`moderndive`

: Datasets and
functions for `tidyverse`

-friendly introductory linear
regressionWe 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.

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.

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

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 courses^{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.`score`

is the outcome variable of interest: average professor evaluation score out of 5 as given by the students in this course.- 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 |

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).

```
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:

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

`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).

Get a tidy regression table

**with confidence intervals**: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 × 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 ## # ℹ 453 more rows`

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

`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")
```

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 .

```
# 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")
```

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

`moderndive`

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

package

`get_regression_table()`

`get_regression_points()`

`get_regression_summaries()`

`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.

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:

- 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.

- 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 × 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 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:

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 |

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 × 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 .

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

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")
```

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 × 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).

This Kaggle competition requires you to fit/train a model to the
provided `train.csv`

training set to make predictions of
house prices in the provided `test.csv`

test set. We present
an application of the `get_regression_points()`

function
allowing students to participate in this Kaggle competition. It
will:

- Read in the training and test data.
- Fit a naive model of house sale price as a function of year sold to the training data.
- Make predictions on the test data and write them to a
`submission.csv`

file that can be submitted to Kaggle using`get_regression_points()`

. Note the use of the`ID`

argument to use the`id`

variable in`test`

to identify the rows (a requirement of Kaggle competition submissions).

```
library(readr)
library(dplyr)
library(moderndive)
# Load in training and test set
train <- read_csv("https://moderndive.com/data/train.csv")
test <- read_csv("https://moderndive.com/data/test.csv")
# Fit model:
house_model <- lm(SalePrice ~ YrSold, data = train)
# Make predictions and save in appropriate data frame format:
submission <- house_model %>%
get_regression_points(newdata = test, ID = "Id") %>%
select(Id, SalePrice = SalePrice_hat)
# Write predictions to csv:
write_csv(submission, "submission.csv")
```

After submitting `submission.csv`

to the leaderboard for
this Kaggle competition, we obtain a “root mean squared logarithmic
error” (RMSLE) score of 0.42918 as seen in .

The fifth common student question:

“What is all this other stuff at the bottom?”

Recall the output of the standard `summary.lm()`

from
earlier:

```
##
## 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
```

Say we wanted to extract the scalar model summaries at the bottom of
this output, such as \(R^2\), \(R^2_{adj}\), and the \(F\)-statistic. We can do so using the
`get_regression_summaries()`

function.

```
get_regression_summaries(score_model)
## # A tibble: 1 × 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
## # ℹ 1 more variable: nobs <dbl>
```

We’ve supplemented the standard scalar summaries output yielded by
`summary()`

with the mean squared error `mse`

and
root mean squared error `rmse`

given their popularity in
machine learning settings.

Finally, the last common student question:

“When would you ever use a parallel slopes model?”

For example, recall the earlier visualizations of the interaction and parallel slopes models for teaching score as a function of age and ethnicity we saw in Figures \(\ref{fig:interaction-model}\) and \(\ref{fig:parallel-slopes-model}\). Let’s present both visualizations side-by-side in .

Students might be wondering “Why would you use the parallel slopes
model on the right when the data clearly form an”X” pattern as seen in
the interaction model on the left?” This is an excellent opportunity to
gently introduce the notion of *model selection* and *Occam’s
Razor*: an interaction model should only be used over a parallel
slopes model **if the additional complexity of the interaction
model is warranted**. Here, we define model
“complexity/simplicity” in terms of the number of parameters in the
corresponding regression tables:

```
# Regression table for interaction model:
interaction_evals <- lm(score ~ age * ethnicity, data = evals)
get_regression_table(interaction_evals)
## # A tibble: 4 × 7
## term estimate std_error statistic p_value lower_ci upper_ci
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 intercept 2.61 0.518 5.04 0 1.59 3.63
## 2 age 0.032 0.011 2.84 0.005 0.01 0.054
## 3 ethnicity: n… 2.00 0.534 3.74 0 0.945 3.04
## 4 age:ethnicit… -0.04 0.012 -3.51 0 -0.063 -0.018
# Regression table for parallel slopes model:
parallel_slopes_evals <- lm(score ~ age + ethnicity, data = evals)
get_regression_table(parallel_slopes_evals)
## # A tibble: 3 × 7
## term estimate std_error statistic p_value lower_ci upper_ci
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 intercept 4.37 0.136 32.1 0 4.1 4.63
## 2 age -0.006 0.003 -2.5 0.013 -0.012 -0.001
## 3 ethnicity: n… 0.138 0.073 1.89 0.059 -0.005 0.282
```

The interaction model is “more complex” as evidenced by its regression table involving 4 rows of parameter estimates whereas the parallel slopes model is “simpler” as evidenced by its regression table involving only 3 parameter estimates. It can be argued however that this additional complexity is warranted given the clearly different slopes in the left-hand plot of .

We now present a contrasting example, this time from Chapter 6 of the
online version of ModernDive
Subsection 6.3.1 involving Massachusetts USA public high schools.^{3} Let’s plot
both the interaction and parallel slopes models in .

```
# Code to plot interaction and parallel slopes models for MA_schools
ggplot(
MA_schools,
aes(x = perc_disadvan, y = average_sat_math, color = size)
) +
geom_point(alpha = 0.25) +
labs(
x = "% economically disadvantaged",
y = "Math SAT Score",
color = "School size"
) +
geom_smooth(method = "lm", se = FALSE)
ggplot(
MA_schools,
aes(x = perc_disadvan, y = average_sat_math, color = size)
) +
geom_point(alpha = 0.25) +
labs(
x = "% economically disadvantaged",
y = "Math SAT Score",
color = "School size"
) +
geom_parallel_slopes(se = FALSE)
```

In terms of the corresponding regression tables, observe that the corresponding regression table for the parallel slopes model has 4 rows as opposed to the 6 for the interaction model, reflecting its higher degree of “model simplicity.”

```
# Regression table for interaction model:
interaction_MA <-
lm(average_sat_math ~ perc_disadvan * size, data = MA_schools)
get_regression_table(interaction_MA)
## # A tibble: 6 × 7
## term estimate std_error statistic p_value lower_ci upper_ci
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 intercept 594. 13.3 44.7 0 568. 620.
## 2 perc_disadvan -2.93 0.294 -9.96 0 -3.51 -2.35
## 3 size: medium -17.8 15.8 -1.12 0.263 -48.9 13.4
## 4 size: large -13.3 13.8 -0.962 0.337 -40.5 13.9
## 5 perc_disadva… 0.146 0.371 0.393 0.694 -0.585 0.877
## 6 perc_disadva… 0.189 0.323 0.586 0.559 -0.446 0.824
# Regression table for parallel slopes model:
parallel_slopes_MA <-
lm(average_sat_math ~ perc_disadvan + size, data = MA_schools)
get_regression_table(parallel_slopes_MA)
## # A tibble: 4 × 7
## term estimate std_error statistic p_value lower_ci upper_ci
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 intercept 588. 7.61 77.3 0 573. 603.
## 2 perc_disadvan -2.78 0.106 -26.1 0 -2.99 -2.57
## 3 size: medium -11.9 7.54 -1.58 0.115 -26.7 2.91
## 4 size: large -6.36 6.92 -0.919 0.359 -20.0 7.26
```

Unlike our earlier comparison of interaction and parallel slopes
models in , in this case it could be argued that the additional
complexity of the interaction model is *not* warranted since the
3 three regression lines in the left-hand interaction are already
somewhat parallel. Therefore the simpler parallel slopes model should be
favored.

Going one step further, notice how the three regression lines in the
visualization of the parallel slopes model in the right-hand plot of
have similar intercepts. In can thus be argued that the additional model
complexity induced by introducing the categorical variable school
`size`

is not warranted. Therefore, a simple linear
regression model using only `perc_disadvan`

percent of the
student body that are economically disadvantaged should be favored.

While many students will inevitably find these results depressing, in our opinion, it is important to additionally emphasize that such regression analyses can be used as an empowering tool to bring to light inequities in access to education and inform policy decisions.

`broom`

functionsAs we mentioned earlier, the three `get_regression_*`

functions are wrappers of functions from the `broom`

package
for converting statistical analysis objects into tidy tibbles along with
a few added tweaks, but with the introductory statistics student in mind
(Robinson and Hayes 2019):

`get_regression_table()`

is a wrapper for`broom::tidy()`

.`get_regression_points()`

is a wrapper for`broom::augment()`

.`get_regression_summaries`

is a wrapper for`broom::glance()`

.

Why did we take this approach to address the initial 5 common student questions at the outset of the article?

- By writing wrappers to pre-existing functions, instead of creating new custom functions, there is minimal re-inventing of the wheel necessary.
- In our experience, novice R users had a hard time understanding the
`broom`

package function names`tidy()`

,`augment()`

, and`glance()`

. To make them more user-friendly, the`moderndive`

package wrappers have much more intuitively named`get_regression_table()`

,`get_regression_points()`

, and`get_regression_summaries()`

. - The variables included in the outputs of the above 3
`broom`

functions are not all applicable to an introductory statistics students and of those that were, we found they were not intuitive for new R users. We therefore cut out some of the variables from the output and renamed some of the remaining variables. For example, compare the outputs of the`get_regression_points()`

wrapper function and the parent`broom::augment()`

function.

```
get_regression_points(score_model)
## # A tibble: 463 × 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
## # ℹ 453 more rows
broom::augment(score_model)
## # A tibble: 463 × 8
## score age .fitted .resid .hat .sigma .cooksd .std.resid
## <dbl> <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 4.7 36 4.25 0.452 0.00560 0.542 0.00197 0.837
## 2 4.1 36 4.25 -0.148 0.00560 0.542 0.000212 -0.274
## 3 3.9 36 4.25 -0.348 0.00560 0.542 0.00117 -0.645
## 4 4.8 36 4.25 0.552 0.00560 0.541 0.00294 1.02
## 5 4.6 59 4.11 0.488 0.00471 0.541 0.00193 0.904
## 6 4.3 59 4.11 0.188 0.00471 0.542 0.000288 0.349
## 7 2.8 59 4.11 -1.31 0.00471 0.538 0.0139 -2.43
## 8 4.1 51 4.16 -0.0591 0.00232 0.542 0.0000139 -0.109
## 9 3.4 51 4.16 -0.759 0.00232 0.541 0.00229 -1.40
## 10 4.5 40 4.22 0.276 0.00374 0.542 0.000488 0.510
## # ℹ 453 more rows
```

The source code for these three `get_regression_*`

functions can be found on GitHub.

The `geom_parallel_slopes()`

is a custom built
`geom`

extension to the `ggplot2`

package. For
example, the `ggplot2`

webpage page gives instructions on how to create such extensions. The
source code for `geom_parallel_slopes()`

written by Evgeni
Chasnovski can be found on GitHub.

Many thanks to Jenny Smetzer @smetzer180, Luke W. Johnston @lwjohnst86,
and Lisa Rosenthal @lisamr for their helpful feedback for this vignette
and to Evgeni Chasnovski @echasnovski for contributing the
`geom_parallel_slopes()`

function via GitHub pull request. The authors do not have any financial
support to disclose.

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