Equivalence Between Conditional and Mixture Approaches to the Rasch Model and Matched Case-Control Studies, With Applications

Analyses of data using Rasch models, including the special case of matched case-control studies, are common applications of conditional likelihood in which the usual inferential procedures are applied only after conditioning on an approximately ancillary statistic. Another common approach to the analysis of Rasch models is to integrate the nuisance parameters over a mixing distribution, using the marginal likelihood obtained as the basis for inference. We show that the full conditional likelihood can always be obtained exactly via the marginal approach, given a particular choice of mixing distribution, and derive necessary and sufŽ cient conditions for the two approaches to agree. Previous work has shown that with sufŽ cient  exibility in the mixing distribution, the maxima of the marginal and conditional likelihoods will be equivalent under concordance criteria. Our argument requires no such criteria, and for any dataset guarantees the equivalence of the whole of the two likelihoods, not just their maxima. This substantially enhances the previous results and provides an alternative derivation for any existing conditional analysis. We give examples of mixing distributions that guarantee the agreement of the two approaches, and explore equivalence classes of such distributions, together with some of their attractive symmetry properties. Our argument also allows for the adaption and extension of analytic techniques already widely used with Rasch data, and in particular with matched case-control studies; potential applications of these advances are illustrated with several examples. These include new numerical algorithms for evaluating the conditional likelihood without directly specifying its computationally awkward functional form, inferences about complex functions of the parameters of interest obtained using existing Markov chain Monte Carlo methods, powerful measures of goodness of Ž t derived from likelihood contributions that are ignored by the conditional approach, and the justiŽ able addition of prior knowledge to existing conditional analyses.