Unique variance with regression analysis

Question

I have some questions about unique variance and hope some of you can help.

For instance, let say I have 3 predictors and 1 dependent variable (DV).

I ran a regression analysis with a sequence of 3 regression models using:

1. Predictor A
2. Predictor A + Predictor B
3. Predictor A + Predictor B + Predictor C

I want to find out whether Predictor A explains the unique variance in the DV.

Assuming Model 1 shows significant contribution to the DV, and after adding Model 2 and Model 3, it still shows significant contribution to the DV.

Does this mean that Predictor A does explain the unique variance of the DV? Or is it the other way around, that is, it does not explain the unique variance of the DV?

Answer 1

I suspect you are asking about the different kinds of sums-of-squares and nested hypothesis tests. The two primary kinds of SS that people worry about are type I SS and type III SS. I have written about this topic several times on CV; you may want to read some of these answers (primarily here and here, but also here and here), to get more information about this issue. In a nutshell, this is about how the sums of squares are partitioned and what SS gets used for the numerator of an F (actually F-change) test. (I discuss the F-change test in a different context here.) Specifically, is the SS determined by dropping each predictor from the final model, with all the other terms still included, or are they dropped in order such that (for example) the first predictor is still out when the second one is dropped? In my previous answers about SS, I emphasized whether all of the information is being used, but, perhaps even more important, we should also notice that these two are answers to different substantive questions. If you want to know if A is related to the DV even after having taken account of the relationships between B and C and the DV, then you should either use type III SS to test it, or (which amounts to the same thing) type I SS where A is dropped first and B & C are still in the model. Moreover, note that the type of SS used is irrelevant if the predictors are perfectly orthogonal. Lastly, it's important to recognize that the F test associated with A for a model in which A is the only predictor, is substantively different from the F-change test in which A is dropped last, because the denominator of F differs.