Problems with infinite solutions in logistic regression

In teaching logistic regression for case–control studies, I ask master’s students in epidemiology to assess an interaction between a 2-level exposure and a 4-level exposure using a likelihood-ratio test. Theory suggests that the test statistic has 3 degrees of freedom, but Stata uses 2 degrees of freedom. The explanation turns out to be that one exposure combination contains controls but no cases, so that one parameter goes to infinity. It is hard to convince the students (and myself) that this combination contributes no degrees of freedom. I will review how Stata handles situations in which parameters go to infinity. Although asymptotics for likelihood-ratio tests may not work well in this situation, I will argue that lrtest should be modified to reflect the true number of degrees of freedom.