Solution. Small stones: A \(0.931\), B \(0.867\). Large stones: A \(0.730\), B \(0.688\). Overall: A \(273/350 = 0.780\), B \(289/350 = 0.826\). A is better for both sizes but worse overall. Treatment A was used mostly for large stones (\(263\) of its \(350\)), which are harder to treat, and B mostly for small ones. Stone size is a confounder: it affects success and is associated with the treatment given. The stratified comparison is the valid one.

Solution. Men: \(\widehat {\OR } = (20 \times 200)/(80 \times 40) = 1.25\), \(v = 0.0925\), interval \((0.69,\ 2.27)\). Women: \(\widehat {\OR } = (40 \times 150)/(20 \times 30) = 10.0\), \(v = 0.115\), interval \((5.1,\ 19.4)\). The pooled log odds ratio has weights \(10.81\) and \(8.70\); \(Q = 20.8\) on \(1\) degree of freedom, \(P < 0.0001\). The crude odds ratio is \((60 \times 350)/(100 \times 70) = 3.0\). The odds ratio is strongly heterogeneous: the exposure is associated with the disease in women but hardly in men. Neither the crude \(3.0\) nor a pooled estimate (about \(3.2\)) describes either sex, and the results should be reported separately.

Solution. Suppose \(\OR = \psi \) and \(\RR = \theta \) in both strata. With baseline risk \(p_0\) and \(p_1 = \theta p_0\), \(\psi = \theta (1 - p_0)/(1 - \theta p_0)\). Solving for \(p_0\): \(\psi - \psi \theta p_0 = \theta - \theta p_0\), so \(p_0(\theta - \psi \theta ) = \theta - \psi \) and \(p_0 = (\psi - \theta )/\{\theta (\psi - 1)\}\), a single value. Two strata with different baseline risks cannot share both \(\psi \) and \(\theta \) unless \(\psi = \theta = 1\).

Solution. \(\hat \beta _W = (12.86 \times 0.693 + 10.63 \times 0.901)/23.49 = 0.787\), an odds ratio of \(2.20\), with \(\text {SE} = 1/\sqrt {23.49} = 0.206\). The estimate hardly moves, but the standard error rises from \(0.184\) to \(0.206\), as Theorem 9.3.1 implies: every stratum adds to \(\sum w_k\), however small its weight.

Solution. \(\RR _{10} = 2\), \(\RR _{01} = 2.5\), \(\RR _{11} = 6\). On the multiplicative scale the joint ratio expected without interaction is \(5\), against \(6\) observed, a modest departure. On the additive scale \(\text {RERI} = 6 - 2 - 2.5 + 1 = 2.5\): the joint excess risk (\(0.10\)) is much larger than the sum of the separate excesses (\(0.02 + 0.03 = 0.05\)). There is strong additive interaction; preventing either exposure in people who have both prevents more cases than the separate effects suggest.