Effect of a preliminary test of homogeneity of stratum-specific odds ratios on their confidence intervals

Consider a case-control study in which the aim is to assess a factor's effect on disease occurrence. We suppose that this factor is dichotomous. Also suppose that the data consists of two strata, each stratum summarized by a two-by-two table. A commonly-proposed two-stage analysis of this type of data is the following. We carry out a preliminary test of homogeneity of the stratum-specific odds ratios. If the null hypothesis of homogeneity is accepted then we find a confidence interval for the assumed common value (across strata) of the odds ratio. We examine the statistical properties of this two-stage analysis, based on the Woolf method, on confidence intervals constructed for the stratum-specific odds ratios, for large numbers of cases and controls for each stratum. We provide both a Monte Carlo simulation method and an elegant large-sample method for this examination. These methods are applied to obtain numerical results in the context of the large numbers of cases and controls for each stratum that arose in a real-life dataset. In this context, we find that the preliminary test of homogeneity of the stratum-specific odds ratios has a very harmful effect on the coverage probabilities of these confidence intervals.