Solution. By Theorem 7.2.1 with \(s_{11} = s_{10}\) and \(s_{01} = 1.5\,s_{00}\), the bias factor is \((s_{11}s_{00})/(s_{10} \times 1.5\,s_{00}) = 1/1.5 = 0.67\). The study will tend to find an odds ratio of \(0.67\), suggesting that coffee protects. The controls over-represent coffee drinkers, so the cases look less exposed than the base they came from.

Solution. The proportions of the population who are in hospital in each category are: both, \(0.1 \times 0.2 \times 0.65 = 0.013\); A only, \(0.1 \times 0.8 \times 0.5 = 0.040\); B only, \(0.9 \times 0.2 \times 0.3 = 0.054\); neither, \(0.9 \times 0.8 \times 0.05 = 0.036\). The odds ratio among patients is \((0.013 \times 0.036)/(0.040 \times 0.054) = 0.22\). Two independent conditions appear strongly negatively associated in hospital, because a patient admitted without one of them must usually have the other. Admission is a collider.

Solution. The bias factor is \((1 + 0.5 \times 2)/(1 + 0.2 \times 2) = 2/1.4 = 1.43\), so by Theorem 7.4.2 the adjusted risk ratio is \(2.0/1.43 = 1.40\). The confounder explains part of the association but not all of it. Cornfield’s conditions, \(R_C \geq 2\) and \(\pi _1/\pi _0 \geq 2\), both hold here (\(3\) and \(2.5\)), yet the confounder does not explain the association away: the conditions are necessary for a confounder to explain an association fully, not sufficient.

Solution. With weights \(w_k = P(C = k)\) common to exposed and unexposed, the crude risk ratio is \[ \frac {\sum _k w_k p_{1k}}{\sum _k w_k p_{0k}} = \sum _k \frac {w_k p_{0k}}{\sum _l w_l p_{0l}} \cdot \frac {p_{1k}}{p_{0k}} , \] a weighted average of the stratum ratios \(p_{1k}/p_{0k}\) with weights proportional to the expected cases among the unexposed in each stratum. When the stratum ratios differ there is effect modification, a separate matter taken up with stratified analysis; the crude ratio is a meaningful summary of them, not a confounded one.

Solution. (a) Obesity raises blood pressure, which raises the risk of heart attack: a mediator. Adjusting for it estimates only the effect of obesity not acting through blood pressure, which is a different question; for the total effect, do not adjust. (b) Age affects both smoking habits and cancer risk and is not affected by smoking: a confounder; adjust. (c) Live birth is affected by both the drug and the defect: a collider. Restricting to live births conditions on it and can bias the association, typically towards a protective-looking effect, since exposed pregnancies with defects are especially likely to be lost.

Solution. The stratum risk ratios are \(0.8/0.5 = 1.6\) and \(0.5/0.2 = 2.5\), and the crude is \(0.65/0.35 = 1.86\). The risk ratio is not constant across strata (the odds ratio is), so there is effect modification on the ratio scale. The crude \(1.86\) is the weighted average of the problem before, with weights proportional to \(0.5\) and \(0.2\), the unexposed risks: \((0.5 \times 1.6 + 0.2 \times 2.5)/0.7 = 1.86\). With no confounding the crude risk ratio is a fair summary of the stratum ratios; the crude odds ratio \(3.45\), by contrast, is smaller than both stratum odds ratios.