# when to analyze full data set vs. residuals

Say I have a data set, and I want to model the dependent variable y, as a function of x1,x2,x3 and x4. I specify my model as:

y ~ x1 + x2 + x3 + x4 + x1*x2


I specify an interaction between x1 and x2 based on prior knowledge. However, the r2 value of the single regression model y ~ x1 is ~0.90. Is it fair to still test for this interaction, as well as effects of x2 and x3 on y? Or is it better to do an analysis of the residuals of the model y~x1? The problem I forsee with the analysis of the residuals is that it can only test for a main effect of x2 on y, it cannot test for an interaction with x1. Any thoughts on how to best handle this type of problem would be greatly appreciated.

• Why do you think it might not be "fair" to test for an interaction or the other main effects beyond $x_1$? And why do you think you can't examine residuals in a model with an interaction term? Quality control of regressions typically involves a plot of residuals against fitted values based on all predictors in the model.
– EdM
Oct 6 '15 at 16:58
• I understand how 'qc' of model regression works. I do believe you can plot residuals vs fitted to examine heteroscedasticity, and examine qqnorm plots, etc. I mean taking the residuals of the model y~ x1 and modeling them as a function of x2 + x3 + x4. In terms of 'fairness' I mean, is the relationship between y and x1 too strong to warrant coding interactions, or is some other type of analysis more appropriate. Oct 6 '15 at 17:21

There's no magical cutoff for an $r^2$ value in regression (except maybe for 1.000), and what's an extraordinarily good fit in a biological study might be absurdly poor in a physics experiment. Unless your sole interest is in doing predictions of $y$ based on $x_1$ values and you have no remaining interest in how the other variables work together with $x_1$ to influence $y$, you still have some work to do and some variance to explain.
If, based on prior knowledge of your subject matter, you expect an interaction between $x_1$ and $x_2$ then you certainly should include that in your model, along with a main effect of $x_2$. If subject-matter knowledge indicates that $x_3$ and $x_4$ also influence $y$, include them too.
The multiple regression will be informative whether the other variables are correlated with $x_1$ or not. If they aren't, they may improve the fit even further. If they are, the nature of the interrelations might help inform your understanding of the underlying processes.