Motivating likelihood ratio test vs Wald test for paper reviewer

Question

I've got back reviews for a paper I've submitted, with the following problem.

I have two logistic regression models, say y ~ A, and y ~ A + B, where B is a factor with several levels. I have performed a likelihood ratio test between them, and it is highly significant. My goal is to show that B has some independent information about the response and thus improves model fit "signficantly" (loaded term) above and beyond A, and I think the LRT shows this very strongly.

The reviewer is not happy with this, they suggest using a different approach of comparing the "change of significance of B between two models": y ~ B vs y ~ A + B, which to me sounds like the Wald test for B in the A + B model, even though the reviewer doesn't realise it.

Given that B has multiple levels and thus multiple Wald tests, it seems to me the LRT is more useful and more powerful (also no multiple testing issue).

Am I right? Are there any nice (not too abstract) references I can use to back these claims?

Answer 1

Wald vs likelihood ratio is secondary to the fact that comparing $y\sim B$ to $y\sim A+B$ is a test of $A$, not a test of $B$. I hope this is a typo in the referee report.

Agresti’s Foundations of Linear and Generalized Linear Models gives some reasons to prefer likelihood ratio testing to Wald testing in logistic regression, though the difference usually is not considerable. However, the reviewer is all kinds of wrong to have you testing $A$ to say how good of a predictor $B$ is. That is roughly equivalent to running ANCOVA by testing the slope instead of the factor—complete nonsense.