A note on the prospective analysis of outcome-dependent samples

Two likelihood representations corresponding to the prospective and retrospective analyses of the case-control design are derived for general outcome-dependent samples with arbitrary discrete or continuous outcomes and possibly non-multiplicative models. Parameter identification in the general outcome-dependent design is reduced to the simple problem of parameter identification in the general odds ratio function. Both likelihoods are shown to generate the same profile likelihood for the common parameter of interest. Maximum likelihood estimators based on either likelihood are semiparametric efficient for the identifiable parameters.