10.3 The Mantel–Haenszel test
Theorem 10.3.1. Under the hypothesis of no association in any stratum, conditionally on all the margins, \[ E\Bigl (\sum _k a_k\Bigr ) = \sum _k \frac {n_{1k}m_{1k}}{n_k}, \qquad \Var \Bigl (\sum _k a_k\Bigr ) = \sum _k \frac {n_{1k}n_{0k}m_{1k}m_{0k}}{n_k^2(n_k - 1)} , \] and the Mantel–Haenszel statistic \[ X^2_{\text {MH}} = \frac {\bigl \{\sum _k a_k - \sum _k E(a_k)\bigr \}^2}{\sum _k \Var (a_k)} \] is approximately \(\chi ^2_1\) when \(\sum _k \Var (a_k)\) is large.
Proof. By Theorem 4.4.2 each \(a_k\), given its margins, is hypergeometric with the stated mean and variance, and the strata are independent. Sums of independent variables have summed means and variances. A sum of many independent bounded variables with large total variance is approximately normal by the Lindeberg central limit theorem, which applies whether the strata are few and large or many and small; the square of the standardised sum is then approximately \(\chi ^2_1\). □
The statistic compares the total number of exposed cases with its expectation under no association, summed across strata before squaring. Associations in the same direction in different strata reinforce each other, and the test is most powerful against a common odds ratio. A continuity correction, subtracting \(\tfrac 12\) from \(|\sum a_k - \sum E(a_k)|\) before squaring, is often used.
Example 10.3.2. In Example 9.1.1, \(\sum _k a_k = 103\). The expected values are \(5.98\), \(21.43\) and \(54.00\), totalling \(81.40\), and the variances \(4.63\), \(11.70\) and \(11.43\), totalling \(27.76\). Then \(X^2_{\text {MH}} = (103 - 81.40)^2/27.76 = 16.8\) (\(16.0\) with the continuity correction), \(P < 0.0001\). The age-adjusted association is highly significant.
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