  Consider a case-control study in which the aim is to assess the effect of a
factor 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.
