Suppose I have a regression model with several predictors ind interactions thereof. For concreteness, suppose we are studying a company's data on salaries and we have predictors
exper (so there are 3 factors and two covariates). We settle on a model expressed SAS-style as
h(salary) = dept title sex age exper dept*sex dept*age title*exper title*age sex*age sex*exper age*exper dept*sex*age sex*age*exper
This model includes many, but not all, interactions up through third-order, and suppose we find that for a suitable transformation
h, the fit statistics and residual plots all look good.
The question to be answered is "does sex affect salary?" I think many people would answer based on, say, the "Type II" $F$ test for
sex. But suppose I make the question more specific by asking "if sex were not taken into account at all, would it make any difference?" This question suggests comparing the above model with the reduced model where all terms involving
sex are excluded:
h(salary) = dept title age exper dept*age title*exper title*age age*exper
A test comparing these models would have quite a few numerator degrees of freedom, whereas the Type II test of
sex has only one. Of course, I do think a lot of people would also look at the individual tests of each of the terms that involve
sex. But it is possible that an overall test of all of these terms together could have a smaller $P$ value than any of the individual tests.
My question is: Is there a name for this kind of reduction-of-model test, where the full model is compared with a reduced model where a predictor is completely excluded? And do lots of people do this kind of thing routinely, and I should cringe in embarrassment for not having seen it much? I'd think people might be wise to consider such a test more often. Any comments?