Is it necessary to control for the interaction of a covariate with an independent variable, after controlling for the covariate itself?

Question

Let's say that I want to look at the difference in average income between men and women working the same job, by using an ANCOVA.

My independent variable is therefore gender, while the dependent variable is income.

I want to control for the age covariate, to account for any differences in the average age between men and women in my sample.

If I see that there is an interaction between the gender and age variables, so that (for example), gender has a bigger impact on income for men than it does for women, do I need to add this interaction as an additional covariate, in addition to the age covariate?

I would think that the answer is 'no', since by adding age as a covariate, I'm already accounting for any age-related differences in the sample, but I'm not completely sure.

Answer 1

Yes and no. Primarily, it depends on what question you are trying to answer. If you believe age is a confounding variable, the no-interaction model is appropriate for testing the hypothesis: "comparing men and women of the same age, what is their expected earnings difference?". If that difference varies across age, then the no-interaction model averages up those differences.

A second defense of the original model: You didn't plan on testing the interaction in the model, and there is an issue of multiple testing that hasn't been addressed. How can we trust that the 0.05 level reflects an reported false positive rate of 0.05 if the finding itself generated the hypothesis?

It is worth reporting as a hypothesis generating claim, rather--and after reporting the results of the original confirmatory analysis--that you observed a statistically significant interaction and that may be evidence that there are age or calendar year effects which underlie earnings differences (or lack thereof) in men and women.