4.3 — Categorical Data and Interactions — Practice

Answers

Required Packages & Data

Load all the required packages we will use (note I have installed them already into the cloud project) by running (clicking the green play button) the chunk below:

library(tidyverse) # your friend and mine
library(broom) # for tidy regression
library(modelsummary) # for nice regression tables

We are returning to the speeding tickets data that we began to explore in R Practice 4.1 on Multivariate Regression. Download and read in (read_csv) the data below.

speeding_tickets.csv

# run or edit this chunk (if you want to rename the data)

# read in data from url 
# or you could download and upload it to this project instead
speed <- read_csv("https://metricsf22.classes.ryansafner.com/files/data/speeding_tickets.csv")

This data comes from a paper by Makowsky and Strattman (2009) that we will examine later. Even though state law sets a formula for tickets based on how fast a person was driving, police officers in practice often deviate from that formula. This dataset includes information on all traffic stops. An amount for the fine is given only for observations in which the police officer decided to assess a fine. There are a number of variables in this dataset, but the one’s we’ll look at are:

Variable	Description
`Amount`	Amount of fine (in dollars) assessed for speeding
`Age`	Age of speeding driver (in years)
`MPHover`	Miles per hour over the speed limit
`Black`	Dummy \(=1\) if driver was black, \(=0\) if not
`Hispanic`	Dummy \(=1\) if driver was Hispanic, \(=0\) if not
`Female`	Dummy \(=1\) if driver was female, \(=0\) if not
`OutTown`	Dummy \(=1\) if driver was not from local town, \(=0\) if not
`OutState`	Dummy \(=1\) if driver was not from local state, \(=0\) if not
`StatePol`	Dummy \(=1\) if driver was stopped by State Police, \(=0\) if stopped by other (local)

We want to explore who gets fines, and how much. We’ll come back to the other variables (which are categorical) in this dataset in later lessons.

Question 1

We will have to do a little more cleaning to get some of the data into a more usable form.

Part A

Inspect the data with str() or head() or glimpse() to see what it looks like.

What class of variable are Black, Hispanic, Female, OutTown, and OutState?

Part B

Notice that when importing the data from the .csv file, R interpreted these variables as numeric (num) or double (dbl), but we want them to be factor (fct) variables, to ensure R recognizes that there are two groups (categories), 0 and 1.

You could convert the variables one at a time to factors using as.factor() inside a mutate() command. But there is a special mutate() command that allows you to apply a transformation (like changing a variable’s class to factor), which you can run the following chunk to execute:

# run or edit this chunk
speed <- speed %>%
  mutate_at(c("Black", "Hispanic", "Female", "OutTown", "OutState"), factor)

speed

Confirm that these are now factor (fct) variables.

Part C

Finally, recall from the last time we worked with this data that there are many NAs for Amount (these are people that were stopped but did not receive a fine). Let’s filter() only those observations for which Amount is a positive number, and save this in your dataframe (assign and overwrite it, or make a new dataframe).

# run or edit this chunk

# see the NAs 
speed %>%
  select(Amount) %>%
  summary()

# overwrite data to keep only positive Amounts
speed <- speed %>%
  filter(Amount > 0)

# see there are no more NAs
speed %>%
  select(Amount) %>%
  summary()

Question 2

Does the sex of the driver affect the fine? There’s already a dummy variable Female in the dataset, so create a scatterplot between Amount (as y) and Female (as x).

Hint: Use geom_jitter() instead of geom_point() to better see the points, and play around with width settings inside geom_jitter()

As an aside, if you had not made Female a factor, and kept it numeric, it might alter the plot.

You can make Female a factor just for plotting purposes by setting aes(x = as.factor(Female)) inside the aes() layer of your ggplot() code.

Question 3

Now let’s start looking at the distribution conditionally to find the different group means.

Part A

Find the mean and standard deviation of Amount for male drivers and again for female drivers.

Hint: properly filter() the data and then use the summarize() command.

Part B

What is the difference between the average Amount for Males and Females?

Part C

We did not go over how to do this in class, but you can run a t-test for the difference in group means to see if the difference is statistically significant. The syntax is similar for a regression:

# run or edit this chunk
t.test(Amount ~ Female,
       data = speed)

Is there a statistically significant difference between Amount for male and female drivers? Hint: this is like any hypothesis test. Here \(H_0: \text{difference}=0\). A \(t\)-value needs to be large enough to be greater than a critical value of \(t\). Check the \(p\)-value and see if it is less than our standard of \(\alpha=0.05.\)

Question 4

Part A

Now run the following regression to ensure we get the same result as the t-test.

\[\widehat{\text{Amount}}_i=\hat{\beta_0}+\hat{\beta_1} \, \text{Female}_i\]

Part B

Write out the estimated regression equation.

Part C

Use the regression coefficients to find

the average Amount for men
the average Amount for women
the difference in average Amount between men and women

Question 5

Let’s recode the sex variable to Male instead of Female.

Part A

Make a new variable called Male and save it in your dataframe using the ifelse() command:

# run or edit this chunk
speed <- speed %>% # overwrite or save as another dataframe
  mutate(Male = ifelse(test = Female == 0, # test indiv. to see if Female is 0
                       yes = 1, # if yes (a Male), code Male as 1
                       no  = 0), # if no (a Female), code Male as 0
         )

# Verify it worked
speed %>%
  select(Female, Male)

Part B

Run the same regression as in question 4, but use Male instead of Female.

\[\widehat{\text{Amount}}_i=\hat{\beta_0}+\hat{\beta_1} \, \text{Male}_i\]

Part C

Write out the estimated regression equation.

Part D

Use the regression coefficients to find

the average Amount for men
the average Amount for women
the difference in average Amount between men and women

Question 6

Run a regression of Amount on Male and Female. What happens, and why?

\[\widehat{\text{Amount}}_i=\hat{\beta_0}+\hat{\beta_1} \, \text{Male}_i + \hat{\beta_2} \, \text{Female}_i\]

Question 7

Age probably has a lot to do with differences in fines, perhaps also age affects fines differences between males and females.

Part A

Run a regression of Amount on Age and Female. How does the coefficient on Female change?

Part B

Now let’s see if the difference in fine between men and women are different depending on the driver’s age. Run the regression again, but add an interaction term between Female and Age, using Female*Age or Female:Age.

\[\widehat{\text{Amount}}_i=\hat{\beta_0}+\hat{\beta_1} \, \text{Age}_i + \hat{\beta_2} \, \text{Female}_i + \hat{\beta_3} \, (\text{Age}_i \times \text{Female}_i)\]

Part C

Write out your estimated regression equation.

Part D

Interpret the interaction effect. Is it statistically significant?

Part E

Plugging in 0 or 1 as necessary, rewrite (on your paper) this regression as two separate equations, one for Males and one for Females.

Part F

Let’s try to visualize this. Make a scatterplot of Age (X) and Amount (Y) and include a regression line.

Try adding color = Female inside your original (global) aes() layer. This will produce two sets of points and regression lines colored by Female.

By the way, if it isn’t a factor variable already, we can ensure that it is with as.factor(Female). We shouldn’t need to in this case because we already reset Female as a faction in question 1.

Part G

Add a final facet layer to the plot make two different sub-plots by sex with facet_wrap( ~ Female).

Question 8

Now let’s look at the possible interaction between sex (Male or Female) and whether a driver is from In-State or Out-of-State (OutState).

Part A

Use R to examine the data and find the average fine for:

Males In-State
Males Out-of-State
Females In-State
Females Out-of-State

Part B

Now run a regression of the following model:

\[\widehat{\text{Amount}}_i=\hat{\beta_0}+\hat{\beta_1} \, \text{Female}_i+\hat{\beta_2} \, \text{OutState}_i+\hat{\beta_3} \, (\text{Female}_i \times \text{OutState}_i)\]

Part C

Write out the estimated regression equation.

Part D

What does each coefficient mean?

Part E

Using the regression equation, what are the average fine for

Males In-State
Males Out-of-State
Females In-State
Females Out-of-State

Compare to your answers in part A.

Question 9

Collect your regressions from questions 4, 5b, 7a, 7b, and 8b and output them in a regression table with modelsummary().