4.3 Inference for the odds ratio
Theorem 4.3.1 (Woolf). In either a cohort study or a case-control study, \[ \Var (\log \widehat {\OR }) \approx \frac {1}{a} + \frac {1}{b} + \frac {1}{c} + \frac {1}{d} , \] where \(\widehat {\OR } = ad/(bc)\) and the cell counts on the right are the observed ones.
Proof. In a cohort study \(\log \widehat {\OR } = \operatorname {logit}\hat p_1 - \operatorname {logit}\hat p_0\). The derivative of \(\operatorname {logit} p = \log \{p/(1 - p)\}\) is \(1/\{p(1 - p)\}\), so by Lemma 4.2.1 \[ \Var (\operatorname {logit}\hat p_1) \approx \frac {1}{p_1^2(1 - p_1)^2} \cdot \frac {p_1(1 - p_1)}{n_1} = \frac {1}{n_1 p_1(1 - p_1)} . \] Substituting \(\hat p_1 = a/n_1\), \(1 - \hat p_1 = c/n_1\) gives \(n_1/(ac) = (a + c)/(ac) = 1/a + 1/c\). Likewise the unexposed contribute \(1/b + 1/d\).
In a case-control study the binomial samples are the cases, of whom \(a\) of \(m_1\) are exposed, and the controls, \(c\) of \(m_0\). The estimate \(ad/(bc)\) is the ratio of the exposure odds \(a/b\) and \(c/d\), and the same argument applied to \(\operatorname {logit}(a/m_1)\) and \(\operatorname {logit}(c/m_0)\) gives \(1/a + 1/b\) and \(1/c + 1/d\). The sum is the same. □
The symmetry of the variance in the four cells reflects the symmetry of the odds ratio between exposure and disease. If any cell is zero, \(\widehat {\OR }\) is \(0\) or infinite and the variance is infinite; the usual remedy is to add \(\tfrac 12\) to every cell (Haldane’s correction) before computing both. Exact methods, which avoid the problem, come with the conditional distribution of Section 4.4.
Example 4.3.2. A case-control study of \(150\) cases and \(300\) controls finds:
| Exposed | Unexposed | Total | |
| Cases | \(60\) | \(90\) | \(150\) |
| Controls | \(66\) | \(234\) | \(300\) |
Then \(\widehat {\OR } = (60 \times 234)/(90 \times 66) = 2.36\), and \[ \text {SE}(\log \widehat {\OR }) = \sqrt {\tfrac {1}{60} + \tfrac {1}{90} + \tfrac {1}{66} + \tfrac {1}{234}} = 0.217 , \] so the \(95\%\) interval is \(2.36\,e^{\pm 1.96 \times 0.217} = (1.54,\ 3.62)\). In the cohort of Example 4.2.3 the odds ratio is \((60 \times 2352)/(48 \times 1140) = 2.58\), with interval \((1.75,\ 3.79)\), a little further from \(1\) than the risk ratio \(2.5\), as it must be (Problem 1.6.3).
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