5.3 Estimating the PAF from a case-control study
Theorem 5.3.1. In a case-control study with cells \(a, b\) (exposed and unexposed cases) and \(c, d\) (exposed and unexposed controls), the estimate of the PAF obtained from Miettinen’s formula with \(\widehat {\OR }\) satisfies \[ 1 - \widehat {\text {PAF}} = \frac {b/m_1}{d/m_0} , \] the proportion of cases unexposed divided by the proportion of controls unexposed, and \[ \Var \{\log (1 - \widehat {\text {PAF}})\} \approx \frac {a}{m_1 b} + \frac {c}{m_0 d} . \]
Proof. With \(\hat P_c = a/m_1\) and \(\widehat {\OR } = ad/(bc)\), \[ \widehat {\text {PAF}} = \hat P_c - \frac {\hat P_c}{\widehat {\OR }} = \frac {a}{m_1} - \frac {a}{m_1} \cdot \frac {bc}{ad} = \frac {a}{m_1} - \frac {bc}{m_1 d} . \] So \(1 - \widehat {\text {PAF}} = (m_1 - a)/m_1 + bc/(m_1 d) = b/m_1 + bc/(m_1 d) = (b/m_1)(d + c)/d\), and \((d + c)/d = m_0/d\).
For the variance, \(\log (1 - \widehat {\text {PAF}}) = \log (b/m_1) - \log (d/m_0)\), a difference of the logs of two independent binomial proportions. By Lemma 4.2.1, as in Proposition 4.2.2, the log of a proportion \(\hat \pi \) from \(m\) trials has variance about \((1 - \pi )/(m\pi )\). For \(b/m_1\) this is estimated by \((a/m_1)/b = a/(m_1 b)\), and for \(d/m_0\) by \(c/(m_0 d)\). □
The interval is computed for \(1 - \text {PAF}\) on the log scale and transformed back, which keeps the upper limit of the PAF below \(1\).
Example 5.3.2. In Example 4.3.2, \(a = 60\), \(b = 90\), \(c = 66\), \(d = 234\), \(m_1 = 150\), \(m_0 = 300\). Then \[ 1 - \widehat {\text {PAF}} = \frac {90/150}{234/300} = \frac {0.600}{0.780} = 0.769 , \] so \(\widehat {\text {PAF}} = 0.231\), which agrees with \(\hat P_c(\widehat {\OR } - 1)/\widehat {\OR } = 0.4 \times 1.364/2.364\). The standard error is \[ \text {SE} = \sqrt {\frac {60}{150 \times 90} + \frac {66}{300 \times 234}} = 0.0734 , \] so \(1 - \text {PAF}\) lies in \(0.769\,e^{\pm 0.144} = (0.666,\ 0.888)\) and the PAF in \((0.112,\ 0.334)\). Removing the exposure would prevent between about a ninth and a third of the cases. The estimate assumes the controls represent the source population and that the odds ratio is unconfounded and close to the risk ratio.
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