Using iris data for example
head(iris) Sepal.Length Sepal.Width Petal.Length Petal.Width Species
1 5.1 3.5 1.4 0.2 setosa
2 4.9 3.0 1.4 0.2 setosa
3 4.7 3.2 1.3 0.2 setosa
4 4.6 3.1 1.5 0.2 setosa
5 5.0 3.6 1.4 0.2 setosa
6 5.4 3.9 1.7 0.4 setosaWe want to fit a multivariate linear regression with Sepal.Length as the outcome variable and Sepal.Width and Species as predictors. What is the difference between Sepal.Length ~ Sepal.Width + Species and Sepal.Length ~ Sepal.Width * Species?
The second formula includes an interaction term. When we use an interaction term, we assume that the effect of sepal width differs across species; that is, the slope of sepal width is different for each species. When we use the first formula, we assume that the effect of sepal width is the same across all species.
R code to ilustrate this
library(patchwork)
library(ggplot2)
library(dplyr)
## no interaction between width and species assumption
lm1 <- lm(Sepal.Length ~ Sepal.Width + Species, data=iris)
p1 <- iris |>
mutate(pred = predict(lm1)) |>
ggplot(aes(x = Sepal.Width, y = pred))+
geom_line()+
facet_wrap(~Species)+
theme_bw()+
scale_y_continuous(limits = c(4,8))
## interaction between width and species assumption
lm2 <- lm(Sepal.Length ~ Sepal.Width*Species, data=iris)
p2 <- iris |>
mutate(pred = predict(lm2)) |>
ggplot(aes(x = Sepal.Width, y = pred))+
geom_line()+
facet_wrap(~Species)+
theme_bw()+
scale_y_continuous(limits = c(4,8))
p1/
p2
As we can see, the slope for versicolor in the bottom figure clearly shows a difference between species, whereas the slopes for the three species in the top figure are the same.