3.7 — Regression with Interaction Effects — R Practice

Part A

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

Part B

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

Hint: use the class(df$variable) command to ask what class something is, where df is your dataframe, and variable is the name of a variable. Alternatively, it should also show the class automatically from the commands from Part A.

Part C

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. Convert each of these variables into factors by reassigning it according to the format:

df <- df %>% # where df is your data
  mutate(my_var = as.factor(my_var) # where my_var is the variable
  )

You could do this for each variable, or all at once, using mutate_at():

df <- df %>%
  mutate_at(c("Black", "Hispanic", "Female", "OutTown", "OutState"), factor)

Recall the class on Data Wrangling for all the mutate possibilities!

Part D

Confirm they are each now factors by checking their class or looking at the data again.

Part E

Get a summary() of Amount using:

df %>% # or whatever your dataframe is called
  select(Amount) %>%
  summary()

Note that there are a lot of NA’s (these are people that were stopped but did not receive a fine)! Let’s filter() to use 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).

df <- df %>% # overwrite or make a new dataframe
  filter(Amount > 0) 

# verify it worked
df %>% 
  select(Amount) %>%
  summar()

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:

t.test(Amount ~ Female,
       data = df)

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 \(\alpha=0.05.\)

Part A

Write out the estimated regression equation.

Part B

Use the regression coefficients to find

• 1. the average Amount for men
• 2. the average Amount for women
• 3. the difference in average Amount between men and women

Part A

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

df <- df %>% # overwrite or save as another dataframe
  mutate(Male = ifelse(Female == 0, # test observation 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
df %>%
  select(Female, Male)

Part B

Run the same regression as in question 5, but use Male instead of Female.

Part C

Write out the estimated regression equation.

Part D

Use the regression coefficients to find

• 1. the average Amount for men
• 2. the average Amount for women
• 3. the difference in average Amount between men and women

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.

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 aes() layer. This will produce two lines and color the points by Female.

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).

Part A

Use R to examine the data and find the mean for (i) Males In-State, (ii) Males Out-of-State, (iii) Females In-State, and (iv) Females Out-of-State. Hint: do what you did in Question 4A.

Part B

Now run a regression of the following model:

\[\text{Amount}_i=\hat{\beta_0}+\hat{\beta_1}Female_i+\hat{\beta_2}OutState_i+\hat{\beta_3}Female_i*OutState_i\]

Part C

Write out the estimated regression equation.

Part D

What does each coefficient mean?

Part E

Using the regression equation, what are the means for

• 1. Males In-State
• 2. Males Out-of-State
• 3. Females In-State
• 4. Females Out-of-State?

Compare to your answers in part A.