# Partial residual plot with interactions?

The NIST website's description of the partial residual plot says that it plots

$$\text{Res}+\hat\beta_iX_i\text{ versus } X_i$$

where

• $\text{Res}$ = residuals from the full model
• $\hat\beta_i$ = regression coefficient of $X_i$
• $X_i$ = the $i$th independent variable in the full model

This decription is clear enough when the full model is of the form

$$Y=\beta_0+\beta_1X_1+\beta_2X_2+\cdots+\beta_iX_i+\cdots$$

However, consider the following full models which include interaction effects:

$$(1)\qquad\qquad Y=\beta_0+\beta_1X_1+\cdots+\beta_iX_i+\cdots+\beta_kX_i^2+\cdots$$ $$(2)\qquad\qquad Y=\beta_0+\beta_1X_1+\cdots+\beta_iX_i+\cdots+\beta_kX_iX_j+\cdots$$ $$(3)\qquad\qquad Y=\beta_0+\beta_1X_1+\cdots+\beta_iX_i+\cdots+\beta_kX_iD+\cdots$$

where $D$ is a categorical dummy variable (i.e. it takes the values 0 or 1) in the second model. Would the partial residual plot in both cases neglect the $\beta_k$ (i.e. interaction part) or would you plot (these are my own suggestions):

$$(1)\qquad\qquad \text{Res}+\hat\beta_iX_i+\hat\beta_kX_i^2\text{ versus }X_i$$ $$(2)\qquad\qquad \text{Res}+\hat\beta_iX_i+\hat\beta_kX_iX_j\text{ versus }X_i$$ $$(3)\qquad\qquad \text{Res}+\hat\beta_iX_i+\hat\beta_kX_iD\text{ versus }X_i$$

(1) makes sense to me, but (2) and (3) do not - since what would you take as the values of $X_j$ and $D$ in the plot? The values that correspond to the $X_i$ in the data set?

As far as I know, partial residual plots in R do not even support interaction terms. So I'd like someone to confirm how cases (1), (2) and (3) should be properly handled for a partial residual plot.

There is a simple procedure (for your case (2) ) illustrated at https://stackoverflow.com/a/24964685, but there the x-axis shows the product $X_i X_j$, not $X_i$. The underlying logic, I assume, is that if you are concerned about the linearity of the relationship captured by the $\beta_k something$ then you construct the partial residual with that "something", the regressor $k$ which, in (2) will be the regressor formed by the product of $X_i X_j$, etc. I guess that could also be used for case (1) (interestingly, plotting $e + \hat{b_2} x_2 + \hat{b_{22}} x_2^2$ against $x_2$ is already discussed in R. Dennis Cook's paper "Exploring partial residual plots"; see section 3., p. 354).
Your case (3) is handled in the effects package by evaluating that sum as you indicate (each case at its observed values of $X_i$ and $D_i$) and then plotting the partial residual vs $X_i$ conditioning on $D$ (i.e., with different panels for different values of $D$).