Chi-Square Tests with One Degree of Freedom; Extensions of the Mantel-Haenszel Procedure

TLDR

The authors extend the Mantel–Haenszel method to analyze multiple 2×2 contingency tables in retrospective disease studies. They generalize the method to multi‑row/column tables with orderable or continuous dimensions, assign scores to rows and columns, compute deviations of cross‑product sums from expectation, and derive a chi‑square statistic from the grand total of deviations. The extended procedure is shown to generalize or match the asymptotic forms of numerous existing non‑parametric methods.

Abstract

Abstract A published method for analyzing multiple 2×2 contingency tables arising in retrospective studies of disease is extended in application and form. Extensions of application include comparisons of age-adjusted death rates, life-table analyses, comparisons of two sets of quantal dosage-response data, and miscellaneous laboratory applications as appropriate. Extensions in form involve considering multiple contingency tables with arbitrarily many rows and/or columns, where rows and columns are orderable, and may even be on a continuous scale. The assignment of some score for each row or column is essential to use of the method. With scores assigned, a deviation of the sum of cross products from expectation, and its variance conditioned on all marginal totals, are computed for each table and a chi square is determined corresponding to the grand total of the deviations. For various specific instances and for various scoring procedures, the procedure extends or is equivalent to the asymptotic form of many known non-parametric techniques.