Relative risk, risk difference and rate difference models for sparse stratified data: a pseudo likelihood approach.
We consider a relative risk and a risk difference model for binomial data, and a rate difference model for Poisson (person year) data. It is assumed that the data are stratified in a large number of small strata. If each stratum has its own parameter in the model, then, due to the large number of parameters, straightforward maximum likelihood leads to inconsistent estimates of the relevant parameters. By contrast to the logistic model, conditioning on the number of events per stratum does not help in eliminating the stratum nuisance parameters. We propose a pseudo likelihood method to overcome these consistency problems. The resulting pseudo maximum likelihood estimates can easily be computed with standard statistical software. Our approach gives a more general framework for the Mantel-Haenszel type estimators proposed in the literature. In the special case of a series of 2 x 2 tables, for the risk and rate difference models, our approach yields exactly these ad hoc Mantel-Haenszel estimators, while for the relative risk model it gives a close approximation of the Mantel-Haenszel relative risk estimator. For the regression models corresponding to the association measures relative risk, risk difference and rate difference, our method provides analogues of conditional logistic regression, which were not previously available.