An interaction model is a Second-Order Model that is used to test whether the outcome of one variable is affected by another variable.

For this presentation, a data set of US arrests is used to show the relationship between how Assault rate and Urban population percentage affect the murder rate with and without an interaction term.

It’s a linear regression model with an additional product term of the two variables that are assumed to be dependent on each other.

\[Y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3(x_1 \cdot x_2) + \epsilon\]

In this study:

• \(Y\) = Murder rate

• \(x_1\) = Assault rate

• \(x_2\) = Urban population percentage

• \(\beta_3(x_1 \cdot x_2)\) = Interaction term

• More accurately predicts any outcome/result
• Captures any relationship that a typical linear regression model might miss
• Cleaner graphs

library(scatterplot3d) model <- lm(Murder ~ Assault * UrbanPop, data = USArrests)

scatterplot3d(USArrests\(Assault, USArrests\)UrbanPop, USArrests$Murder, pch = 16, color = “darkgreen”,

xlab = “Assault”, ylab = “Urban Pop”, zlab = “Murder”,

main = “3D Perspective”)

It seems like as \(x_1\) (Assault) increases, \(Y\) (Murder) increases, which confirms a positive relationship between them

It also seems like the urban population alone doesn’t do much to the murder rate, which is captured by

\[\frac{\partial Y}{\partial x_1} = \beta_1 + \beta_3 x_2\]