How do the ML estimation commands (e.g., logit and probit) compute the model chi-squared test when the estimate robust standard errors on clustered data?

When you specify vce(robust), specify vce(cluster clustvar), or use pweights for a maximum likelihood estimation command that allows these options, the model chi-squared test is a Wald test rather than a likelihood-ratio test.

When you have clusters or pweights, the likelihood used for estimation is not a true likelihood; i.e., it is not the distribution of the sample. For clustering, observations are no longer independent. For pweights, the likelihood does not reflect the "randomness" of the sampling weights. Thus, here, one should not use the conventional likelihood-ratio test.

When you only have a few clusters (say, <100), an adjusted Wald test is better than the standard Wald test. The svy commands use the adjusted Wald test by default, as does the test command when used after svy estimation. For more information, see [R] test also Korn and Graubard (1990).

Reference

Korn, E. L., and B. I. Graubard. 1990.

Simultaneous testing of regression coefficients with complex survey data: Use of Bonferroni t statistics. American Statistician 44: 270–276.