Multiple linear regression

Types of predictors cont’d

Published

February 03, 2026

Announcements

Topics

  • Interaction terms

Computing setup

# load packages
library(tidyverse)
library(tidymodels)
library(openintro)
library(patchwork)
library(knitr)
library(kableExtra)
library(viridis) #adjust color palette

# set default theme and larger font size for ggplot2
ggplot2::theme_set(ggplot2::theme_minimal(base_size = 16))

Data: Peer-to-peer lender

Today’s data is a sample of 50 loans made through a peer-to-peer lending club. The data is in the loan50 data frame in the openintro R package.

# A tibble: 50 × 4
   annual_income_th debt_to_income verified_income interest_rate
              <dbl>          <dbl> <fct>                   <dbl>
 1             59           0.558  Not Verified            10.9 
 2             60           1.31   Not Verified             9.92
 3             75           1.06   Verified                26.3 
 4             75           0.574  Not Verified             9.92
 5            254           0.238  Not Verified             9.43
 6             67           1.08   Source Verified          9.92
 7             28.8         0.0997 Source Verified         17.1 
 8             80           0.351  Not Verified             6.08
 9             34           0.698  Not Verified             7.97
10             80           0.167  Source Verified         12.6 
# ℹ 40 more rows

Variables

Predictors:

  • annual_income_th: Annual income (in $1000s)
  • debt_to_income: Debt-to-income ratio, i.e. the percentage of a borrower’s total debt divided by their total income
  • verified_income: Whether borrower’s income source and amount have been verified (Not Verified, Source Verified, Verified)

Response: interest_rate: Interest rate for the loan

Interaction terms

Interaction terms

  • Sometimes the relationship between a predictor variable and the response depends on the value of another predictor variable.
  • This is an interaction effect.
  • To account for this, we can include interaction terms in the model.

Interest rate vs. annual income

The lines are not parallel indicating there is a potential interaction effect. The slope of annual income differs based on the income verification.

Model with interaction terms

int_fit_2 <- lm(interest_rate ~ debt_to_income + verified_income + annual_income_th + verified_income * annual_income_th,
      data = loan50)
term estimate std.error statistic p.value
(Intercept) 9.560 2.034 4.700 0.000
debt_to_income 0.691 0.685 1.009 0.319
verified_incomeSource Verified 3.577 2.539 1.409 0.166
verified_incomeVerified 9.923 3.654 2.716 0.009
annual_income_th -0.007 0.020 -0.341 0.735
verified_incomeSource Verified:annual_income_th -0.016 0.026 -0.643 0.523
verified_incomeVerified:annual_income_th -0.032 0.033 -0.979 0.333

Model with interaction terms

  • Write the estimated regression equation for the people with Not Verified income.

  • Write the estimated regression equation for people with Verified income.

Interpreting interaction terms

  • What the interaction means: The effect of annual income on the interest rate differs by -0.016 when the income is source verified compared to when it is not verified, holding all else constant.
  • Interpreting annual_income for source verified: If the income is source verified, we expect the interest rate to decrease by 0.023% (-0.007 + -0.016) for each additional thousand dollars in annual income, holding all else constant.

Indicators and interactions

In general, how do

  • indicators for categorical predictors impact the model equation?

  • interaction terms impact the model equation?

Recap

  • Interpreted interaction terms