diff --git a/lessons/univariate_models.Rmd b/lessons/univariate_models.Rmd index c70661e..cb8ef1e 100644 --- a/lessons/univariate_models.Rmd +++ b/lessons/univariate_models.Rmd @@ -66,7 +66,7 @@ These different goals of model building require different approaches and modes of model evaluation. Hypothesis testing is typically geared towards testing a small handful of carefully crafted ***A PRIORI*** hypotheses. -In this context they model is typically judged useful if it is statistically +In this context the model is typically judged useful if it is statistically significant. However, many times though the investigator does not have a clear *_a priori_* hypothesis [(see post at Small Pond Science)](http://smallpondscience.com/2013/06/04/pretending-you-planned-to-test-that-hypothesis-the-whole-time/) @@ -94,14 +94,13 @@ re-frame their analyses as if they are confirmatory rather than exploratory. And of course there is pressure during peer-review to only report on statistics that are significant. -You might be wondering why this is a big deal. The reason is that you will inevitably -get good fitting models (high R^2) and statistically significant results (p < 0.05) -if you keep adding variables to a model even if those variables by definition are -independent of the response variable [(Freedman 1983)](http://amstat.tandfonline.com/doi/abs/10.1080/00031305.1983.10482729#.Ul17gVAkJPQ). - -The solution to Freedman's paradox is to approach model comparison as carefully -and intentionally as possible with a small number of deliberately chosen models. -Burnham and Anderson (2002, p19) advocate for: +You might be wondering why this is a big deal. The reason is Freedman's paradox +[(Freedman 1983)](http://amstat.tandfonline.com/doi/abs/10.1080/00031305.1983.10482729#.Ul17gVAkJPQ) which demonstrates that you will inevitably get good fitting models +(high R^2) and statistically significant results (p < 0.05) if you keep adding +variables to a model even if those variables by definition are independent of +the response variable. The solution to Freedman's paradox is to approach model +comparison as carefully and intentionally as possible with a small number of +deliberately chosen models. Burnham and Anderson (2002, p19) advocate for:
... a conservative approach to the overall issue of *strategy* in the @@ -219,7 +218,7 @@ intuition which will guide our modeling and interpretation of the statistics. >A quick note on graphics. I will primarily use base graphics in this course but increasingly the R community is moving towards using `ggplot` for graphics. -`ggplot` is a really great set of tools for making bueatiful graphics but it can +`ggplot` is a really great set of tools for making beautiful graphics but it can be more difficult to understand exactly how it works and how to custumize graphics. Therefore, I want to expose you to both methods of producing graphics the simple and clunky base graphics and the elegent and shiny `ggplot` graphics. @@ -263,13 +262,18 @@ ggplot(data = weeds) + Technically this is the same graphic as produced by base but you can see that the R code is a bit more cryptic. -Let's break it down line by line: `ggplot(data = weeds) +` spawns a `ggplot` graphic and specifies that the data will be provided by the object `weeds`. Now we can simply refer to column names -of `weeds` in the remainder of the plotting call. +Let's break it down line by line: `ggplot(data = weeds) +` spawns a `ggplot` +graphic and specifies that the data will be provided by the object `weeds`. +Now we can simply refer to column names of `weeds` in the remainder of the +plotting call. -The next line: `geom_boxplot(mapping = aes(x = trt, y = fruit_mass_mg)) +` specifies the geometry of the graphic in this case a boxplot, there are many other that can be specified in `ggplot`. The geometry function requires a -few arguments including how to map data on to the graphic using the argument -`mapping`. The mapping is usually wrapped in the function `aes` which provides an aesthetically pleasing rendering of the data on the graphic, `aes` requires -and `x` and `y` variables. +The next line: `geom_boxplot(mapping = aes(x = trt, y = fruit_mass_mg)) +` +specifies the geometry of the graphic in this case a boxplot, there are many +other that can be specified in `ggplot`. The geometry function requires a few +arguments including how to map data on to the graphic using the argument +`mapping`. The mapping is usually wrapped in the function `aes` which provides +an aesthetically pleasing rendering of the data on the graphic, `aes` requires +and `x` and `y` variables. The last line: `labs(x = 'Treatment', y = 'Fruit mass (mg)')` provides the axis labels. @@ -372,9 +376,8 @@ A few quick points about this `ggplot` call. * rather than specify the mapping of the `x` and `y` for each `geom_*` we simply specify them once when setting up the graphic in the first `ggplot()` call -* Note the usage of `manual_color()` - this is generally note needed as -`ggplot`'s default colors are usually pretty attractive. I'm not sure why the -packages is showing 'blue' as 'purple' +* Note the usage of `manual_color()` - this is generally not needed as +`ggplot`'s default colors are usually pretty attractive. ##### Excercise Modify the `ggplot` call so that the `x` and `y` axes are properly labeled. @@ -450,7 +453,7 @@ There is the minimal intercept only model $$\mathbf{y} = \beta_0 + \varepsilon$$ This model essentially just uses the mean of *y* as a predictor of *y*. This may seem silly but this is essentially what you compare all more complex models -against. +against - it is our null model. ```{r null model} null_mod <- lm(fruit_mass_mg ~ 1, data = weeds) @@ -498,7 +501,8 @@ Let's take a closer look at the `trt_mod` which includes the main effect due to treatment. Note that only one of the levels of treatment is provided as a coefficient. In this case it is `unfertilized`. To better understand this you need to consider how factor variables are included in regression models. A -categorical variable is encoded in R into a set of orthogonal contrasts. +categorical variable is encoded in R into a set of orthogonal contrasts (aka +[dummy variables](https://ordination.okstate.edu/envvar.htm)). ```{r} levels(weeds$trt) @@ -513,9 +517,11 @@ factor as a set of orthogonal contrasts. This explains why the treatment variabl only requires a single regression coefficient. Sometimes we have factors that are ranked such as low, medium, high. In this -case the variable is called **ordinal** as opposed to our **nomial** treatment -variable. The contrasts of ordinal variables are not as simple to specify and -typically a Helmert polynomial contrasts are used. +case the variable is called **ordinal** as opposed to our **nominal** treatment +variable which did not contain ranks. +The contrasts of ordinal variables are not as simple to specify and typically a +Helmert polynomial contrasts are used (if you want to know more and possibly a +better solution see [Gullickson's 2020 blog post](https://aarongullickson.netlify.app/post/better-contrasts-for-ordinal-variables-in-r/)) Let's examine these models graphically. @@ -530,15 +536,15 @@ abline(ht_mod) ``` -The first panel of this graphic doesn't quite look correct because the regression -line is not intersecting the center of the boxplots which is what we would -expect. This is because by default -R assigns the first level of the treatment factor an x-value of 1 and the second level of the factor a value of 2. Therefore when the regression line is added to the -plot it is plotting the y-intercept (which is the mean value of the fertilized -group) off the graph to the left one unit. To correct this we have to -build the graph from scratch a bit more carefully making sure that the fertilized -group is plotted an an x-axis value of 0 so that the regression line intersects that -groups properly. +The first panel of this graphic doesn't quite look correct because the +regression line is not intersecting the center of the boxplots which is what we +would expect. This is because by default R assigns the first level of the +treatment factor an x-value of 1 and the second level of the factor a value of +2. Therefore when the regression line is added to the plot it is plotting the +y-intercept (which is the mean value of the fertilized group) off the graph to +the left one unit. To correct this we have to build the graph from scratch a bit +more carefully making sure that the fertilized group is plotted an an x-axis +value of 0 so that the regression line intersects that groups properly. ```{r} par(mfrow=c(1,2)) diff --git a/lessons/univariate_models.html b/lessons/univariate_models.html index e72dfe8..d3ca17c 100644 --- a/lessons/univariate_models.html +++ b/lessons/univariate_models.html @@ -30,7 +30,7 @@ !function(e,t){"use strict";"object"==typeof module&&"object"==typeof module.exports?module.exports=e.document?t(e,!0):function(e){if(!e.document)throw new Error("jQuery requires a window with a document");return t(e)}:t(e)}("undefined"!=typeof window?window:this,function(C,e){"use strict";var t=[],r=Object.getPrototypeOf,s=t.slice,g=t.flat?function(e){return t.flat.call(e)}:function(e){return t.concat.apply([],e)},u=t.push,i=t.indexOf,n={},o=n.toString,v=n.hasOwnProperty,a=v.toString,l=a.call(Object),y={},m=function(e){return"function"==typeof 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