Combining case-control studies for identifiability and efficiency improvement in logistic regression

Abstract

Can two separate case-control studies, one about Hepatitis disease and the other about Fibrosis, for example, be combined together? It would be hugely beneficial if two or more separately conducted case-control studies, even for entirely irrelevant purposes, can be merged together with a unified analysis that produce better statistical properties, e.g., more accurate estimation of parameters. In this paper, we show that, when using the popular logistic regression model, the combined/integrative analysis produces more accurate estimation of the slope parameters than the single case-control study. It is known that, in a single logistic case-control study, the intercept is not identifiable, contrary to prospective studies. In combined case-control studies, however, the intercepts are proved to be identifiable under mild conditions. The resulting maximum likelihood estimates of the intercepts and slopes are proved to be consistent and asymptotically normal, with asymptotic variances achieving the semiparametric efficiency lower bound.

Semi‐supervised inference for nonparametric logistic regression

“…Recall that for the classical logistic regression under case-control study, 24 the intercept term and the density function of X * , denoted by f (⋅), are not estimable, as the score function of the intercept term lies in the linear space spanned by the score function of f (⋅). The identifiability issue of the classical logistic regression under case-control studies was carefully studied by Tang et al 46 With the availability of the unlabeled samples in a semi-supervised setting, f (⋅) is always identifiable. We propose a two-stage procedure to estimate g(⋅): first, the unlabeled data are used to estimate the density function of X * ; then a likelihood-based estimator for g(⋅) based on B-spline is obtained at the second stage.…”