A note on estimating crude odds ratios in case-control studies with differentially misclassified exposure

Morrissey and Spiegelman (1999, Biometrics 55, 338 344) provided a comparative study of adjustment methods for exposure misclassification in case-control studies equipped with an internal validation sample. In addition to the maximum likelihood (ML) approach, they considered two intuitive procedures based on proposals in the literature. Despite appealing ease of computation associated with the latter two methods, efficiency calculations suggested that ML was often to be recommended for the analyst with access to a numerical routine to facilitate it. Here, a reparameterization of the likelihood reveals that one of the intuitive approaches, the inverse matrix method, is in fact ML under differential misclassification. This correction is intended to alert readers to the existence of a simple closed-form ML estimator for the odds ratio in this setting so that they may avoid assuming that a commercially inaccessible optimization routine must be sought to implement ML.