moderndive: Datasets and functions for tidyverse-friendly introductory linear regression

Albert Y. Kim, Chester Ismay, and Max Kuhn

2021-07-21

1 Summary

We present the moderndive R package of datasets and functions for tidyverse-friendly introductory linear regression (Wickham, Averick, et al. 2019). These tools leverage the well-developed tidyverse and broom packages to facilitate 1) working with regression tables that include confidence intervals, 2) accessing regression outputs on an observation level (e.g. fitted/predicted values and residuals), 3) inspecting scalar summaries of regression fit (e.g. \(R^2\), \(R^2_{adj}\), and mean squared error), and 4) visualizing parallel slopes regression models using ggplot2-like syntax (Wickham, Chang, et al. 2019; Robinson and Hayes 2019). This R package is designed to supplement the book “Statistical Inference via Data Science: A ModernDive into R and the Tidyverse” (Ismay and Kim 2019). Note that the book is also available online at https://moderndive.com and is referred to as “ModernDive” for short.

2 Statement of Need

Linear regression has long been a staple of introductory statistics courses. While the curricula of introductory statistics courses has much evolved of late, the overall importance of regression remains the same (American Statistical Association Undergraduate Guidelines Workgroup 2016). Furthermore, while the use of the R statistical programming language for statistical analysis is not new, recent developments such as the tidyverse suite of packages have made statistical computation with R accessible to a broader audience (Wickham, Averick, et al. 2019). We go one step further by leveraging the tidyverse and the broom packages to make linear regression accessible to students taking an introductory statistics course (Robinson and Hayes 2019). Such students are likely to be new to statistical computation with R; we designed moderndive with these students in mind.

3 Introduction

Let’s load all the R packages we are going to need.

library(moderndive)
library(ggplot2)
library(dplyr)
library(knitr)
library(broom)

Let’s consider data gathered from end of semester student evaluations for a sample of 463 courses taught by 94 professors from the University of Texas at Austin (Diez, Barr, and Çetinkaya-Rundel 2015). This data is included in the evals data frame from the moderndive package.

In the following table, we present a subset of 9 of the 14 variables included for a random sample of 5 courses1:

  1. ID uniquely identifies the course whereas prof_ID identifies the professor who taught this course. This distinction is important since many professors taught more than one course.
  2. score is the outcome variable of interest: average professor evaluation score out of 5 as given by the students in this course.
  3. The remaining variables are demographic variables describing that course’s instructor, including bty_avg (average “beauty” score) for that professor as given by a panel of 6 students.2
ID prof_ID score age bty_avg gender ethnicity language rank
129 23 3.7 62 3.000 male not minority english tenured
109 19 4.7 46 4.333 female not minority english tenured
28 6 4.8 62 5.500 male not minority english tenured
434 88 2.8 62 2.000 male not minority english tenured
330 66 4.0 64 2.333 male not minority english tenured

3.1 Regression analysis the “good old-fashioned” way

Let’s fit a simple linear regression model of teaching score as a function of instructor age using the lm() function.

Let’s now study the output of the fitted model score_model “the good old-fashioned way”: using summary() which calls summary.lm() behind the scenes (we’ll refer to them interchangeably throughout this paper).

Here are 5 common student questions we’ve heard over the years in our introductory statistics courses based on this output:

  1. “Wow! Look at those p-value stars! Stars are good, so I should try to get many stars, right?”
  2. “How do we extract the values in the regression table?”
  3. “Where are the fitted/predicted values and residuals?”
  4. “How do I apply this model to a new set of data to make predictions?”
  5. “What is all this other stuff at the bottom?”

3.2 Regression analysis using moderndive

To address these questions, we’ve included three functions in the moderndive package that take a fitted model object as input and return the same information as summary.lm(), but output them in tidyverse-friendly format (Wickham, Averick, et al. 2019). As we’ll see later, while these three functions are thin wrappers to existing functions in the broom package for converting statistical objects into tidy tibbles, we modified them with the introductory statistics student in mind (Robinson and Hayes 2019).

  1. Get a tidy regression table with confidence intervals:

  2. Get information on each point/observation in your regression, including fitted/predicted values and residuals, in a single data frame:

  3. Get scalar summaries of a regression fit including \(R^2\) and \(R^2_{adj}\) but also the (root) mean-squared error:

3.3 Bonus: Visualizing parallel slopes models with moderndive

Furthermore, say you would like to create a visualization of the relationship between two numerical variables and a third categorical variable with \(k\) levels. Let’s create this using a colored scatterplot via the ggplot2 package for data visualization (Wickham, Chang, et al. 2019). Using geom_smooth(method = "lm", se = FALSE) yields a visualization of an interaction model where each of the \(k\) regression lines has their own intercept and slope. For example in , we extend our previous regression model by now mapping the categorical variable ethnicity to the color aesthetic.

Visualization of interaction model.

Visualization of interaction model.

However, many introductory statistics courses start with the easier to teach “common slope, different intercepts” regression model, also known as the parallel slopes model. However, no argument to plot such models exists within geom_smooth().

Evgeni Chasnovski thus wrote a custom geom_ extension to ggplot2 called geom_parallel_slopes(); this extension is included in the moderndive package. Much like geom_smooth() from the ggplot2 package, you add geom_parallel_slopes() as a layer to the code, resulting in .

Visualization of parallel slopes model.

Visualization of parallel slopes model.

At this point however, students will inevitably ask a sixth question: “When would you ever use a parallel slopes model?”

3.4 Why should you use the moderndive package?

To recap this introduction, we believe that the following functions included in the moderndive package

  1. get_regression_table()
  2. get_regression_points()
  3. get_regression_summaries()
  4. geom_parallel_slopes()

are effective pedagogical tools that can help address the six common questions posed by students about introductory linear regression performed in R. We now argue why.

4 Features

4.1 1. Less p-value stars, more confidence intervals

The first common student question:

“Wow! Look at those p-value stars! Stars are good, so I should try to get many stars, right?”

We argue that the summary.lm() output is deficient in an introductory statistics setting because:

  1. The Signif. codes: 0 '' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 only encourage p-hacking. In case you have not yet been convinced of the perniciousness of p-hacking, perhaps comedian John Oliver can convince you.
  2. While not a silver bullet for eliminating misinterpretations of statistical inference, confidence intervals present students with a sense of the associated effect sizes of any explanatory variables. Thus, practical as well as statistical significance is emphasized. These are not included by default in the output of summary.lm().

Instead of summary(), let’s use the get_regression_table() function:

Observe how the p-value stars are omitted and confidence intervals for the point estimates of all regression parameters are included by default. By including them in the output, we can then emphasize to students that they “surround” the point estimates in the estimate column. Note the confidence level is defaulted to 95%; this default can be changed using the conf.level argument:

4.2 2. Outputs as tibbles

The second common student question:

“How do we extract the values in the regression table?”

While one might argue that extracting the intercept and slope coefficients can be “simply” done using coefficients(score_model), what about the standard errors? For example, a Google query of “how do I extract standard errors from lm in R” yielded results from the R mailing list and from Cross Validated suggesting we run:

We argue that this task shouldn’t be this hard, especially in an introductory statistics setting. To rectify this, the three get_regression_* functions all return data frames in the tidyverse-style tibble (tidy table) format (Müller and Wickham 2019). Therefore you can extract columns using the pull() function from the dplyr package (Wickham et al. 2020):

or equivalently you can use the $ sign operator from base R:

Furthermore, by piping the above get_regression_table(score_model) output into the kable() function from the knitr package (Xie 2020), you can obtain aesthetically pleasing regression tables in R Markdown documents, instead of tables written in jarring computer output font:

term estimate std_error statistic p_value lower_ci upper_ci
intercept 4.462 0.127 35.195 0.000 4.213 4.711
age -0.006 0.003 -2.311 0.021 -0.011 -0.001

4.3 3. Birds of a feather should flock together: Fitted values & residuals

The third common student question:

“Where are the fitted/predicted values and residuals?”

How can we extract point-by-point information from a regression model, such as the fitted/predicted values and the residuals? (Note we only display the first 10 out of 463 of such values for brevity’s sake.)

##        1        2        3        4        5        6        7 
## 4.248156 4.248156 4.248156 4.248156 4.111577 4.111577 4.111577 
##        8        9       10 
## 4.159083 4.159083 4.224403
##           1           2           3           4           5 
##  0.45184376 -0.14815624 -0.34815624  0.55184376  0.48842294 
##           6           7           8           9          10 
##  0.18842294 -1.31157706 -0.05908286 -0.75908286  0.27559666

But why have the original explanatory/predictor age and outcome variable score in evals, the fitted/predicted values score_hat, and residual floating around in separate vectors? Since each observation relates to the same course, we argue it makes more sense to organize them together in the same data frame using get_regression_points():

## # A tibble: 10 x 5
##       ID score   age score_hat residual
##    <int> <dbl> <int>     <dbl>    <dbl>
##  1     1   4.7    36      4.25    0.452
##  2     2   4.1    36      4.25   -0.148
##  3     3   3.9    36      4.25   -0.348
##  4     4   4.8    36      4.25    0.552
##  5     5   4.6    59      4.11    0.488
##  6     6   4.3    59      4.11    0.188
##  7     7   2.8    59      4.11   -1.31 
##  8     8   4.1    51      4.16   -0.059
##  9     9   3.4    51      4.16   -0.759
## 10    10   4.5    40      4.22    0.276

Observe that the original outcome variable score and explanatory/predictor variable age are now supplemented with the fitted/predicted values score_hat and residual columns. By putting the fitted values, predicted values, and residuals next to the original data, we argue that the computation of these values is less opaque. For example, instructors can emphasize how all values in the first row of output are computed.

Furthermore, recall that since all outputs in the moderndive package are tibble data frames, custom residual analysis plots can be created instead of relying on the default plots yielded by plot.lm(). For example, we can check for the normality of residuals using the histogram of residuals shown in .

Histogram visualizing distribution of residuals.

Histogram visualizing distribution of residuals.

As another example, we can investigate potential relationships between the residuals and all explanatory/predictor variables and the presence of heteroskedasticity using partial residual plots, like the partial residual plot over age shown in . If the term “heteroskedasticity” is new to you, it corresponds to the variability of one variable being unequal across the range of values of another variable. The presence of heteroskedasticity violates one of the assumptions of inference for linear regression.

Partial residual residual plot over age.

Partial residual residual plot over age.

4.4 4. A quick-and-easy Kaggle predictive modeling competition submission!

The fourth common student question:

“How do I apply this model to a new set of data to make predictions?”

With the fields of machine learning and artificial intelligence gaining prominence, the importance of predictive modeling cannot be understated. Therefore, we’ve designed the get_regression_points() function to allow for a newdata argument to quickly apply a previously fitted model to new observations.

Let’s create an artificial “new” dataset consisting of two instructors of age 39 and 42 and save it in a tibble data frame called new_prof. We then set the newdata argument to get_regression_points() to apply our previously fitted model score_model to this new data, where score_hat holds the corresponding fitted/predicted values.

Let’s do another example, this time using the Kaggle House Prices: Advanced Regression Techniques practice competition ( displays the homepage for this competition).