Tests and limits for the common odds ratio of several 2 × 2 contingency tables: methods in analogy with the Mantel-Haenszel procedure

For a postulated common odds ratio for several 2 × 2 contingency tables one may, by conditioning on the marginals of the seperate tables, determine the exact expectation and variance of the entry in a particular cell of each table, hence for the total of such cells across all tables. This makes it feasible to determine limiting values, via single-degree-of-freedom, continuity-corrected chi-square tests on the common odds ratio–one determines lower and upper limits corresponding to just barely significant chi-square values. The Mantel-Haenszel approach can be viewed as a special application of this, but directed specifically to the case of unity for the odds ratio, for which the expectation and variance formulas are particularly simple. Computation of exact expectations and variances may be feasible only for 2 × 2 tables of limited size, but asymptotic formulas can be applied in other instances.Illustration is given for a particular set of four 2 × 2 tables in which both exact limits and limits by the proposed method could be applied, the two methods giving reasonably good agreement. Both procedures are directed at the distribution of the total over the designated cells, the proposed method treating that distribution as being asymptotically normal. Especially good agreement of proposed with exact limits could be anticipated in more asymptotic situations (overall, not for individual tables) but in practice this may not be demonstrable as the computation of exact limits is then unfeasible.