Fitting prospective regression models to case-control data

SUMMARY We consider fitting prospective regression models to data obtained by case-control or response selective sampling from a finite population with known population totals in each response category. Maximum likelihood estimation is developed and compared with two pseudo-likelihood approaches. The relative efficiencies of the methods are explored in the special case of estimating the parameters of the proportional odds model for ordinal responses. For such applications the method Hsieh, Manski & McFadden (1985) call 'conditional maximum likelihood' is shown to be essentially as efficient as maximum likelihood; the latter is considerably more difficult to implement. In contrast the use of a weighted estimate of the prospective likelihood can lead to a substantial loss of efficiency.