Combining 2 × 2 tables that contain structural zeros

When one cell or more of a contingency table is necessarily zero, the table is said to contain structural zeros. These structural voids may be inherent to the problem at hand, or, in some applications, they may be introduced intentionally through the experimental design. With data classified by a third stratification variable, one needs a method for analysing a set of independent 2 x 2 tables. We propose Cochran-Mantel-Haenszel type analyses that combine observations across strata. We also propose tests of hypotheses pertaining to the marginal and conditional probabilities.