Fitting binary regression models with case-augmented samples

S In a case-augmented study, measurements on a random sample from a population are augmented by information from an independent sample of cases, that is units with some characteristic of interest. We show that inferences about the effect of the covariates on the probability of being a case can be made by fitting a modified prospective likelihood. We also show that this procedure is fully efficient. 1. I This paper is concerned with the problem of fitting a regression model, pr(Y =1|x)=p(x; b) (1) say, relating the mean of a binary response variable, Y , to a p-dimensional vector of covariates, x, in situations where observations from a random sample drawn from the whole population are augmented by data from an independent sample of cases, i.e. units with Y =1. There are two distinct situations to be considered. In the first, Design 1, we have information about the covariate values, x, but not the responses, Y , for individuals in the original random sample; we shall call this the reference sample. In the second, Design 2, we have information about both Y and x for members of the reference sample. Cosslet (1981) uses the terms case-supplemented sampling to describe the first situation and case-enriched sampling for the second. We shall use these terms and the general term case-augmented sampling to cover both situations. Both designs can be regarded as variants of the standard, unmatched case-control design, except that here controls are drawn at random from the whole population rather than from the noncases, i.e. units with Y =0. Obviously there is very little difference between the designs if cases are rare in the population, but one of the advantages of the case-augmented designs is that they allow us to estimate relative risks without invoking the 'rare disease' assumption, for, with a simple application of Bayes' Theorem, we can write the risk at covariate value x relative to that at some baseline, x 0 , as pr(Y =1|x) pr(Y =1|x 0) = g(x|Y =1) g(x 0 |Y =1)N g(x) g(x 0) ,