1.1 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.

score_model <- lm(score ~ age, data = evals)

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).

summary(score_model)
## 
## Call:
## lm(formula = score ~ age, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.9185 -0.3531  0.1172  0.4172  0.8825 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  4.461932   0.126778  35.195   <2e-16 ***
## age         -0.005938   0.002569  -2.311   0.0213 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5413 on 461 degrees of freedom
## Multiple R-squared:  0.01146,    Adjusted R-squared:  0.009311 
## F-statistic: 5.342 on 1 and 461 DF,  p-value: 0.02125

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?”

1.2 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 merely 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:

   get_regression_table(score_model)
   ## # A tibble: 2 x 7
   ##   term      estimate std_error statistic p_value lower_ci upper_ci
   ##   <chr>        <dbl>     <dbl>     <dbl>   <dbl>    <dbl>    <dbl>
   ## 1 intercept    4.46      0.127     35.2    0        4.21     4.71 
   ## 2 age         -0.006     0.003     -2.31   0.021   -0.011   -0.001

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

   get_regression_points(score_model)
   ## # A tibble: 463 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
   ## # … with 453 more rows

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

   get_regression_summaries(score_model)
   ## # A tibble: 1 x 9
   ##   r_squared adj_r_squared   mse  rmse sigma statistic p_value    df
   ##       <dbl>         <dbl> <dbl> <dbl> <dbl>     <dbl>   <dbl> <dbl>
   ## 1     0.011         0.009 0.292 0.540 0.541      5.34   0.021     1
   ## # … with 1 more variable: nobs <dbl>

1.3 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.

# Code to visualize interaction model:
ggplot(evals, aes(x = age, y = score, color = ethnicity)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE) +
  labs(x = "Age", y = "Teaching score", color = "Ethnicity")

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 merely add a geom_parallel_slopes() layer to the code, resulting in .

# Code to visualize parallel slopes model:
ggplot(evals, aes(x = age, y = score, color = ethnicity)) +
  geom_point() +
  geom_parallel_slopes(se = FALSE) +
  labs(x = "Age", y = "Teaching score", color = "Ethnicity")

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?”

1.4 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.

2.1 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:

get_regression_table(score_model)
## # A tibble: 2 x 7
##   term      estimate std_error statistic p_value lower_ci upper_ci
##   <chr>        <dbl>     <dbl>     <dbl>   <dbl>    <dbl>    <dbl>
## 1 intercept    4.46      0.127     35.2    0        4.21     4.71 
## 2 age         -0.006     0.003     -2.31   0.021   -0.011   -0.001

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 easily emphasize to students that they “surround” the point estimates in the estimate column. Note the confidence level is defaulted to 95%.

2.2 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:

sqrt(diag(vcov(score_model)))
## (Intercept)         age 
## 0.126778499 0.002569157

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 easily extract columns using the pull() function from the dplyr package (Wickham et al. 2020):

get_regression_table(score_model) %>%
  pull(std_error)
## [1] 0.127 0.003

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

get_regression_table(score_model)$std_error
## [1] 0.127 0.003

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:

get_regression_table(score_model) %>%
  kable()

2.3 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.)

fitted(score_model)

##        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

residuals(score_model)

##           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():

score_model_points <- get_regression_points(score_model)
score_model_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 easily 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 .

# Code to visualize distribution of residuals:
ggplot(score_model_points, aes(x = residual)) +
  geom_histogram(bins = 20) +
  labs(x = "Residual", y = "Count")

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.

# Code to visualize partial residual plot over age:
ggplot(score_model_points, aes(x = age, y = residual)) +
  geom_point() +
  labs(x = "Age", y = "Residual")

Partial residual residual plot over age.

2.4 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.

new_prof <- tibble(age = c(39, 42))
get_regression_points(score_model, newdata = new_prof)
## # A tibble: 2 x 3
##      ID   age score_hat
##   <int> <dbl>     <dbl>
## 1     1    39      4.23
## 2     2    42      4.21

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