From d3534b8c9078064028c0682e3bb97bac02fdfb42 Mon Sep 17 00:00:00 2001 From: hardin47 Date: Tue, 26 Sep 2023 16:17:10 -0700 Subject: [PATCH 1/6] first draft of new chapter 3 --- 03-data-applications-old.qmd | 350 ++++++++++++++++++ 03-data-applications.qmd | 326 ++++++++-------- .../execute-results/html.json | 4 +- .../figure-html/fig-paralympic-cat-1.png | Bin 0 -> 440038 bytes .../figure-html/fig-paralympic-grouped-1.png | Bin 0 -> 164529 bytes .../fig-paralympic-ungrouped-1.png | Bin 0 -> 123042 bytes .../execute-results/html.json | 4 +- .../execute-results/html.json | 4 +- book.bib | 12 + 9 files changed, 545 insertions(+), 155 deletions(-) create mode 100644 03-data-applications-old.qmd create mode 100644 _freeze/03-data-applications/figure-html/fig-paralympic-cat-1.png create mode 100644 _freeze/03-data-applications/figure-html/fig-paralympic-grouped-1.png create mode 100644 _freeze/03-data-applications/figure-html/fig-paralympic-ungrouped-1.png diff --git a/03-data-applications-old.qmd b/03-data-applications-old.qmd new file mode 100644 index 00000000..d216e40c --- /dev/null +++ b/03-data-applications-old.qmd @@ -0,0 +1,350 @@ +# Applications: Data {#sec-data-applications} + +```{r} +#| include: false +source("_common.R") +``` + +## Case study: Passwords {#case-study-passwords} + +Stop for a second and think about how many passwords you've used so far today. +You've probably used one to unlock your phone, one to check email, and probably at least one to log on to a social media account. +Made a debit purchase? +You've probably entered a password there too. + +If you're reading this book, and particularly if you're reading it online, chances are you have had to create a password once or twice in your life. +And if you are diligent about your safety and privacy, you've probably chosen passwords that would be hard for others to guess, or *crack*. + +In this case study we introduce a dataset on passwords. +The goal of the case study is to walk you through what a data scientist does when they first get a hold of a dataset as well as to provide some "foreshadowing" of concepts and techniques we'll introduce in the next few chapters on exploratory data analysis. + +::: {.data data-latex=""} +The [`passwords`](https://github.com/rfordatascience/tidytuesday/blob/master/data/2020/2020-01-14/readme.md) data can be found in the [**tidytuesdayR**](https://thebioengineer.github.io/tidytuesdayR/) R package. +::: + +@tbl-passwords-df-head shows the first ten rows from the dataset, which are the ten most common passwords. +Perhaps unsurprisingly, "password" tops the list, followed by "123456". + +```{r} +#| label: tbl-passwords-df-head +#| tbl-cap: Top ten rows of the `passwords` dataset. +# https://raw.githubusercontent.com/rfordatascience/tidytuesday/master/data/2020/2020-01-14/passwords.csv +passwords <- readr::read_csv("data/passwords.csv") +passwords <- passwords |> + select(-font_size, -rank_alt) |> + filter(!is.na(category)) |> + mutate(time_unit = fct_relevel(time_unit, "seconds", "minutes", "hours", "days", "weeks", "months", "years")) + +passwords |> + slice_head(n = 10) |> + kbl(linesep = "", booktabs = TRUE, + row.names = FALSE) |> + kable_styling(bootstrap_options = c("striped", "condensed"), + latex_options = c("striped", "hold_position")) +``` + +When you encounter a new dataset, taking a peek at the first few rows as we did in @tbl-passwords-df-head is almost instinctual. +It can often be helpful to look at the last few rows of the data as well to get a sense of the size of the data as well as potentially discover any characteristics that may not be apparent in the top few rows. +@tbl-passwords-df-tail shows the bottom ten rows of the passwords dataset, which reveals that we are looking at a dataset of 500 passwords. + +```{r} +#| label: tbl-passwords-df-tail +#| tbl-cap: Bottom ten rows of the `passwords` dataset. +passwords |> + slice_tail(n = 10) |> + kbl(linesep = "", booktabs = TRUE, row.names = FALSE) |> + kable_styling(bootstrap_options = c("striped", "condensed"), + latex_options = c("striped", "hold_position")) +``` + +At this stage it's also useful to think about how these data were collected, as that will inform the scope of any inference you can make based on your analysis of the data. + +::: {.guidedpractice data-latex=""} +Do these data come from an observational study or an experiment?[^03-data-applications-1] +::: + +[^03-data-applications-1]: This is an observational study. + Researchers collected data on existing passwords in use and identified most common ones to put together this dataset. + +::: {.guidedpractice data-latex=""} +There are `r nrow(passwords)` rows and `r ncol(passwords)` columns in the dataset. +What does each row and each column represent?[^03-data-applications-2] +::: + +[^03-data-applications-2]: Each row represents a password and each column represents a variable which contains information on each password. + +Once you've identified the rows and columns, it's useful to review the data dictionary to learn about what each column in the dataset represents. +This is provided in @tbl-passwords-var-def. + +```{r} +#| label: tbl-passwords-var-def +#| tbl-cap: Variables and their descriptions for the `passwords` dataset. +passwords_var_def <- tribble( + ~variable, ~description, + "rank", "Popularity in the database of released passwords.", + "password", "Actual text of the password.", + "category", "Category password falls into.", + "value", "Time to crack by online guessing.", + "time_unit", "Time unit to match with value.", + "offline_crack_sec", "Time to crack offline in seconds.", + "strength", "Strength of password, relative only to passwords in this dataset. Lower values indicate weaker passwords." +) + +passwords_var_def |> + kbl(linesep = "", booktabs = TRUE, + col.names = c("Variable", "Description")) |> + kable_styling(bootstrap_options = c("striped", "condensed"), + latex_options = c("striped", "hold_position"), full_width = TRUE) |> + column_spec(1, monospace = TRUE) |> + column_spec(2, width = "30em") +``` + +We now have a better sense of what each column represents, but we do not yet know much about the characteristics of each of the variables. + +::: {.workedexample data-latex=""} +Determine whether each variable in the passwords dataset is numerical or categorical. +For numerical variables, further classify them as continuous or discrete. +For categorical variables, determine if the variable is ordinal. + +------------------------------------------------------------------------ + +The numerical variables in the dataset are `rank` (discrete), `value` (continuous), and `offline_crack_sec` (continuous). +The categorical variables are `password`, `time_unit`. +The strength variable is trickier to classify -- we can think of it as discrete numerical or as an ordinal variable as it takes on numerical values, however it's used to categorize the passwords on an ordinal scale. +One way of approaching this is thinking about whether the values the variable takes vary linearly, e.g., is the difference in strength between passwords with strength levels 8 and 9 the same as the difference with those with strength levels 9 and 10. +If this is not necessarily the case, we would classify the variable as ordinal. +Determining the classification of this variable requires understanding of how `strength` values were determined, which is a very typical workflow for working with data. +Sometimes the data dictionary (presented in @tbl-passwords-var-def) isn't sufficient, and we need to go back to the data source and try to understand the data better before we can proceed with the analysis meaningfully. +::: + +Next, let's try to get to know each variable a little bit better. +For categorical variables, this involves figuring out what their levels are and how commonly represented they are in the data. +@fig-passwords-cat shows the distributions of the categorical variables in this dataset. +We can see that password strengths of 0-10 are more common than higher values. +The most common password category is name (e.g. michael, jennifer, jordan, etc.) and the least common is food (e.g., pepper, cheese, coffee, etc.). +Many passwords can be cracked in the matter of days by online cracking with some taking as little as seconds and some as long as years to break. +Each of these visualizations is a bar plot, which you will learn more about in @sec-explore-categorical. + +```{r} +#| label: fig-passwords-cat +#| fig-cap: Distributions of the categorical variables in the `passwords` dataset. +#| Plot A shows the distribution of password strengths, Plot B password +#| categories, and Plot C length of time it takes to crack the passwords by +#| online guessing. +#| fig-asp: 1.0 +#| out.width: 100% +p_category <- passwords |> + count(category, sort = TRUE) |> + mutate(category = fct_reorder(category, n)) |> + ggplot(aes(y = category, x = n, fill = fct_rev(category))) + + geom_col(show.legend = FALSE) + + scale_fill_openintro() + + labs( + x = "Count", + y = NULL, + title = "Categories" + ) + + theme(plot.title.position = "plot") + +p_time_unit <- passwords |> + count(time_unit) |> + ggplot(aes(y = time_unit, x = n)) + + geom_col(show.legend = FALSE) + + labs( + x = "Count", + y = NULL, + title = "Length of time to crack", + subtitle = "By online guessing" + ) + + theme(plot.title.position = "plot") + +p_strength <- passwords |> + ggplot(aes(y = strength)) + + geom_histogram(binwidth = 1, show.legend = FALSE) + + scale_y_continuous(breaks = seq(0, 50, 5), trans = "reverse") + + labs( + x = "Count", + y = NULL, + title = "Strengths" + ) + + theme(plot.title.position = "plot") + +patchwork <- p_strength | (p_category / p_time_unit) + +patchwork + + plot_annotation( + title = "Strengths, categories, and cracking time\nof 500 most common passwords", + tag_levels = "A" + ) & + theme(plot.tag = element_text(size = 12, color = "darkgray")) +``` + +Similarly, we can examine the distributions of the numerical variables as well. +We already know that rank ranges between 1 and 500 in this dataset, based on @tbl-passwords-df-head and @tbl-passwords-df-tail. +The value variable is slightly more complicated to consider since the numerical values in that column are meaningless without the time unit that accompanies them. +@tbl-passwords-online-crack-summary shows the minimum and maximum amount of time it takes to crack a password by online guessing. +For example, there are 11 passwords in the dataset that can be broken in a matter of seconds, and each of them take 11.11 seconds to break, since the minimum and the maximum of observations in this group are exactly equal to this value. +And there are 65 passwords that take years to break, ranging from 2.56 years to 92.27 years. + +```{r} +#| label: tbl-passwords-online-crack-summary +#| tbl-cap: Minimum and maximum amount of time it takes to crack a password by +#| online guessing as well as the number of observations that fall into each +#| time unit category. +passwords |> + group_by(time_unit) |> + summarise( + n = n(), + min = min(value), + max = max(value) + ) |> + kbl(linesep = "", booktabs = TRUE) |> + kable_styling(bootstrap_options = c("striped", "condensed"), + latex_options = c("striped", "hold_position")) +``` + +Even though passwords that take a large number of years to crack can seem like good options (see @tbl-passwords-long-crack for a list of them), now that you've seen them here (and the fact that they are in a dataset of 500 most common passwords), you should not use them as secure passwords! + +```{r} +#| label: tbl-passwords-long-crack +#| tbl-cap: Passwords that take the longest amount of time to crack by online +#| guessing. +passwords |> + filter(value == 92.27) |> + kbl(linesep = "", booktabs = TRUE, row.names = FALSE) |> + kable_styling(bootstrap_options = c("striped", "condensed"), + latex_options = c("striped", "hold_position")) +``` + +\clearpage + +The last numerical variable in the dataset is `offline_crack_sec`. +@fig-password-offline-crack-hist shows the distribution of this variable, which reveals that all of these passwords can be cracked offline in under 30 seconds, with a large number of them being crackable in just a few seconds. + +```{r} +#| label: fig-password-offline-crack-hist +#| fig-cap: Histogram of the length of time it takes to crack passwords offline. +ggplot(passwords, aes(x = offline_crack_sec)) + + geom_histogram(binwidth = 1) + + labs( + x = "Length of time (seconds)", + y = "Count", + title = "Length of time to crack passwords offline" + ) +``` + +So far we examined the distributions of each individual variable, but it would be more interesting to explore relationships between multiple variables. +@fig-password-strength-rank-category shows the relationship between rank and strength of passwords by category, where more common passwords (those with higher rank) are plotted higher on the y-axis than those that are less common in this dataset. +The stronger the password, the larger text it's represented with on the plot. +While this visualization reveals some passwords that are less common, and stronger than others, we should reiterate that you should not use any of these passwords. +And if you already do, it's time to go change it! + +```{r} +#| label: fig-password-strength-rank-category +#| fig-cap: Rank vs. strength of 500 most common passwords by category. +#| fig-asp: 1.2 +#| out.width: 100% +passwords |> + mutate(category = fct_relevel(category, "name", "cool-macho", "simple-alphanumeric", "fluffy", "sport", "nerdy-pop", "animal", "password-related", "rebellious-rude", "food")) |> + ggplot(aes(x = strength, y = rank, color = category)) + + geom_text(aes(label = password, size = strength), + check_overlap = TRUE, show.legend = FALSE) + + facet_wrap(vars(category), ncol = 3) + + coord_cartesian(ylim = c(525, -10)) + + scale_y_continuous(breaks = c(1, 100, 200, 300, 400, 500), minor_breaks = NULL, trans = "reverse") + + scale_color_openintro() + + labs( + x = "Strength of password", + y = "Rank of popularity", + title = "500 most common passwords by category", + caption = "Data: Information is beautiful, via TidyTuesday" + ) +``` + +In this case study, we introduced you to the very first steps a data scientist takes when they start working with a new dataset. +In the next few chapters, we will introduce exploratory data analysis and you'll learn more about the various types of data visualizations and summary statistics you can make to get to know your data better. + +Before you move on, we encourage you to think about whether the following questions can be answered with this dataset, and if yes, how you might go about answering them. +It's okay if your answer is "I'm not sure", we simply want to get your exploratory juices flowing to prime you for what's to come! + +1. What characteristics are associated with a strong vs. a weak password? +2. Do more popular passwords take shorter or longer to crack compared to less popular passwords? +3. Are passwords that start with letters or numbers more common among the list of top 500 most common passwords? + +\clearpage + +## Interactive R tutorials {#data-tutorials} + +Navigate the concepts you've learned in this chapter in R using the following self-paced tutorials. +All you need is your browser to get started! + +::: {.alltutorials data-latex=""} +[Tutorial 1: Introduction to data](https://openintrostat.github.io/ims-tutorials/01-data/) + +::: {.content-hidden unless-format="pdf"} +https://openintrostat.github.io/ims-tutorials/01-data +::: +::: + +::: {.singletutorial data-latex=""} +[Tutorial 1 - Lesson 1: Language of data](https://openintro.shinyapps.io/ims-01-data-01/) + +::: {.content-hidden unless-format="pdf"} +https://openintro.shinyapps.io/ims-01-data-01 +::: +::: + +::: {.singletutorial data-latex=""} +[Tutorial 1 - Lesson 2: Types of studies](https://openintro.shinyapps.io/ims-01-data-02/) + +::: {.content-hidden unless-format="pdf"} +https://openintro.shinyapps.io/ims-01-data-02 +::: +::: + +::: {.singletutorial data-latex=""} +[Tutorial 1 - Lesson 3: Sampling strategies and experimental design](https://openintro.shinyapps.io/ims-01-data-03/) + +::: {.content-hidden unless-format="pdf"} +https://openintro.shinyapps.io/ims-01-data-03 +::: +::: + +::: {.singletutorial data-latex=""} +[Tutorial 1 - Lesson 4: Case study](https://openintro.shinyapps.io/ims-01-data-04/) + +::: {.content-hidden unless-format="pdf"} +https://openintro.shinyapps.io/ims-01-data-04 +::: +::: + +::: {.content-hidden unless-format="pdf"} +You can also access the full list of tutorials supporting this book at\ +. +::: + +::: {.content-visible when-format="html"} +You can also access the full list of tutorials supporting this book [here](https://openintrostat.github.io/ims-tutorials). +::: + +## R labs {#data-labs} + +Further apply the concepts you've learned in this part in R with computational labs that walk you through a data analysis case study. + +::: {.singlelab data-latex=""} +[Intro to R - Birth rates](https://www.openintro.org/go?id=ims-r-lab-intro-to-r) + +::: {.content-hidden unless-format="pdf"} +https://www.openintro.org/go?i +d=ims-r-lab-intro-to-r +::: +::: + +::: {.content-hidden unless-format="pdf"} +You can also access the full list of labs supporting this book at\ +. +::: + +::: {.content-visible when-format="html"} +You can also access the full list of labs supporting this book [here](https://www.openintro.org/go?id=ims-r-labs). +::: diff --git a/03-data-applications.qmd b/03-data-applications.qmd index d216e40c..5cc92a8b 100644 --- a/03-data-applications.qmd +++ b/03-data-applications.qmd @@ -3,94 +3,98 @@ ```{r} #| include: false source("_common.R") +paralympic_1500_raw <- read_csv("~/Desktop/openintro/data-raw/paralympic_1500/paralympic_1500.csv") + +paralympic_1500 <- paralympic_1500_raw |> + mutate(time_min = time) |> + mutate(time = paste(min, sec, sep = ":")) |> + select(-min, -sec) + ``` -## Case study: Passwords {#case-study-passwords} +## Case study: Olympic 1500m {#case-study-passwords} -Stop for a second and think about how many passwords you've used so far today. -You've probably used one to unlock your phone, one to check email, and probably at least one to log on to a social media account. -Made a debit purchase? -You've probably entered a password there too. +While many of you may be glued to the Olympic Games every four years (or every two years if you fancy both summer and winter sports), the Paralympic Games are less popular than the Olympic Games, even if they hold the same competitive thrills. -If you're reading this book, and particularly if you're reading it online, chances are you have had to create a password once or twice in your life. -And if you are diligent about your safety and privacy, you've probably chosen passwords that would be hard for others to guess, or *crack*. +The Paralympic Games began as a way to support soldiers who had been wounded in World War II as a way to help soldiers use sport to rehabilitate. +The first Paralympic Games were held in Rome, Italy in 1960. +Since 1988 (Seoul, South Korea), the Paralymic Games have been held a few weeks later than the Olympic Games in the same city, in both the summer and winter. -In this case study we introduce a dataset on passwords. -The goal of the case study is to walk you through what a data scientist does when they first get a hold of a dataset as well as to provide some "foreshadowing" of concepts and techniques we'll introduce in the next few chapters on exploratory data analysis. +In this case study we introduce a dataset comparing Olympic and Paralympic gold medal finishers in the 1500m running competition (the Olympic "mile", if a bit shorter than a full mile). +The goal of the case study is to walk you through what a data scientist does when they first get a hold of a dataset. +We also provide some "foreshadowing" of concepts and techniques we'll introduce in the next few chapters on exploratory data analysis. +Last, we introduce [Simpson's paradox](https://en.wikipedia.org/wiki/Simpson%27s_paradox) and discuss the importance of understanding the impact of multiple variables in an analysis. ::: {.data data-latex=""} -The [`passwords`](https://github.com/rfordatascience/tidytuesday/blob/master/data/2020/2020-01-14/readme.md) data can be found in the [**tidytuesdayR**](https://thebioengineer.github.io/tidytuesdayR/) R package. +The [`paralympic_1500`](http://openintrostat.github.io/openintro/reference/paralympic_1500.html) data can be found in the [openintro](http://openintrostat.github.io/openintro/) R package. ::: -@tbl-passwords-df-head shows the first ten rows from the dataset, which are the ten most common passwords. -Perhaps unsurprisingly, "password" tops the list, followed by "123456". +@tbl-paralympic-df-tail shows the last ten rows from the dataset, which are the ten most recent 1500m races. +Notice that there are racers from both the Men's and Women's divisions as well as those of varying visual impairment (T11, T12, T13, and Olympic). +The T11 athletes have almost complete visual impairment, run with a black-out blindfold, and are allowed to run with a guide-runner. +T12 and T13 athletes have some visual impairment, and Olympic runners do not have visual impairment. ```{r} -#| label: tbl-passwords-df-head -#| tbl-cap: Top ten rows of the `passwords` dataset. -# https://raw.githubusercontent.com/rfordatascience/tidytuesday/master/data/2020/2020-01-14/passwords.csv -passwords <- readr::read_csv("data/passwords.csv") -passwords <- passwords |> - select(-font_size, -rank_alt) |> - filter(!is.na(category)) |> - mutate(time_unit = fct_relevel(time_unit, "seconds", "minutes", "hours", "days", "weeks", "months", "years")) - -passwords |> - slice_head(n = 10) |> +#| label: tbl-paralympic-df-tail +#| tbl-cap: Last ten rows of the `paralympic_1500` dataset. + +paralympic_1500 |> + slice_tail(n = 10) |> kbl(linesep = "", booktabs = TRUE, row.names = FALSE) |> kable_styling(bootstrap_options = c("striped", "condensed"), latex_options = c("striped", "hold_position")) ``` -When you encounter a new dataset, taking a peek at the first few rows as we did in @tbl-passwords-df-head is almost instinctual. -It can often be helpful to look at the last few rows of the data as well to get a sense of the size of the data as well as potentially discover any characteristics that may not be apparent in the top few rows. -@tbl-passwords-df-tail shows the bottom ten rows of the passwords dataset, which reveals that we are looking at a dataset of 500 passwords. +When you encounter a new dataset, taking a peek at the last few rows as we did in @tbl-paralympic-df-tail is almost instinctual. +It can often be helpful to look at the first few rows of the data as well to get a sense for other aspects of the data which may not be apparent int he last few rows. +@tbl-paralympic-df-head shows the top ten rows of the `paralympic_1500` dataset, which reveals that for at least the first 10 Olympiads, there were no runners in the Women's division or in the Paralympics. ```{r} -#| label: tbl-passwords-df-tail -#| tbl-cap: Bottom ten rows of the `passwords` dataset. -passwords |> - slice_tail(n = 10) |> +#| label: tbl-paralympic-df-head +#| tbl-cap: First ten rows of the `paralympic_1500` dataset. +paralympic_1500 |> + slice_head(n = 10) |> kbl(linesep = "", booktabs = TRUE, row.names = FALSE) |> kable_styling(bootstrap_options = c("striped", "condensed"), latex_options = c("striped", "hold_position")) ``` -At this stage it's also useful to think about how these data were collected, as that will inform the scope of any inference you can make based on your analysis of the data. +At this stage it's also useful to think about how the data were collected, as that will inform the scope of any inference you can make based on your analysis of the data. ::: {.guidedpractice data-latex=""} Do these data come from an observational study or an experiment?[^03-data-applications-1] ::: [^03-data-applications-1]: This is an observational study. - Researchers collected data on existing passwords in use and identified most common ones to put together this dataset. + Researchers collected data on past gold medal race times in both Olympic and Paralympic Games. ::: {.guidedpractice data-latex=""} -There are `r nrow(passwords)` rows and `r ncol(passwords)` columns in the dataset. +There are `r nrow(paralympic_1500)` rows and `r ncol(paralympic_1500)` columns in the dataset. What does each row and each column represent?[^03-data-applications-2] ::: -[^03-data-applications-2]: Each row represents a password and each column represents a variable which contains information on each password. +[^03-data-applications-2]: Each row represents a 1500m gold medal race and each column represents a variable containing information on each race. Once you've identified the rows and columns, it's useful to review the data dictionary to learn about what each column in the dataset represents. -This is provided in @tbl-passwords-var-def. +The column information is provided in @tbl-paralympic-var-def. ```{r} -#| label: tbl-passwords-var-def -#| tbl-cap: Variables and their descriptions for the `passwords` dataset. -passwords_var_def <- tribble( +#| label: tbl-paralympic-var-def +#| tbl-cap: Variables and their descriptions for the `paralympic_1500` dataset. +paralympic_var_def <- tribble( ~variable, ~description, - "rank", "Popularity in the database of released passwords.", - "password", "Actual text of the password.", - "category", "Category password falls into.", - "value", "Time to crack by online guessing.", - "time_unit", "Time unit to match with value.", - "offline_crack_sec", "Time to crack offline in seconds.", - "strength", "Strength of password, relative only to passwords in this dataset. Lower values indicate weaker passwords." + "year", "Year the Games took place.", + "city", "City of the Games.", + "country_of_games", "Country of the Games.", + "division", "Division: `Men` or `Women`.", + "name", "Name of the athlete.", + "country_of_athlete", "Country of athlete.", + "time", "Time of gold medal race, in m:s.", + "time_min", "Time of gold medal race, in decimal minutes (min + sec/60)." ) -passwords_var_def |> +paralympic_var_def |> kbl(linesep = "", booktabs = TRUE, col.names = c("Variable", "Description")) |> kable_styling(bootstrap_options = c("striped", "condensed"), @@ -102,174 +106,198 @@ passwords_var_def |> We now have a better sense of what each column represents, but we do not yet know much about the characteristics of each of the variables. ::: {.workedexample data-latex=""} -Determine whether each variable in the passwords dataset is numerical or categorical. +Determine whether each variable in the `paralympic_1500` dataset is numerical or categorical. For numerical variables, further classify them as continuous or discrete. For categorical variables, determine if the variable is ordinal. ------------------------------------------------------------------------ -The numerical variables in the dataset are `rank` (discrete), `value` (continuous), and `offline_crack_sec` (continuous). -The categorical variables are `password`, `time_unit`. -The strength variable is trickier to classify -- we can think of it as discrete numerical or as an ordinal variable as it takes on numerical values, however it's used to categorize the passwords on an ordinal scale. -One way of approaching this is thinking about whether the values the variable takes vary linearly, e.g., is the difference in strength between passwords with strength levels 8 and 9 the same as the difference with those with strength levels 9 and 10. -If this is not necessarily the case, we would classify the variable as ordinal. -Determining the classification of this variable requires understanding of how `strength` values were determined, which is a very typical workflow for working with data. -Sometimes the data dictionary (presented in @tbl-passwords-var-def) isn't sufficient, and we need to go back to the data source and try to understand the data better before we can proceed with the analysis meaningfully. +The numerical variables in the dataset are `year` (discrete), and `time_min` (continuous). +The categorical variables are `city`, `country_of_games`, `division`, `type`, `name`, and `country_of_athlete`. +The `time` variable is trickier to classify -- we can think of it as numerical, but it is classified as categorical. +The categorical classification is due to the colon `:` which separates the hours from the seconds. +The lubridate R package is fantastic for working with date and time variables. +Sometimes the data dictionary (presented in @tbl-paralympic-var-def) isn't sufficient, and we need to go back to the data source and try to understand the data better before we can proceed with the analysis meaningfully. ::: Next, let's try to get to know each variable a little bit better. For categorical variables, this involves figuring out what their levels are and how commonly represented they are in the data. -@fig-passwords-cat shows the distributions of the categorical variables in this dataset. -We can see that password strengths of 0-10 are more common than higher values. -The most common password category is name (e.g. michael, jennifer, jordan, etc.) and the least common is food (e.g., pepper, cheese, coffee, etc.). -Many passwords can be cracked in the matter of days by online cracking with some taking as little as seconds and some as long as years to break. -Each of these visualizations is a bar plot, which you will learn more about in @sec-explore-categorical. +@fig-paralympic-cat shows the distributions of two of the categorical variables in this dataset. +We can see that the United States has hosted the Games most often, but runners from Great Britain and Kenya have won the 1500m most often. +There are a large number of coutries who have had a single gold medal winner of the 1500m. +Similarly, there are a large number of countries who have hosted the Games only once. +Over the last century, the name describing the country for athletes from one particular region has changed and includes Russian Federaion, Unified Team, and Russian Paralympic Committee. +Both of the visualizations are bar plots, which you will learn more about in @sec-explore-categorical. ```{r} -#| label: fig-passwords-cat -#| fig-cap: Distributions of the categorical variables in the `passwords` dataset. -#| Plot A shows the distribution of password strengths, Plot B password -#| categories, and Plot C length of time it takes to crack the passwords by -#| online guessing. +#| label: fig-paralympic-cat +#| fig-cap: Distributions of categorical variables in the `paralympic_1500` dataset. +#| Plot A shows the distribution of the country of origin of the athlete; Plot B +#| shows the distribution of the country in which the Games gook place. #| fig-asp: 1.0 #| out.width: 100% -p_category <- passwords |> - count(category, sort = TRUE) |> - mutate(category = fct_reorder(category, n)) |> - ggplot(aes(y = category, x = n, fill = fct_rev(category))) + +p_country_games <- paralympic_1500 |> + group_by(country_of_games, year) |> + sample_n(size = 1) |> + ungroup() |> + group_by(country_of_games) |> + count(country_of_games, sort = TRUE) |> + ungroup() |> + mutate(country_of_games = fct_reorder(country_of_games, n)) |> + ggplot(aes(y = country_of_games, x = n, + fill = fct_rev(country_of_games))) + geom_col(show.legend = FALSE) + scale_fill_openintro() + labs( x = "Count", y = NULL, - title = "Categories" + title = "Country of Games" ) + theme(plot.title.position = "plot") -p_time_unit <- passwords |> - count(time_unit) |> - ggplot(aes(y = time_unit, x = n)) + +p_country_athlete <- paralympic_1500 |> + group_by(country_of_athlete, year) |> + sample_n(size = 1) |> + ungroup() |> + group_by(country_of_athlete) |> + count(country_of_athlete, sort = TRUE) |> + ungroup() |> + mutate(country_of_athlete = fct_reorder(country_of_athlete, n)) |> + ggplot(aes(y = country_of_athlete, x = n)) + geom_col(show.legend = FALSE) + - labs( - x = "Count", - y = NULL, - title = "Length of time to crack", - subtitle = "By online guessing" - ) + - theme(plot.title.position = "plot") - -p_strength <- passwords |> - ggplot(aes(y = strength)) + - geom_histogram(binwidth = 1, show.legend = FALSE) + - scale_y_continuous(breaks = seq(0, 50, 5), trans = "reverse") + labs( x = "Count", y = NULL, - title = "Strengths" + title = "Country of athlete" ) + theme(plot.title.position = "plot") -patchwork <- p_strength | (p_category / p_time_unit) +patchwork <- p_country_athlete | p_country_games patchwork + plot_annotation( - title = "Strengths, categories, and cracking time\nof 500 most common passwords", + title = "Olympic and Paralympic Games, Men's division", tag_levels = "A" ) & theme(plot.tag = element_text(size = 12, color = "darkgray")) ``` Similarly, we can examine the distributions of the numerical variables as well. -We already know that rank ranges between 1 and 500 in this dataset, based on @tbl-passwords-df-head and @tbl-passwords-df-tail. -The value variable is slightly more complicated to consider since the numerical values in that column are meaningless without the time unit that accompanies them. -@tbl-passwords-online-crack-summary shows the minimum and maximum amount of time it takes to crack a password by online guessing. -For example, there are 11 passwords in the dataset that can be broken in a matter of seconds, and each of them take 11.11 seconds to break, since the minimum and the maximum of observations in this group are exactly equal to this value. -And there are 65 passwords that take years to break, ranging from 2.56 years to 92.27 years. +We already know that the 1500m times are mostly between 3.5min and 4.5min, based on @tbl-paralympic-df-tail and @tbl-paralympic-df-head. We can break down the 1500m time by division and type of race. +@tbl-paralympic-summary shows the mean, minimum, and maximum 1500m times broken down by division and race type. +Recall that the Men's Olympic division has taken place since 1896, whereas the Men's Paralympic division has happened only since 1960. +The maximum race time, therefore, should be taken into context in terms of the year of the Games. ```{r} -#| label: tbl-passwords-online-crack-summary -#| tbl-cap: Minimum and maximum amount of time it takes to crack a password by -#| online guessing as well as the number of observations that fall into each -#| time unit category. -passwords |> - group_by(time_unit) |> +#| label: tbl-paralympic-summary +#| tbl-cap: Mean, minimum, and maximum of the gold medal times for the 1500m race +#| broken down by division and type of race. +paralympic_1500 |> + group_by(division, type) |> summarise( - n = n(), - min = min(value), - max = max(value) + mean = round(mean(time_min),3), + min = round(min(time_min),3), + max = round(max(time_min),3) ) |> kbl(linesep = "", booktabs = TRUE) |> kable_styling(bootstrap_options = c("striped", "condensed"), latex_options = c("striped", "hold_position")) ``` -Even though passwords that take a large number of years to crack can seem like good options (see @tbl-passwords-long-crack for a list of them), now that you've seen them here (and the fact that they are in a dataset of 500 most common passwords), you should not use them as secure passwords! +\clearpage -```{r} -#| label: tbl-passwords-long-crack -#| tbl-cap: Passwords that take the longest amount of time to crack by online -#| guessing. -passwords |> - filter(value == 92.27) |> - kbl(linesep = "", booktabs = TRUE, row.names = FALSE) |> - kable_styling(bootstrap_options = c("striped", "condensed"), - latex_options = c("striped", "hold_position")) -``` +So far we examined aspects of some of the individual variables, and we have broken down the 1500m race times in terms of division and race type. +You might have already wondered how the race times vary across year. +The `paralymic_1500` dataset will provide us with an ability to explore an important statistical concept, Simpson's paradox. -\clearpage +## Simpson's paradox + +Simpson's paradox \index{Simpson's paradox} is a description of three (or more) variables. +The paradox happens when a third variable reverses the relationship between the first two variables. -The last numerical variable in the dataset is `offline_crack_sec`. -@fig-password-offline-crack-hist shows the distribution of this variable, which reveals that all of these passwords can be cracked offline in under 30 seconds, with a large number of them being crackable in just a few seconds. +Let's start by considering how the 1500m gold medal race times have changed over year. +@fig-paralympic-ungrouped shows a scatterplot describing 1500m race times and year for Men's Olympic and Paralympic (T11) athletes with a line of best fit (to the entire dataset) superimposed. +Notice that the line of best fit shows a **positive** relationship between race time and year. +That is, for later years, the predicted gold medal time is higher than in earlier years. +The line of best fit will be further explored in @sec-model-slr. ```{r} -#| label: fig-password-offline-crack-hist -#| fig-cap: Histogram of the length of time it takes to crack passwords offline. -ggplot(passwords, aes(x = offline_crack_sec)) + - geom_histogram(binwidth = 1) + +#| label: fig-paralympic-ungrouped +#| fig-cap: 1500m race time for Men's Olympic and Paralympic (T11) athletes. The line represents a line of best fit to the entire dataset. +#| fig-asp: 1.2 +#| out.width: 100% + +paralympic_1500 |> + filter(division == "Men", type == "Olympic" | type == "T11") |> + filter(year > 1950) |> + ggplot(aes(x = year, y = time_min)) + + geom_point() + + geom_smooth(method = "lm", se = FALSE) + labs( - x = "Length of time (seconds)", - y = "Count", - title = "Length of time to crack passwords offline" - ) + x = "Year", + y = NULL, + title = "1500m race time, in minutes" + ) + + theme(plot.title.position = "plot") ``` -So far we examined the distributions of each individual variable, but it would be more interesting to explore relationships between multiple variables. -@fig-password-strength-rank-category shows the relationship between rank and strength of passwords by category, where more common passwords (those with higher rank) are plotted higher on the y-axis than those that are less common in this dataset. -The stronger the password, the larger text it's represented with on the plot. -While this visualization reveals some passwords that are less common, and stronger than others, we should reiterate that you should not use any of these passwords. -And if you already do, it's time to go change it! +Of course, both your eye and your intuition are likely telling you that it wouldn't make any sense to try to model all of the athletes together. +Instead, a separate model should be run for each of the two types of Games: Olympic and Paralympic (T11). +@fig-paralympic-grouped shows a scatterplot describing 1500m race times and year for Men's Olympic and Paralympic (T11) athletes with a line of best fit superimposed separately for each of the two types of races. +Notice that within each type of race, the relationship between 1500m race time and year is now **negative**. ```{r} -#| label: fig-password-strength-rank-category -#| fig-cap: Rank vs. strength of 500 most common passwords by category. +#| label: fig-paralympic-grouped +#| fig-cap: 1500m race time for Men's Olympic and Paralympic (T11) athletes. The best fit line is now fit separately to the Olympic and Paralympic athletes. #| fig-asp: 1.2 #| out.width: 100% -passwords |> - mutate(category = fct_relevel(category, "name", "cool-macho", "simple-alphanumeric", "fluffy", "sport", "nerdy-pop", "animal", "password-related", "rebellious-rude", "food")) |> - ggplot(aes(x = strength, y = rank, color = category)) + - geom_text(aes(label = password, size = strength), - check_overlap = TRUE, show.legend = FALSE) + - facet_wrap(vars(category), ncol = 3) + - coord_cartesian(ylim = c(525, -10)) + - scale_y_continuous(breaks = c(1, 100, 200, 300, 400, 500), minor_breaks = NULL, trans = "reverse") + - scale_color_openintro() + + +paralympic_1500 |> + filter(division == "Men", type == "Olympic" | type == "T11") |> + filter(year > 1950) |> + ggplot(aes(x = year, y = time_min, group = type, color = type)) + + geom_point() + + geom_smooth(method = "lm", se = FALSE) + labs( - x = "Strength of password", - y = "Rank of popularity", - title = "500 most common passwords by category", - caption = "Data: Information is beautiful, via TidyTuesday" - ) + x = "Year", + y = NULL, + color = "Race type", + title = "1500m race time, in minutes" + ) + + theme(plot.title.position = "plot") ``` +::: {.important data-latex=""} +**Simpson's paradox.** + +An association or relationship between two variables in one direction (e.g., positive) that reverses (e.g., becomes negative) when broken down by a third variable is known as Simpson's paradox. +::: + +```{r} +#| include: false +terms_chp_3 <- c("Simpson's paradox") +``` + +Simpson's paradox was seen in the 1500m race data because the aggregate data showed a positive relationship (positive slope) between year and race time but a negative relationship (negative slope) between year and race time when broken down by the type of race. + +Simpson's paradox can be seen with categorical or numeric data. +Oftentimes the paradox happens because the third variable (here, race type) is imbalanced. +There are either more observations in one group or the observations happen at different intervals across the two groups. +In the 1500m data, we saw that the T11 runners had fewer observations and their times were both generally slower and more recent than the Olympic runners. + +In the 1500m analysis, it would be most prudent to report the trends separately for the Olympic and the T11 athletes. +However, in other situations, it might be better to aggregate the data and report the overall trend. +Many additional examples of Simpson's paradox and a further exploration is given in @Witmer:2021. + In this case study, we introduced you to the very first steps a data scientist takes when they start working with a new dataset. -In the next few chapters, we will introduce exploratory data analysis and you'll learn more about the various types of data visualizations and summary statistics you can make to get to know your data better. +In the next few chapters, we will introduce exploratory data analysis, and you'll learn more about the various types of data visualizations and summary statistics you can make to get to know your data better. -Before you move on, we encourage you to think about whether the following questions can be answered with this dataset, and if yes, how you might go about answering them. +Before you move on, we encourage you to think about whether the following questions can be answered with this dataset, and if yes, how you might go about answering them? It's okay if your answer is "I'm not sure", we simply want to get your exploratory juices flowing to prime you for what's to come! -1. What characteristics are associated with a strong vs. a weak password? -2. Do more popular passwords take shorter or longer to crack compared to less popular passwords? -3. Are passwords that start with letters or numbers more common among the list of top 500 most common passwords? +1. Has there every been a year when a visually impaired paralympic gold medal athlete beat the olympic gold medal athlete? +2. When comparing the paralympic and olympic 1500m gold medal athletes, does Simpson's paradox hold in the Women's division? +3. Is there a biological boundary which establishes a time under which no human could run 1500m? \clearpage diff --git a/_freeze/03-data-applications/execute-results/html.json b/_freeze/03-data-applications/execute-results/html.json index 591e16ef..51391c37 100644 --- a/_freeze/03-data-applications/execute-results/html.json +++ b/_freeze/03-data-applications/execute-results/html.json @@ -1,7 +1,7 @@ { - "hash": "1011f7566a5c43b3419d3a2e8bfe6adf", + "hash": "055aff82c14a6ba9c74472da5be44373", "result": { - "markdown": "# Applications: Data {#sec-data-applications}\n\n\n\n\n\n## Case study: Passwords {#case-study-passwords}\n\nStop for a second and think about how many passwords you've used so far today.\nYou've probably used one to unlock your phone, one to check email, and probably at least one to log on to a social media account.\nMade a debit purchase?\nYou've probably entered a password there too.\n\nIf you're reading this book, and particularly if you're reading it online, chances are you have had to create a password once or twice in your life.\nAnd if you are diligent about your safety and privacy, you've probably chosen passwords that would be hard for others to guess, or *crack*.\n\nIn this case study we introduce a dataset on passwords.\nThe goal of the case study is to walk you through what a data scientist does when they first get a hold of a dataset as well as to provide some \"foreshadowing\" of concepts and techniques we'll introduce in the next few chapters on exploratory data analysis.\n\n::: {.data data-latex=\"\"}\nThe [`passwords`](https://github.com/rfordatascience/tidytuesday/blob/master/data/2020/2020-01-14/readme.md) data can be found in the [**tidytuesdayR**](https://thebioengineer.github.io/tidytuesdayR/) R package.\n:::\n\n@tbl-passwords-df-head shows the first ten rows from the dataset, which are the ten most common passwords.\nPerhaps unsurprisingly, \"password\" tops the list, followed by \"123456\".\n\n\n::: {#tbl-passwords-df-head .cell tbl-cap='Top ten rows of the `passwords` dataset.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
rank password category value time_unit offline_crack_sec strength
1 password password-related 6.91 years 2.170 8
2 123456 simple-alphanumeric 18.52 minutes 0.000 4
3 12345678 simple-alphanumeric 1.29 days 0.001 4
4 1234 simple-alphanumeric 11.11 seconds 0.000 4
5 qwerty simple-alphanumeric 3.72 days 0.003 8
6 12345 simple-alphanumeric 1.85 minutes 0.000 4
7 dragon animal 3.72 days 0.003 8
8 baseball sport 6.91 years 2.170 4
9 football sport 6.91 years 2.170 7
10 letmein password-related 3.19 months 0.084 8
\n\n`````\n:::\n:::\n\n\nWhen you encounter a new dataset, taking a peek at the first few rows as we did in @tbl-passwords-df-head is almost instinctual.\nIt can often be helpful to look at the last few rows of the data as well to get a sense of the size of the data as well as potentially discover any characteristics that may not be apparent in the top few rows.\n@tbl-passwords-df-tail shows the bottom ten rows of the passwords dataset, which reveals that we are looking at a dataset of 500 passwords.\n\n\n::: {#tbl-passwords-df-tail .cell tbl-cap='Bottom ten rows of the `passwords` dataset.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
rank password category value time_unit offline_crack_sec strength
491 natasha name 3.19 months 0.084 7
492 sniper cool-macho 3.72 days 0.003 8
493 chance name 3.72 days 0.003 7
494 genesis nerdy-pop 3.19 months 0.084 7
495 hotrod cool-macho 3.72 days 0.003 7
496 reddog cool-macho 3.72 days 0.003 6
497 alexande name 6.91 years 2.170 9
498 college nerdy-pop 3.19 months 0.084 7
499 jester name 3.72 days 0.003 7
500 passw0rd password-related 92.27 years 29.020 28
\n\n`````\n:::\n:::\n\n\nAt this stage it's also useful to think about how these data were collected, as that will inform the scope of any inference you can make based on your analysis of the data.\n\n::: {.guidedpractice data-latex=\"\"}\nDo these data come from an observational study or an experiment?[^03-data-applications-1]\n:::\n\n[^03-data-applications-1]: This is an observational study.\n Researchers collected data on existing passwords in use and identified most common ones to put together this dataset.\n\n::: {.guidedpractice data-latex=\"\"}\nThere are 500 rows and 7 columns in the dataset.\nWhat does each row and each column represent?[^03-data-applications-2]\n:::\n\n[^03-data-applications-2]: Each row represents a password and each column represents a variable which contains information on each password.\n\nOnce you've identified the rows and columns, it's useful to review the data dictionary to learn about what each column in the dataset represents.\nThis is provided in @tbl-passwords-var-def.\n\n\n::: {#tbl-passwords-var-def .cell tbl-cap='Variables and their descriptions for the `passwords` dataset.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
Variable Description
rank Popularity in the database of released passwords.
password Actual text of the password.
category Category password falls into.
value Time to crack by online guessing.
time_unit Time unit to match with value.
offline_crack_sec Time to crack offline in seconds.
strength Strength of password, relative only to passwords in this dataset. Lower values indicate weaker passwords.
\n\n`````\n:::\n:::\n\n\nWe now have a better sense of what each column represents, but we do not yet know much about the characteristics of each of the variables.\n\n::: {.workedexample data-latex=\"\"}\nDetermine whether each variable in the passwords dataset is numerical or categorical.\nFor numerical variables, further classify them as continuous or discrete.\nFor categorical variables, determine if the variable is ordinal.\n\n------------------------------------------------------------------------\n\nThe numerical variables in the dataset are `rank` (discrete), `value` (continuous), and `offline_crack_sec` (continuous).\nThe categorical variables are `password`, `time_unit`.\nThe strength variable is trickier to classify -- we can think of it as discrete numerical or as an ordinal variable as it takes on numerical values, however it's used to categorize the passwords on an ordinal scale.\nOne way of approaching this is thinking about whether the values the variable takes vary linearly, e.g., is the difference in strength between passwords with strength levels 8 and 9 the same as the difference with those with strength levels 9 and 10.\nIf this is not necessarily the case, we would classify the variable as ordinal.\nDetermining the classification of this variable requires understanding of how `strength` values were determined, which is a very typical workflow for working with data.\nSometimes the data dictionary (presented in @tbl-passwords-var-def) isn't sufficient, and we need to go back to the data source and try to understand the data better before we can proceed with the analysis meaningfully.\n:::\n\nNext, let's try to get to know each variable a little bit better.\nFor categorical variables, this involves figuring out what their levels are and how commonly represented they are in the data.\n@fig-passwords-cat shows the distributions of the categorical variables in this dataset.\nWe can see that password strengths of 0-10 are more common than higher values.\nThe most common password category is name (e.g. michael, jennifer, jordan, etc.) and the least common is food (e.g., pepper, cheese, coffee, etc.).\nMany passwords can be cracked in the matter of days by online cracking with some taking as little as seconds and some as long as years to break.\nEach of these visualizations is a bar plot, which you will learn more about in @sec-explore-categorical.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Distributions of the categorical variables in the `passwords` dataset. Plot A shows the distribution of password strengths, Plot B password categories, and Plot C length of time it takes to crack the passwords by online guessing.](03-data-applications_files/figure-html/fig-passwords-cat-1.png){#fig-passwords-cat width=100%}\n:::\n:::\n\n\nSimilarly, we can examine the distributions of the numerical variables as well.\nWe already know that rank ranges between 1 and 500 in this dataset, based on @tbl-passwords-df-head and @tbl-passwords-df-tail.\nThe value variable is slightly more complicated to consider since the numerical values in that column are meaningless without the time unit that accompanies them.\n@tbl-passwords-online-crack-summary shows the minimum and maximum amount of time it takes to crack a password by online guessing.\nFor example, there are 11 passwords in the dataset that can be broken in a matter of seconds, and each of them take 11.11 seconds to break, since the minimum and the maximum of observations in this group are exactly equal to this value.\nAnd there are 65 passwords that take years to break, ranging from 2.56 years to 92.27 years.\n\n\n::: {#tbl-passwords-online-crack-summary .cell tbl-cap='Minimum and maximum amount of time it takes to crack a password by online guessing as well as the number of observations that fall into each time unit category.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
time_unit n min max
seconds 11 11.11 11.11
minutes 51 1.85 18.52
hours 43 3.09 17.28
days 238 1.29 3.72
weeks 5 1.84 3.70
months 87 3.19 3.19
years 65 2.56 92.27
\n\n`````\n:::\n:::\n\n\nEven though passwords that take a large number of years to crack can seem like good options (see @tbl-passwords-long-crack for a list of them), now that you've seen them here (and the fact that they are in a dataset of 500 most common passwords), you should not use them as secure passwords!\n\n\n::: {#tbl-passwords-long-crack .cell tbl-cap='Passwords that take the longest amount of time to crack by online guessing.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
rank password category value time_unit offline_crack_sec strength
26 trustno1 simple-alphanumeric 92.3 years 29.0 25
336 rush2112 nerdy-pop 92.3 years 29.0 48
406 jordan23 sport 92.3 years 29.3 34
500 passw0rd password-related 92.3 years 29.0 28
\n\n`````\n:::\n:::\n\n\n\\clearpage\n\nThe last numerical variable in the dataset is `offline_crack_sec`.\n@fig-password-offline-crack-hist shows the distribution of this variable, which reveals that all of these passwords can be cracked offline in under 30 seconds, with a large number of them being crackable in just a few seconds.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Histogram of the length of time it takes to crack passwords offline.](03-data-applications_files/figure-html/fig-password-offline-crack-hist-1.png){#fig-password-offline-crack-hist width=90%}\n:::\n:::\n\n\nSo far we examined the distributions of each individual variable, but it would be more interesting to explore relationships between multiple variables.\n@fig-password-strength-rank-category shows the relationship between rank and strength of passwords by category, where more common passwords (those with higher rank) are plotted higher on the y-axis than those that are less common in this dataset.\nThe stronger the password, the larger text it's represented with on the plot.\nWhile this visualization reveals some passwords that are less common, and stronger than others, we should reiterate that you should not use any of these passwords.\nAnd if you already do, it's time to go change it!\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Rank vs. strength of 500 most common passwords by category.](03-data-applications_files/figure-html/fig-password-strength-rank-category-1.png){#fig-password-strength-rank-category width=100%}\n:::\n:::\n\n\nIn this case study, we introduced you to the very first steps a data scientist takes when they start working with a new dataset.\nIn the next few chapters, we will introduce exploratory data analysis and you'll learn more about the various types of data visualizations and summary statistics you can make to get to know your data better.\n\nBefore you move on, we encourage you to think about whether the following questions can be answered with this dataset, and if yes, how you might go about answering them.\nIt's okay if your answer is \"I'm not sure\", we simply want to get your exploratory juices flowing to prime you for what's to come!\n\n1. What characteristics are associated with a strong vs. a weak password?\n2. Do more popular passwords take shorter or longer to crack compared to less popular passwords?\n3. Are passwords that start with letters or numbers more common among the list of top 500 most common passwords?\n\n\\clearpage\n\n## Interactive R tutorials {#data-tutorials}\n\nNavigate the concepts you've learned in this chapter in R using the following self-paced tutorials.\nAll you need is your browser to get started!\n\n::: {.alltutorials data-latex=\"\"}\n[Tutorial 1: Introduction to data](https://openintrostat.github.io/ims-tutorials/01-data/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintrostat.github.io/ims-tutorials/01-data\n:::\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 1 - Lesson 1: Language of data](https://openintro.shinyapps.io/ims-01-data-01/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-01-data-01\n:::\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 1 - Lesson 2: Types of studies](https://openintro.shinyapps.io/ims-01-data-02/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-01-data-02\n:::\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 1 - Lesson 3: Sampling strategies and experimental design](https://openintro.shinyapps.io/ims-01-data-03/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-01-data-03\n:::\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 1 - Lesson 4: Case study](https://openintro.shinyapps.io/ims-01-data-04/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-01-data-04\n:::\n:::\n\n::: {.content-hidden unless-format=\"pdf\"}\nYou can also access the full list of tutorials supporting this book at\\\n.\n:::\n\n::: {.content-visible when-format=\"html\"}\nYou can also access the full list of tutorials supporting this book [here](https://openintrostat.github.io/ims-tutorials).\n:::\n\n## R labs {#data-labs}\n\nFurther apply the concepts you've learned in this part in R with computational labs that walk you through a data analysis case study.\n\n::: {.singlelab data-latex=\"\"}\n[Intro to R - Birth rates](https://www.openintro.org/go?id=ims-r-lab-intro-to-r)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://www.openintro.org/go?i\nd=ims-r-lab-intro-to-r\n:::\n:::\n\n::: {.content-hidden unless-format=\"pdf\"}\nYou can also access the full list of labs supporting this book at\\\n.\n:::\n\n::: {.content-visible when-format=\"html\"}\nYou can also access the full list of labs supporting this book [here](https://www.openintro.org/go?id=ims-r-labs).\n:::\n", + "markdown": "# Applications: Data {#sec-data-applications}\n\n\n\n\n\n## Case study: Olympic 1500m {#case-study-passwords}\n\nWhile many of you may be glued to the Olympic Games every four years (or every two years if you fancy both summer and winter sports), the Paralympic Games are less popular than the Olympic Games, even if they hold the same competitive thrills.\n\nThe Paralympic Games began as a way to support soldiers who had been wounded in World War II as a way to help soldiers use sport to rehabilitate.\nThe first Paralympic Games were held in Rome, Italy in 1960.\nSince 1988 (Seoul, South Korea), the Paralymic Games have been held a few weeks later than the Olympic Games in the same city, in both the summer and winter.\n\nIn this case study we introduce a dataset comparing Olympic and Paralympic gold medal finishers in the 1500m running competition (the Olympic \"mile\", if a bit shorter than a full mile).\nThe goal of the case study is to walk you through what a data scientist does when they first get a hold of a dataset.\nWe also provide some \"foreshadowing\" of concepts and techniques we'll introduce in the next few chapters on exploratory data analysis.\nLast, we introduce [Simpson's paradox](https://en.wikipedia.org/wiki/Simpson%27s_paradox) and discuss the importance of understanding the impact of multiple variables in an analysis.\n\n::: {.data data-latex=\"\"}\nThe [`paralympic_1500`](http://openintrostat.github.io/openintro/reference/paralympic_1500.html) data can be found in the [openintro](http://openintrostat.github.io/openintro/) R package.\n:::\n\n@tbl-paralympic-df-tail shows the last ten rows from the dataset, which are the ten most recent 1500m races.\nNotice that there are racers from both the Men's and Women's divisions as well as those of varying visual impairment (T11, T12, T13, and Olympic).\nThe T11 athletes have almost complete visual impairment, run with a black-out blindfold, and are allowed to run with a guide-runner.\nT12 and T13 athletes have some visual impairment, and Olympic runners do not have visual impairment.\n\n\n::: {#tbl-paralympic-df-tail .cell tbl-cap='Last ten rows of the `paralympic_1500` dataset.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
year city country_of_games division type name country_of_athlete time time_min
2016 Rio de Janeiro Brazil Men T13 Abdellatif Baka Algeria 3:48.29 3.81
2016 Rio de Janeiro Brazil Women Olympic Faith Chepngetich Kipyegon Kenya 4:8.92 4.15
2016 Rio de Janeiro Brazil Women T11 Jin Zheng China 4:38.92 4.65
2016 Rio de Janeiro Brazil Women T13 Somaya Bousaid Tunisia 4:21.45 4.36
2020 Tokyo Japan Men Olympic Jakob Ingebrigtsen Norway 3:28.32 3.47
2020 Tokyo Japan Men T11 Yeltsin Jacques Brazil 3:57.6 3.96
2020 Tokyo Japan Men T13 Anton Kuliatin Russian Paralympic Committee 3:54.04 3.90
2020 Tokyo Japan Women Olympic Faith Chepngetich Kipyegon Kenya 3:53.11 3.88
2020 Tokyo Japan Women T11 Monica Olivia Rodriguez Saavedra Mexico 4:37.4 4.62
2020 Tokyo Japan Women T13 Tigist Gezahagn Menigstu Ethiopia 4:23.24 4.39
\n\n`````\n:::\n:::\n\n\nWhen you encounter a new dataset, taking a peek at the last few rows as we did in @tbl-paralympic-df-tail is almost instinctual.\nIt can often be helpful to look at the first few rows of the data as well to get a sense for other aspects of the data which may not be apparent int he last few rows.\n@tbl-paralympic-df-head shows the top ten rows of the `paralympic_1500` dataset, which reveals that for at least the first 10 Olympiads, there were no runners in the Women's division or in the Paralympics.\n\n\n::: {#tbl-paralympic-df-head .cell tbl-cap='First ten rows of the `paralympic_1500` dataset.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
year city country_of_games division type name country_of_athlete time time_min
1896 Athens Greece Men Olympic Edwin Flack Australia 4:33.2 4.55
1900 Paris France Men Olympic Charles Bennett Great Britain 4:6.2 4.10
1904 St Louis USA Men Olympic Jim Lightbody USA 4:5.4 4.09
1908 London United Kingdom Men Olympic Mel Sheppard USA 4:3.4 4.06
1912 Stockholm Sweden Men Olympic Arnold Jackson Great Britain 3:56.8 3.95
1920 Antwerp Belgium Men Olympic Albert Hill Great Britain 4:1.8 4.03
1924 Paris France Men Olympic Paavo Nurmi Finland 3:53.6 3.89
1928 Amsterdam Netherlands Men Olympic Harri Larva Finland 3:53.2 3.89
1932 Los Angeles USA Men Olympic Luigi Beccali Italy 3:51.2 3.85
1936 Berlin Germany Men Olympic Jack Lovelock New Zealand 3:47.8 3.80
\n\n`````\n:::\n:::\n\n\nAt this stage it's also useful to think about how the data were collected, as that will inform the scope of any inference you can make based on your analysis of the data.\n\n::: {.guidedpractice data-latex=\"\"}\nDo these data come from an observational study or an experiment?[^03-data-applications-1]\n:::\n\n[^03-data-applications-1]: This is an observational study.\n Researchers collected data on past gold medal race times in both Olympic and Paralympic Games.\n\n::: {.guidedpractice data-latex=\"\"}\nThere are 82 rows and 9 columns in the dataset.\nWhat does each row and each column represent?[^03-data-applications-2]\n:::\n\n[^03-data-applications-2]: Each row represents a 1500m gold medal race and each column represents a variable containing information on each race.\n\nOnce you've identified the rows and columns, it's useful to review the data dictionary to learn about what each column in the dataset represents.\nThe column information is provided in @tbl-paralympic-var-def.\n\n\n::: {#tbl-paralympic-var-def .cell tbl-cap='Variables and their descriptions for the `paralympic_1500` dataset.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
Variable Description
year Year the Games took place.
city City of the Games.
country_of_games Country of the Games.
division Division: `Men` or `Women`.
name Name of the athlete.
country_of_athlete Country of athlete.
time Time of gold medal race, in m:s.
time_min Time of gold medal race, in decimal minutes (min + sec/60).
\n\n`````\n:::\n:::\n\n\nWe now have a better sense of what each column represents, but we do not yet know much about the characteristics of each of the variables.\n\n::: {.workedexample data-latex=\"\"}\nDetermine whether each variable in the `paralympic_1500` dataset is numerical or categorical.\nFor numerical variables, further classify them as continuous or discrete.\nFor categorical variables, determine if the variable is ordinal.\n\n------------------------------------------------------------------------\n\nThe numerical variables in the dataset are `year` (discrete), and `time_min` (continuous).\nThe categorical variables are `city`, `country_of_games`, `division`, `type`, `name`, and `country_of_athlete`.\nThe `time` variable is trickier to classify -- we can think of it as numerical, but it is classified as categorical.\nThe categorical classification is due to the colon `:` which separates the hours from the seconds.\nThe lubridate R package is fantastic for working with date and time variables.\nSometimes the data dictionary (presented in @tbl-paralympic-var-def) isn't sufficient, and we need to go back to the data source and try to understand the data better before we can proceed with the analysis meaningfully.\n:::\n\nNext, let's try to get to know each variable a little bit better.\nFor categorical variables, this involves figuring out what their levels are and how commonly represented they are in the data.\n@fig-paralympic-cat shows the distributions of two of the categorical variables in this dataset.\nWe can see that the United States has hosted the Games most often, but runners from Great Britain and Kenya have won the 1500m most often.\nThere are a large number of coutries who have had a single gold medal winner of the 1500m.\nSimilarly, there are a large number of countries who have hosted the Games only once.\nOver the last century, the name describing the country for athletes from one particular region has changed and includes Russian Federaion, Unified Team, and Russian Paralympic Committee.\nBoth of the visualizations are bar plots, which you will learn more about in @sec-explore-categorical.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Distributions of categorical variables in the `paralympic_1500` dataset. Plot A shows the distribution of the country of origin of the athlete; Plot B shows the distribution of the country in which the Games gook place.](03-data-applications_files/figure-html/fig-paralympic-cat-1.png){#fig-paralympic-cat width=100%}\n:::\n:::\n\n\nSimilarly, we can examine the distributions of the numerical variables as well.\nWe already know that the 1500m times are mostly between 3.5min and 4.5min, based on @tbl-paralympic-df-tail and @tbl-paralympic-df-head. We can break down the 1500m time by division and type of race.\n@tbl-paralympic-summary shows the mean, minimum, and maximum 1500m times broken down by division and race type.\nRecall that the Men's Olympic division has taken place since 1896, whereas the Men's Paralympic division has happened only since 1960.\nThe maximum race time, therefore, should be taken into context in terms of the year of the Games.\n\n\n::: {#tbl-paralympic-summary .cell tbl-cap='Mean, minimum, and maximum of the gold medal times for the 1500m race broken down by division and type of race.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
division type mean min max
Men Olympic 3.76 3.47 4.55
Men T11 4.14 3.96 4.31
Men T12 4.11 3.94 4.25
Men T13 3.98 3.81 4.24
Women Olympic 4.02 3.88 4.18
Women T11 5.05 4.62 5.63
Women T12 4.88 4.61 5.57
Women T13 4.55 4.23 5.24
\n\n`````\n:::\n:::\n\n\n\\clearpage\n\nSo far we examined aspects of some of the individual variables, and we have broken down the 1500m race times in terms of division and race type.\nYou might have already wondered how the race times vary across year.\nThe `paralymic_1500` dataset will provide us with an ability to explore an important statistical concept, Simpson's paradox.\n\n## Simpson's paradox\n\nSimpson's paradox \\index{Simpson's paradox} is a description of three (or more) variables.\nThe paradox happens when a third variable reverses the relationship between the first two variables.\n\nLet's start by considering how the 1500m gold medal race times have changed over year.\n@fig-paralympic-ungrouped shows a scatterplot describing 1500m race times and year for Men's Olympic and Paralympic (T11) athletes with a line of best fit (to the entire dataset) superimposed.\nNotice that the line of best fit shows a **positive** relationship between race time and year.\nThat is, for later years, the predicted gold medal time is higher than in earlier years.\nThe line of best fit will be further explored in @sec-model-slr.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![1500m race time for Men's Olympic and Paralympic (T11) athletes. The line represents a line of best fit to the entire dataset.](03-data-applications_files/figure-html/fig-paralympic-ungrouped-1.png){#fig-paralympic-ungrouped width=100%}\n:::\n:::\n\n\nOf course, both your eye and your intuition are likely telling you that it wouldn't make any sense to try to model all of the athletes together.\nInstead, a separate model should be run for each of the two types of Games: Olympic and Paralympic (T11).\n@fig-paralympic-grouped shows a scatterplot describing 1500m race times and year for Men's Olympic and Paralympic (T11) athletes with a line of best fit superimposed separately for each of the two types of races.\nNotice that within each type of race, the relationship between 1500m race time and year is now **negative**.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![1500m race time for Men's Olympic and Paralympic (T11) athletes. The best fit line is now fit separately to the Olympic and Paralympic athletes.](03-data-applications_files/figure-html/fig-paralympic-grouped-1.png){#fig-paralympic-grouped width=100%}\n:::\n:::\n\n\n::: {.important data-latex=\"\"}\n**Simpson's paradox.**\n\nAn association or relationship between two variables in one direction (e.g., positive) that reverses (e.g., becomes negative) when broken down by a third variable is known as Simpson's paradox.\n:::\n\n\n\n\n\nSimpson's paradox was seen in the 1500m race data because the aggregate data showed a positive relationship (positive slope) between year and race time but a negative relationship (negative slope) between year and race time when broken down by the type of race.\n\nSimpson's paradox can be seen with categorical or numeric data.\nOftentimes the paradox happens because the third variable (here, race type) is imbalanced.\nThere are either more observations in one group or the observations happen at different intervals across the two groups.\nIn the 1500m data, we saw that the T11 runners had fewer observations and their times were both generally slower and more recent than the Olympic runners.\n\nIn the 1500m analysis, it would be most prudent to report the trends separately for the Olympic and the T11 athletes.\nHowever, in other situations, it might be better to aggregate the data and report the overall trend.\nMany additional examples of Simpson's paradox and a further exploration is given in @Witmer:2021.\n\nIn this case study, we introduced you to the very first steps a data scientist takes when they start working with a new dataset.\nIn the next few chapters, we will introduce exploratory data analysis, and you'll learn more about the various types of data visualizations and summary statistics you can make to get to know your data better.\n\nBefore you move on, we encourage you to think about whether the following questions can be answered with this dataset, and if yes, how you might go about answering them?\nIt's okay if your answer is \"I'm not sure\", we simply want to get your exploratory juices flowing to prime you for what's to come!\n\n1. Has there every been a year when a visually impaired paralympic gold medal athlete beat the olympic gold medal athlete?\n2. When comparing the paralympic and olympic 1500m gold medal athletes, does Simpson's paradox hold in the Women's division?\n3. Is there a biological boundary which establishes a time under which no human could run 1500m?\n\n\\clearpage\n\n## Interactive R tutorials {#data-tutorials}\n\nNavigate the concepts you've learned in this chapter in R using the following self-paced tutorials.\nAll you need is your browser to get started!\n\n::: {.alltutorials data-latex=\"\"}\n[Tutorial 1: Introduction to data](https://openintrostat.github.io/ims-tutorials/01-data/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintrostat.github.io/ims-tutorials/01-data\n:::\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 1 - Lesson 1: Language of data](https://openintro.shinyapps.io/ims-01-data-01/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-01-data-01\n:::\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 1 - Lesson 2: Types of studies](https://openintro.shinyapps.io/ims-01-data-02/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-01-data-02\n:::\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 1 - Lesson 3: Sampling strategies and experimental design](https://openintro.shinyapps.io/ims-01-data-03/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-01-data-03\n:::\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 1 - Lesson 4: Case study](https://openintro.shinyapps.io/ims-01-data-04/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-01-data-04\n:::\n:::\n\n::: {.content-hidden unless-format=\"pdf\"}\nYou can also access the full list of tutorials supporting this book at\\\n.\n:::\n\n::: {.content-visible when-format=\"html\"}\nYou can also access the full list of tutorials supporting this book [here](https://openintrostat.github.io/ims-tutorials).\n:::\n\n## R labs {#data-labs}\n\nFurther apply the concepts you've learned in this part in R with computational labs that walk you through a data analysis case study.\n\n::: {.singlelab data-latex=\"\"}\n[Intro to R - Birth rates](https://www.openintro.org/go?id=ims-r-lab-intro-to-r)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://www.openintro.org/go?i\nd=ims-r-lab-intro-to-r\n:::\n:::\n\n::: {.content-hidden unless-format=\"pdf\"}\nYou can also access the full list of labs supporting this book at\\\n.\n:::\n\n::: {.content-visible when-format=\"html\"}\nYou can also access the full list of labs supporting this book [here](https://www.openintro.org/go?id=ims-r-labs).\n:::\n", "supporting": [ "03-data-applications_files" ], diff --git a/_freeze/03-data-applications/figure-html/fig-paralympic-cat-1.png b/_freeze/03-data-applications/figure-html/fig-paralympic-cat-1.png new file mode 100644 index 0000000000000000000000000000000000000000..f4f5c356977c242f839a87d6c4376e7f985e99df GIT binary patch literal 440038 zcmeEu1zVNt)-@^^m>{8~(jlmbfPkPF@6=Uq`;AYR#HdHIGl5s|hu;oof(K@t~j6mk-Vtt 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zL7BqR+EM3B+#q%0i938}E`Hxo3})L#z(Jz3TFbS8GzaUUbc_PLI)L&s>B7F$$NqWL zLu4VkVCtvjfo~Byk=fsRe#8lrLPIIJZ=p{y871P#B@ZOHV#NyJ>X8b+it6fcaQjHI z;pwO2G%+jAk4&;6jcgWW|2z<-l-&Wj0cSkWlKv5oQuQ*SK_L%M5{d&s(#F9{UnI)K z2OohwK?9f(%D;FGRm=TgOJ4Y~m%!XG7bL7JPQ=wWN@a5#z*EjF0_!l&DSOV3|N1P5Pu1|=Jeuz>x! z@mW5R=u=JhCXpr)_izAol$wQ`yvgQA9#}Jr{Pl3mU??fq{mxl)=4a)X-tFI^%gKhy{ zZ$<7;${63e8bA~(!^*Mzy4W?is=kJY^$8j==-3)S0|iM<;SboA#vm4tEPD6+4wC<- zxC^ZxkFcRqLDcWhU0OZ=3e@iuZ5Q^2`klGzyhEmb!%?;18K~dIiv7p$PDmg`t-Pj)a@a1C=bf3%BAi How well or badly do you think the government are doing at handling Britain's exit from the European Union?\n>\n> - Very well\n> - Fairly well\n> - Fairly badly\n> - Very badly\n> - Don't know\n\n\n::: {.cell}\n\n:::\n\n::: {.cell}\n::: {.cell-output-display}\n![Bar plot three different ways. Plot A: Alphabetic ordering of levels, Plot B: Bars ordered in descending order of frequency, Plot C: Bars ordered in the same order as they were presented in the survey question.](06-explore-applications_files/figure-html/fig-brexit-bars-1.png){#fig-brexit-bars width=100%}\n:::\n:::\n\n\n### Make the labels as easy to read as possible\n\nThe Brexit survey results were additionally broken down by region in Great Britain.\nThe stacked bar plot allows for comparison of Brexit opinion across the five regions.\nIn @fig-brexit-region the bars are vertical in Plot A and horizontal in Plot B. While the quantitative information in the two graphics is identical, flipping the graph and creating horizontal bars provides more space for the **axis labels**.\nThe easier the categories are to read, the more the reader will learn from the visualization.\nRemember, the goal is to convey as much information as possible in a succinct and clear manner.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Stacked bar plots vertically and horizontally. The horizontal orientation makes the region labels easier to read.](06-explore-applications_files/figure-html/fig-brexit-region-1.png){#fig-brexit-region width=100%}\n:::\n:::\n\n\n\n\n### Pick a purpose\n\nEvery graphical decision should be made with a **purpose**.\nAs previously mentioned, sticking with default options is not always best for conveying the narrative of your data story.\nStacked bar plots tell one part of a story.\nDepending on your research question, they may not tell the part of the story most important to the research.\n@fig-seg-three-ways provides three different ways of representing the same information.\nIf the most important comparison across regions is proportion, you might prefer Plot A. If the most important comparison across regions also considers the total number of individuals in the region, you might prefer Plot B. If a separate bar plot for each region makes the point you'd like, use Plot C, which has been **faceted** by region.\n\n\n\n\n\nPlot C in @fig-seg-three-ways also provides full titles and a succinct URL with the data source.\nOther deliberate decisions to consider include using informative labels and avoiding redundancy.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Three different representations of the two variables including survey opinion and region. Use the graphic that best conveys the data narrative at hand.](06-explore-applications_files/figure-html/fig-seg-three-ways-1.png){#fig-seg-three-ways width=90%}\n:::\n:::\n\n\n\n\n### Select meaningful colors\n\n\n\nOne last consideration for building graphs is to consider color choices.\nDefault or rainbow colors are not always the choice which will best distinguish the level of your variables.\nMuch research has been done to find color combinations which are distinct and which are clear for differently sighted individuals.\nThe cividis scale works well with ordinal data.\n[@Nunez:2018] @fig-brexit-viridis shows the same plot with two different color themes.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Identical bar plots with two different coloring options. Plot A uses a default color scale, Plot B uses colors from the cividis scale.](06-explore-applications_files/figure-html/fig-brexit-viridis-1.png){#fig-brexit-viridis width=90%}\n:::\n:::\n\n\nIn this chapter different representations are contrasted to demonstrate best practices in creating graphs.\nThe fundamental principle is that your graph should provide maximal information succinctly and clearly.\nLabels should be clear and oriented horizontally for the reader.\nDon't forget titles and, if possible, include the source of the data.\n\n\\clearpage\n\n## Interactive R tutorials {#explore-tutorials}\n\nNavigate the concepts you've learned in this chapter in R using the following self-paced tutorials.\nAll you need is your browser to get started!\n\n::: {.alltutorials data-latex=\"\"}\n[Tutorial 2: Exploratory data analysis](https://openintrostat.github.io/ims-tutorials/02-explore/)\\\n::: {.content-hidden unless-format=\"pdf\"} https://openintrostat.github.io/ims-tutorials/02-explore\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 2 - Lesson 1: Visualizing categorical data](https://openintro.shinyapps.io/ims-02-explore-01/)\\\n::: {.content-hidden unless-format=\"pdf\"} https://openintro.shinyapps.io/ims-02-explore-01\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 2 - Lesson 2: Visualizing numerical data](https://openintro.shinyapps.io/ims-02-explore-02/)\\\n::: {.content-hidden unless-format=\"pdf\"} https://openintro.shinyapps.io/ims-02-explore-02\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 2 - Lesson 3: Summarizing with statistics](https://openintro.shinyapps.io/ims-02-explore-03/)\\\n::: {.content-hidden unless-format=\"pdf\"} https://openintro.shinyapps.io/ims-02-explore-03\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 2 - Lesson 4: Case study](https://openintro.shinyapps.io/ims-02-explore-04/)\\\n::: {.content-hidden unless-format=\"pdf\"} https://openintro.shinyapps.io/ims-02-explore-04\n:::\n\n:::\n\n::: {.content-hidden unless-format=\"pdf\"}\nYou can also access the full list of tutorials supporting this book at\\\n.\n:::\n\n::: {.content-visible when-format=\"html\"}\nYou can also access the full list of tutorials supporting this book [here](https://openintrostat.github.io/ims-tutorials).\n:::\n\n## R labs {#explore-labs}\n\nFurther apply the concepts you've learned in this part in R with computational labs that walk you through a data analysis case study.\n\n::: {.singlelab data-latex=\"\"}\n[Intro to data - Flight delays](https://www.openintro.org/go?id=ims-r-lab-intro-to-data)\\\n::: {.content-hidden unless-format=\"pdf\"} https://www.openintro.org/go?i\nd=ims-r-lab-intro-to-data\n:::\n\n:::\n\n::: {.content-hidden unless-format=\"pdf\"}\nYou can also access the full list of labs supporting this book at\\\n.\n:::\n\n::: {.content-visible when-format=\"html\"}\nYou can also access the full list of labs supporting this book [here](https://www.openintro.org/go?id=ims-r-labs).\n:::\n", + "markdown": "# Applications: Explore {#sec-explore-applications}\n\n\n\n\n\n## Case study: Effective communication of exploratory results {#case-study-effective-comms}\n\nGraphs can powerfully communicate ideas directly and quickly.\nWe all know, after all, that \"a picture is worth 1000 words.\" Unfortunately, however, there are times when an image conveys a message which is inaccurate or misleading.\n\nThis chapter focuses on how graphs can best be utilized to present data accurately and effectively.\nAlong with data modeling, creative visualization is somewhat of an art.\nHowever, even with an art, there are recommended guiding principles.\nWe provide a few best practices for creating data visualizations.\n\n### Keep it simple\n\nWhen creating a graphic, keep in mind what it is that you'd like your reader to see.\nColors should be used to group items or differentiate levels in meaningful ways.\nColors can be distracting when they are only used to brighten up the plot.\n\nConsider a manufacturing company that has summarized its costs into five different categories.\nIn the two graphics provided in @fig-pie-to-bar, notice that the magnitudes in the pie chart in @fig-pie-to-bar-1 are difficult for the eye to compare.\nThat is, can your eye tell how different \"Buildings and administration\" is from \"Workplace materials\" when looking at the slices of pie?\nAdditionally, the colors in the pie chart do not mean anything and are therefore distracting.\nLastly, the three-dimensional aspect of the image does not improve the reader's ability to understand the data presented.\n\nAs an alternative, a bar plot is been provided in @fig-pie-to-bar-2. Notice how much easier it is to identify the magnitude of the differences across categories while not being distracted by other aspects of the image.\nTypically, a bar plot will be easier for the reader to digest than a pie chart, especially if the categorical data being plotted has more than just a few levels.\n\n\n::: {.cell}\n\n:::\n\n::: {#fig-pie-to-bar .cell layout-ncol=\"2\"}\n::: {.cell-output-display}\n![A three-dimensional pie chart.](images/pie-3d.jpg){#fig-pie-to-bar-1 width=50%}\n:::\n\n::: {.cell-output-display}\n![A bar plot.](06-explore-applications_files/figure-html/fig-pie-to-bar-2.png){#fig-pie-to-bar-2 width=50%}\n:::\n\nSame information displayed with two very different visualizations.\n:::\n\n\n### Use color to draw attention\n\nThere are many reasons why you might choose to add **color** to your plots.\nAn important principle to keep in mind is to use color to draw attention.\nOf course, you should still think about how visually pleasing your visualization is, and if you're adding color for making it visually pleasing without drawing attention to a particular feature, that might be fine.\nHowever, you should be critical of default coloring and explicitly decide whether to include color and how.\nNotice that in Plot B in @fig-red-bar the coloring is done in such a way to draw the reader's attention to one particular piece of information.\nThe default coloring in Plot A can be distracting and makes the reader question, for example, is there something similar about the red and purple bars?\nAlso note that not everyone sees color the same way, often it's useful to add color and one more feature (e.g., pattern) so that you can refer to the features you're drawing attention to in multiple ways.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![The default coloring in the first bar plot does nothing for the understanding of the data. In the second plot, the color draws attention directly to the bar on Buildings and Administration.](06-explore-applications_files/figure-html/fig-red-bar-1.png){#fig-red-bar width=90%}\n:::\n:::\n\n\n\n\n### Tell a story\n\nFor many graphs, an important aspect is the inclusion of information which is not provided in the dataset that is being plotted.\nThe external information serves to contextualize the data and helps communicate the narrative of the research.\nIn @fig-duke-hires, the graph on the right is **annotated** with information about the start of the university's fiscal year which contextualizes the information provided by the data.\nSometimes the additional information may be a diagonal line given by $y = x$, points above the line quickly show the reader which values have a $y$ coordinate larger than the $x$ coordinate; points below the line show the opposite.\n\n\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Credit: Angela Zoss and Eric Monson, Duke Data Visualization Services](images/time-series-story.png){#fig-duke-hires width=100%}\n:::\n:::\n\n\n### Order matters\n\nMost software programs have built in methods for some of the plot details.\nFor example, the default option for the software program used in this text, R, is to order the bars in a bar plot alphabetically.\nAs seen in @fig-brexit-bars, the alphabetical ordering isn't particularly meaningful for describing the data.\nSometimes it makes sense to **order** the bars from tallest to shortest (or vice versa).\nBut in this case, the best ordering is probably the one in which the questions were asked.\nAn ordering which does not make sense in the context of the problem (e.g., alphabetically here), can mislead the reader who might take a quick glance at the axes and not read the bar labels carefully.\n\n\n\n\n\nIn September 2019, YouGov survey asked 1,639 Great Britain adults the following question[^06-explore-applications-1]:\n\n[^06-explore-applications-1]: Source: [YouGov Survey Results](https://d25d2506sfb94s.cloudfront.net/cumulus_uploads/document/x0msmggx08/YouGov%20-%20Brexit%20and%202019%20election.pdf), retrieved Oct 7, 2019.\n\n> How well or badly do you think the government are doing at handling Britain's exit from the European Union?\n>\n> - Very well\n> - Fairly well\n> - Fairly badly\n> - Very badly\n> - Don't know\n\n\n::: {.cell}\n\n:::\n\n::: {.cell}\n::: {.cell-output-display}\n![Bar plot three different ways. Plot A: Alphabetic ordering of levels, Plot B: Bars ordered in descending order of frequency, Plot C: Bars ordered in the same order as they were presented in the survey question.](06-explore-applications_files/figure-html/fig-brexit-bars-1.png){#fig-brexit-bars width=100%}\n:::\n:::\n\n\n### Make the labels as easy to read as possible\n\nThe Brexit survey results were additionally broken down by region in Great Britain.\nThe stacked bar plot allows for comparison of Brexit opinion across the five regions.\nIn @fig-brexit-region the bars are vertical in Plot A and horizontal in Plot B. While the quantitative information in the two graphics is identical, flipping the graph and creating horizontal bars provides more space for the **axis labels**.\nThe easier the categories are to read, the more the reader will learn from the visualization.\nRemember, the goal is to convey as much information as possible in a succinct and clear manner.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Stacked bar plots vertically and horizontally. The horizontal orientation makes the region labels easier to read.](06-explore-applications_files/figure-html/fig-brexit-region-1.png){#fig-brexit-region width=100%}\n:::\n:::\n\n\n\n\n### Pick a purpose\n\nEvery graphical decision should be made with a **purpose**.\nAs previously mentioned, sticking with default options is not always best for conveying the narrative of your data story.\nStacked bar plots tell one part of a story.\nDepending on your research question, they may not tell the part of the story most important to the research.\n@fig-seg-three-ways provides three different ways of representing the same information.\nIf the most important comparison across regions is proportion, you might prefer Plot A. If the most important comparison across regions also considers the total number of individuals in the region, you might prefer Plot B. If a separate bar plot for each region makes the point you'd like, use Plot C, which has been **faceted** by region.\n\n\n\n\n\nPlot C in @fig-seg-three-ways also provides full titles and a succinct URL with the data source.\nOther deliberate decisions to consider include using informative labels and avoiding redundancy.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Three different representations of the two variables including survey opinion and region. Use the graphic that best conveys the data narrative at hand.](06-explore-applications_files/figure-html/fig-seg-three-ways-1.png){#fig-seg-three-ways width=90%}\n:::\n:::\n\n\n\n\n### Select meaningful colors\n\n\n\nOne last consideration for building graphs is to consider color choices.\nDefault or rainbow colors are not always the choice which will best distinguish the level of your variables.\nMuch research has been done to find color combinations which are distinct and which are clear for differently sighted individuals.\nThe cividis scale works well with ordinal data.\n[@Nunez:2018] @fig-brexit-viridis shows the same plot with two different color themes.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Identical bar plots with two different coloring options. Plot A uses a default color scale, Plot B uses colors from the cividis scale.](06-explore-applications_files/figure-html/fig-brexit-viridis-1.png){#fig-brexit-viridis width=90%}\n:::\n:::\n\n\nIn this chapter different representations are contrasted to demonstrate best practices in creating graphs.\nThe fundamental principle is that your graph should provide maximal information succinctly and clearly.\nLabels should be clear and oriented horizontally for the reader.\nDon't forget titles and, if possible, include the source of the data.\n\n\\clearpage\n\n## Interactive R tutorials {#explore-tutorials}\n\nNavigate the concepts you've learned in this chapter in R using the following self-paced tutorials.\nAll you need is your browser to get started!\n\n::: {.alltutorials data-latex=\"\"}\n[Tutorial 2: Exploratory data analysis](https://openintrostat.github.io/ims-tutorials/02-explore/)\\\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintrostat.github.io/ims-tutorials/02-explore\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 2 - Lesson 1: Visualizing categorical data](https://openintro.shinyapps.io/ims-02-explore-01/)\\\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-02-explore-01\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 2 - Lesson 2: Visualizing numerical data](https://openintro.shinyapps.io/ims-02-explore-02/)\\\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-02-explore-02\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 2 - Lesson 3: Summarizing with statistics](https://openintro.shinyapps.io/ims-02-explore-03/)\\\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-02-explore-03\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 2 - Lesson 4: Case study](https://openintro.shinyapps.io/ims-02-explore-04/)\\\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-02-explore-04\n:::\n\n:::\n\n::: {.content-hidden unless-format=\"pdf\"}\nYou can also access the full list of tutorials supporting this book at\\\n.\n:::\n\n::: {.content-visible when-format=\"html\"}\nYou can also access the full list of tutorials supporting this book [here](https://openintrostat.github.io/ims-tutorials).\n:::\n\n## R labs {#explore-labs}\n\nFurther apply the concepts you've learned in this part in R with computational labs that walk you through a data analysis case study.\n\n::: {.singlelab data-latex=\"\"}\n[Intro to data - Flight delays](https://www.openintro.org/go?id=ims-r-lab-intro-to-data)\\\n::: {.content-hidden unless-format=\"pdf\"} \nhttps://www.openintro.org/go?id=ims-r-lab-intro-to-data\n:::\n\n:::\n\n::: {.content-hidden unless-format=\"pdf\"}\nYou can also access the full list of labs supporting this book at\\\n.\n:::\n\n::: {.content-visible when-format=\"html\"}\nYou can also access the full list of labs supporting this book [here](https://www.openintro.org/go?id=ims-r-labs).\n:::\n", "supporting": [ "06-explore-applications_files" ], diff --git a/_freeze/10-model-applications/execute-results/html.json b/_freeze/10-model-applications/execute-results/html.json index 80726e68..bd21e3df 100644 --- a/_freeze/10-model-applications/execute-results/html.json +++ b/_freeze/10-model-applications/execute-results/html.json @@ -1,7 +1,7 @@ { - "hash": "2433e66b0bfe2c87bcb7c0fb4b0c3d56", + "hash": "cfc9b5ca57db4c76be7a10c7394b3709", "result": { - "markdown": "# Applications: Model {#model-application}\n\n\n\n\n\n## Case study: Houses for sale {#model-case-study}\n\nTake a walk around your neighborhood and you'll probably see a few houses for sale.\nIf you find a house for sale, you can probably go online and look up its price.\nYou'll quickly note that the prices seem a bit arbitrary -- the homeowners get to decide what the amount they want to list their house for, and many criteria factor into this decision, e.g., what do comparable houses (\"comps\" in real estate speak) sell for, how quickly they need to sell the house, etc.\n\nIn this case study we'll formalize the process of figuring out how much to list a house for by using data on current home sales In November of 2020, information on 98 houses in the Duke Forest neighborhood of Durham, NC were scraped from [Zillow](https://www.zillow.com).\nThe homes were all recently sold at the time of data collection, and the goal of the project was to build a model for predicting the sale price based on a particular home's characteristics.\nThe first four homes are shown in Table \\@ref(tab:duke-data-frame), and descriptions for each variable are shown in Table \\@ref(tab:duke-variables).\n\n::: {.data data-latex=\"\"}\nThe [`duke_forest`](http://openintrostat.github.io/openintro/reference/duke_forest.html) data can be found in the [**openintro**](http://openintrostat.github.io/openintro) R package.\n:::\n\n\\vspace{-4mm}\n\n\n::: {.cell}\n::: {.cell-output-display}\n`````{=html}\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
Top four rows of the data describing homes for sale in the Duke Forest neighborhood of Durham, NC.
price bed bath area year_built cooling lot
1,520,000 3 4 6,040 1,972 central 0.97
1,030,000 5 4 4,475 1,969 central 1.38
420,000 2 3 1,745 1,959 central 0.51
680,000 4 3 2,091 1,961 central 0.84
\n\n`````\n:::\n:::\n\n\n\\vspace{-4mm}\n\n\n::: {.cell}\n::: {.cell-output-display}\n`````{=html}\n\n\n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
Variables and their descriptions for the `duke_forest` dataset.
Variable Description
price Sale price, in USD
bed Number of bedrooms
bath Number of bathrooms
area Area of home, in square feet
year_built Year the home was built
cooling Cooling system: central or other (other is baseline)
lot Area of the entire property, in acres
\n\n`````\n:::\n:::\n\n\n\\clearpage\n\n### Correlating with `price`\n\nAs mentioned, the goal of the data collection was to build a model for the sale price of homes.\nWhile using multiple predictor variables is likely preferable to using only one variable, we start by learning about the variables themselves and their relationship to price.\nFigure \\@ref(fig:single-scatter) shows scatterplots describing price as a function of each of the predictor variables.\nAll of the variables seem to be positively associated with price (higher values of the variable are matched with higher price values).\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Scatter plots describing six different predictor variables' relationship with the price of a home.](10-model-applications_files/figure-html/single-scatter-1.png){width=100%}\n:::\n:::\n\n\n::: {.guidedpractice data-latex=\"\"}\nIn Figure \\@ref(fig:single-scatter) there does not appear to be a correlation value calculated for the predictor variable, `cooling`.\nWhy not?\nCan the variable still be used in the linear model?\nExplain.[^10-model-applications-1]\n:::\n\n[^10-model-applications-1]: The correlation coefficient can only be calculated to describe the relationship between two numerical variables.\n The predictor variable `cooling` is categorical, not numerical.\n It *can*, however, be used in the linear model as a binary indicator variable coded, for example, with a `1` for central and `0` for other.\n\n::: {.workedexample data-latex=\"\"}\nIn Figure \\@ref(fig:single-scatter) which variable seems to be most informative for predicting house price?\nProvide two reasons for your answer.\n\n------------------------------------------------------------------------\n\nThe `area` of the home is the variable which is most highly correlated with `price`.\nAdditionally, the scatterplot for `price` vs. `area` seems to show a strong linear relationship between the two variables.\nNote that the correlation coefficient and the scatterplot linearity will often give the same conclusion.\nHowever, recall that the correlation coefficient is very sensitive to outliers, so it is always wise to look at the scatterplot even when the variables are highly correlated.\n:::\n\n### Modeling `price` with `area`\n\nA linear model was fit to predict `price` from `area`.\nThe resulting model information is given in Table \\@ref(tab:price-slr).\n\n\n::: {.cell}\n::: {.cell-output-display}\n`````{=html}\n\n\n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
Summary of least squares fit for price on area.
term estimate std.error statistic p.value
(Intercept) 116,652 53,302 2.19 0.0311
area 159 18 8.78 <0.0001
Adjusted R-sq = 0.4394
df = 96
\n\n`````\n:::\n:::\n\n\n::: {.guidedpractice data-latex=\"\"}\nInterpret the value of $b_1$ = 159 in the context of the problem.[^10-model-applications-2]\n:::\n\n[^10-model-applications-2]: For each additional square foot of house, we would expect such houses to cost, on average, \\$159 more.\n\n::: {.guidedpractice data-latex=\"\"}\nUsing the output in Table \\@ref(tab:price-slr), write out the model for predicting `price` from `area`.[^10-model-applications-3]\n:::\n\n[^10-model-applications-3]: $\\widehat{\\texttt{price}} = 116,652 + 159 \\times \\texttt{area}$\n\nThe residuals from the linear model can be used to assess whether a linear model is appropriate.\nFigure \\@ref(fig:price-resid-slr) plots the residuals $e_i = y_i - \\hat{y}_i$ on the $y$-axis and the fitted (or predicted) values $\\hat{y}_i$ on the $x$-axis.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Residuals versus predicted values for the model predicting sale price from area of home.](10-model-applications_files/figure-html/price-resid-slr-1.png){width=70%}\n:::\n:::\n\n\n::: {.guidedpractice data-latex=\"\"}\nWhat aspect(s) of the residual plot indicate that a linear model is appropriate?\nWhat aspect(s) of the residual plot seem concerning when fitting a linear model?[^10-model-applications-4]\n:::\n\n[^10-model-applications-4]: The residual plot shows that the relationship between `area` and the average `price` of a home is indeed linear.\n However, the residuals are quite large for expensive homes.\n The large residuals indicate potential outliers or increasing variability, either of which could warrant more involved modeling techniques than are presented in this text.\n\n### Modeling `price` with multiple variables\n\nIt seems as though the predictions of home price might be more accurate if more than one predictor variable was used in the linear model.\nTable \\@ref(tab:price-mlr) displays the output from a linear model of `price` regressed on `area`, `bed`, `bath`, `year_built`, `cooling`, and `lot`.\n\n\n::: {.cell}\n::: {.cell-output-display}\n`````{=html}\n\n\n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
Summary of least squares fit for price on multiple predictor variables.
term estimate std.error statistic p.value
(Intercept) -2,910,715 1,787,934 -1.63 0.107
area 102 23 4.42 <0.0001
bed -13,692 25,928 -0.53 0.5987
bath 41,076 24,662 1.67 0.0993
year_built 1,459 914 1.60 0.1139
coolingcentral 84,065 30,338 2.77 0.0068
lot 356,141 75,940 4.69 <0.0001
Adjusted R-sq = 0.5896
df = 90
\n\n`````\n:::\n:::\n\n\n::: {.workedexample data-latex=\"\"}\nUsing Table \\@ref(tab:price-mlr), write out the linear model of price on the six predictor variables.\n\n------------------------------------------------------------------------\n\n$$\n\\begin{aligned}\n\\widehat{\\texttt{price}} &= -2,910,715 \\\\\n&+ 102 \\times \\texttt{area} - 13,692 \\times \\texttt{bed} \\\\\n&+ 41,076 \\times \\texttt{bath} + 1,459 \\times \\texttt{year_built}\\\\\n&+ 84,065 \\times \\texttt{cooling}_{\\texttt{central}} + 356,141 \\times \\texttt{lot}\n\\end{aligned}\n$$\n:::\n\n::: {.guidedpractice data-latex=\"\"}\nThe value of the estimated coefficient on $\\texttt{cooling}_{\\texttt{central}}$ is $b_5 = 84,065.$ Interpret the value of $b_5$ in the context of the problem.[^10-model-applications-5]\n:::\n\n[^10-model-applications-5]: The coefficient indicates that if all the other variables are kept constant, homes with central air conditioning cost \\$84,065 more, on average.\n\nA friend suggests that maybe you do not need all six variables to have a good model for `price`.\nYou consider taking a variable out, but you aren't sure which one to remove.\n\n\n::: {.cell}\n\n:::\n\n\n::: {.workedexample data-latex=\"\"}\nResults corresponding to the full model for the housing data are shown in Table \\@ref(tab:price-mlr).\nHow should we proceed under the backward elimination strategy?\n\n------------------------------------------------------------------------\n\nOur baseline adjusted $R^2$ from the full model is 0.59, and we need to determine whether dropping a predictor will improve the adjusted $R^2$.\nTo check, we fit models that each drop a different predictor, and we record the adjusted $R^2$:\n\n- Excluding `area`: 0.506\n- Excluding `bed`: 0.593\n- Excluding `bath`: 0.582\n- Excluding `year_built`: 0.583\n- Excluding `cooling`: 0.559\n- Excluding `lot`: 0.489\n\nThe model without `bed` has the highest adjusted $R^2$ of 0.593, higher than the adjusted $R^2$ for the full model.\nBecause eliminating `bed` leads to a model with a higher adjusted $R^2$ than the full model, we drop `bed` from the model.\n\nIt might seem counter-intuitive to exclude information on number of bedrooms from the model.\nAfter all, we would expect homes with more bedrooms to cost more, and we can see a clear relationship between number of bedrooms and sale price in Figure \\@ref(fig:single-scatter).\nHowever, note that `area` is still in the model, and it's quite likely that the area of the home and the number of bedrooms are highly associated.\nTherefore, the model already has information on \"how much space is available in the house\" with the inclusion of `area`.\n\nSince we eliminated a predictor from the model in the first step, we see whether we should eliminate any additional predictors.\nOur baseline adjusted $R^2$ is now 0.593.\nWe fit another set of new models, which consider eliminating each of the remaining predictors in addition to `bed`:\n\n\n::: {.cell}\n\n:::\n\n\n- Excluding `bed` and `area`: 0.51\n- Excluding `bed` and `bath`: 0.586\n- Excluding `bed` and `year_built`: 0.586\n- Excluding `bed` and `cooling`: 0.563\n- Excluding `bed` and `lot`: 0.493\n\nNone of these models lead to an improvement in adjusted $R^2$, so we do not eliminate any of the remaining predictors.\n:::\n\n\\clearpage\n\nThat is, after backward elimination, we are left with the model that keeps all predictors except `bed`, which we can summarize using the coefficients from Table \\@ref(tab:price-full-except-bed).\n\n\n::: {.cell}\n::: {.cell-output-display}\n`````{=html}\n\n\n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
Summary of least squares fit for price on multiple predictor variables, excluding number of bedrooms.
term estimate std.error statistic p.value
(Intercept) -2,952,641 1,779,079 -1.66 0.1004
area 99 22 4.44 <0.0001
bath 36,228 22,799 1.59 0.1155
year_built 1,466 910 1.61 0.1107
coolingcentral 83,856 30,215 2.78 0.0067
lot 357,119 75,617 4.72 <0.0001
Adjusted R-sq = 0.5929
df = 91
\n\n`````\n:::\n:::\n\n\n\\vspace{-4mm}\n\nThen, the linear model for predicting sale price based on this model is as follows:\n\n$$ \n\\begin{aligned}\n\\widehat{\\texttt{price}} &= -2,952,641 + 99 \\times \\texttt{area}\\\\ \n&+ 36,228 \\times \\texttt{bath} + 1,466 \\times \\texttt{year_built}\\\\\n&+ 83,856 \\times \\texttt{cooling}_{\\texttt{central}} + 357,119 \\times \\texttt{lot}\n\\end{aligned}\n$$\n\n::: {.workedexample data-latex=\"\"}\nThe residual plot for the model with all of the predictor variables except `bed` is given in Figure \\@ref(fig:price-resid-mlr-nobed).\nHow do the residuals in Figure \\@ref(fig:price-resid-mlr-nobed) compare to the residuals in Figure \\@ref(fig:price-resid-slr)?\n\n------------------------------------------------------------------------\n\nThe residuals, for the most part, are randomly scattered around 0.\nHowever, there is one extreme outlier with a residual of -\\$750,000, a house whose actual sale price is a lot lower than its predicted price.\nAlso, we observe again that the residuals are quite large for expensive homes.\n:::\n\n\\vspace{-4mm}\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Residuals versus predicted values for the model predicting sale price from all predictors except for number of bedrooms.](10-model-applications_files/figure-html/price-resid-mlr-nobed-1.png){width=70%}\n:::\n:::\n\n::: {.cell}\n\n:::\n\n\n::: {.guidedpractice data-latex=\"\"}\nConsider a house with 1,803 square feet, 2.5 bathrooms, 0.145 acres, built in 1941, that has central air conditioning.\nWhat is the predicted price of the home?[^10-model-applications-6]\n:::\n\n[^10-model-applications-6]: $\\widehat{\\texttt{price}} = -2,952,641 + 99 \\times 1803\\\\ + 36,228 \\times 2.5 + 1,466 \\times 1941\\\\ + 83,856 \\times 1 + 357,119 \\times 0.145\\\\ = \\$297,570.$\n\n::: {.guidedpractice data-latex=\"\"}\nIf you later learned that the house (with a predicted price of \\$297,570) had recently sold for \\$804,133, would you think the model was terrible?\nWhat if you learned that the house was in California?[^10-model-applications-7]\n:::\n\n[^10-model-applications-7]: A residual of \\$506,563 is reasonably big.\n Note that the large residuals (except a few homes) in Figure \\@ref(fig:price-resid-mlr-nobed) are closer to \\$250,000 (about half as big).\n After we learn that the house is in California, we realize that the model shouldn't be applied to the new home at all!\n The original data are from Durham, NC, and models based on the Durham, NC data should be used only to explore patterns in prices for homes in Durham, NC.\n\n\\clearpage\n\n## Interactive R tutorials {#sec-model-tutorials}\n\nNavigate the concepts you've learned in this chapter in R using the following self-paced tutorials.\nAll you need is your browser to get started!\n\n::: {.alltutorials data-latex=\"\"}\n[Tutorial 3: Regression modeling](https://openintrostat.github.io/ims-tutorials/03-model/)\\\n::: {.content-hidden unless-format=\"pdf\"} https://openintrostat.github.io/ims-tutorials/03-model\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 1: Visualizing two variables](https://openintro.shinyapps.io/ims-03-model-01/)\\\n::: {.content-hidden unless-format=\"pdf\"} https://openintro.shinyapps.io/ims-03-model-01\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 2: Correlation](https://openintro.shinyapps.io/ims-03-model-02/)\\\n::: {.content-hidden unless-format=\"pdf\"} https://openintro.shinyapps.io/ims-03-model-02\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 3: Simple linear regression](https://openintro.shinyapps.io/ims-03-model-03/)\\\n::: {.content-hidden unless-format=\"pdf\"} https://openintro.shinyapps.io/ims-03-model-03\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 4: Interpreting regression models](https://openintro.shinyapps.io/ims-03-model-04/)\\\n::: {.content-hidden unless-format=\"pdf\"} https://openintro.shinyapps.io/ims-03-model-04\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 5: Model fit](https://openintro.shinyapps.io/ims-03-model-05/)\\\n::: {.content-hidden unless-format=\"pdf\"} https://openintro.shinyapps.io/ims-03-model-05\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 6: Parallel slopes](https://openintro.shinyapps.io/ims-03-model-06/)\\\n::: {.content-hidden unless-format=\"pdf\"} https://openintro.shinyapps.io/ims-03-model-06\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 7: Evaluating and extending parallel slopes model](https://openintro.shinyapps.io/ims-03-model-07/)\\\n::: {.content-hidden unless-format=\"pdf\"} https://openintro.shinyapps.io/ims-03-model-07\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 8: Multiple regression](https://openintro.shinyapps.io/ims-03-model-08/)\\\n::: {.content-hidden unless-format=\"pdf\"} https://openintro.shinyapps.io/ims-03-model-08\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 9: Logistic regression](https://openintro.shinyapps.io/ims-03-model-09/)\\\n::: {.content-hidden unless-format=\"pdf\"} https://openintro.shinyapps.io/ims-03-model-09\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 10: Case study: Italian restaurants in NYC](https://openintro.shinyapps.io/ims-03-model-10/)\\\n::: {.content-hidden unless-format=\"pdf\"} https://openintro.shinyapps.io/ims-03-model-10\n:::\n\n:::\n\n::: {.content-hidden unless-format=\"pdf\"}\nYou can also access the full list of tutorials supporting this book at\\\n.\n:::\n\n::: {.content-visible when-format=\"html\"}\nYou can also access the full list of tutorials supporting this book [here](https://openintrostat.github.io/ims-tutorials).\n:::\n\n## R labs {#model-labs}\n\nFurther apply the concepts you've learned in this part in R with computational labs that walk you through a data analysis case study.\n\n::: {.singlelab data-latex=\"\"}\n[Introduction to linear regression - Human Freedom Index](https://www.openintro.org/go?id=ims-r-lab-model)\\\n::: {.content-hidden unless-format=\"pdf\"} https://www.openintro.org/go?i\nd=ims-r-lab-model\n:::\n\n:::\n\n::: {.content-hidden unless-format=\"pdf\"}\nYou can also access the full list of labs supporting this book at\\\n.\n:::\n\n::: {.content-visible when-format=\"html\"}\nYou can also access the full list of labs supporting this book [here](https://www.openintro.org/go?id=ims-r-labs).\n:::\n", + "markdown": "# Applications: Model {#model-application}\n\n\n\n\n\n## Case study: Houses for sale {#model-case-study}\n\nTake a walk around your neighborhood and you'll probably see a few houses for sale.\nIf you find a house for sale, you can probably go online and look up its price.\nYou'll quickly note that the prices seem a bit arbitrary -- the homeowners get to decide what the amount they want to list their house for, and many criteria factor into this decision, e.g., what do comparable houses (\"comps\" in real estate speak) sell for, how quickly they need to sell the house, etc.\n\nIn this case study we'll formalize the process of figuring out how much to list a house for by using data on current home sales In November of 2020, information on 98 houses in the Duke Forest neighborhood of Durham, NC were scraped from [Zillow](https://www.zillow.com).\nThe homes were all recently sold at the time of data collection, and the goal of the project was to build a model for predicting the sale price based on a particular home's characteristics.\nThe first four homes are shown in Table \\@ref(tab:duke-data-frame), and descriptions for each variable are shown in Table \\@ref(tab:duke-variables).\n\n::: {.data data-latex=\"\"}\nThe [`duke_forest`](http://openintrostat.github.io/openintro/reference/duke_forest.html) data can be found in the [**openintro**](http://openintrostat.github.io/openintro) R package.\n:::\n\n\\vspace{-4mm}\n\n\n::: {.cell}\n::: {.cell-output-display}\n`````{=html}\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
Top four rows of the data describing homes for sale in the Duke Forest neighborhood of Durham, NC.
price bed bath area year_built cooling lot
1,520,000 3 4 6,040 1,972 central 0.97
1,030,000 5 4 4,475 1,969 central 1.38
420,000 2 3 1,745 1,959 central 0.51
680,000 4 3 2,091 1,961 central 0.84
\n\n`````\n:::\n:::\n\n\n\\vspace{-4mm}\n\n\n::: {.cell}\n::: {.cell-output-display}\n`````{=html}\n\n\n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
Variables and their descriptions for the `duke_forest` dataset.
Variable Description
price Sale price, in USD
bed Number of bedrooms
bath Number of bathrooms
area Area of home, in square feet
year_built Year the home was built
cooling Cooling system: central or other (other is baseline)
lot Area of the entire property, in acres
\n\n`````\n:::\n:::\n\n\n\\clearpage\n\n### Correlating with `price`\n\nAs mentioned, the goal of the data collection was to build a model for the sale price of homes.\nWhile using multiple predictor variables is likely preferable to using only one variable, we start by learning about the variables themselves and their relationship to price.\nFigure \\@ref(fig:single-scatter) shows scatterplots describing price as a function of each of the predictor variables.\nAll of the variables seem to be positively associated with price (higher values of the variable are matched with higher price values).\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Scatter plots describing six different predictor variables' relationship with the price of a home.](10-model-applications_files/figure-html/single-scatter-1.png){width=100%}\n:::\n:::\n\n\n::: {.guidedpractice data-latex=\"\"}\nIn Figure \\@ref(fig:single-scatter) there does not appear to be a correlation value calculated for the predictor variable, `cooling`.\nWhy not?\nCan the variable still be used in the linear model?\nExplain.[^10-model-applications-1]\n:::\n\n[^10-model-applications-1]: The correlation coefficient can only be calculated to describe the relationship between two numerical variables.\n The predictor variable `cooling` is categorical, not numerical.\n It *can*, however, be used in the linear model as a binary indicator variable coded, for example, with a `1` for central and `0` for other.\n\n::: {.workedexample data-latex=\"\"}\nIn Figure \\@ref(fig:single-scatter) which variable seems to be most informative for predicting house price?\nProvide two reasons for your answer.\n\n------------------------------------------------------------------------\n\nThe `area` of the home is the variable which is most highly correlated with `price`.\nAdditionally, the scatterplot for `price` vs. `area` seems to show a strong linear relationship between the two variables.\nNote that the correlation coefficient and the scatterplot linearity will often give the same conclusion.\nHowever, recall that the correlation coefficient is very sensitive to outliers, so it is always wise to look at the scatterplot even when the variables are highly correlated.\n:::\n\n### Modeling `price` with `area`\n\nA linear model was fit to predict `price` from `area`.\nThe resulting model information is given in Table \\@ref(tab:price-slr).\n\n\n::: {.cell}\n::: {.cell-output-display}\n`````{=html}\n\n\n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
Summary of least squares fit for price on area.
term estimate std.error statistic p.value
(Intercept) 116,652 53,302 2.19 0.0311
area 159 18 8.78 <0.0001
Adjusted R-sq = 0.4394
df = 96
\n\n`````\n:::\n:::\n\n\n::: {.guidedpractice data-latex=\"\"}\nInterpret the value of $b_1$ = 159 in the context of the problem.[^10-model-applications-2]\n:::\n\n[^10-model-applications-2]: For each additional square foot of house, we would expect such houses to cost, on average, \\$159 more.\n\n::: {.guidedpractice data-latex=\"\"}\nUsing the output in Table \\@ref(tab:price-slr), write out the model for predicting `price` from `area`.[^10-model-applications-3]\n:::\n\n[^10-model-applications-3]: $\\widehat{\\texttt{price}} = 116,652 + 159 \\times \\texttt{area}$\n\nThe residuals from the linear model can be used to assess whether a linear model is appropriate.\nFigure \\@ref(fig:price-resid-slr) plots the residuals $e_i = y_i - \\hat{y}_i$ on the $y$-axis and the fitted (or predicted) values $\\hat{y}_i$ on the $x$-axis.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Residuals versus predicted values for the model predicting sale price from area of home.](10-model-applications_files/figure-html/price-resid-slr-1.png){width=70%}\n:::\n:::\n\n\n::: {.guidedpractice data-latex=\"\"}\nWhat aspect(s) of the residual plot indicate that a linear model is appropriate?\nWhat aspect(s) of the residual plot seem concerning when fitting a linear model?[^10-model-applications-4]\n:::\n\n[^10-model-applications-4]: The residual plot shows that the relationship between `area` and the average `price` of a home is indeed linear.\n However, the residuals are quite large for expensive homes.\n The large residuals indicate potential outliers or increasing variability, either of which could warrant more involved modeling techniques than are presented in this text.\n\n### Modeling `price` with multiple variables\n\nIt seems as though the predictions of home price might be more accurate if more than one predictor variable was used in the linear model.\nTable \\@ref(tab:price-mlr) displays the output from a linear model of `price` regressed on `area`, `bed`, `bath`, `year_built`, `cooling`, and `lot`.\n\n\n::: {.cell}\n::: {.cell-output-display}\n`````{=html}\n\n\n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
Summary of least squares fit for price on multiple predictor variables.
term estimate std.error statistic p.value
(Intercept) -2,910,715 1,787,934 -1.63 0.107
area 102 23 4.42 <0.0001
bed -13,692 25,928 -0.53 0.5987
bath 41,076 24,662 1.67 0.0993
year_built 1,459 914 1.60 0.1139
coolingcentral 84,065 30,338 2.77 0.0068
lot 356,141 75,940 4.69 <0.0001
Adjusted R-sq = 0.5896
df = 90
\n\n`````\n:::\n:::\n\n\n::: {.workedexample data-latex=\"\"}\nUsing Table \\@ref(tab:price-mlr), write out the linear model of price on the six predictor variables.\n\n------------------------------------------------------------------------\n\n$$\n\\begin{aligned}\n\\widehat{\\texttt{price}} &= -2,910,715 \\\\\n&+ 102 \\times \\texttt{area} - 13,692 \\times \\texttt{bed} \\\\\n&+ 41,076 \\times \\texttt{bath} + 1,459 \\times \\texttt{year_built}\\\\\n&+ 84,065 \\times \\texttt{cooling}_{\\texttt{central}} + 356,141 \\times \\texttt{lot}\n\\end{aligned}\n$$\n:::\n\n::: {.guidedpractice data-latex=\"\"}\nThe value of the estimated coefficient on $\\texttt{cooling}_{\\texttt{central}}$ is $b_5 = 84,065.$ Interpret the value of $b_5$ in the context of the problem.[^10-model-applications-5]\n:::\n\n[^10-model-applications-5]: The coefficient indicates that if all the other variables are kept constant, homes with central air conditioning cost \\$84,065 more, on average.\n\nA friend suggests that maybe you do not need all six variables to have a good model for `price`.\nYou consider taking a variable out, but you aren't sure which one to remove.\n\n\n::: {.cell}\n\n:::\n\n\n::: {.workedexample data-latex=\"\"}\nResults corresponding to the full model for the housing data are shown in Table \\@ref(tab:price-mlr).\nHow should we proceed under the backward elimination strategy?\n\n------------------------------------------------------------------------\n\nOur baseline adjusted $R^2$ from the full model is 0.59, and we need to determine whether dropping a predictor will improve the adjusted $R^2$.\nTo check, we fit models that each drop a different predictor, and we record the adjusted $R^2$:\n\n- Excluding `area`: 0.506\n- Excluding `bed`: 0.593\n- Excluding `bath`: 0.582\n- Excluding `year_built`: 0.583\n- Excluding `cooling`: 0.559\n- Excluding `lot`: 0.489\n\nThe model without `bed` has the highest adjusted $R^2$ of 0.593, higher than the adjusted $R^2$ for the full model.\nBecause eliminating `bed` leads to a model with a higher adjusted $R^2$ than the full model, we drop `bed` from the model.\n\nIt might seem counter-intuitive to exclude information on number of bedrooms from the model.\nAfter all, we would expect homes with more bedrooms to cost more, and we can see a clear relationship between number of bedrooms and sale price in Figure \\@ref(fig:single-scatter).\nHowever, note that `area` is still in the model, and it's quite likely that the area of the home and the number of bedrooms are highly associated.\nTherefore, the model already has information on \"how much space is available in the house\" with the inclusion of `area`.\n\nSince we eliminated a predictor from the model in the first step, we see whether we should eliminate any additional predictors.\nOur baseline adjusted $R^2$ is now 0.593.\nWe fit another set of new models, which consider eliminating each of the remaining predictors in addition to `bed`:\n\n\n::: {.cell}\n\n:::\n\n\n- Excluding `bed` and `area`: 0.51\n- Excluding `bed` and `bath`: 0.586\n- Excluding `bed` and `year_built`: 0.586\n- Excluding `bed` and `cooling`: 0.563\n- Excluding `bed` and `lot`: 0.493\n\nNone of these models lead to an improvement in adjusted $R^2$, so we do not eliminate any of the remaining predictors.\n:::\n\n\\clearpage\n\nThat is, after backward elimination, we are left with the model that keeps all predictors except `bed`, which we can summarize using the coefficients from Table \\@ref(tab:price-full-except-bed).\n\n\n::: {.cell}\n::: {.cell-output-display}\n`````{=html}\n\n\n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
Summary of least squares fit for price on multiple predictor variables, excluding number of bedrooms.
term estimate std.error statistic p.value
(Intercept) -2,952,641 1,779,079 -1.66 0.1004
area 99 22 4.44 <0.0001
bath 36,228 22,799 1.59 0.1155
year_built 1,466 910 1.61 0.1107
coolingcentral 83,856 30,215 2.78 0.0067
lot 357,119 75,617 4.72 <0.0001
Adjusted R-sq = 0.5929
df = 91
\n\n`````\n:::\n:::\n\n\n\\vspace{-4mm}\n\nThen, the linear model for predicting sale price based on this model is as follows:\n\n$$ \n\\begin{aligned}\n\\widehat{\\texttt{price}} &= -2,952,641 + 99 \\times \\texttt{area}\\\\ \n&+ 36,228 \\times \\texttt{bath} + 1,466 \\times \\texttt{year_built}\\\\\n&+ 83,856 \\times \\texttt{cooling}_{\\texttt{central}} + 357,119 \\times \\texttt{lot}\n\\end{aligned}\n$$\n\n::: {.workedexample data-latex=\"\"}\nThe residual plot for the model with all of the predictor variables except `bed` is given in Figure \\@ref(fig:price-resid-mlr-nobed).\nHow do the residuals in Figure \\@ref(fig:price-resid-mlr-nobed) compare to the residuals in Figure \\@ref(fig:price-resid-slr)?\n\n------------------------------------------------------------------------\n\nThe residuals, for the most part, are randomly scattered around 0.\nHowever, there is one extreme outlier with a residual of -\\$750,000, a house whose actual sale price is a lot lower than its predicted price.\nAlso, we observe again that the residuals are quite large for expensive homes.\n:::\n\n\\vspace{-4mm}\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Residuals versus predicted values for the model predicting sale price from all predictors except for number of bedrooms.](10-model-applications_files/figure-html/price-resid-mlr-nobed-1.png){width=70%}\n:::\n:::\n\n::: {.cell}\n\n:::\n\n\n::: {.guidedpractice data-latex=\"\"}\nConsider a house with 1,803 square feet, 2.5 bathrooms, 0.145 acres, built in 1941, that has central air conditioning.\nWhat is the predicted price of the home?[^10-model-applications-6]\n:::\n\n[^10-model-applications-6]: $\\widehat{\\texttt{price}} = -2,952,641 + 99 \\times 1803\\\\ + 36,228 \\times 2.5 + 1,466 \\times 1941\\\\ + 83,856 \\times 1 + 357,119 \\times 0.145\\\\ = \\$297,570.$\n\n::: {.guidedpractice data-latex=\"\"}\nIf you later learned that the house (with a predicted price of \\$297,570) had recently sold for \\$804,133, would you think the model was terrible?\nWhat if you learned that the house was in California?[^10-model-applications-7]\n:::\n\n[^10-model-applications-7]: A residual of \\$506,563 is reasonably big.\n Note that the large residuals (except a few homes) in Figure \\@ref(fig:price-resid-mlr-nobed) are closer to \\$250,000 (about half as big).\n After we learn that the house is in California, we realize that the model shouldn't be applied to the new home at all!\n The original data are from Durham, NC, and models based on the Durham, NC data should be used only to explore patterns in prices for homes in Durham, NC.\n\n\\clearpage\n\n## Interactive R tutorials {#sec-model-tutorials}\n\nNavigate the concepts you've learned in this chapter in R using the following self-paced tutorials.\nAll you need is your browser to get started!\n\n::: {.alltutorials data-latex=\"\"}\n[Tutorial 3: Regression modeling](https://openintrostat.github.io/ims-tutorials/03-model/)\\\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintrostat.github.io/ims-tutorials/03-model\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 1: Visualizing two variables](https://openintro.shinyapps.io/ims-03-model-01/)\\\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-03-model-01\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 2: Correlation](https://openintro.shinyapps.io/ims-03-model-02/)\\\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-03-model-02\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 3: Simple linear regression](https://openintro.shinyapps.io/ims-03-model-03/)\\\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-03-model-03\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 4: Interpreting regression models](https://openintro.shinyapps.io/ims-03-model-04/)\\\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-03-model-04\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 5: Model fit](https://openintro.shinyapps.io/ims-03-model-05/)\\\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-03-model-05\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 6: Parallel slopes](https://openintro.shinyapps.io/ims-03-model-06/)\\\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-03-model-06\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 7: Evaluating and extending parallel slopes model](https://openintro.shinyapps.io/ims-03-model-07/)\\\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-03-model-07\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 8: Multiple regression](https://openintro.shinyapps.io/ims-03-model-08/)\\\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-03-model-08\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 9: Logistic regression](https://openintro.shinyapps.io/ims-03-model-09/)\\\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-03-model-09\n:::\n\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 3 - Lesson 10: Case study: Italian restaurants in NYC](https://openintro.shinyapps.io/ims-03-model-10/)\\\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-03-model-10\n:::\n\n:::\n\n::: {.content-hidden unless-format=\"pdf\"}\nYou can also access the full list of tutorials supporting this book at\\\n.\n:::\n\n::: {.content-visible when-format=\"html\"}\nYou can also access the full list of tutorials supporting this book [here](https://openintrostat.github.io/ims-tutorials).\n:::\n\n## R labs {#model-labs}\n\nFurther apply the concepts you've learned in this part in R with computational labs that walk you through a data analysis case study.\n\n::: {.singlelab data-latex=\"\"}\n[Introduction to linear regression - Human Freedom Index](https://www.openintro.org/go?id=ims-r-lab-model)\\\n::: {.content-hidden unless-format=\"pdf\"} \nhttps://www.openintro.org/go?id=ims-r-lab-model\n:::\n\n:::\n\n::: {.content-hidden unless-format=\"pdf\"}\nYou can also access the full list of labs supporting this book at\\\n.\n:::\n\n::: {.content-visible when-format=\"html\"}\nYou can also access the full list of labs supporting this book [here](https://www.openintro.org/go?id=ims-r-labs).\n:::\n", "supporting": [ "10-model-applications_files" ], diff --git a/book.bib b/book.bib index 616cdbd6..ed57e8c0 100644 --- a/book.bib +++ b/book.bib @@ -196,4 +196,16 @@ @article{Hesterbeg:2015 year={2015}, publisher={Taylor \& Francis}, doi={10.1080/00031305.2015.1089789} +} + +@article{Witmer:2021, +author = {Jeff Witmer}, +title = {Simpson's Paradox, Visual Displays, and Causal Diagrams}, +journal = {The American Mathematical Monthly}, +volume = {128}, +number = {7}, +pages = {598-610}, +year = {2021}, +publisher = {Taylor & Francis}, +doi = {10.1080/00029890.2021.1932237} } \ No newline at end of file From f6ea58cb646a204684885f068a3a9a4ad0be9a1c Mon Sep 17 00:00:00 2001 From: Jo Hardin Date: Wed, 11 Oct 2023 13:22:28 -0700 Subject: [PATCH 2/6] Update 03-data-applications.qmd Co-authored-by: Mine Cetinkaya-Rundel --- 03-data-applications.qmd | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/03-data-applications.qmd b/03-data-applications.qmd index 5cc92a8b..3e0ac747 100644 --- a/03-data-applications.qmd +++ b/03-data-applications.qmd @@ -46,7 +46,7 @@ paralympic_1500 |> latex_options = c("striped", "hold_position")) ``` -When you encounter a new dataset, taking a peek at the last few rows as we did in @tbl-paralympic-df-tail is almost instinctual. +When you encounter a new dataset, taking a peek at the last few rows as we did in @tbl-paralympic-df-tail should be almost instinctual. It can often be helpful to look at the first few rows of the data as well to get a sense for other aspects of the data which may not be apparent int he last few rows. @tbl-paralympic-df-head shows the top ten rows of the `paralympic_1500` dataset, which reveals that for at least the first 10 Olympiads, there were no runners in the Women's division or in the Paralympics. From 1a9275c0209d885bb6c4598090c83909346d47ce Mon Sep 17 00:00:00 2001 From: Jo Hardin Date: Wed, 11 Oct 2023 13:23:07 -0700 Subject: [PATCH 3/6] Update 03-data-applications.qmd Co-authored-by: Mine Cetinkaya-Rundel --- 03-data-applications.qmd | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/03-data-applications.qmd b/03-data-applications.qmd index 3e0ac747..67d34082 100644 --- a/03-data-applications.qmd +++ b/03-data-applications.qmd @@ -281,7 +281,7 @@ terms_chp_3 <- c("Simpson's paradox") Simpson's paradox was seen in the 1500m race data because the aggregate data showed a positive relationship (positive slope) between year and race time but a negative relationship (negative slope) between year and race time when broken down by the type of race. Simpson's paradox can be seen with categorical or numeric data. -Oftentimes the paradox happens because the third variable (here, race type) is imbalanced. +Often the paradox happens because the third variable (here, race type) is imbalanced. There are either more observations in one group or the observations happen at different intervals across the two groups. In the 1500m data, we saw that the T11 runners had fewer observations and their times were both generally slower and more recent than the Olympic runners. From 31f0a1c98c139b57adcd107e832368b818645e1d Mon Sep 17 00:00:00 2001 From: Jo Hardin Date: Wed, 11 Oct 2023 13:24:03 -0700 Subject: [PATCH 4/6] Update 03-data-applications.qmd Co-authored-by: Mine Cetinkaya-Rundel --- 03-data-applications.qmd | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/03-data-applications.qmd b/03-data-applications.qmd index 67d34082..f85a6a2a 100644 --- a/03-data-applications.qmd +++ b/03-data-applications.qmd @@ -47,7 +47,7 @@ paralympic_1500 |> ``` When you encounter a new dataset, taking a peek at the last few rows as we did in @tbl-paralympic-df-tail should be almost instinctual. -It can often be helpful to look at the first few rows of the data as well to get a sense for other aspects of the data which may not be apparent int he last few rows. +It can often be helpful to look at the first few rows of the data as well to get a sense of other aspects of the data which may not be apparent int he last few rows. @tbl-paralympic-df-head shows the top ten rows of the `paralympic_1500` dataset, which reveals that for at least the first 10 Olympiads, there were no runners in the Women's division or in the Paralympics. ```{r} From df9883fc025a913123fca9a6ab19809444c3e53f Mon Sep 17 00:00:00 2001 From: Jo Hardin Date: Wed, 11 Oct 2023 13:24:24 -0700 Subject: [PATCH 5/6] Update 03-data-applications.qmd Co-authored-by: Mine Cetinkaya-Rundel --- 03-data-applications.qmd | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/03-data-applications.qmd b/03-data-applications.qmd index f85a6a2a..a36ee30c 100644 --- a/03-data-applications.qmd +++ b/03-data-applications.qmd @@ -77,7 +77,7 @@ What does each row and each column represent?[^03-data-applications-2] [^03-data-applications-2]: Each row represents a 1500m gold medal race and each column represents a variable containing information on each race. Once you've identified the rows and columns, it's useful to review the data dictionary to learn about what each column in the dataset represents. -The column information is provided in @tbl-paralympic-var-def. +The data dictionary is provided in @tbl-paralympic-var-def. ```{r} #| label: tbl-paralympic-var-def From 4fab28dc5703e600a12a3d4d0ef9358e67c147c0 Mon Sep 17 00:00:00 2001 From: hardin47 Date: Wed, 11 Oct 2023 14:40:16 -0700 Subject: [PATCH 6/6] updates to the paralympic chapter --- 03-data-applications.qmd | 83 ++++++++++++------ .../execute-results/html.json | 4 +- .../figure-html/fig-paralympic-cat-1.png | Bin 440038 -> 535857 bytes .../figure-html/fig-paralympic-grouped-1.png | Bin 164529 -> 211653 bytes .../figure-html/fig-paralympic-rio-1.png | Bin 0 -> 183659 bytes .../fig-paralympic-ungrouped-1.png | Bin 123042 -> 165467 bytes introduction-to-data.qmd | 2 +- 7 files changed, 58 insertions(+), 31 deletions(-) create mode 100644 _freeze/03-data-applications/figure-html/fig-paralympic-rio-1.png diff --git a/03-data-applications.qmd b/03-data-applications.qmd index a36ee30c..0fec42b2 100644 --- a/03-data-applications.qmd +++ b/03-data-applications.qmd @@ -3,22 +3,17 @@ ```{r} #| include: false source("_common.R") -paralympic_1500_raw <- read_csv("~/Desktop/openintro/data-raw/paralympic_1500/paralympic_1500.csv") - -paralympic_1500 <- paralympic_1500_raw |> - mutate(time_min = time) |> - mutate(time = paste(min, sec, sep = ":")) |> - select(-min, -sec) - +library(openintro) +data("paralympic_1500") ``` -## Case study: Olympic 1500m {#case-study-passwords} +## Case study: Olympic 1500m {#case-study-paralympics} While many of you may be glued to the Olympic Games every four years (or every two years if you fancy both summer and winter sports), the Paralympic Games are less popular than the Olympic Games, even if they hold the same competitive thrills. -The Paralympic Games began as a way to support soldiers who had been wounded in World War II as a way to help soldiers use sport to rehabilitate. +The Paralympic Games began as a way to support soldiers who had been wounded in World War II as a way to help them rehabilitate. The first Paralympic Games were held in Rome, Italy in 1960. -Since 1988 (Seoul, South Korea), the Paralymic Games have been held a few weeks later than the Olympic Games in the same city, in both the summer and winter. +Since 1988 (Seoul, South Korea), the Paralympic Games have been held a few weeks later than the Olympic Games in the same city, in both the summer and winter. In this case study we introduce a dataset comparing Olympic and Paralympic gold medal finishers in the 1500m running competition (the Olympic "mile", if a bit shorter than a full mile). The goal of the case study is to walk you through what a data scientist does when they first get a hold of a dataset. @@ -32,7 +27,7 @@ The [`paralympic_1500`](http://openintrostat.github.io/openintro/reference/paral @tbl-paralympic-df-tail shows the last ten rows from the dataset, which are the ten most recent 1500m races. Notice that there are racers from both the Men's and Women's divisions as well as those of varying visual impairment (T11, T12, T13, and Olympic). The T11 athletes have almost complete visual impairment, run with a black-out blindfold, and are allowed to run with a guide-runner. -T12 and T13 athletes have some visual impairment, and Olympic runners do not have visual impairment. +T12 and T13 athletes have some visual impairment, and the visual acuity of Olympic runners is not determined. ```{r} #| label: tbl-paralympic-df-tail @@ -116,17 +111,16 @@ The numerical variables in the dataset are `year` (discrete), and `time_min` (co The categorical variables are `city`, `country_of_games`, `division`, `type`, `name`, and `country_of_athlete`. The `time` variable is trickier to classify -- we can think of it as numerical, but it is classified as categorical. The categorical classification is due to the colon `:` which separates the hours from the seconds. -The lubridate R package is fantastic for working with date and time variables. -Sometimes the data dictionary (presented in @tbl-paralympic-var-def) isn't sufficient, and we need to go back to the data source and try to understand the data better before we can proceed with the analysis meaningfully. +Sometimes the data dictionary (presented in @tbl-paralympic-var-def) isn't sufficient for a complete analysis, and we need to go back to the data source and try to understand the data better before we can proceed with the analysis meaningfully. ::: Next, let's try to get to know each variable a little bit better. For categorical variables, this involves figuring out what their levels are and how commonly represented they are in the data. @fig-paralympic-cat shows the distributions of two of the categorical variables in this dataset. We can see that the United States has hosted the Games most often, but runners from Great Britain and Kenya have won the 1500m most often. -There are a large number of coutries who have had a single gold medal winner of the 1500m. +There are a large number of countries who have had a single gold medal winner of the 1500m. Similarly, there are a large number of countries who have hosted the Games only once. -Over the last century, the name describing the country for athletes from one particular region has changed and includes Russian Federaion, Unified Team, and Russian Paralympic Committee. +Over the last century, the name describing the country for athletes from one particular region has changed and includes Russian Federation, Unified Team, and Russian Paralympic Committee. Both of the visualizations are bar plots, which you will learn more about in @sec-explore-categorical. ```{r} @@ -134,8 +128,9 @@ Both of the visualizations are bar plots, which you will learn more about in @se #| fig-cap: Distributions of categorical variables in the `paralympic_1500` dataset. #| Plot A shows the distribution of the country of origin of the athlete; Plot B #| shows the distribution of the country in which the Games gook place. -#| fig-asp: 1.0 -#| out.width: 100% +#| fig-alt: Two separate bar plots. The left panel shows a bar plot counting the number of gold medal athletes from each country. Great Britain has had 8 top finishers, Kenya has had 7 top finishers, and Tunisia and Algeria have both had 5. The right panel shows a bar plot counting the number of games which have happened in each country. The USA has hosted 4 games, the UK has hosted 3 games, and each of Japan, Greece, Germany, France, and Australia have hosted the games twice. +#| fig-asp: 1.4 +#| out-width: 100% p_country_games <- paralympic_1500 |> group_by(country_of_games, year) |> sample_n(size = 1) |> @@ -183,7 +178,8 @@ patchwork + ``` Similarly, we can examine the distributions of the numerical variables as well. -We already know that the 1500m times are mostly between 3.5min and 4.5min, based on @tbl-paralympic-df-tail and @tbl-paralympic-df-head. We can break down the 1500m time by division and type of race. +We already know that the 1500m times are mostly between 3.5min and 4.5min, based on @tbl-paralympic-df-tail and @tbl-paralympic-df-head. +We can break down the 1500m time by division and type of race. @tbl-paralympic-summary shows the mean, minimum, and maximum 1500m times broken down by division and race type. Recall that the Men's Olympic division has taken place since 1896, whereas the Men's Paralympic division has happened only since 1960. The maximum race time, therefore, should be taken into context in terms of the year of the Games. @@ -204,7 +200,37 @@ paralympic_1500 |> latex_options = c("striped", "hold_position")) ``` -\clearpage + +### Fun fact {-} + +Sometimes playing around with the dataset will uncover interesting elements about the context in which the data were collected. +A scatterplot of the Men's 1500m broken down by race type shows that, in each given year, the Olympic runner is substantially faster than the Paralympic runners, with one exception. +In the Rio de Janeiro 2016 games, the [T13 gold medal athlete ran faster (3:48.29) than the Olympic gold medal athlete (3:50.00)](https://www.paralympic.org/news/remarkable-finish-1500m-rio-2016) (see @fig-paralympic-rio). +In fact, some internet sleuthing tells you that the **top four** T13 finishers all finished the 1500m under 3:50.00! + +```{r} +#| label: fig-paralympic-rio +#| fig-cap: 1500m race time for Men's Olympic and Paralympic athletes. Dashed grey line represents the Rio games in 2016. +#| fig-alt: A scatterplot with year on the x-axis and gold medal 1500m time on the y-axis. The points are colored by which group the athlete is in - T11, T12, T13, or Olympic. A vertical line at 2016 show that in the Rio games the T13 gold medal athlete was faster than the Olympic gold medal athlete. +#| fig-asp: 1.2 +#| out-width: 100% + +paralympic_1500 |> + filter(division == "Men") |> + filter(year > 1950) |> + ggplot(aes(x = year, y = time_min, group = type, color = type)) + + geom_point() + + geom_vline(xintercept = 2016, color = "darkgrey", lty = 2) + + scale_color_openintro() + + labs( + x = "Year", + y = NULL, + color = "Race type", + title = "1500m race time, in minutes" + ) + + theme(plot.title.position = "plot") +``` + So far we examined aspects of some of the individual variables, and we have broken down the 1500m race times in terms of division and race type. You might have already wondered how the race times vary across year. @@ -216,16 +242,16 @@ Simpson's paradox \index{Simpson's paradox} is a description of three (or more) The paradox happens when a third variable reverses the relationship between the first two variables. Let's start by considering how the 1500m gold medal race times have changed over year. -@fig-paralympic-ungrouped shows a scatterplot describing 1500m race times and year for Men's Olympic and Paralympic (T11) athletes with a line of best fit (to the entire dataset) superimposed. +@fig-paralympic-ungrouped shows a scatterplot describing 1500m race times and year for Men's Olympic and Paralympic (T11) athletes with a line of best fit (to the entire dataset) superimposed (see @sec-model-slr where we will present fitting a line to a scatterplot). Notice that the line of best fit shows a **positive** relationship between race time and year. That is, for later years, the predicted gold medal time is higher than in earlier years. -The line of best fit will be further explored in @sec-model-slr. ```{r} #| label: fig-paralympic-ungrouped #| fig-cap: 1500m race time for Men's Olympic and Paralympic (T11) athletes. The line represents a line of best fit to the entire dataset. +#| fig-alt: A scatterplot with year on the x-axis and gold medal 1500m time on the y-axis. A line of best fit is drawn over the points. #| fig-asp: 1.2 -#| out.width: 100% +#| out-width: 100% paralympic_1500 |> filter(division == "Men", type == "Olympic" | type == "T11") |> @@ -249,8 +275,9 @@ Notice that within each type of race, the relationship between 1500m race time a ```{r} #| label: fig-paralympic-grouped #| fig-cap: 1500m race time for Men's Olympic and Paralympic (T11) athletes. The best fit line is now fit separately to the Olympic and Paralympic athletes. +#| fig-alt: A scatterplot with year on the x-axis and gold medal 1500m time on the y-axis. The points are colored by the type of athlete - T11 or Olympic. Lines of best fit are drawn separately for the two groups (T11 and Olympic). #| fig-asp: 1.2 -#| out.width: 100% +#| out-width: 100% paralympic_1500 |> filter(division == "Men", type == "Olympic" | type == "T11") |> @@ -258,6 +285,7 @@ paralympic_1500 |> ggplot(aes(x = year, y = time_min, group = type, color = type)) + geom_point() + geom_smooth(method = "lm", se = FALSE) + + scale_color_openintro() + labs( x = "Year", y = NULL, @@ -270,7 +298,7 @@ paralympic_1500 |> ::: {.important data-latex=""} **Simpson's paradox.** -An association or relationship between two variables in one direction (e.g., positive) that reverses (e.g., becomes negative) when broken down by a third variable is known as Simpson's paradox. +Simpson's paradox happens when an association or relationship between two variables in one direction (e.g., positive) reverses (e.g., becomes negative) when a third variable is considered. ::: ```{r} @@ -280,7 +308,7 @@ terms_chp_3 <- c("Simpson's paradox") Simpson's paradox was seen in the 1500m race data because the aggregate data showed a positive relationship (positive slope) between year and race time but a negative relationship (negative slope) between year and race time when broken down by the type of race. -Simpson's paradox can be seen with categorical or numeric data. +Simpson's paradox is observed with categorical data and with numeric data. Often the paradox happens because the third variable (here, race type) is imbalanced. There are either more observations in one group or the observations happen at different intervals across the two groups. In the 1500m data, we saw that the T11 runners had fewer observations and their times were both generally slower and more recent than the Olympic runners. @@ -295,11 +323,10 @@ In the next few chapters, we will introduce exploratory data analysis, and you'l Before you move on, we encourage you to think about whether the following questions can be answered with this dataset, and if yes, how you might go about answering them? It's okay if your answer is "I'm not sure", we simply want to get your exploratory juices flowing to prime you for what's to come! -1. Has there every been a year when a visually impaired paralympic gold medal athlete beat the olympic gold medal athlete? -2. When comparing the paralympic and olympic 1500m gold medal athletes, does Simpson's paradox hold in the Women's division? +1. Has there every been a year when a visually impaired paralympic gold medal athlete beat the Olympic gold medal athlete? +2. When comparing the paralympic and Olympic 1500m gold medal athletes, does Simpson's paradox hold in the Women's division? 3. Is there a biological boundary which establishes a time under which no human could run 1500m? -\clearpage ## Interactive R tutorials {#data-tutorials} diff --git a/_freeze/03-data-applications/execute-results/html.json b/_freeze/03-data-applications/execute-results/html.json index 51391c37..a494c95b 100644 --- a/_freeze/03-data-applications/execute-results/html.json +++ b/_freeze/03-data-applications/execute-results/html.json @@ -1,7 +1,7 @@ { - "hash": "055aff82c14a6ba9c74472da5be44373", + "hash": "f3578d82bbdcf85c82b70c2ef4b3db17", "result": { - "markdown": "# Applications: Data {#sec-data-applications}\n\n\n\n\n\n## Case study: Olympic 1500m {#case-study-passwords}\n\nWhile many of you may be glued to the Olympic Games every four years (or every two years if you fancy both summer and winter sports), the Paralympic Games are less popular than the Olympic Games, even if they hold the same competitive thrills.\n\nThe Paralympic Games began as a way to support soldiers who had been wounded in World War II as a way to help soldiers use sport to rehabilitate.\nThe first Paralympic Games were held in Rome, Italy in 1960.\nSince 1988 (Seoul, South Korea), the Paralymic Games have been held a few weeks later than the Olympic Games in the same city, in both the summer and winter.\n\nIn this case study we introduce a dataset comparing Olympic and Paralympic gold medal finishers in the 1500m running competition (the Olympic \"mile\", if a bit shorter than a full mile).\nThe goal of the case study is to walk you through what a data scientist does when they first get a hold of a dataset.\nWe also provide some \"foreshadowing\" of concepts and techniques we'll introduce in the next few chapters on exploratory data analysis.\nLast, we introduce [Simpson's paradox](https://en.wikipedia.org/wiki/Simpson%27s_paradox) and discuss the importance of understanding the impact of multiple variables in an analysis.\n\n::: {.data data-latex=\"\"}\nThe [`paralympic_1500`](http://openintrostat.github.io/openintro/reference/paralympic_1500.html) data can be found in the [openintro](http://openintrostat.github.io/openintro/) R package.\n:::\n\n@tbl-paralympic-df-tail shows the last ten rows from the dataset, which are the ten most recent 1500m races.\nNotice that there are racers from both the Men's and Women's divisions as well as those of varying visual impairment (T11, T12, T13, and Olympic).\nThe T11 athletes have almost complete visual impairment, run with a black-out blindfold, and are allowed to run with a guide-runner.\nT12 and T13 athletes have some visual impairment, and Olympic runners do not have visual impairment.\n\n\n::: {#tbl-paralympic-df-tail .cell tbl-cap='Last ten rows of the `paralympic_1500` dataset.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
year city country_of_games division type name country_of_athlete time time_min
2016 Rio de Janeiro Brazil Men T13 Abdellatif Baka Algeria 3:48.29 3.81
2016 Rio de Janeiro Brazil Women Olympic Faith Chepngetich Kipyegon Kenya 4:8.92 4.15
2016 Rio de Janeiro Brazil Women T11 Jin Zheng China 4:38.92 4.65
2016 Rio de Janeiro Brazil Women T13 Somaya Bousaid Tunisia 4:21.45 4.36
2020 Tokyo Japan Men Olympic Jakob Ingebrigtsen Norway 3:28.32 3.47
2020 Tokyo Japan Men T11 Yeltsin Jacques Brazil 3:57.6 3.96
2020 Tokyo Japan Men T13 Anton Kuliatin Russian Paralympic Committee 3:54.04 3.90
2020 Tokyo Japan Women Olympic Faith Chepngetich Kipyegon Kenya 3:53.11 3.88
2020 Tokyo Japan Women T11 Monica Olivia Rodriguez Saavedra Mexico 4:37.4 4.62
2020 Tokyo Japan Women T13 Tigist Gezahagn Menigstu Ethiopia 4:23.24 4.39
\n\n`````\n:::\n:::\n\n\nWhen you encounter a new dataset, taking a peek at the last few rows as we did in @tbl-paralympic-df-tail is almost instinctual.\nIt can often be helpful to look at the first few rows of the data as well to get a sense for other aspects of the data which may not be apparent int he last few rows.\n@tbl-paralympic-df-head shows the top ten rows of the `paralympic_1500` dataset, which reveals that for at least the first 10 Olympiads, there were no runners in the Women's division or in the Paralympics.\n\n\n::: {#tbl-paralympic-df-head .cell tbl-cap='First ten rows of the `paralympic_1500` dataset.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
year city country_of_games division type name country_of_athlete time time_min
1896 Athens Greece Men Olympic Edwin Flack Australia 4:33.2 4.55
1900 Paris France Men Olympic Charles Bennett Great Britain 4:6.2 4.10
1904 St Louis USA Men Olympic Jim Lightbody USA 4:5.4 4.09
1908 London United Kingdom Men Olympic Mel Sheppard USA 4:3.4 4.06
1912 Stockholm Sweden Men Olympic Arnold Jackson Great Britain 3:56.8 3.95
1920 Antwerp Belgium Men Olympic Albert Hill Great Britain 4:1.8 4.03
1924 Paris France Men Olympic Paavo Nurmi Finland 3:53.6 3.89
1928 Amsterdam Netherlands Men Olympic Harri Larva Finland 3:53.2 3.89
1932 Los Angeles USA Men Olympic Luigi Beccali Italy 3:51.2 3.85
1936 Berlin Germany Men Olympic Jack Lovelock New Zealand 3:47.8 3.80
\n\n`````\n:::\n:::\n\n\nAt this stage it's also useful to think about how the data were collected, as that will inform the scope of any inference you can make based on your analysis of the data.\n\n::: {.guidedpractice data-latex=\"\"}\nDo these data come from an observational study or an experiment?[^03-data-applications-1]\n:::\n\n[^03-data-applications-1]: This is an observational study.\n Researchers collected data on past gold medal race times in both Olympic and Paralympic Games.\n\n::: {.guidedpractice data-latex=\"\"}\nThere are 82 rows and 9 columns in the dataset.\nWhat does each row and each column represent?[^03-data-applications-2]\n:::\n\n[^03-data-applications-2]: Each row represents a 1500m gold medal race and each column represents a variable containing information on each race.\n\nOnce you've identified the rows and columns, it's useful to review the data dictionary to learn about what each column in the dataset represents.\nThe column information is provided in @tbl-paralympic-var-def.\n\n\n::: {#tbl-paralympic-var-def .cell tbl-cap='Variables and their descriptions for the `paralympic_1500` dataset.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
Variable Description
year Year the Games took place.
city City of the Games.
country_of_games Country of the Games.
division Division: `Men` or `Women`.
name Name of the athlete.
country_of_athlete Country of athlete.
time Time of gold medal race, in m:s.
time_min Time of gold medal race, in decimal minutes (min + sec/60).
\n\n`````\n:::\n:::\n\n\nWe now have a better sense of what each column represents, but we do not yet know much about the characteristics of each of the variables.\n\n::: {.workedexample data-latex=\"\"}\nDetermine whether each variable in the `paralympic_1500` dataset is numerical or categorical.\nFor numerical variables, further classify them as continuous or discrete.\nFor categorical variables, determine if the variable is ordinal.\n\n------------------------------------------------------------------------\n\nThe numerical variables in the dataset are `year` (discrete), and `time_min` (continuous).\nThe categorical variables are `city`, `country_of_games`, `division`, `type`, `name`, and `country_of_athlete`.\nThe `time` variable is trickier to classify -- we can think of it as numerical, but it is classified as categorical.\nThe categorical classification is due to the colon `:` which separates the hours from the seconds.\nThe lubridate R package is fantastic for working with date and time variables.\nSometimes the data dictionary (presented in @tbl-paralympic-var-def) isn't sufficient, and we need to go back to the data source and try to understand the data better before we can proceed with the analysis meaningfully.\n:::\n\nNext, let's try to get to know each variable a little bit better.\nFor categorical variables, this involves figuring out what their levels are and how commonly represented they are in the data.\n@fig-paralympic-cat shows the distributions of two of the categorical variables in this dataset.\nWe can see that the United States has hosted the Games most often, but runners from Great Britain and Kenya have won the 1500m most often.\nThere are a large number of coutries who have had a single gold medal winner of the 1500m.\nSimilarly, there are a large number of countries who have hosted the Games only once.\nOver the last century, the name describing the country for athletes from one particular region has changed and includes Russian Federaion, Unified Team, and Russian Paralympic Committee.\nBoth of the visualizations are bar plots, which you will learn more about in @sec-explore-categorical.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Distributions of categorical variables in the `paralympic_1500` dataset. Plot A shows the distribution of the country of origin of the athlete; Plot B shows the distribution of the country in which the Games gook place.](03-data-applications_files/figure-html/fig-paralympic-cat-1.png){#fig-paralympic-cat width=100%}\n:::\n:::\n\n\nSimilarly, we can examine the distributions of the numerical variables as well.\nWe already know that the 1500m times are mostly between 3.5min and 4.5min, based on @tbl-paralympic-df-tail and @tbl-paralympic-df-head. We can break down the 1500m time by division and type of race.\n@tbl-paralympic-summary shows the mean, minimum, and maximum 1500m times broken down by division and race type.\nRecall that the Men's Olympic division has taken place since 1896, whereas the Men's Paralympic division has happened only since 1960.\nThe maximum race time, therefore, should be taken into context in terms of the year of the Games.\n\n\n::: {#tbl-paralympic-summary .cell tbl-cap='Mean, minimum, and maximum of the gold medal times for the 1500m race broken down by division and type of race.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
division type mean min max
Men Olympic 3.76 3.47 4.55
Men T11 4.14 3.96 4.31
Men T12 4.11 3.94 4.25
Men T13 3.98 3.81 4.24
Women Olympic 4.02 3.88 4.18
Women T11 5.05 4.62 5.63
Women T12 4.88 4.61 5.57
Women T13 4.55 4.23 5.24
\n\n`````\n:::\n:::\n\n\n\\clearpage\n\nSo far we examined aspects of some of the individual variables, and we have broken down the 1500m race times in terms of division and race type.\nYou might have already wondered how the race times vary across year.\nThe `paralymic_1500` dataset will provide us with an ability to explore an important statistical concept, Simpson's paradox.\n\n## Simpson's paradox\n\nSimpson's paradox \\index{Simpson's paradox} is a description of three (or more) variables.\nThe paradox happens when a third variable reverses the relationship between the first two variables.\n\nLet's start by considering how the 1500m gold medal race times have changed over year.\n@fig-paralympic-ungrouped shows a scatterplot describing 1500m race times and year for Men's Olympic and Paralympic (T11) athletes with a line of best fit (to the entire dataset) superimposed.\nNotice that the line of best fit shows a **positive** relationship between race time and year.\nThat is, for later years, the predicted gold medal time is higher than in earlier years.\nThe line of best fit will be further explored in @sec-model-slr.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![1500m race time for Men's Olympic and Paralympic (T11) athletes. The line represents a line of best fit to the entire dataset.](03-data-applications_files/figure-html/fig-paralympic-ungrouped-1.png){#fig-paralympic-ungrouped width=100%}\n:::\n:::\n\n\nOf course, both your eye and your intuition are likely telling you that it wouldn't make any sense to try to model all of the athletes together.\nInstead, a separate model should be run for each of the two types of Games: Olympic and Paralympic (T11).\n@fig-paralympic-grouped shows a scatterplot describing 1500m race times and year for Men's Olympic and Paralympic (T11) athletes with a line of best fit superimposed separately for each of the two types of races.\nNotice that within each type of race, the relationship between 1500m race time and year is now **negative**.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![1500m race time for Men's Olympic and Paralympic (T11) athletes. The best fit line is now fit separately to the Olympic and Paralympic athletes.](03-data-applications_files/figure-html/fig-paralympic-grouped-1.png){#fig-paralympic-grouped width=100%}\n:::\n:::\n\n\n::: {.important data-latex=\"\"}\n**Simpson's paradox.**\n\nAn association or relationship between two variables in one direction (e.g., positive) that reverses (e.g., becomes negative) when broken down by a third variable is known as Simpson's paradox.\n:::\n\n\n\n\n\nSimpson's paradox was seen in the 1500m race data because the aggregate data showed a positive relationship (positive slope) between year and race time but a negative relationship (negative slope) between year and race time when broken down by the type of race.\n\nSimpson's paradox can be seen with categorical or numeric data.\nOftentimes the paradox happens because the third variable (here, race type) is imbalanced.\nThere are either more observations in one group or the observations happen at different intervals across the two groups.\nIn the 1500m data, we saw that the T11 runners had fewer observations and their times were both generally slower and more recent than the Olympic runners.\n\nIn the 1500m analysis, it would be most prudent to report the trends separately for the Olympic and the T11 athletes.\nHowever, in other situations, it might be better to aggregate the data and report the overall trend.\nMany additional examples of Simpson's paradox and a further exploration is given in @Witmer:2021.\n\nIn this case study, we introduced you to the very first steps a data scientist takes when they start working with a new dataset.\nIn the next few chapters, we will introduce exploratory data analysis, and you'll learn more about the various types of data visualizations and summary statistics you can make to get to know your data better.\n\nBefore you move on, we encourage you to think about whether the following questions can be answered with this dataset, and if yes, how you might go about answering them?\nIt's okay if your answer is \"I'm not sure\", we simply want to get your exploratory juices flowing to prime you for what's to come!\n\n1. Has there every been a year when a visually impaired paralympic gold medal athlete beat the olympic gold medal athlete?\n2. When comparing the paralympic and olympic 1500m gold medal athletes, does Simpson's paradox hold in the Women's division?\n3. Is there a biological boundary which establishes a time under which no human could run 1500m?\n\n\\clearpage\n\n## Interactive R tutorials {#data-tutorials}\n\nNavigate the concepts you've learned in this chapter in R using the following self-paced tutorials.\nAll you need is your browser to get started!\n\n::: {.alltutorials data-latex=\"\"}\n[Tutorial 1: Introduction to data](https://openintrostat.github.io/ims-tutorials/01-data/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintrostat.github.io/ims-tutorials/01-data\n:::\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 1 - Lesson 1: Language of data](https://openintro.shinyapps.io/ims-01-data-01/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-01-data-01\n:::\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 1 - Lesson 2: Types of studies](https://openintro.shinyapps.io/ims-01-data-02/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-01-data-02\n:::\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 1 - Lesson 3: Sampling strategies and experimental design](https://openintro.shinyapps.io/ims-01-data-03/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-01-data-03\n:::\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 1 - Lesson 4: Case study](https://openintro.shinyapps.io/ims-01-data-04/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-01-data-04\n:::\n:::\n\n::: {.content-hidden unless-format=\"pdf\"}\nYou can also access the full list of tutorials supporting this book at\\\n.\n:::\n\n::: {.content-visible when-format=\"html\"}\nYou can also access the full list of tutorials supporting this book [here](https://openintrostat.github.io/ims-tutorials).\n:::\n\n## R labs {#data-labs}\n\nFurther apply the concepts you've learned in this part in R with computational labs that walk you through a data analysis case study.\n\n::: {.singlelab data-latex=\"\"}\n[Intro to R - Birth rates](https://www.openintro.org/go?id=ims-r-lab-intro-to-r)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://www.openintro.org/go?i\nd=ims-r-lab-intro-to-r\n:::\n:::\n\n::: {.content-hidden unless-format=\"pdf\"}\nYou can also access the full list of labs supporting this book at\\\n.\n:::\n\n::: {.content-visible when-format=\"html\"}\nYou can also access the full list of labs supporting this book [here](https://www.openintro.org/go?id=ims-r-labs).\n:::\n", + "markdown": "# Applications: Data {#sec-data-applications}\n\n\n\n\n\n## Case study: Olympic 1500m {#case-study-paralympics}\n\nWhile many of you may be glued to the Olympic Games every four years (or every two years if you fancy both summer and winter sports), the Paralympic Games are less popular than the Olympic Games, even if they hold the same competitive thrills.\n\nThe Paralympic Games began as a way to support soldiers who had been wounded in World War II as a way to help them rehabilitate.\nThe first Paralympic Games were held in Rome, Italy in 1960.\nSince 1988 (Seoul, South Korea), the Paralympic Games have been held a few weeks later than the Olympic Games in the same city, in both the summer and winter.\n\nIn this case study we introduce a dataset comparing Olympic and Paralympic gold medal finishers in the 1500m running competition (the Olympic \"mile\", if a bit shorter than a full mile).\nThe goal of the case study is to walk you through what a data scientist does when they first get a hold of a dataset.\nWe also provide some \"foreshadowing\" of concepts and techniques we'll introduce in the next few chapters on exploratory data analysis.\nLast, we introduce [Simpson's paradox](https://en.wikipedia.org/wiki/Simpson%27s_paradox) and discuss the importance of understanding the impact of multiple variables in an analysis.\n\n::: {.data data-latex=\"\"}\nThe [`paralympic_1500`](http://openintrostat.github.io/openintro/reference/paralympic_1500.html) data can be found in the [openintro](http://openintrostat.github.io/openintro/) R package.\n:::\n\n@tbl-paralympic-df-tail shows the last ten rows from the dataset, which are the ten most recent 1500m races.\nNotice that there are racers from both the Men's and Women's divisions as well as those of varying visual impairment (T11, T12, T13, and Olympic).\nThe T11 athletes have almost complete visual impairment, run with a black-out blindfold, and are allowed to run with a guide-runner.\nT12 and T13 athletes have some visual impairment, and the visual acuity of Olympic runners is not determined.\n\n\n::: {#tbl-paralympic-df-tail .cell tbl-cap='Last ten rows of the `paralympic_1500` dataset.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
year city country_of_games division type name country_of_athlete time time_min
2016 Rio de Janeiro Brazil Men T13 Abdellatif Baka Algeria 3:48.29 3.81
2016 Rio de Janeiro Brazil Women Olympic Faith Chepngetich Kipyegon Kenya 4:8.92 4.15
2016 Rio de Janeiro Brazil Women T11 Jin Zheng China 4:38.92 4.65
2016 Rio de Janeiro Brazil Women T13 Somaya Bousaid Tunisia 4:21.45 4.36
2020 Tokyo Japan Men Olympic Jakob Ingebrigtsen Norway 3:28.32 3.47
2020 Tokyo Japan Men T11 Yeltsin Jacques Brazil 3:57.6 3.96
2020 Tokyo Japan Men T13 Anton Kuliatin Russian Paralympic Committee 3:54.04 3.90
2020 Tokyo Japan Women Olympic Faith Chepngetich Kipyegon Kenya 3:53.11 3.88
2020 Tokyo Japan Women T11 Monica Olivia Rodriguez Saavedra Mexico 4:37.4 4.62
2020 Tokyo Japan Women T13 Tigist Gezahagn Menigstu Ethiopia 4:23.24 4.39
\n\n`````\n:::\n:::\n\n\nWhen you encounter a new dataset, taking a peek at the last few rows as we did in @tbl-paralympic-df-tail should be almost instinctual.\nIt can often be helpful to look at the first few rows of the data as well to get a sense of other aspects of the data which may not be apparent int he last few rows.\n@tbl-paralympic-df-head shows the top ten rows of the `paralympic_1500` dataset, which reveals that for at least the first 10 Olympiads, there were no runners in the Women's division or in the Paralympics.\n\n\n::: {#tbl-paralympic-df-head .cell tbl-cap='First ten rows of the `paralympic_1500` dataset.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
year city country_of_games division type name country_of_athlete time time_min
1896 Athens Greece Men Olympic Edwin Flack Australia 4:33.2 4.55
1900 Paris France Men Olympic Charles Bennett Great Britain 4:6.2 4.10
1904 St Louis USA Men Olympic Jim Lightbody USA 4:5.4 4.09
1908 London United Kingdom Men Olympic Mel Sheppard USA 4:3.4 4.06
1912 Stockholm Sweden Men Olympic Arnold Jackson Great Britain 3:56.8 3.95
1920 Antwerp Belgium Men Olympic Albert Hill Great Britain 4:1.8 4.03
1924 Paris France Men Olympic Paavo Nurmi Finland 3:53.6 3.89
1928 Amsterdam Netherlands Men Olympic Harri Larva Finland 3:53.2 3.89
1932 Los Angeles USA Men Olympic Luigi Beccali Italy 3:51.2 3.85
1936 Berlin Germany Men Olympic Jack Lovelock New Zealand 3:47.8 3.80
\n\n`````\n:::\n:::\n\n\nAt this stage it's also useful to think about how the data were collected, as that will inform the scope of any inference you can make based on your analysis of the data.\n\n::: {.guidedpractice data-latex=\"\"}\nDo these data come from an observational study or an experiment?[^03-data-applications-1]\n:::\n\n[^03-data-applications-1]: This is an observational study.\n Researchers collected data on past gold medal race times in both Olympic and Paralympic Games.\n\n::: {.guidedpractice data-latex=\"\"}\nThere are 82 rows and 9 columns in the dataset.\nWhat does each row and each column represent?[^03-data-applications-2]\n:::\n\n[^03-data-applications-2]: Each row represents a 1500m gold medal race and each column represents a variable containing information on each race.\n\nOnce you've identified the rows and columns, it's useful to review the data dictionary to learn about what each column in the dataset represents.\nThe data dictionary is provided in @tbl-paralympic-var-def.\n\n\n::: {#tbl-paralympic-var-def .cell tbl-cap='Variables and their descriptions for the `paralympic_1500` dataset.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
Variable Description
year Year the Games took place.
city City of the Games.
country_of_games Country of the Games.
division Division: `Men` or `Women`.
name Name of the athlete.
country_of_athlete Country of athlete.
time Time of gold medal race, in m:s.
time_min Time of gold medal race, in decimal minutes (min + sec/60).
\n\n`````\n:::\n:::\n\n\nWe now have a better sense of what each column represents, but we do not yet know much about the characteristics of each of the variables.\n\n::: {.workedexample data-latex=\"\"}\nDetermine whether each variable in the `paralympic_1500` dataset is numerical or categorical.\nFor numerical variables, further classify them as continuous or discrete.\nFor categorical variables, determine if the variable is ordinal.\n\n------------------------------------------------------------------------\n\nThe numerical variables in the dataset are `year` (discrete), and `time_min` (continuous).\nThe categorical variables are `city`, `country_of_games`, `division`, `type`, `name`, and `country_of_athlete`.\nThe `time` variable is trickier to classify -- we can think of it as numerical, but it is classified as categorical.\nThe categorical classification is due to the colon `:` which separates the hours from the seconds.\nSometimes the data dictionary (presented in @tbl-paralympic-var-def) isn't sufficient for a complete analysis, and we need to go back to the data source and try to understand the data better before we can proceed with the analysis meaningfully.\n:::\n\nNext, let's try to get to know each variable a little bit better.\nFor categorical variables, this involves figuring out what their levels are and how commonly represented they are in the data.\n@fig-paralympic-cat shows the distributions of two of the categorical variables in this dataset.\nWe can see that the United States has hosted the Games most often, but runners from Great Britain and Kenya have won the 1500m most often.\nThere are a large number of countries who have had a single gold medal winner of the 1500m.\nSimilarly, there are a large number of countries who have hosted the Games only once.\nOver the last century, the name describing the country for athletes from one particular region has changed and includes Russian Federation, Unified Team, and Russian Paralympic Committee.\nBoth of the visualizations are bar plots, which you will learn more about in @sec-explore-categorical.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![Distributions of categorical variables in the `paralympic_1500` dataset. Plot A shows the distribution of the country of origin of the athlete; Plot B shows the distribution of the country in which the Games gook place.](03-data-applications_files/figure-html/fig-paralympic-cat-1.png){#fig-paralympic-cat fig-alt='Two separate bar plots. The left panel shows a bar plot counting the number of gold medal athletes from each country. Great Britain has had 8 top finishers, Kenya has had 7 top finishers, and Tunisia and Algeria have both had 5. The right panel shows a bar plot counting the number of games which have happened in each country. The USA has hosted 4 games, the UK has hosted 3 games, and each of Japan, Greece, Germany, France, and Australia have hosted the games twice.' width=100%}\n:::\n:::\n\n\nSimilarly, we can examine the distributions of the numerical variables as well.\nWe already know that the 1500m times are mostly between 3.5min and 4.5min, based on @tbl-paralympic-df-tail and @tbl-paralympic-df-head.\nWe can break down the 1500m time by division and type of race.\n@tbl-paralympic-summary shows the mean, minimum, and maximum 1500m times broken down by division and race type.\nRecall that the Men's Olympic division has taken place since 1896, whereas the Men's Paralympic division has happened only since 1960.\nThe maximum race time, therefore, should be taken into context in terms of the year of the Games.\n\n\n::: {#tbl-paralympic-summary .cell tbl-cap='Mean, minimum, and maximum of the gold medal times for the 1500m race broken down by division and type of race.'}\n::: {.cell-output-display}\n`````{=html}\n\n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n
division type mean min max
Men Olympic 3.76 3.47 4.55
Men T11 4.14 3.96 4.31
Men T12 4.11 3.94 4.25
Men T13 3.98 3.81 4.24
Women Olympic 4.02 3.88 4.18
Women T11 5.05 4.62 5.63
Women T12 4.88 4.61 5.57
Women T13 4.55 4.23 5.24
\n\n`````\n:::\n:::\n\n\n\n### Fun fact {-}\n\nSometimes playing around with the dataset will uncover interesting elements about the context in which the data were collected.\nA scatterplot of the Men's 1500m broken down by race type shows that, in each given year, the Olympic runner is substantially faster than the Paralympic runners, with one exception.\nIn the Rio de Janeiro 2016 games, the [T13 gold medal athlete ran faster (3:48.29) than the Olympic gold medal athlete (3:50.00)](https://www.paralympic.org/news/remarkable-finish-1500m-rio-2016) (see @fig-paralympic-rio).\nIn fact, some internet sleuthing tells you that the **top four** T13 finishers all finished the 1500m under 3:50.00!\n\n\n::: {.cell}\n::: {.cell-output-display}\n![1500m race time for Men's Olympic and Paralympic athletes. Dashed grey line represents the Rio games in 2016.](03-data-applications_files/figure-html/fig-paralympic-rio-1.png){#fig-paralympic-rio fig-alt='A scatterplot with year on the x-axis and gold medal 1500m time on the y-axis. The points are colored by which group the athlete is in - T11, T12, T13, or Olympic. A vertical line at 2016 show that in the Rio games the T13 gold medal athlete was faster than the Olympic gold medal athlete.' width=100%}\n:::\n:::\n\n\n\nSo far we examined aspects of some of the individual variables, and we have broken down the 1500m race times in terms of division and race type.\nYou might have already wondered how the race times vary across year.\nThe `paralymic_1500` dataset will provide us with an ability to explore an important statistical concept, Simpson's paradox.\n\n## Simpson's paradox\n\nSimpson's paradox \\index{Simpson's paradox} is a description of three (or more) variables.\nThe paradox happens when a third variable reverses the relationship between the first two variables.\n\nLet's start by considering how the 1500m gold medal race times have changed over year.\n@fig-paralympic-ungrouped shows a scatterplot describing 1500m race times and year for Men's Olympic and Paralympic (T11) athletes with a line of best fit (to the entire dataset) superimposed (see @sec-model-slr where we will present fitting a line to a scatterplot). \nNotice that the line of best fit shows a **positive** relationship between race time and year.\nThat is, for later years, the predicted gold medal time is higher than in earlier years.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![1500m race time for Men's Olympic and Paralympic (T11) athletes. The line represents a line of best fit to the entire dataset.](03-data-applications_files/figure-html/fig-paralympic-ungrouped-1.png){#fig-paralympic-ungrouped fig-alt='A scatterplot with year on the x-axis and gold medal 1500m time on the y-axis. A line of best fit is drawn over the points.' width=100%}\n:::\n:::\n\n\nOf course, both your eye and your intuition are likely telling you that it wouldn't make any sense to try to model all of the athletes together.\nInstead, a separate model should be run for each of the two types of Games: Olympic and Paralympic (T11).\n@fig-paralympic-grouped shows a scatterplot describing 1500m race times and year for Men's Olympic and Paralympic (T11) athletes with a line of best fit superimposed separately for each of the two types of races.\nNotice that within each type of race, the relationship between 1500m race time and year is now **negative**.\n\n\n::: {.cell}\n::: {.cell-output-display}\n![1500m race time for Men's Olympic and Paralympic (T11) athletes. The best fit line is now fit separately to the Olympic and Paralympic athletes.](03-data-applications_files/figure-html/fig-paralympic-grouped-1.png){#fig-paralympic-grouped fig-alt='A scatterplot with year on the x-axis and gold medal 1500m time on the y-axis. The points are colored by the type of athlete - T11 or Olympic. Lines of best fit are drawn separately for the two groups (T11 and Olympic).' width=100%}\n:::\n:::\n\n\n::: {.important data-latex=\"\"}\n**Simpson's paradox.**\n\nSimpson's paradox happens when an association or relationship between two variables in one direction (e.g., positive) reverses (e.g., becomes negative) when a third variable is considered.\n:::\n\n\n\n\n\nSimpson's paradox was seen in the 1500m race data because the aggregate data showed a positive relationship (positive slope) between year and race time but a negative relationship (negative slope) between year and race time when broken down by the type of race.\n\nSimpson's paradox is observed with categorical data and with numeric data.\nOften the paradox happens because the third variable (here, race type) is imbalanced.\nThere are either more observations in one group or the observations happen at different intervals across the two groups.\nIn the 1500m data, we saw that the T11 runners had fewer observations and their times were both generally slower and more recent than the Olympic runners.\n\nIn the 1500m analysis, it would be most prudent to report the trends separately for the Olympic and the T11 athletes.\nHowever, in other situations, it might be better to aggregate the data and report the overall trend.\nMany additional examples of Simpson's paradox and a further exploration is given in @Witmer:2021.\n\nIn this case study, we introduced you to the very first steps a data scientist takes when they start working with a new dataset.\nIn the next few chapters, we will introduce exploratory data analysis, and you'll learn more about the various types of data visualizations and summary statistics you can make to get to know your data better.\n\nBefore you move on, we encourage you to think about whether the following questions can be answered with this dataset, and if yes, how you might go about answering them?\nIt's okay if your answer is \"I'm not sure\", we simply want to get your exploratory juices flowing to prime you for what's to come!\n\n1. Has there every been a year when a visually impaired paralympic gold medal athlete beat the Olympic gold medal athlete?\n2. When comparing the paralympic and Olympic 1500m gold medal athletes, does Simpson's paradox hold in the Women's division?\n3. Is there a biological boundary which establishes a time under which no human could run 1500m?\n\n\n## Interactive R tutorials {#data-tutorials}\n\nNavigate the concepts you've learned in this chapter in R using the following self-paced tutorials.\nAll you need is your browser to get started!\n\n::: {.alltutorials data-latex=\"\"}\n[Tutorial 1: Introduction to data](https://openintrostat.github.io/ims-tutorials/01-data/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintrostat.github.io/ims-tutorials/01-data\n:::\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 1 - Lesson 1: Language of data](https://openintro.shinyapps.io/ims-01-data-01/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-01-data-01\n:::\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 1 - Lesson 2: Types of studies](https://openintro.shinyapps.io/ims-01-data-02/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-01-data-02\n:::\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 1 - Lesson 3: Sampling strategies and experimental design](https://openintro.shinyapps.io/ims-01-data-03/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-01-data-03\n:::\n:::\n\n::: {.singletutorial data-latex=\"\"}\n[Tutorial 1 - Lesson 4: Case study](https://openintro.shinyapps.io/ims-01-data-04/)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://openintro.shinyapps.io/ims-01-data-04\n:::\n:::\n\n::: {.content-hidden unless-format=\"pdf\"}\nYou can also access the full list of tutorials supporting this book at\\\n.\n:::\n\n::: {.content-visible when-format=\"html\"}\nYou can also access the full list of tutorials supporting this book [here](https://openintrostat.github.io/ims-tutorials).\n:::\n\n## R labs {#data-labs}\n\nFurther apply the concepts you've learned in this part in R with computational labs that walk you through a data analysis case study.\n\n::: {.singlelab data-latex=\"\"}\n[Intro to R - Birth rates](https://www.openintro.org/go?id=ims-r-lab-intro-to-r)\n\n::: {.content-hidden unless-format=\"pdf\"}\nhttps://www.openintro.org/go?i\nd=ims-r-lab-intro-to-r\n:::\n:::\n\n::: {.content-hidden unless-format=\"pdf\"}\nYou can also access the full list of labs supporting this book at\\\n.\n:::\n\n::: {.content-visible when-format=\"html\"}\nYou can also access the full list of labs supporting this book [here](https://www.openintro.org/go?id=ims-r-labs).\n:::\n", "supporting": [ "03-data-applications_files" ], diff --git a/_freeze/03-data-applications/figure-html/fig-paralympic-cat-1.png b/_freeze/03-data-applications/figure-html/fig-paralympic-cat-1.png index f4f5c356977c242f839a87d6c4376e7f985e99df..a5300f1155aa5ea61123f54a320da1b12c9fa08a 100644 GIT binary patch literal 535857 zcmeEu1zVNt)-@{V7Hflql1hi5A_4+}N~x5z5@G<-Azh-Fw1g-n3Ifv74WiN|Al-s= 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