interaction effect

In multiple linear regression, it is assumed that the relationship between the response variable and the predictor variables is additive. But, in reality that is not the case.

The standard linear model with two variables is

\begin{equation} Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 \end{equation}

The influence of \(X_1\) on \(Y\) is determined solely by \(\beta_1\), regardless of \(X_2\). However, what happens when the effect of \(X_1\) varies with \(X_2\)? For instance, the glucose spike response is affected not only by the quantity (in grams) of dal and rice but also by whether they are consumed together, leading to a different response.

To tackle this issue, we can introduce a new predictor, which is the product of \( X_1 \) and \( X_2 \), along with an associated coefficient.

\begin{equation} \( Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 (X_1 X_2) \) \end{equation}

Rewriting this provides a simpler interpretation,

\begin{equation} Y = \beta_0 + (\beta_1 + \beta_3 X_2) X_1 + \beta_2 X_2 \end{equation}

This equation indicates that the contribution of \(X_1\) depends on the value of \(X_2\).