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Why should I not do a likelihood-ratio test after an ML estimation (e.g., logit, probit) with clustering or pweights?

Title   Likelihood-ratio test after survey/robust ML estimation
Author William Sribney, StataCorp

The “likelihood” for pweighted or clustered MLEs is not a true likelihood; i.e., it is NOT the distribution of the sample. When there is clustering, individual observations are no longer independent, and the “likelihood” does not reflect this. Where there are pweights, the “likelihood” does not fully account for the “randomness” of the weighted sampling.

The “likelihood” for pweighted or clustered MLEs is used only for the computation of the point estimates and should not be used for variance estimation using standard formulas. Thus the standard likelihood-ratio test should NOT be used after estimating pweighted or clustered MLEs.

Instead of likelihood-ratio tests (the lrtest command), Wald tests (the test command) should be used.

The svy commands allow the use of the test command, which computes an adjusted Wald test. This adjustment is useful when the total number of clusters is small (<∼100).

The test command also has a mtest(bonferroni) option. Some statisticians argue that the mtest(bonferroni) option of test gives a better test than an adjusted Wald test. See

Korn, E. L., and B. I. Graubard. 1990.
Simultaneous testing of regression coefficients with complex survey
data: use of Bonferroni t statistics.
The American Statistician 44: 270–276.

for such an argument.

The Bonferroni adjustment carries the tacit assumption that the multidimensional test (let k = the dimension) being conducted consists of k hypotheses that individually make sense as different research questions—or said more precisely, a priori you should have no knowledge that the individual hypotheses are highly collinear. If the individual hypotheses ARE highly collinear, then the Bonferroni adjustment can be overly conservative. So if you suspect this in advance, you might want to stay away from the Bonferroni adjustment. But, in statistics at least, being conservative is safe, so doing the Bonferroni adjustment should be fine.