The Logit Model and Response-Based Samples

It is well-known that, under the logit model for binary response, the random sampling and response-based sampling maximum likelihood estimators coincide for all parameters except the intercept. Citing this coincidence, many researchers have assumed the logit model and analyzed data from response-based samples as if those data were obtained by random sampling. We argue that this practice should be avoided unless the researcher really believes the logit specification. One preferable alternative is the weighted maximum likelihood estimator of Manski and Lerman (1977). Random sampling maximum likelihood analysis does not have a natural interpretation when the true response function is not logit. Weighted maximum likelihood analysis estimates a constrained best predictor of the binary response and so remains interpretable.