology to justify any particular distributional form for the SSD other than that its domain be restricted to the positive real line [@newman_2000], [@Zajdlik_2005], [@chapman_2007], [@fox_2016]. -Indeed, [@chapman_2007] described the identification of a suitable probability model as one of the most important and difficult choices in the use of SSDs. -Compounding this lack of clarity about the functional form of the SSD is the omnipresent, and equally vexatious issue of small sample size, meaning that any plausible candidate model is unlikely to be rejected [@fox_recent_2021]. -The ssdtools R package uses a model averaging procedure to avoid the need to a-priori select a candidate distribution and instead uses a measure of ‘fit’ for each model to compute weights to be applied to an initial set of candidate distributions. -The method, as applied in the SSD context is described in detail in [@fox_recent_2021], and potentially provides a level of flexibility and parsimony that is difficult to achieve with a single SSD distribution. - -[@fox_methodologies_2022] - -## Preliminaries - -Before we jump into model averaging and in particular, SSD Model Averaging, let's backup a little and consider why we average and the advantages and disadvantages of averaging. - -#### The pros and cons of averaging - -We're all familiar with the process of averaging. -Indeed, *averages* are pervasive in everyday life - we talk of average income; mean sea level; average global temperature; average height, weight, age etc. etc. So what's the obsession with *averaging*? -It's simple really - it's what statisticians call data reduction which is just a fancy name to describe the process of summarising a lot of *raw data* using a small number of (hopefully) representative summary statistics such as the mean and the standard deviation. -Clearly, it's a lot easier to work with just a single mean than all the individual data values. -That's the upside. -The downside is that the process of data reduction decimates your original data - you lose information in the process. -Nevertheless, the benefits tend to outweigh this information loss. -Indeed, much of 'conventional' statistical theory and practice is focused on the mean. -Examples include T-tests, ANOVA, regression, and clustering. -When we talk of an 'average' we are usually referring to the simple, *arithmetic mean*:\[\bar{X}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{X}_{i}}}\] although we recognize there are other types of mean including the geometric mean, the harmonic mean and the weighted mean. -The last of these is particularly pertinent to model averaging. - -### Weighted Averages - -For the simple arithmetic mean, all of the individual values receive the same weighting - they each contribute $\frac{1}{n}$ to the summation. -While this is appropriate in many cases, it's not useful when the components contribute to varying degrees. -An example familiar to ecotoxicologists is that of a *time-varying* concentration as shown in the figure below.
- -![](images/Figure1.jpg){width=90% align="center"} - -From the figure we see there are 5 concentrations going from left to to right: $\left\{ 0.25,0.95,0.25,0.12,0.5 \right\}$. -If we were to take the simple arithmetic mean of these concentrations we get $\bar{X}=0.414$. -But this ignores the different *durations* of these 5 concentrations. -Of the 170 hours, 63 were at concentration 0.25, 25 at concentration 0.95, 23 at concentration 0.25, 23 at concentration 0.12, and 36 at concentration 0.50. -So if we were to *weight* these concentrations by time have:
\[{{{\bar{X}}}_{TW}}=\frac{\left( 63\cdot 0.25+25\cdot 0.95+23\cdot 0.25+23\cdot 0.12+36\cdot 0.50 \right)}{\left( 63+25+23+23+36 \right)}=\frac{56.01}{170}=0.33\]
So, our formula for a *weighted average* is:\[\bar{X}=\sum\limits_{i=1}^{n}{{{w}_{i}}{{X}_{i}}}\] with $0\le {{w}_{i}}\le 1$ and $\sum\limits_{i=1}^{n}{{{w}_{i}}=1}$. -
Note, the simple arithmetic mean is just a special case of the weighted mean with $\sum\limits_{i=1}^{n}{{{w}_{i}}=\frac{1}{n}}$ ; $\forall i=1,\ldots ,n$ - -## Model Averaging -The *weighted average* acknowledges that the elements in the computation are not of equal 'importance'. -In the example above, this importance was based on the *proportion of time* that the concentration was at a particular level. -Bayesians are well-versed in this concept - the elicitation of *prior distributions* for model parameters provides a mechanism for weighting the degree to which the analysis is informed by existing knowledge versus using a purely data-driven approach. -Model averaging is usually used in the context of estimating model parameters or quantities derived from a fitted model - for example an EC50 derived from a C-R model. -Let's motivate the discussion using the following small dataset of toxicity estimates for some chemical. - - -```{r echo=FALSE,warning=FALSE, message=FALSE,class.output="scroll-100"} -samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -print(samp) -# knitr::kable(samp,caption="Some toxicity data (concentrations)") -``` - -Now, suppose we have only two possibilities for fitting an SSD - both lognormal distributions. -Model 1 is the LN(-1.067,0.414) distribution while Model 2 is the LN(-0.387,0.617) distribution. -A plot of the empirical *cdf* and Models 1 and 2 is shown below. - -```{r echo=FALSE,fig.cap="Emprirical cdf (black); Model 1(green); and Model 2 (blue)", fig.width=7,fig.height=4.5} -samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -samp <- sort(samp) -plot(ecdf(samp), main = "Empirical and fitted SSDs", xlab = "Concentration", ylab = "Probability") -xx <- seq(0.01, 3, by = 0.01) -lines(xx, plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10]))), col = "#77d408") -lines(xx, plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))), col = "#08afd4") -# lines(xx,(0.4419*plnorm(xx,meanlog=mean(log(samp[5:15])),sd=sd(log(samp[5:15])))+ -# 0.5581*plnorm(xx,meanlog=mean(log(samp[1:10])),sd=sd(log(samp[1:10])))) ,col="#d40830") -``` - -
We see that Model 1 fits well in the lower, left region and poorly in the upper region, while the reverse is true for Model 2. -So using *either* Model 1 **or** Model 2 is going to result in a poor fit overall. -However, the obvious thing to do is to **combine** both models. -We could just try using 50% of Model 1 and 50% of Model 2, but that may be sub-optimal. -It turns out that the best fit is obtained by using 44% of Model 1 and 56% of Model 2. Redrawing the plot and adding the *weighted average* of Models 1 and 2 is shown below. - -```{r echo=FALSE,fig.cap="Empirical cdf (black); Model 1(green); Model 2 (blue); and averaged Model (red)",fig.width=7,fig.height=4.5} -samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -samp <- sort(samp) -plot(ecdf(samp), main = "Empirical and fitted SSDs", xlab = "Concentration", ylab = "Probability") -xx <- seq(0.01, 3, by = 0.01) -lines(xx, plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10]))), col = "#77d408") -lines(xx, plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))), col = "#08afd4") -lines(xx, (0.4419 * plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))) + - 0.5581 * plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10])))), col = "#d40830") -``` - -
Clearly the strategy has worked - we now have an excellent fitting SSD. -What about estimation of an *HC20*? -It's a simple matter to work out the *individual* *HC20* values for Models 1&2 using the appropriate `qlnorm()` function in `R`. -Thus we have: - -```{r, echo=TRUE} -# Model 1 HC20 -cat("Model 1 HC20 =", qlnorm(0.2, -1.067, 0.414)) - -# Model 2 HC20 -cat("Model 2 HC20 =", qlnorm(0.2, -0.387, 0.617)) -``` -What about the averaged distribution? -An intuitively appealing approach would be to apply the same weights to the individual *HC20* values as was applied to the respective models. That is `0.44*0.2428209 + 0.56*0.4040243 = 0.33`. - -So our model-averaged *HC20* estimate is 0.33. -As a check, we can determine the *fraction affected* at concentration = 0.33 - it should of course be 20%. Let's take a look at the plot. - -```{r echo=FALSE,fig.width=7,fig.height=5} -samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -samp <- sort(samp) -xx <- seq(0.01, 3, by = 0.01) - - -plot(xx, (0.4419 * plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))) + - 0.5581 * plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10])))), -col = "#d40830", type = "l", xlab = "Concentration", ylab = "Probability" -) -segments(0.33, -1, 0.33, 0.292, col = "blue", lty = 21) -segments(-1, 0.292, 0.33, 0.292, col = "blue", lty = 21) -mtext("0.3", side = 2, at = 0.3, cex = 0.8, col = "blue") -mtext("0.33", side = 1, at = 0.33, cex = 0.8, col = "blue") -``` - -Something's wrong - the fraction affected at concentration 0.33 is 30% - **not the required 20%**. -This issue is taken up in the next section - -## Model Averaged SSDs -As we've just seen, applying the model weights to component *HCx* values and summing does **not** produce the correct result. -The reason for this can be explained mathematically as follows (*if your not interested in the mathematical explanation - skip ahead to the next section*). - -

The fallacy of weighting individual *HCx* values

- -The correct expression for a model-averaged SSD is: $$G\left( x \right) = \sum\limits_{i = 1}^k {{w_i}} {F_i}\left( x \right)$$ where ${F_i}\left( \cdot \right)$ is the *i^th^* component SSD (i.e. *cdf*) and *w~i~* is the weight assigned to ${F_i}\left( \cdot \right)$. -
Notice that the function $G\left( x \right)$ is a proper *cumulative distribution function* (*cdf*) which means for a given quantile, *x*, $G\left( x \right)$ returns the *cumulative probability*: $$P\left[ {X \leqslant x} \right]$$ - -
Now, the *incorrect* approach takes a weighted sum of the component *inverse cdf's*, that is: - -$$H\left( p \right) = \sum\limits_{i = 1}^k {{w_i}} {F_i}^{ - 1}\left( p \right)$$ -where ${F_i}^{ - 1}\left( \cdot \right)$ is the *i^th^* *inverse cdf*. -Notice that ${G_i}\left( \cdot \right)$ is a function of a *quantile* and returns a **probability** while ${H_i}\left( \cdot \right)$ is a function of a *probability* and returns an **quantile**. -

Now, the correct method of determining the *HCx* is to work with the proper model-averaged *cdf* $G\left( x \right)$. -This means finding the **inverse** function ${G^{ - 1}}\left( p \right)$. -We'll address how we do this in a moment.

-The reason why $H\left( p \right)$ does **not** return the correct result is because of the implicit assumption that the inverse of $G\left( x \right)$ is equivalent to $H\left( p \right)$. -This is akin to stating the inverse of a *sum* is equal to the sum of the inverses i.e. -$$\sum\limits_{i = 1}^n {\frac{1}{{{X_i}}}} = \frac{1}{{\sum\limits_{i = 1}^n {{X_i}} }}{\text{ ???}}$$ -
-For the mathematical nerds: -There are some very special cases where the above identity does in fact hold, but for that you need to use **complex numbers**. -

For example, consider two complex numbers $${\text{a = }}\frac{{\left( {5 - i} \right)}}{2}{\text{ and }}b = - 1.683 - 1.915i$$ -It can be shown that $$\frac{1}{{a + b}} = \frac{1}{a} + \frac{1}{b} = 0.126 + 0.372i$$ -

- -

Back to the issue at hand, and since we're not dealing with complex numbers, it's safe to say:$${G^{ - 1}}\left( p \right) \ne H\left( p \right)$$ -

-If you need a visual demonstration, we can plot $G\left( x \right)$ and the *inverse* of $H\left( p \right)$ both as functions of *x* (a quantile) for our two-component lognormal distribution above. - - -```{r echo=FALSE, fig.width=8,fig.height=6} -t <- seq(0.01, 0.99, by = 0.001) - - -F <- 0.4419 * qlnorm(t, -1.067, 0.414) + 0.5581 * qlnorm(t, -0.387, 0.617) - -plot(xx, (0.4419 * plnorm(xx, meanlog = mean(log(samp[5:15])), sd = sd(log(samp[5:15]))) + - 0.5581 * plnorm(xx, meanlog = mean(log(samp[1:10])), sd = sd(log(samp[1:10])))), -col = "#d40830", type = "l", xlab = "Concentration", ylab = "Probability" -) - -lines(F, t, col = "#51c157", lwd = 1.75) - -segments(-1, 0.2, 0.34, 0.2, col = "black", lty = 21, lwd = 2) -segments(0.28, 0.2, 0.28, -1, col = "red", lty = 21, lwd = 2) -segments(0.34, 0.2, 0.34, -1, col = "#51c157", lty = 21, lwd = 2) -segments(1.12, -1, 1.12, 0.9, col = "grey", lty = 21, lwd = 1.7) - -text(0.25, 0.6, "Correct MA-SSD", col = "red", cex = 0.75) -text(0.75, 0.4, "Erroneous MA-SSD", col = "#51c157", cex = 0.75) -mtext("1.12", side = 1, at = 1.12, cex = 0.8, col = "grey") -``` - -Clearly, the two functions are **not** the same and thus *HCx* values derived from each will nearly always be different (as indicated by the positions of the vertical red and green dashed lines in the Figure above corresponding to the 2 values of the *HC20*). -(Note: The two curves do cross over at a concentration of about 1.12 corresponding to the 90^th^ percentile, but in the region of ecotoxicological interest, there is no such cross-over and so the two approaches will **always** yield different *HCx* values with this difference → 0 as x → 0). -

We next discuss the use of a model-averaged SSD to obtain the *correct* model-averaged *HCx*. - -## Computing a model-averaged *HCx* -A proper *HCx* needs to satisfy what David Fox refers to as **the inversion principle**. -

More formally, the inversion principle states that an *HCx* (denoted as ${\varphi _x}$) must satisfy the following:

$$df\left( {{\varphi _x}} \right) = x\quad \quad and\quad \quad qf\left( x \right) = {\varphi _x}$$

-where $df\left( \cdot \right)$ is a model-averaged *distribution function* (i.e. SSD) and $qf\left( \cdot \right)$ is a model-averaged *quantile function*. -For this equality to hold, it is necessary that $qf\left( p \right) = d{f^{ - 1}}\left( p \right)$.

-
So, in our example above, the green curve was taken to be $qf\left( x \right)$ and this was used to derive ${\varphi _x}$ but the *fraction affected* $\left\{ { = df\left( {{\varphi _x}} \right)} \right\}$ at ${\varphi _x}$ is computed using the red curve. -

In `ssdtools` the following is a check that the inversion principle holds:

-
- -``` -# Obtain a model-averaged HCx using the ssd_hc() function -hcp<-ssd_hc(x, p = p) -# Check that the inversion principle holds -ssd_hp(x, hcp, multi_est = TRUE) == p # this should result in logical `TRUE` -``` -*Note: if the `multi_est` argument is set to `FALSE` the test will fail*. -
- -

The *inversion principle* ensures that we only use a **single** distribution function to compute both the *HCx* *and* the fraction affected. -Referring to the figure below, the *HCx* is obtained from the MA-SSD (red curve) by following the → arrows while the fraction affected is obtained by following the ← arrows. - -![](images/Figure2.jpg){width=100% align="center"} - -

Finally, we'll briefly discuss how the *HCx* is computed in `R` using the same method as has been implemented in `ssdtools`.

- -### Computing the *HCx* in `R`/`ssdtools` -Recall, our MA-SSD was given as -$$G\left( x \right) = \sum\limits_{i = 1}^k {{w_i}} {F_i}\left( x \right)$$ -and an *HCx* is obtained from the MA-SSD by essentially working 'in reverse' by starting at a value of $x$ on the **vertical** scale in the Figure above and following the → arrows and reading off the corresponding value on the horizontal scale. -

Obviously, we need to be able to 'codify' this process in `R` (or any other computer language).
Mathematically this is equivalent to seeking a solution to the following equation:$${x:G\left( x \right) = p}$$ -or, equivalently:$$x:G\left( x \right) - p = 0$$ -for some fraction affected, $p$. -

-

Finding the solution to this last equation is referred to as *finding the root(s)* of the function $G\left( x \right)$ or, as is made clear in the figure below, *finding the zero-crossing* of the function $G\left( x \right)$ for the case $p=0.2$. - -![](images/uniroot.jpg){width=100% align="center"} - -
-

In `R` finding the roots of $x:G\left( x \right) - p = 0$ is achieved using the `uniroot()` function.

- Help on the `uniroot` function can be found [here](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/uniroot.html) - -## Where do the model-averaged weights come from? -This is a little more complex, although we'll try to provide a non-mathematical explanation. -For those interested in going deeper, a more comprehensive treatment can be found in [@model_averaging] and [@fletcher]. - -

This time, we'll look at fitting a gamma, lognormal, and pareto distribution to our sample data: - -```{r echo=FALSE,warning=FALSE, results="markup",message=FALSE,class.output="scroll-100"} -samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -print(samp) -# knitr::kable(samp,caption="Some toxicity data (concentrations)") -``` -
The adequacy (or otherwise) of a fitted model can be assessed using a variety of numerical measures known as **goodness-of-fit** or GoF statistics. -These are invariably based on a measure of discrepancy between the emprical data and the hypothesized model. -Common GoF statistics used to test whether the hypothesis of some specified theoretical probability distribution is plausible for a given data set include: *Kolmogorov-Smirnov test; Anderson-Darling test; Shapiro-Wilk test;and Cramer-von Mises test*. - [The Cramer-von Mises](https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion) test is a good choice and is readily performed using the `cvm.test()` function in the `goftest` package in `R` as follows: - -```{r, echo=TRUE,results='hide',warning=FALSE,message=FALSE} -dat <- data.frame(Conc = c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59)) -library(goftest) -library(EnvStats) # this is required for the Pareto cdf (ppareto) - -# Examine the fit for the gamma distribution (NB: parameters estimated from the data) -cvm.test(dat$Conc, null = "pgamma", shape = 2.0591977, scale = 0.3231032, estimated = TRUE) - -# Examine the fit for the lognormal distribution (NB: parameters estimated from the data) -cvm.test(dat$Conc, null = "plnorm", meanlog = -0.6695120, sd = 0.7199573, estimated = TRUE) - -# Examine the fit for the Pareto distribution (NB: parameters estimated from the data) -cvm.test(dat$Conc, null = "ppareto", location = 0.1800000, shape = 0.9566756, estimated = TRUE) -``` - -``` - Cramer-von Mises test of goodness-of-fit - Braun's adjustment using 4 groups - Null hypothesis: Gamma distribution - with parameters shape = 2.0591977, scale = 0.3231032 - Parameters assumed to have been estimated from data - -data: dat$Conc -omega2max = 0.34389, p-value = 0.3404 - - - Cramer-von Mises test of goodness-of-fit - Braun's adjustment using 4 groups - Null hypothesis: log-normal distribution - with parameter meanlog = -0.669512 - Parameters assumed to have been estimated from data - -data: dat$Conc -omega2max = 0.32845, p-value = 0.3719 - - - Cramer-von Mises test of goodness-of-fit - Braun's adjustment using 4 groups - Null hypothesis: distribution ‘ppareto’ - with parameters location = 0.18, shape = 0.9566756 - Parameters assumed to have been estimated from data - -data: dat$Conc -omega2max = 0.31391, p-value = 0.4015 -``` - -From this output and using a level of significance of $p = 0.05$, we see that none of the distributions is implausible. -However, if *forced* to choose just one distribution, we would choose the *Pareto* distribution (smaller values of the `omega2max` statistic are better). -However, this does not mean that the gamma and lognormal distributions are of no value in describing the data. -We can see from the plot below, that in fact both the gamma and lognormal distributions do a reasonable job over the range of toxicity values. -The use of the Pareto may be a questionable choice given it is truncated at 0.18 (which is the minimum value of our toxicity data). -
- -```{r echo=FALSE,fig.cap="Emprirical cdf (black); lognormal (green); gamma (blue); and Pareeto (red)", fig.width=7,fig.height=4.5} -library(EnvStats) -samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -samp <- sort(samp) -plot(ecdf(samp), main = "Empirical and fitted SSDs", xlab = "Concentration", ylab = "Probability") -xx <- seq(0.01, 3, by = 0.01) -lines(xx, plnorm(xx, meanlog = -0.6695120, sd = 0.7199573), col = "#77d408") -lines(xx, pgamma(xx, shape = 2.0591977, scale = 0.3231032), col = "#08afd4") -lines(xx, ppareto(xx, location = 0.1800000, shape = 0.9566756), col = "red") -# lines(xx,(0.4419*plnorm(xx,meanlog=mean(log(samp[5:15])),sd=sd(log(samp[5:15])))+ -# 0.5581*plnorm(xx,meanlog=mean(log(samp[1:10])),sd=sd(log(samp[1:10])))) ,col="#d40830") -``` -

- -


As in the earlier example, we might expect to find a better fitting distribution by combining *all three distributions* using a *weighted SSD*. -The issue we face now is *how do we choose the weights* to reflect the relative fits of the three distributions? -Like all tests of statistical significance, a *p-value* is computed from the value of the relevant *test statistic* - in this case, the value of the `omega2max` test statistic. -For this particular test, it's a case of the smaller the better. -From the output above we see that the `omega2max` values are $0.344$ for the gamma distribution, $0.328$ for the lognormal distribution, and $0.314$ for the Pareto distribution.

- -

We might somewhat naively compute the relative weights as:
${w_1} = \frac{{{{0.344}^{ - 1}}}}{{\left( {{{0.344}^{ - 1}} + {{0.328}^{ - 1}} + {{0.314}^{ - 1}}} \right)}} = 0.318$         ${w_2} = \frac{{{{0.328}^{ - 1}}}}{{\left( {{{0.344}^{ - 1}} + {{0.328}^{ - 1}} + {{0.314}^{ - 1}}} \right)}} = 0.333$     and ${w_3} = \frac{{{{0.314}^{ - 1}}}}{{\left( {{{0.344}^{ - 1}} + {{0.328}^{ - 1}} + {{0.314}^{ - 1}}} \right)}} = 0.349$
   (we use *reciprocals* since smaller values of `omega2max` represent better fits). -As will be seen shortly - these are incorrect. -

However, being based on a simplistic measure of discrepancy between the *observed* and *hypothesized* distributions, the `omega2max` statistic is a fairly 'blunt instrument' and has no grounding in information theory which *is* the basis for determining the weights that we seek. -

- -
A discussion of *information theoretic* methods for assessing goodness-of-fit is beyond the scope of this vignette. Interested readers should consult [@model_averaging]. -A commonly used metric to determine the model-average weights is the **Akaike Information Criterion** or [AIC](https://en.wikipedia.org/wiki/Akaike_information_criterion). -The formula for the $AIC$ is: -$$AIC = 2k - 2\ln \left( \ell \right)$$ -where $k$ is the number of model parameters and $\ell$ is the *likelihood* for that model. -Again, a full discussion of statistical likelihood is beyond the present scope. -A relatively gentle introduction can be found [here](https://ep-news.web.cern.ch/what-likelihood-function-and-how-it-used-particle-physics). -

The likelihood for our three distributions can be computed in `R` as follows: - - -```{r, echo=TRUE} -sum(log(dgamma(dat$Conc, shape = 2.0591977, scale = 0.3231032))) -sum(log(dlnorm(dat$Conc, meanlog = -0.6695120, sdlog = 0.7199573))) -sum(log(EnvStats::dpareto(dat$Conc, location = 0.1800000, shape = 0.9566756))) -``` - -From which the *AIC* values readily follow: - -```{r echo=FALSE,results='markup'} -dat <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -k <- 2 # number of parameters for each of the distributions -# Gamma distribution -aic1 <- 2 * k - 2 * sum(log(dgamma(dat, shape = 2.0591977, scale = 0.3231032))) -cat("AIC for gamma distribution =", aic1, "\n") - -# lognormal distribution -aic2 <- 2 * k - 2 * sum(log(dlnorm(dat, meanlog = -0.6695120, sdlog = 0.7199573))) -cat("AIC for lognormal distribution =", aic2, "\n") - -# Pareto distribution -aic3 <- 2 * k - 2 * sum(log(EnvStats::dpareto(dat, location = 0.1800000, shape = 0.9566756))) -cat("AIC for Pareto distribution =", aic3, "\n") -``` -

- -

As with the `omega2max` statistic, **smaller** values of *AIC* are better. Thus, a comparison of the AIC values above gives the ranking of distributional fits (best to worst) as: *Pareto > lognormal > gamma* - -

- -### Computing model weights from the `AIC` -We will simply provide a formula for computing the model weights from the `AIC` values. -More detailed information can be found [here](https://training.visionanalytix.com/ssd-model-averaging/). - -The *AIC* for the *i^th^* distribution fitted to the data is -$$AI{C_i} = 2{k_i} - 2\ln \left( {{L_i}} \right)$$ -where ${L_i}$ is the *i^th^ likelihood* and ${k_i}$ is the *number of parameters* for the *i^th^ distribution*. -Next, we form the differences: -$${\Delta _i} = AI{C_i} - AI{C_0}$$ -where $AI{C_0}$ is the *AIC* for the **best-fitting** model (i.e.$AI{C_0} = \mathop {\min }\limits_i \left\{ {AI{C_i}} \right\}$ ). -The *model-averaged weights* ${w_i}$ are then computed as - -
-AIC Model averaging weights - -$${w}_{i}=\frac{\exp \left\{ -\frac{1}{2}{{\Delta }_{i}} \right\}}{\sum{\exp \left\{ -\frac{1}{2}{{\Delta }_{i}} \right\}}}$$ - -
-
- -

The model-averaged weights for the gamma, lognormal, and Pareto distributions used in the previous example can be computed 'manually' in `R` as follows: - - -```{r echo=TRUE,results='hide'} -dat <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -aic <- NULL -k <- 2 # number of parameters for each of the distributions - - -aic[1] <- 2 * k - 2 * sum(log(dgamma(dat, shape = 2.0591977, scale = 0.3231032))) # Gamma distribution - -aic[2] <- 2 * k - 2 * sum(log(dlnorm(dat, meanlog = -0.6695120, sdlog = 0.7199573))) # lognormal distribution - -aic[3] <- 2 * k - 2 * sum(log(EnvStats::dpareto(dat, location = 0.1800000, shape = 0.9566756))) # Pareto distribution - -delta <- aic - min(aic) # compute the delta values - -aic.w <- exp(-0.5 * delta) -aic.w <- round(aic.w / sum(aic.w), 4) - -cat( - " AIC weight for gamma distribution =", aic.w[1], "\n", - "AIC weight for lognormal distribution =", aic.w[2], "\n", - "AIC weight for pareto distribution =", aic.w[3], "\n" -) -``` -``` - AIC weight for gamma distribution = 0.1191 - AIC weight for lognormal distribution = 0.3985 - AIC weight for pareto distribution = 0.4824 -``` -

- -Finally, let's look at the fitted *model-averaged SSD*: - -```{r echo=FALSE,fig.cap="Empirical cdf (black) and model-averaged fit (magenta)",fig.width=8,fig.height=5} -samp <- c(1.73, 0.57, 0.33, 0.28, 0.3, 0.29, 2.15, 0.8, 0.76, 0.54, 0.42, 0.83, 0.21, 0.18, 0.59) -samp <- sort(samp) -plot(ecdf(samp), main = "Empirical and fitted SSDs", xlab = "Concentration", ylab = "Probability") -xx <- seq(0.01, 3, by = 0.005) - -lines(xx, plnorm(xx, meanlog = -0.6695120, sd = 0.7199573), col = "#959495", lty = 2) -lines(xx, pgamma(xx, shape = 2.0591977, scale = 0.3231032), col = "#959495", lty = 3) -lines(xx, ppareto(xx, location = 0.1800000, shape = 0.9566756), col = "#959495", lty = 4) -lines(xx, 0.1191 * pgamma(xx, shape = 2.0591977, scale = 0.3231032) + - 0.3985 * plnorm(xx, meanlog = -0.6695120, sd = 0.7199573) + - 0.4824 * ppareto(xx, location = 0.1800000, shape = 0.9566756), col = "#FF33D5", lwd = 1.5) -``` - -As can be seen from the figure above, the model-averaged fit provides a very good fit to the empirical data. - -### Correcting for distributions having differing numbers of parameters - -In deriving the AIC, Akaike had to make certain, strong assumptions. -In addition, the bias factor (the $2k$ term) was derived from theoretical considerations (such as mathematical *expectation*) that relate to *infinite* sample sizes. -For small sample sizes, the AIC is likely to select models having too many parameters (i.e models which *over-fit*). -In 1978, Sugiura proposed a modification to the AIC to address this problem, although it too relied on a number of assumptions. -This 'correction' to the AIC for small samples (referred to as $AI{{C}_{c}}$) is - -
-Corrected Akaike Information Criterion (AICc) - -$$AI{{C}_{c}}=AIC+\frac{2{{k}^{2}}+2k}{n-k-1}$$ - -where n is the sample size and k is the number of parameters. - -
- -
It is clear from the formula for $AI{{C}_{c}}$ that for   $n\gg k$,    $AI{{C}_{c}}\simeq AIC$. The issue of sample size is ubiquitous in statistics, but even more so in ecotoxicology where logistical and practical limitations invariably mean we are dealing with (pathologically) small sample sizes. There are no hard and fast rules as to what constitutes an *appropriate* sample size for SSD modelling. However, Professor David Fox's personal rule of thumb which works quite well is: - - -
Sample size rule-of-thumb for SSD modelling $$n\ge 5k+1$$ - -where n is the sample size and k is the number of parameters. -
- -
Since most of the common SSD models are 2-parameter, we should be aiming to have a sample size of at least 11. -For 3-parameter models (like the Burr III), the minimum sample size is 16 and if we wanted to fit a mixture of two, 2-parameter models (eg. *logNormal-logNormal* or *logLogistic-logLogistic*) the sample size should be *at least* 26. -Sadly, this is rarely the case in practice! Jurisdictional guidance material may specify minimum sample size requirements, and should be adhered to where available and relevant. - -### Model-Averaging in `ssdtools` - -Please see the [Getting started with ssdtools](https://bcgov.github.io/ssdtools/articles/ssdtools.html) vignette for examples of obtaining model-averaged *HCx* values and predictions using `ssdtools`. - -## References - -
- -```{r, results = "asis", echo = FALSE} -cat(ssdtools::ssd_licensing_md()) -``` diff --git a/doc/small-sample-bias.R b/doc/small-sample-bias.R deleted file mode 100644 index 49425038..00000000 --- a/doc/small-sample-bias.R +++ /dev/null @@ -1 +0,0 @@ -### This is an R script tangled from 'small-sample-bias.pdf.asis' diff --git a/doc/ssdtools.R b/doc/ssdtools.R deleted file mode 100644 index 06f80ff1..00000000 --- a/doc/ssdtools.R +++ /dev/null @@ -1,94 +0,0 @@ -## ----include = FALSE---------------------------------------------------------- -knitr::opts_chunk$set( - collapse = TRUE, - comment = "#>", - fig.width = 6, - fig.height = 4 -) - -## ----eval = FALSE------------------------------------------------------------- -# install.packages(c("ssdtools", "tidyverse")) - -## ----message = FALSE---------------------------------------------------------- -library(ssdtools) -library(ggplot2) - -## ----eval = FALSE------------------------------------------------------------- -# data <- read_csv(file = "path/to/file.csv") - -## ----------------------------------------------------------------------------- -ssddata::ccme_boron - -## ----------------------------------------------------------------------------- -ssd_dists_all() - -## ----------------------------------------------------------------------------- -fits <- ssd_fit_dists(ssddata::ccme_boron, dists = c("llogis", "lnorm", "gamma")) - -## ----------------------------------------------------------------------------- -tidy(fits) - -## ----fig.alt="A plot of the CCME boron dataset with the gamma, log-logistic and log-normal distributions with a simple black and white background color scheme."---- -theme_set(theme_bw()) # set plot theme - -autoplot(fits) + - ggtitle("Species Sensitivity Distributions for Boron") + - scale_colour_ssd() - -## ----------------------------------------------------------------------------- -ssd_gof(fits) - -## ----eval = FALSE------------------------------------------------------------- -# set.seed(99) -# boron_pred <- predict(fits, ci = TRUE) - -## ----------------------------------------------------------------------------- -boron_pred - -## ----fig.alt="A plot of the CCME boron dataset species colored by group and the model average species sensitivity distribution with a simple black and white background color scheme."---- -ssd_plot(ssddata::ccme_boron, boron_pred, - color = "Group", label = "Species", - xlab = "Concentration (mg/L)", ribbon = TRUE -) + - expand_limits(x = 5000) + # to ensure the species labels fit - ggtitle("Species Sensitivity for Boron") + - scale_colour_ssd() - -## ----------------------------------------------------------------------------- -set.seed(99) -boron_hc5 <- ssd_hc(fits, proportion = 0.05, ci = TRUE) -print(boron_hc5) -boron_pc <- ssd_hp(fits, conc = boron_hc5$est, ci = TRUE) -print(boron_pc) - -## ----------------------------------------------------------------------------- -boron_censored <- ssddata::ccme_boron |> - dplyr::mutate(left = Conc, right = Conc) - -boron_censored$left[c(3, 6, 8)] <- NA - -## ----------------------------------------------------------------------------- -dists <- ssd_fit_dists(boron_censored, - dists = ssd_dists_bcanz(n = 2), - left = "left", right = "right" -) - -## ----------------------------------------------------------------------------- -ssd_gof(dists) - -## ----------------------------------------------------------------------------- -ssd_hc(dists, average = FALSE) -ssd_hc(dists) - -## ----fig.alt="A plot of the left censored CCME boron dataset with the model average species sensitivity distribution and arrows indicating the censoring."---- -set.seed(99) -pred <- predict(dists, ci = TRUE, parametric = FALSE) - -ssd_plot(boron_censored, pred, - left = "left", right = "right", - xlab = "Concentration (mg/L)" -) - -## ----results = "asis", echo = FALSE------------------------------------------- -cat(ssdtools::ssd_licensing_md()) - diff --git a/doc/ssdtools.Rmd b/doc/ssdtools.Rmd deleted file mode 100644 index a553fa05..00000000 --- a/doc/ssdtools.Rmd +++ /dev/null @@ -1,267 +0,0 @@ ---- -title: "Getting Started with ssdtools" -author: "ssdtools Team" -date: '`r format(Sys.time(), "%Y-%m-%d", tz = "UTC")`' -bibliography: references.bib -mathfont: Courier -latex_engine: MathJax -output: rmarkdown::html_vignette -vignette: > - %\VignetteIndexEntry{Getting Started with ssdtools} - %\VignetteEngine{knitr::rmarkdown} - %\VignetteEncoding{UTF-8} ---- - -```{r, include = FALSE} -knitr::opts_chunk$set( - collapse = TRUE, - comment = "#>", - fig.width = 6, - fig.height = 4 -) -``` - -## Introduction - -`ssdtools` is an R package to fit Species Sensitivity Distributions (SSDs) using Maximum Likelihood and model averaging. - -SSDs are cumulative probability distributions that are used to estimate the percent of species that are affected and/or protected by a given concentration of a chemical. -The concentration that affects 5% of the species is referred to as the 5% Hazard Concentration (*HC~5~*). This is equivalent to a 95% protection value (*PC~95~*). -For more information on SSDs the reader is referred to @posthuma_species_2001. - -In order to use `ssdtools` you need to install R (see below) or use the Shiny [app](https://bcgov-env.shinyapps.io/ssdtools/). -The shiny app includes a user guide. -This vignette is a user manual for the R package. - -## Philosophy - -`ssdtools` provides the key functionality required to fit SSDs using Maximum Likelihood and [model averaging](https://bcgov.github.io/ssdtools/articles/A_model_averaging.html) in R. -It is intended to be used in conjunction with [tidyverse](https://www.tidyverse.org) packages such as `readr` to input data, `tidyr` and `dplyr` to group and manipulate data and `ggplot2` [@ggplot2] to plot data. -As such it endeavors to fulfill the tidyverse [manifesto](https://tidyverse.tidyverse.org/articles/manifesto.html). - -## Installing - -In order to install R [@r] the appropriate binary for the users operating system should be downloaded from [CRAN](https://cran.r-project.org) and then installed. - -Once R is installed, the `ssdtools` package can be installed (together with the tidyverse) by executing the following code at the R console - -```{r, eval = FALSE} -install.packages(c("ssdtools", "tidyverse")) -``` - -The `ssdtools` package (and ggplot2 package) can then be loaded into the current session using - -```{r, message = FALSE} -library(ssdtools) -library(ggplot2) -``` - -## Getting Help - -To get additional information on a particular function just type `?` followed by the name of the function at the R console. -For example `?ssd_gof` brings up the R documentation for the `ssdtools` goodness of fit function. - -For more information on using R the reader is referred to [R for Data Science](https://r4ds.had.co.nz) [@wickham_r_2016]. - -If you discover a bug in `ssdtools` please file an issue with a [reprex](https://reprex.tidyverse.org/articles/reprex-dos-and-donts.html) (repeatable example) at . - -## Inputting Data - -Once the `ssdtools` package has been loaded the next task is to input some data. -An easy way to do this is to save the concentration data for a *single* chemical as a column called `Conc` in a comma separated file (`.csv`). -Each row should be the sensitivity concentration for a separate species. -If species and/or group information is available then this can be saved as `Species` and `Group` columns. -The `.csv` file can then be read into R using the following - -```{r, eval = FALSE} -data <- read_csv(file = "path/to/file.csv") -``` - -For the purposes of this manual we use the CCME dataset for boron from the [`ssddata`](https://github.com/open-AIMS/ssddata) package. - -```{r} -ssddata::ccme_boron -``` - -## Fitting Distributions - -The function `ssd_fit_dists()` inputs a data frame and fits one or more distributions. -The user can specify a subset of the following `r length(ssd_dists_all())` distributions. Please see the [distributions](https://bcgov.github.io/ssdtools/articles/B_distributions.html) and [model averaging](https://bcgov.github.io/ssdtools/articles/A_model_averaging.html) vignettes for more information regarding appropriate use of distributions and the use of model-averaged SSDs. - -```{r} -ssd_dists_all() -``` - -using the `dists` argument. - -```{r} -fits <- ssd_fit_dists(ssddata::ccme_boron, dists = c("llogis", "lnorm", "gamma")) -``` - -## Coefficients - -The estimates for the various terms can be extracted using the tidyverse generic `tidy` function (or the base R generic `coef` function). - -```{r} -tidy(fits) -``` - -## Plots - -It is generally more informative to plot the fits using the `autoplot` generic function (a wrapper on `ssd_plot_cdf()`). -As `autoplot` returns a `ggplot` object it can be modified prior to plotting. -For more information see the [customising plots](https://bcgov.github.io/ssdtools/articles/A_model_averaging.html) vignette. - -```{r, fig.alt="A plot of the CCME boron dataset with the gamma, log-logistic and log-normal distributions with a simple black and white background color scheme."} -theme_set(theme_bw()) # set plot theme - -autoplot(fits) + - ggtitle("Species Sensitivity Distributions for Boron") + - scale_colour_ssd() -``` - -## Selecting One Distribution - -Given multiple distributions the user is faced with choosing the "best" distribution (or as discussed below averaging the results weighted by the fit). - -```{r} -ssd_gof(fits) -``` - -The `ssd_gof()` function returns three test statistics that can be used to evaluate the fit of the various distributions to the data. - -- [Anderson-Darling](https://en.wikipedia.org/wiki/Anderson–Darling_test) (`ad`) statistic, -- [Kolmogorov-Smirnov](https://en.wikipedia.org/wiki/Kolmogorov–Smirnov_test) (`ks`) statistic and -- [Cramer-von Mises](https://en.wikipedia.org/wiki/Cramér–von_Mises_criterion) (`cvm`) statistic - -and three information criteria - -- Akaike's Information Criterion (`AIC`), -- Akaike's Information Criterion corrected for sample size (`AICc`) and -- Bayesian Information Criterion (`BIC`) - -Note if `ssd_gof()` is called with `pvalue = TRUE` then the p-values rather than the statistics are returned for the ad, ks and cvm tests. - -Following @burnham_model_2002 we recommend the `AICc` for model selection. -The best predictive model is that with the lowest `AICc` (indicated by the model with a `delta` value of 0 in the goodness of fit table). -In the current example the best predictive model is the gamma distribution but both the lnorm and llogis distributions have some support. - -For further information on the advantages of an information theoretic approach in the context of selecting SSDs the reader is referred to @fox_recent_2021. - -## Averaging Multiple Distributions - -Often other distributions will fit the data almost as well as the best distribution as evidenced by `delta` values < 2 [@burnham_model_2002]. -In general, the recommended approach is to estimate the average fit based on the relative weights of the distributions [@burnham_model_2002]. -The `AICc` based weights are indicated by the `weight` column in the goodness of fit table. -A detailed introduction to model averaging can be found in the [Model averaging](https://bcgov.github.io/ssdtools/articles/A_model_averaging.html) vignette. -A discussion on the recommended set of default distributions can be found in the [Distributions](https://bcgov.github.io/ssdtools/articles/B_distributions.html) vignette. - -## Estimating the Fit - -The `predict` function can be used to generate model-averaged estimates (or if `average = FALSE` estimates for each distribution individual) by bootstrapping. -Model averaging is based on `AICc` unless the data censored is which case `AICc` is undefined. -In this situation model averaging is only possible if the distributions have the same number of parameters (so that `AIC` can be used to compare the models). - -```{r, eval = FALSE} -set.seed(99) -boron_pred <- predict(fits, ci = TRUE) -``` - -The resultant object is a data frame of the estimated concentration (`est`) with standard error (`se`) and lower (`lcl`) and upper (`ucl`) 95% confidence limits (CLs) by percent of species affected (`percent`). -The object includes the number of bootstraps (`nboot`) data sets generated as well as the proportion of the data sets that successfully fitted (`pboot`). - -```{r} -boron_pred -``` - -The data frame of the estimates can then be plotted together with the original data using the `ssd_plot()` function to summarize an analysis. -Once again the returned object is a `ggplot` object which can be customized prior to plotting. - -```{r, fig.alt="A plot of the CCME boron dataset species colored by group and the model average species sensitivity distribution with a simple black and white background color scheme."} -ssd_plot(ssddata::ccme_boron, boron_pred, - color = "Group", label = "Species", - xlab = "Concentration (mg/L)", ribbon = TRUE -) + - expand_limits(x = 5000) + # to ensure the species labels fit - ggtitle("Species Sensitivity for Boron") + - scale_colour_ssd() -``` - -In the above plot the model-averaged 95% confidence interval is indicated by the shaded band and the model-averaged 5%/95% Hazard/Protection Concentration (*HC5*/ *PC~95~*) by the dotted line. -Hazard/Protection concentrations are discussed below. - -## Hazard/Protection Concentrations - -The 5% hazard concentration (*HC5*) is the concentration that affects 5% of the species tested. This is equivalent to the 95% protection concentration which protects 95% of species (*PC~95~*). The hazard and protection concentrations are directly interchangeable, and terminology depends simply on user preference. - -The hazard/protection concentrations can be obtained using the `ssd_hc()` function, which can be used to obtain any desired percentage value. The fitted SSD can also be used to determine the percentage of species protected at a given concentration using `ssd_hp()`. - -```{r} -set.seed(99) -boron_hc5 <- ssd_hc(fits, proportion = 0.05, ci = TRUE) -print(boron_hc5) -boron_pc <- ssd_hp(fits, conc = boron_hc5$est, ci = TRUE) -print(boron_pc) -``` - -### Censored Data - -Censored data is that for which only a lower and/or upper limit is known for a particular species. -If the `right` argument in `ssd_fit_dists()` is different to the `left` argument then the data are considered to be censored. - -It is important to note that `ssdtools` doesn't currently support right censored data. - -Let's produce some left censored data. - -```{r} -boron_censored <- ssddata::ccme_boron |> - dplyr::mutate(left = Conc, right = Conc) - -boron_censored$left[c(3, 6, 8)] <- NA -``` - -As the sample size `n` is undefined for censored data, `AICc` cannot be calculated. -However, if all the models have the same number of parameters, the `AIC` `delta` values are identical to those for `AICc`. -For this reason, `ssdtools` only permits model averaging of censored data for distributions with the same number of parameters. -We can call only the default two parameter models using `ssd_dists_bcanz(n = 2)`. - -```{r} -dists <- ssd_fit_dists(boron_censored, - dists = ssd_dists_bcanz(n = 2), - left = "left", right = "right" -) -``` - -There are less goodness-of-fit statistics available for fits to censored data (currently just `AIC` and `BIC`). - -```{r} -ssd_gof(dists) -``` - - -The model-averaged predictions are calculated using `AIC` -```{r} -ssd_hc(dists, average = FALSE) -ssd_hc(dists) -``` - -The confidence intervals can currently only be generated for censored data using non-parametric bootstrapping. -The horizontal lines in the plot indicate the censoring (range of possible values). - -```{r, fig.alt="A plot of the left censored CCME boron dataset with the model average species sensitivity distribution and arrows indicating the censoring."} -set.seed(99) -pred <- predict(dists, ci = TRUE, parametric = FALSE) - -ssd_plot(boron_censored, pred, - left = "left", right = "right", - xlab = "Concentration (mg/L)" -) -``` - -## References - -
- -```{r, results = "asis", echo = FALSE} -cat(ssdtools::ssd_licensing_md()) -``` diff --git a/doc/small-sample-bias.pdf b/vignettes/small-sample-bias.pdf similarity index 100% rename from doc/small-sample-bias.pdf rename to vignettes/small-sample-bias.pdf diff --git a/doc/small-sample-bias.pdf.asis b/vignettes/small-sample-bias.pdf.asis similarity index 100% rename from doc/small-sample-bias.pdf.asis rename to vignettes/small-sample-bias.pdf.asis From f2212fda5dde2c3272afffb2883bd3f645f94586 Mon Sep 17 00:00:00 2001 From: Joe Thorley Date: Tue, 3 Dec 2024 07:50:40 -0800 Subject: [PATCH 04/66] update .gitignore --- .gitignore | 12 ++++-------- 1 file changed, 4 insertions(+), 8 deletions(-) diff --git a/.gitignore b/.gitignore index 8c6473e6..3f4dbc55 100644 --- a/.gitignore +++ b/.gitignore @@ -10,12 +10,8 @@ *.html -/docs/ -/inst/docs/ +docs +inst/docs +scripts + /paper/paper.pdf -/scripts/boron_samples1 -/scripts/checks.md -inst/doc -/ignore -/doc/ -/Meta/ From fdd30c67835134efaa600d4a651c2790f9b04571 Mon Sep 17 00:00:00 2001 From: Joe Thorley Date: Tue, 3 Dec 2024 07:59:54 -0800 Subject: [PATCH 05/66] try fixing log-normal math --- vignettes/distributions.Rmd | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/vignettes/distributions.Rmd b/vignettes/distributions.Rmd index 6cc62b1c..2f31aa73 100644 --- a/vignettes/distributions.Rmd +++ b/vignettes/distributions.Rmd @@ -258,13 +258,13 @@ A random variable, $X$ is lognormally distributed if the *logarithm* of $X$ is n
Lognormal Distribution - $${f_X}\left( {x;\mu ,\sigma } \right) = \frac{1}{{\sqrt {2\pi } {\kern 1pt} \sigma x}}\exp \left[ { - \frac{{{{\left( {\ln x - \mu } \right)}^2}}}{{2{\sigma ^2}}}} \right]; \quad x,\sigma > 0,\; - \infty < \mu < \infty $$ + $${f_X}\left( {x;\mu ,\sigma } \right) = \frac{1}{{\sqrt {2\pi } {\kern 1pt} \sigma x}}\exp \left[ { - \frac{{{{\left( {\ln x - \mu } \right)}^2}}}{{2{\sigma ^2}}}} \right]; \quad x,\sigma > 0; - \infty < \mu < \infty $$

-The lognormal distribution was selected as the starting distribution given the data are for effect concentrations. +The log-normal distribution was selected as the starting distribution given the data are for effect concentrations. The log-normal distribution can be fitted using `ssdtools` supplying the string `lnorm` to the `dists` argument in the `ssd_fit_dists` call. ```{r echo=FALSE,fig.align='center',fig.width=9,fig.height=5,fig.cap="Sample lognormal probability density (A) and cumulative probability (B) functions."} From 38be175ea18536e22dab5b2a231c79fc0529f2cb Mon Sep 17 00:00:00 2001 From: Joe Thorley Date: Tue, 3 Dec 2024 08:19:15 -0800 Subject: [PATCH 06/66] playing with log-normal dist --- vignettes/distributions.Rmd | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/vignettes/distributions.Rmd b/vignettes/distributions.Rmd index 2f31aa73..63566203 100644 --- a/vignettes/distributions.Rmd +++ b/vignettes/distributions.Rmd @@ -258,7 +258,9 @@ A random variable, $X$ is lognormally distributed if the *logarithm* of $X$ is n
Lognormal Distribution - $${f_X}\left( {x;\mu ,\sigma } \right) = \frac{1}{{\sqrt {2\pi } {\kern 1pt} \sigma x}}\exp \left[ { - \frac{{{{\left( {\ln x - \mu } \right)}^2}}}{{2{\sigma ^2}}}} \right]; \quad x,\sigma > 0; - \infty < \mu < \infty $$ + $${f_X}\left( {x;\mu ,\sigma } \right) = \frac{1}{{\sqrt {2\pi } {\kern 1pt} \sigma x}}\exp \left[ { - \frac{{{{\left( {\ln x - \mu } \right)}^2}}}{{2{\sigma^2}}}} \right] $$ + +$$; \quad x,\sigma > 0; - \infty < \mu < \infty$$
From 373bfede9d9d69a11b6f0da09564f860767d30a9 Mon Sep 17 00:00:00 2001 From: Joe Thorley Date: Tue, 3 Dec 2024 09:06:23 -0800 Subject: [PATCH 07/66] distributions --- vignettes/distributions.Rmd | 5 +---- 1 file changed, 1 insertion(+), 4 deletions(-) diff --git a/vignettes/distributions.Rmd b/vignettes/distributions.Rmd index 63566203..22f1c9ca 100644 --- a/vignettes/distributions.Rmd +++ b/vignettes/distributions.Rmd @@ -258,10 +258,7 @@ A random variable, $X$ is lognormally distributed if the *logarithm* of $X$ is n
Lognormal Distribution - $${f_X}\left( {x;\mu ,\sigma } \right) = \frac{1}{{\sqrt {2\pi } {\kern 1pt} \sigma x}}\exp \left[ { - \frac{{{{\left( {\ln x - \mu } \right)}^2}}}{{2{\sigma^2}}}} \right] $$ - -$$; \quad x,\sigma > 0; - \infty < \mu < \infty$$ - + $${f_X}\left( x; \mu, \sigma \right) = \frac{1}{\sqrt{2\pi}\ \sigma x}\exp \left[ - \frac{\left( {\ln x - \mu } \right)^2}{2\sigma^2} \right] ;\ x,\sigma > 0;\ - \infty < \mu < \infty$$

From edf1f216d121c422b181afb0a76c79c36fb0f41e Mon Sep 17 00:00:00 2001 From: Joe Thorley Date: Tue, 3 Dec 2024 10:45:59 -0800 Subject: [PATCH 08/66] log-normal --- vignettes/distributions.Rmd | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/vignettes/distributions.Rmd b/vignettes/distributions.Rmd index 22f1c9ca..64067ff7 100644 --- a/vignettes/distributions.Rmd +++ b/vignettes/distributions.Rmd @@ -256,7 +256,7 @@ The [log-normal](https://en.wikipedia.org/wiki/Log-normal_distribution) distribu A random variable, $X$ is lognormally distributed if the *logarithm* of $X$ is normally distributed. The *pdf* of $X$ is given by
- Lognormal Distribution + Log-normal Distribution $${f_X}\left( x; \mu, \sigma \right) = \frac{1}{\sqrt{2\pi}\ \sigma x}\exp \left[ - \frac{\left( {\ln x - \mu } \right)^2}{2\sigma^2} \right] ;\ x,\sigma > 0;\ - \infty < \mu < \infty$$
From ba30c6801dc8934f1f68b9c432b63c1cdd818eab Mon Sep 17 00:00:00 2001 From: Seb Dalgarno Date: Thu, 5 Dec 2024 22:55:26 -0800 Subject: [PATCH 09/66] add xlimits argument to ssd_plot to set x-axis limits (passed to scale_x_continuous(limits)) --- R/params.R | 4 ++++ R/ssd-plot.R | 10 ++++++---- man/params.Rd | 7 +++++++ man/ssd_plot.Rd | 10 +++++++++- tests/testthat/test-ssd-plot.R | 5 +++++ 5 files changed, 31 insertions(+), 5 deletions(-) diff --git a/R/params.R b/R/params.R index e70ad989..d05c7bdb 100644 --- a/R/params.R +++ b/R/params.R @@ -119,6 +119,10 @@ #' - `NULL` for no breaks #' - `waiver()` for the default breaks #' - A numeric vector of positions +#' @param xlimits The x-axis limits as one of: +#' - `NULL` to use the default scale range +#' - A numeric vector of length two providing the limits. +#' Use NA to refer to the existing minimum or maximum limits. #' @param xintercept The x-value for the intersect #' @param xlab A string of the x-axis label. #' @param yintercept The y-value for the intersect. diff --git a/R/ssd-plot.R b/R/ssd-plot.R index 8ddd8574..e3c2fdb3 100644 --- a/R/ssd-plot.R +++ b/R/ssd-plot.R @@ -18,7 +18,7 @@ #' @export ggplot2::waiver -plot_coord_scale <- function(data, xlab, ylab, trans, big.mark, suffix, xbreaks = waiver()) { +plot_coord_scale <- function(data, xlab, ylab, trans, big.mark, suffix, xbreaks = waiver(), xlimits = NULL) { chk_string(xlab) chk_string(ylab) @@ -30,7 +30,8 @@ plot_coord_scale <- function(data, xlab, ylab, trans, big.mark, suffix, xbreaks coord_trans(x = trans), scale_x_continuous(xlab, breaks = xbreaks, - labels = ssd_label_comma(big.mark = big.mark) + labels = ssd_label_comma(big.mark = big.mark), + limits = xlimits ), scale_y_continuous(ylab, labels = label_percent(suffix = suffix), limits = c(0, 1), @@ -57,7 +58,8 @@ ssd_plot <- function(data, pred, left = "Conc", right = left, ..., shift_x = 3, add_x = 0, bounds = c(left = 1, right = 1), big.mark = ",", suffix = "%", - trans = "log10", xbreaks = waiver()) { + trans = "log10", xbreaks = waiver(), + xlimits = NULL) { .chk_data(data, left, right, weight = NULL, missing = TRUE) chk_unused(...) chk_null_or(label, vld = vld_string) @@ -162,7 +164,7 @@ ssd_plot <- function(data, pred, left = "Conc", right = left, ..., gp <- gp + plot_coord_scale(data, xlab = xlab, ylab = ylab, big.mark = big.mark, suffix = suffix, - trans = trans, xbreaks = xbreaks + trans = trans, xbreaks = xbreaks, xlimits = xlimits ) if (!is.null(label)) { diff --git a/man/params.Rd b/man/params.Rd index bcc8f32f..fb302e0f 100644 --- a/man/params.Rd +++ b/man/params.Rd @@ -202,6 +202,13 @@ remove them with a warning.} \item A numeric vector of positions }} +\item{xlimits}{The x-axis limits as one of: +\itemize{ +\item \code{NULL} to use the default scale range +\item A numeric vector of length two providing the limits. +Use NA to refer to the existing minimum or maximum limits. +}} + \item{xintercept}{The x-value for the intersect} \item{xlab}{A string of the x-axis label.} diff --git a/man/ssd_plot.Rd b/man/ssd_plot.Rd index 572d08de..4178809a 100644 --- a/man/ssd_plot.Rd +++ b/man/ssd_plot.Rd @@ -27,7 +27,8 @@ ssd_plot( big.mark = ",", suffix = "\%", trans = "log10", - xbreaks = waiver() + xbreaks = waiver(), + xlimits = NULL ) } \arguments{ @@ -83,6 +84,13 @@ relative to the extremes for non-missing values.} \item \code{waiver()} for the default breaks \item A numeric vector of positions }} + +\item{xlimits}{The x-axis limits as one of: +\itemize{ +\item \code{NULL} to use the default scale range +\item A numeric vector of length two providing the limits. +Use NA to refer to the existing minimum or maximum limits. +}} } \description{ Plots species sensitivity data and distributions. diff --git a/tests/testthat/test-ssd-plot.R b/tests/testthat/test-ssd-plot.R index 7aba8b27..8442aec0 100644 --- a/tests/testthat/test-ssd-plot.R +++ b/tests/testthat/test-ssd-plot.R @@ -68,3 +68,8 @@ test_that("ssd_plot fills in missing order", { "missing_order" ) }) + +test_that("ssd_plot xlims", { + expect_snapshot_plot(ssd_plot(ssddata::ccme_boron, boron_pred, xlimits = c(NA, 10000)), "boron_limits") +}) + \ No newline at end of file From 92b12c710b18d59b80a2c66cc22653b5fc8b7d6d Mon Sep 17 00:00:00 2001 From: Seb Dalgarno Date: Fri, 6 Dec 2024 18:04:41 -0800 Subject: [PATCH 10/66] add ssd_label_comma_hc label function to offset hc value (appending \n) --- R/scales.R | 26 ++++++++++++++++++++++++++ 1 file changed, 26 insertions(+) diff --git a/R/scales.R b/R/scales.R index aff7388a..9f05df50 100644 --- a/R/scales.R +++ b/R/scales.R @@ -41,3 +41,29 @@ ssd_label_comma <- function(digits = 3, ..., big.mark = ",") { y } } + +#' Label numbers with significant digits and comma and offset hazard concentration value if present in breaks. +#' +#' @inheritParams params +#' +#' @return A "labelling" function that takes a vector x and +#' returns a character vector of `length(x)` giving a label for each input value. +#' @seealso [scales::label_comma()] +#' @export +#' +#' @examples +#' ggplot2::ggplot(data = ssddata::anon_e, ggplot2::aes(x = Conc / 10)) + +#' geom_ssdpoint() + +#' ggplot2::scale_x_log10(labels = ssd_label_comma_hc()) +ssd_label_comma_hc <- function(hc_value, digits = 3, big.mark = ",") { + chk_number(hc_value) + + function(x) { + marked <- ssd_label_comma(digits = digits, big.mark = big.mark)(x) + purrr::map_chr(marked, ~ { + if (!is.na(.x) && .x == signif(hc_value, digits = digits)) + .x <- paste0("\n", .x) + .x + }) + } +} From 95de5d23459342be22d795892ab186bf196af9e7 Mon Sep 17 00:00:00 2001 From: Seb Dalgarno Date: Fri, 6 Dec 2024 18:07:48 -0800 Subject: [PATCH 11/66] add ability in plot_coord_scale to append hc value to xbreaks and offset to new line, restrict transformations to log, log10 and identity --- R/ssd-plot.R | 32 +++++++++++++++++++++++--------- 1 file changed, 23 insertions(+), 9 deletions(-) diff --git a/R/ssd-plot.R b/R/ssd-plot.R index e3c2fdb3..48c47633 100644 --- a/R/ssd-plot.R +++ b/R/ssd-plot.R @@ -18,19 +18,29 @@ #' @export ggplot2::waiver -plot_coord_scale <- function(data, xlab, ylab, trans, big.mark, suffix, xbreaks = waiver(), xlimits = NULL) { +plot_coord_scale <- function(data, xlab, ylab, trans, big.mark, suffix, xbreaks = waiver(), xlimits = NULL, hc_value = NULL) { chk_string(xlab) chk_string(ylab) - - if (is.waive(xbreaks) && trans == "log10") { - xbreaks <- trans_breaks("log10", function(x) 10^x) - } - + + if (is.waive(xbreaks)) { + xbreaks <- switch(trans, + "log10" = function(x) unique(c(scales::log10_trans()$breaks(x), hc_value)), + "log" = function(x) unique(c(scales::log_trans()$breaks(x), hc_value)), + "identity" = function(x) unique(c(scales::identity_trans()$breaks(x), hc_value))) + } else { + xbreaks <- unique(c(xbreaks, hc_value)) + } + + ssd_label_fun <- ssd_label_comma(big.mark = big.mark) + if(!is.null(hc_value)) { + ssd_label_fun <- ssd_label_comma_hc(hc_value, big.mark = big.mark) + } + list( coord_trans(x = trans), scale_x_continuous(xlab, breaks = xbreaks, - labels = ssd_label_comma(big.mark = big.mark), + labels = ssd_label_fun, limits = xlimits ), scale_y_continuous(ylab, @@ -90,7 +100,7 @@ ssd_plot <- function(data, pred, left = "Conc", right = left, ..., chk_string(big.mark) chk_string(suffix) .chk_bounds(bounds) - chk_string(trans) + chk_subset(trans, c("log10", "log", "identity")) data <- process_data(data, left, right, weight = NULL) data <- bound_data(data, bounds) @@ -162,9 +172,13 @@ ssd_plot <- function(data, pred, left = "Conc", right = left, ..., ), stat = "identity") } + hc_value <- NULL + if(!is.null(hc)){ + hc_value <- pred$est[pred$proportion %in% hc] + } gp <- gp + plot_coord_scale(data, xlab = xlab, ylab = ylab, big.mark = big.mark, suffix = suffix, - trans = trans, xbreaks = xbreaks, xlimits = xlimits + trans = trans, xbreaks = xbreaks, xlimits = xlimits, hc_value = hc_value ) if (!is.null(label)) { From 3186e9b0d8e57364fd565b0291e71e15c05a2f4f Mon Sep 17 00:00:00 2001 From: Seb Dalgarno Date: Fri, 6 Dec 2024 18:08:20 -0800 Subject: [PATCH 12/66] add hc_value param for plot_coord_scale, update documentation on accepted trans values, minor typo in xintercept --- R/params.R | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/R/params.R b/R/params.R index d05c7bdb..4e1dbecf 100644 --- a/R/params.R +++ b/R/params.R @@ -42,6 +42,7 @@ #' @param dists A character vector of the distribution names. #' @param fitdists An object of class fitdists. #' @param hc A value between 0 and 1 indicating the proportion hazard concentration (or NULL). +#' @param hc_value A number of the hazard concentration value to offset. #' @param label A string of the column in data with the labels. #' @param left A string of the column in data with the concentrations. #' @param level A number between 0 and 1 of the confidence level of the interval. @@ -112,7 +113,7 @@ #' @param size A number for the size of the labels. #' @param suffix Additional text to display after the number on the y-axis. #' @param tails A flag or NULL specifying whether to only include distributions with both tails. -#' @param trans A string which transformation to use by default `"log10"`. +#' @param trans A string of which transformation to use. Accepted values include `"log10"`, `"log"`, and `"identity"` (`"log10"` by default). #' @param weight A string of the numeric column in data with positive weights less than or equal to 1,000 or NULL. #' @param x The object. #' @param xbreaks The x-axis breaks as one of: @@ -123,7 +124,7 @@ #' - `NULL` to use the default scale range #' - A numeric vector of length two providing the limits. #' Use NA to refer to the existing minimum or maximum limits. -#' @param xintercept The x-value for the intersect +#' @param xintercept The x-value for the intersect. #' @param xlab A string of the x-axis label. #' @param yintercept The y-value for the intersect. #' @param ylab A string of the x-axis label. From 2c86c6d475285d35499ab22cc7e3da1605a64f2a Mon Sep 17 00:00:00 2001 From: Seb Dalgarno Date: Fri, 6 Dec 2024 18:08:34 -0800 Subject: [PATCH 13/66] document --- NAMESPACE | 1 + man/geom_hcintersect.Rd | 2 +- man/params.Rd | 6 ++++-- man/ssd_label_comma_hc.Rd | 30 ++++++++++++++++++++++++++++++ man/ssd_plot.Rd | 2 +- man/ssd_plot_data.Rd | 2 +- 6 files changed, 38 insertions(+), 5 deletions(-) create mode 100644 man/ssd_label_comma_hc.Rd diff --git a/NAMESPACE b/NAMESPACE index 013bfea1..39d0661f 100644 --- a/NAMESPACE +++ b/NAMESPACE @@ -100,6 +100,7 @@ export(ssd_hp) export(ssd_hp_bcanz) export(ssd_is_censored) export(ssd_label_comma) +export(ssd_label_comma_hc) export(ssd_licensing_md) export(ssd_match_moments) export(ssd_min_pmix) diff --git a/man/geom_hcintersect.Rd b/man/geom_hcintersect.Rd index e58e904a..0e3c4180 100644 --- a/man/geom_hcintersect.Rd +++ b/man/geom_hcintersect.Rd @@ -63,7 +63,7 @@ lists which parameters it can accept. \link[ggplot2:draw_key]{key glyphs}, to change the display of the layer in the legend. }} -\item{xintercept}{The x-value for the intersect} +\item{xintercept}{The x-value for the intersect.} \item{yintercept}{The y-value for the intersect.} diff --git a/man/params.Rd b/man/params.Rd index fb302e0f..525b42c5 100644 --- a/man/params.Rd +++ b/man/params.Rd @@ -57,6 +57,8 @@ Distributions with an absolute AIC difference greater than delta are excluded fr \item{hc}{A value between 0 and 1 indicating the proportion hazard concentration (or NULL).} +\item{hc_value}{A number of the hazard concentration value to offset.} + \item{label}{A string of the column in data with the labels.} \item{left}{A string of the column in data with the concentrations.} @@ -189,7 +191,7 @@ remove them with a warning.} \item{tails}{A flag or NULL specifying whether to only include distributions with both tails.} -\item{trans}{A string which transformation to use by default \code{"log10"}.} +\item{trans}{A string of which transformation to use. Accepted values include \code{"log10"}, \code{"log"}, and \code{"identity"} (\code{"log10"} by default).} \item{weight}{A string of the numeric column in data with positive weights less than or equal to 1,000 or NULL.} @@ -209,7 +211,7 @@ remove them with a warning.} Use NA to refer to the existing minimum or maximum limits. }} -\item{xintercept}{The x-value for the intersect} +\item{xintercept}{The x-value for the intersect.} \item{xlab}{A string of the x-axis label.} diff --git a/man/ssd_label_comma_hc.Rd b/man/ssd_label_comma_hc.Rd new file mode 100644 index 00000000..0e4d1f85 --- /dev/null +++ b/man/ssd_label_comma_hc.Rd @@ -0,0 +1,30 @@ +% Generated by roxygen2: do not edit by hand +% Please edit documentation in R/scales.R +\name{ssd_label_comma_hc} +\alias{ssd_label_comma_hc} +\title{Label numbers with significant digits and comma and offset hazard concentration value if present in breaks.} +\usage{ +ssd_label_comma_hc(hc_value, digits = 3, big.mark = ",") +} +\arguments{ +\item{hc_value}{A number of the hazard concentration value to offset.} + +\item{digits}{A whole number specifying the number of significant figures.} + +\item{big.mark}{A string specifying used between every 3 digits to separate thousands on the x-axis.} +} +\value{ +A "labelling" function that takes a vector x and +returns a character vector of \code{length(x)} giving a label for each input value. +} +\description{ +Label numbers with significant digits and comma and offset hazard concentration value if present in breaks. +} +\examples{ +ggplot2::ggplot(data = ssddata::anon_e, ggplot2::aes(x = Conc / 10)) + + geom_ssdpoint() + + ggplot2::scale_x_log10(labels = ssd_label_comma_hc()) +} +\seealso{ +\code{\link[scales:label_number]{scales::label_comma()}} +} diff --git a/man/ssd_plot.Rd b/man/ssd_plot.Rd index 4178809a..d525b6ae 100644 --- a/man/ssd_plot.Rd +++ b/man/ssd_plot.Rd @@ -76,7 +76,7 @@ relative to the extremes for non-missing values.} \item{suffix}{Additional text to display after the number on the y-axis.} -\item{trans}{A string which transformation to use by default \code{"log10"}.} +\item{trans}{A string of which transformation to use. Accepted values include \code{"log10"}, \code{"log"}, and \code{"identity"} (\code{"log10"} by default).} \item{xbreaks}{The x-axis breaks as one of: \itemize{ diff --git a/man/ssd_plot_data.Rd b/man/ssd_plot_data.Rd index 78a2d405..8ced328a 100644 --- a/man/ssd_plot_data.Rd +++ b/man/ssd_plot_data.Rd @@ -57,7 +57,7 @@ ssd_plot_data( uncensored missing (0 and Inf) data in terms of the orders of magnitude relative to the extremes for non-missing values.} -\item{trans}{A string which transformation to use by default \code{"log10"}.} +\item{trans}{A string of which transformation to use. Accepted values include \code{"log10"}, \code{"log"}, and \code{"identity"} (\code{"log10"} by default).} \item{xbreaks}{The x-axis breaks as one of: \itemize{ From 65ad89e7cbf01494457d7b6ef085cc6df24b7c69 Mon Sep 17 00:00:00 2001 From: Seb Dalgarno Date: Fri, 6 Dec 2024 18:09:14 -0800 Subject: [PATCH 14/66] add two new tests for ssd_plot with hc value --- tests/testthat/test-ssd-plot.R | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) diff --git a/tests/testthat/test-ssd-plot.R b/tests/testthat/test-ssd-plot.R index 8442aec0..4a0c3e8d 100644 --- a/tests/testthat/test-ssd-plot.R +++ b/tests/testthat/test-ssd-plot.R @@ -72,4 +72,11 @@ test_that("ssd_plot fills in missing order", { test_that("ssd_plot xlims", { expect_snapshot_plot(ssd_plot(ssddata::ccme_boron, boron_pred, xlimits = c(NA, 10000)), "boron_limits") }) - \ No newline at end of file + +test_that("ssd_plot no hcvalue", { + expect_snapshot_plot(ssd_plot(ssddata::ccme_boron, boron_pred, hc = NULL), "boron_nohc") +}) + +test_that("ssd_plot if hc value also in xbreaks", { + expect_snapshot_plot(ssd_plot(ssddata::ccme_boron, boron_pred, xbreaks = c(1, 1.26, 10, 100)), "boron_hcdup") +}) \ No newline at end of file From 99750172bb7504bd4f582bd32426d0c4eb1e4bb4 Mon Sep 17 00:00:00 2001 From: Seb Dalgarno Date: Fri, 6 Dec 2024 18:10:00 -0800 Subject: [PATCH 15/66] update ssd_plot and plot_data snapshots, hc value now included + with offset by default, breaks slightly different (but look good/better) --- .../_snaps/autoplot/autoplot_bigmark.png | Bin 23030 -> 21785 bytes tests/testthat/_snaps/autoplot/suffix.png | Bin 23106 -> 21888 bytes tests/testthat/_snaps/match-moments/cdf.png | Bin 42867 -> 41123 bytes .../testthat/_snaps/plot-cdf/fits_average.png | Bin 22643 -> 21092 bytes .../_snaps/plot-data/big_mark_comma.png | Bin 17723 -> 17603 bytes .../_snaps/plot-data/big_mark_space.png | Bin 17683 -> 17554 bytes .../testthat/_snaps/plot-data/ccme_boron.png | Bin 17461 -> 17296 bytes tests/testthat/_snaps/plot-data/suffix.png | Bin 17609 -> 17495 bytes .../_snaps/ssd-plot/boron_bigmark.png | Bin 25277 -> 25862 bytes .../testthat/_snaps/ssd-plot/boron_breaks.png | Bin 24415 -> 24596 bytes .../testthat/_snaps/ssd-plot/boron_color.png | Bin 27667 -> 28243 bytes .../testthat/_snaps/ssd-plot/boron_hcdup.png | Bin 0 -> 25515 bytes .../testthat/_snaps/ssd-plot/boron_limits.png | Bin 0 -> 24129 bytes tests/testthat/_snaps/ssd-plot/boron_nohc.png | Bin 0 -> 24653 bytes tests/testthat/_snaps/ssd-plot/boron_pred.png | Bin 25121 -> 25656 bytes .../_snaps/ssd-plot/boron_pred_label.png | Bin 42838 -> 43621 bytes .../_snaps/ssd-plot/boron_pred_shift_x.png | Bin 42919 -> 43700 bytes .../testthat/_snaps/ssd-plot/boron_shape.png | Bin 26019 -> 26347 bytes .../_snaps/ssd-plot/missing_order.png | Bin 25500 -> 24690 bytes tests/testthat/_snaps/ssd-plot/no_ribbon.png | Bin 32348 -> 32501 bytes tests/testthat/_snaps/ssd-plot/ribbon.png | Bin 25121 -> 25656 bytes 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