Interaction effects

2025-10-01

Packages in these notes

library(tidyverse)
library(palmerpenguins)
library(knitr)
library(kableExtra)
library(tidymodels)

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.

Example: penguins

The relationship between penguin bill length and body mass depends on the penguin’s island.

• plot
• code
• discuss

penguins |>
  ggplot(aes(x = bill_length_mm, 
             y = body_mass_g, color = island)) +
  geom_point() +
  theme_bw() +
  geom_smooth(method = 'lm', se = F) +
  labs(x = "Bill length (mm)", 
       y = "Body mass (g)", 
       title = "Island interaction effect")

The lines are not parallel indicating there is a potential interaction effect. The slope for bill length differs based on the penguin’s island.

Interaction term in model

penguin_fit <- lm(body_mass_g ~ bill_length_mm * island,
      data = penguins)

term	estimate	std.error	statistic	p.value
(Intercept)	-1726.02	292.08	-5.91	0
bill_length_mm	142.34	6.42	22.18	0
islandDream	4478.69	395.31	11.33	0
islandTorgersen	2870.56	777.69	3.69	0
bill_length_mm:islandDream	-120.60	8.77	-13.75	0
bill_length_mm:islandTorgersen	-76.57	19.53	-3.92	0

Interaction term in model

• Exercise
• Solution

• Write the regression equation for penguins from island Torgersen

• Write the regression equation for penguins from island Dream

• Write the regression equation for penguins from island Biscoe

Penguins equations for each island:

\[ \begin{aligned} \hat{y}_{\text{torgersen}} &= (-1726.02 + 2870.56) + (142.34 - 76.57)x_1\\ &= 1144.54 + 65.77x_1\\ \hat{y}_{\text{dream}} &= (-1726.02 + 4478.69 ) + (142.34 - 120.60)x_1\\ &= 2752.67 + 21.74x_1\\ \hat{y}_{\text{biscoe}} &= -1726.02 + 142.34 x_1 \end{aligned} \]

where \(\hat{y}\) is the predicted body mass for a given bill length \(x_1\).

Interpreting interaction terms

What the interaction means:

• The effect of bill length on body mass differs by -76.5 when the penguin comes from island Torgersen instead of Biscoe.

• The effect of bill length on body mass differs by -120 when the penguin comes from island Dream instead of Biscoe.