Proof. Let \(r\) be the risk among the unexposed with \(C\) absent. The unexposed have risk \(r\) or \(rR_C\), so their average risk is \(r\{(1 - \pi _0) + \pi _0 R_C\} = r\{1 + \pi _0(R_C - 1)\}\). The exposed have risks \(\theta r\) and \(\theta r R_C\), averaging \(\theta r\{1 + \pi _1(R_C - 1)\}\). The ratio of the averages is the stated expression. □

Proof. With \(\theta = 1\) and \(x = R_C - 1 > 0\), \(\RR _{\text {obs}} = (1 + \pi _1 x)/(1 + \pi _0 x)\), which exceeds \(1\) only if \(\pi _1 > \pi _0\). As a function of \(x\) it is increasing (its derivative has the sign of \(\pi _1 - \pi _0\)) and tends to \(\pi _1/\pi _0\) as \(x \to \infty \), so \(\RR _{\text {obs}} \leq \pi _1/\pi _0\). For the other inequality, \begin {align*} \frac {1 + \pi _1 x}{1 + \pi _0 x} \leq 1 + x &\iff 1 + \pi _1 x \leq (1 + x)(1 + \pi _0 x) \\ &\iff \pi _1 \leq 1 + \pi _0(1 + x) , \end {align*}

which holds because \(\pi _1 \leq 1\). □

Proof. Let \(w_k = P(C = k)\), the same among exposed and unexposed by independence, and let the stratum risks be \(p_{0k}\) and \(p_{1k} = \theta p_{0k}\). The crude risks are \(\sum _k w_k p_{1k} = \theta \sum _k w_k p_{0k}\) and \(\sum _k w_k p_{0k}\), with ratio \(\theta \). For the difference, \(p_{1k} = p_{0k} + \delta \) gives crude risks differing by \(\delta \sum _k w_k = \delta \). □