9.3 Pooling homogeneous estimates
If the measure is homogeneous, say a common log odds ratio \(\beta \), each stratum gives an estimate \(\hat \beta _k = \log \widehat {\OR }_k\) with approximate variance \(v_k\), and the strata are independent. Among all weighted averages, the best uses inverse-variance weights.
Theorem 9.3.1. Let \(\hat \beta _1, \dots , \hat \beta _K\) be independent and unbiased for \(\beta \), with variances \(v_k\). Among estimators \(\sum _k c_k\hat \beta _k\) with \(\sum _k c_k = 1\), the variance is minimised by \(c_k \propto 1/v_k\). With \(w_k = 1/v_k\) the minimum-variance estimator and its variance are \[ \hat \beta _W = \frac {\sum _k w_k\hat \beta _k}{\sum _k w_k}, \qquad \Var (\hat \beta _W) = \frac {1}{\sum _k w_k} . \]
Proof. Any such estimator is unbiased, with variance \(\sum _k c_k^2 v_k\). By the Cauchy–Schwarz inequality, \[ 1 = \Bigl (\sum _k c_k\Bigr )^2 = \Bigl (\sum _k c_k\sqrt {v_k} \cdot \frac {1}{\sqrt {v_k}}\Bigr )^2 \leq \sum _k c_k^2 v_k \sum _k \frac {1}{v_k} , \] so \(\sum _k c_k^2 v_k \geq 1/\sum _k w_k\), with equality when \(c_k\sqrt {v_k}\) is proportional to \(1/\sqrt {v_k}\), that is \(c_k \propto 1/v_k\). Then \(c_k = w_k/\sum _l w_l\) and the variance is \(\sum _k w_k^2 v_k/(\sum _l w_l)^2 = 1/\sum _l w_l\). □
Applied to log odds ratios with Woolf’s variances, this is the Woolf pooled estimator. Its weakness is that the weights and the stratum estimates come from the same small tables: a stratum with a zero cell has an infinite log odds ratio and must be corrected or dropped, and with many sparse strata (as in matched studies, where each set is a stratum) the estimator is badly biased. The Mantel–Haenszel estimator of Section 10 avoids both problems.
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