Maximum likelihood for generalised case-control studies

Abstract

Suppose that we have a population of units with values (x,y), where x is a vector of covariates and y is a response variable, generated according to the model f(y|x;θ)g(x). Both x and y can be multivariate and any elements of x and y can be either discrete or continuous. The first term, f(y|x;θ), is some regression model linking response-variable behaviour to that of the covariates and we are interested in fitting this model. We discuss semiparametric maximum-likelihood estimation of θ for a class of study designs and data structures in which some units are not fully observed and in which it is impossible or undesirable to model the covariate distribution, g(x), which is typically of little interest in its own right. The key feature of the designs is that the probability that a unit is fully observed can depend on the value of the response, y. Applications include a number of useful generalisations of the basic case-control design and a broad class of missing data and measurement-error problems.