10.5 Risk ratios and risk differences

For cohort data with risks, the same idea gives \[ \widehat {\RR }_{\text {MH}} = \frac {\sum _k a_k n_{0k}/n_k}{\sum _k b_k n_{1k}/n_k}, \qquad \widehat {\RD }_{\text {MH}} = \frac {\sum _k (a_k n_{0k} - b_k n_{1k})/n_k}{\sum _k n_{1k}n_{0k}/n_k} . \] The first is a weighted average of the stratum risk ratios with weights \(b_k n_{1k}/n_k\), the second of the stratum risk differences with weights \(n_{1k}n_{0k}/n_k\). Greenland and Robins (1985) gave the variance estimator, consistent in large and sparse strata alike, \[ \widehat {\Var }(\log \widehat {\RR }_{\text {MH}}) = \frac {\sum _k (m_{1k}n_{1k}n_{0k}/n_k^2 - a_k b_k/n_k)}{\bigl (\sum _k a_k n_{0k}/n_k\bigr )\bigl (\sum _k b_k n_{1k}/n_k\bigr )} . \] The test of no association is the same \(X^2_{\text {MH}}\) as before, since conditionally on the margins it does not matter whether the table arose from a cohort or a case-control design.

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