Closure of the class of binary generalized linear models in some non-standard settings

This paper considers fitting generalized linear models to binary data in nonstandard settings such as case–control samples, studies with misclassified responses and misspecified models. We develop simple methods for fitting models to case–control data and show that a closure property holds for generalized linear models in the nonstandard settings, i.e. if the responses follow a generalized linear model in the population of interest, then so will the observed response in the non-standard setting, but with a modified link function. These results imply that we can analyse data and study problems in the non-standard settings by using classical generalized linear model methods such as the iteratively reweighted least squares algorithm. Example data illustrate the results.