Solution. \(\text {PPV} = 0.95p/\{0.95p + 0.02(1 - p)\}\), which is \(0.045\) at \(p = 0.001\), \(0.324\) at \(p = 0.01\) and \(0.841\) at \(p = 0.1\). At low prevalence the \(2\%\) false positive rate applies to almost the whole population, while the \(95\%\) true positive rate applies to very few, so false positives dominate (Proposition 2.2.2). Remedies are to screen only a higher-prevalence group (by age or risk factors), or to follow a positive result with a second, more specific test; testing in series raises the effective specificity to about \(1 - (1 - \text {Sp}_1)(1 - \text {Sp}_2)\) if the two tests err independently.

Solution. \(\hat p^* = 84/600 = 0.140\) and \(J = 0.89\), so \(\hat p = (0.140 - 0.03)/0.89 = 0.124\) and \(\text {SE} = \sqrt {0.14 \times 0.86/600}/0.89 = 0.0159\). The interval is \(0.124 \pm 0.031 = (0.093, 0.155)\). With \(15\) positives, \(\hat p^* = 0.025 < 1 - \text {Sp} = 0.03\), so \(\hat p = (0.025 - 0.03)/0.89 < 0\): fewer people tested positive than the false positive rate alone would produce. The estimate is truncated to \(0\); the data are consistent with no infection at all.

Solution. Each observer calls \(2 + 4 = 6\) positive, so \(p_e = 0.06^2 + 0.94^2 = 0.8872\) and \(p_o = 0.92\). Then \(\kappa = (0.92 - 0.8872)/(1 - 0.8872) = 0.29\). When one category is rare, two observers agree on most subjects by chance, so \(p_e\) is close to \(1\) and the agreement beyond chance is small: the observers agree on only \(2\) of the \(10\) subjects that either of them called positive. Kappa depends on the prevalence of the trait, and should be reported with the marginal proportions.

Solution. (a) \(p_i^* = 0.7p_i\), so \(\RR ^* = 0.014/0.007 = 2\), unbiased (Theorem 2.4.2, \(\text {Sp} = 1\)). (b) \(p_i^* = 0.01 + 0.99p_i\), so \(\RR ^* = (0.01 + 0.0198)/(0.01 + 0.0099) = 0.0298/0.0199 = 1.50\). Losing \(30\%\) of the cases costs only precision; a \(1\%\) false positive rate, applied to a population in which the disease itself has a risk of \(1\)–\(2\%\), adds as many false cases as there are true ones in the unexposed and pulls the ratio a long way towards \(1\). For a rare disease, specificity is what matters.

Solution. Take \(q_1 = 0.4\), \(q_0 = 0.2\) (true \(\OR = 2.67\)) and \(\text {Sp} = 1\) in both groups. If cases report exposure with \(\text {Se} = 1\) and controls with \(\text {Se} = 0.5\), then \(q_1^* = 0.4\), \(q_0^* = 0.1\) and \(\OR ^* = (0.4/0.6)/(0.1/0.9) = 6.0\), biased away from the null. If instead cases under-report with \(\text {Se} = 0.4\) and controls report fully, \(q_1^* = 0.16\), \(q_0^* = 0.2\) and \(\OR ^* = (0.16/0.84)/(0.2/0.8) = 0.76\): a harmful exposure appears protective. Differential error has no predictable direction, which is why it is worse than nondifferential error of the same size.

Solution. The true risk is \(0.8 \times 0.08 + 0.2\,\mu _N = 0.064 + 0.2\,\mu _N\), which ranges from \(0.064\) (if no one lost became ill) to \(0.124\) (if \(30\%\) did). The complete-case estimate \(0.08\) is unbiased only if the risk among those lost equals the risk among those followed: loss to follow-up unrelated to outcome. A range like this, computed from extreme but plausible assumptions about the missing, is a simple and honest sensitivity analysis.