Proof. In the cohort design the \(n_1\) exposed subjects are a random sample from the exposed population, so each one independently has the disease with probability \(p_1\); likewise for the unexposed. In the case-control design each sampled case is exposed with probability \(q_1\) and each control with probability \(q_0\). To see that the risks cannot be estimated in a case-control study, use Bayes’ theorem with \(\pi = P(D)\): \[ p_1 = \frac {q_1 \pi }{q_1 \pi + q_0 (1 - \pi )}, \] which takes every value in \((0,1)\) as \(\pi \) ranges over \((0,1)\) with \(q_1, q_0\) fixed. The distribution of the data depends only on \((q_1, q_0)\), so two populations with the same \((q_1, q_0)\) but different \(\pi \) give the same distribution of data and different \(p_1\). No function of the data can be consistent for \(p_1\). The cohort argument is the same with the roles of \(E\) and \(D\) exchanged. □

Proof. Both sides equal the cross-product ratio of the joint probabilities. Write \(\pi _{DE} = P(D \cap E)\) and similarly for the other three cells. Then \(q_1 = \pi _{DE}/P(D)\) and \(1 - q_1 = \pi _{D\bar E}/P(D)\), so \(q_1/(1 - q_1) = \pi _{DE}/\pi _{D\bar E}\); similarly \(q_0/(1 - q_0) = \pi _{\bar D E}/\pi _{\bar D \bar E}\). The left side is therefore \[ \frac {\pi _{DE}\,\pi _{\bar D \bar E}}{\pi _{D\bar E}\,\pi _{\bar D E}} . \] On the right, \(p_1/(1 - p_1) = \pi _{DE}/\pi _{\bar D E}\) and \(p_0/(1 - p_0) = \pi _{D\bar E}/\pi _{\bar D \bar E}\), whose ratio is the same cross-product. □

Proof. \(a/m_1 \to q_1\) and \(c/m_0 \to q_0\) in probability by the law of large numbers, and \(b/m_1 \to 1 - q_1\), \(d/m_0 \to 1 - q_0\). The sample odds ratio \(ad/(bc) = \{(a/m_1)(d/m_0)\}/\{(b/m_1)(c/m_0)\}\) is a continuous function of these four, so it converges to the exposure odds ratio, which equals the disease odds ratio by Theorem 1.4.2. □

Proof. Each exposed case in the population is included with probability \(f_1\), so \(E(a) = f_1 N_{DE}\), where \(N_{DE}\) is the number of exposed cases; \(a/f_1\) is unbiased for \(N_{DE}\) and \(c/f_0\) for \(N_{\bar D E}\). The risk among the exposed is \(N_{DE}/(N_{DE} + N_{\bar D E})\), estimated by substituting. For the odds ratio, \(\hat p_1/(1 - \hat p_1) = (a/f_1)/(c/f_0)\) and \(\hat p_0/(1 - \hat p_0) = (b/f_1)/(d/f_0)\); the sampling fractions cancel in the ratio. □