10.7 Testing homogeneity: the Breslow–Day test

Woolf’s test of Section 9 breaks down in sparse strata. Breslow and Day (1980) proposed a test that uses the Mantel–Haenszel estimate: in each stratum find the cell count \(\tilde a_k\) which, with the observed margins, has odds ratio \(\widehat {\OR }_{\text {MH}}\) (the root in the admissible range of a quadratic equation), and its variance \(\tilde v_k = (1/\tilde a_k + 1/\tilde b_k + 1/\tilde c_k + 1/\tilde d_k)^{-1}\). Then \[ X^2_{\text {BD}} = \sum _k \frac {(a_k - \tilde a_k)^2}{\tilde v_k} \] is approximately \(\chi ^2_{K - 1}\) under homogeneity. For Example 9.1.1 the fitted counts are \(10.17\), \(30.70\) and \(62.36\), and \(X^2_{\text {BD}} = 0.50\) on \(2\) degrees of freedom, \(P = 0.78\), in agreement with Woolf’s test.

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