# 11 Inference for Regression

Note: This chapter is still under construction. If you would like to contribute, please check us out on GitHub at https://github.com/moderndive/moderndive_book.

## 11.1 Refresher: Professor evaluations data

Let’s revisit the professor evaluations data that we analyzed using multiple regression with one numerical and one categorical predictor. In particular

• $$y$$: outcome variable of instructor evaluation score
• predictor variables
• $$x_1$$: numerical explanatory/predictor variable of age
• $$x_2$$: categorical explanatory/predictor variable of gender
library(ggplot2)
library(dplyr)
library(moderndive)

evals <- evals %>%
select(score, ethnicity, gender, language, age, bty_avg, rank)

First, recall that we had two competing potential models to explain professors’ teaching scores:

1. Model 1: No interaction term. i.e. both male and female profs have the same slope describing the associated effect of age on teaching score
2. Model 2: Includes an interaction term. i.e. we allow for male and female profs to have different slopes describing the associated effect of age on teaching score

### 11.1.1 Refresher: Visualizations

Recall the plots we made for both these models:

### 11.1.2 Refresher: Regression tables

Last, let’s recall the regressions we fit. First, the regression with no interaction effect: note the use of + in the formula.

score_model_2 <- lm(score ~ age + gender, data = evals)
get_regression_table(score_model_2)
Table 11.1: Model 1: Regression table with no interaction effect included
term estimate std_error statistic p_value lower_ci upper_ci
intercept 4.484 0.125 35.79 0.000 4.238 4.730
age -0.009 0.003 -3.28 0.001 -0.014 -0.003
gendermale 0.191 0.052 3.63 0.000 0.087 0.294

Second, the regression with an interaction effect: note the use of * in the formula.

score_model_3 <- lm(score ~ age * gender, data = evals)
get_regression_table(score_model_3)
Table 11.2: Model 2: Regression table with interaction effect included
term estimate std_error statistic p_value lower_ci upper_ci
intercept 4.883 0.205 23.80 0.000 4.480 5.286
age -0.018 0.004 -3.92 0.000 -0.026 -0.009
gendermale -0.446 0.265 -1.68 0.094 -0.968 0.076
age:gendermale 0.014 0.006 2.45 0.015 0.003 0.024

### 11.1.3 Script of R code

An R script file of all R code used in this chapter is available here.