4.2 Inference for the risk ratio and risk difference
In a cohort study \(a \sim \text {Bin}(n_1, p_1)\) and \(b \sim \text {Bin}(n_0, p_0)\) independently, and the estimates are \(\hat p_1 = a/n_1\) and \(\hat p_0 = b/n_0\). The sampling distributions of the ratio measures are much closer to normal on the log scale, and their variances are found by the delta method.
Lemma 4.2.1 (Delta method). If \(\sqrt n(\hat \theta - \theta ) \to N(0, \sigma ^2)\) in distribution and \(g\) is differentiable at \(\theta \) with \(g'(\theta ) \neq 0\), then \(\sqrt n\{g(\hat \theta ) - g(\theta )\} \to N(0, g'(\theta )^2\sigma ^2)\). So \(\Var g(\hat \theta ) \approx g'(\theta )^2 \Var (\hat \theta )\).
Proof. By Taylor’s theorem \(g(\hat \theta ) - g(\theta ) = g'(\theta )(\hat \theta - \theta ) + r(\hat \theta )(\hat \theta - \theta )\), where \(r(\hat \theta ) \to 0\) as \(\hat \theta \to \theta \). Since \(\hat \theta \to \theta \) in probability, \(r(\hat \theta ) \to 0\) in probability, and \(\sqrt n(\hat \theta - \theta )\) is bounded in probability, so the remainder times \(\sqrt n\) tends to \(0\). The result follows from Slutsky’s theorem. □
Proposition 4.2.2. In a cohort study, \[ \Var (\log \widehat {\RR }) \approx \frac {1 - p_1}{n_1 p_1} + \frac {1 - p_0}{n_0 p_0} , \] estimated by \[ \frac {1}{a} - \frac {1}{n_1} + \frac {1}{b} - \frac {1}{n_0} , \] and \(\Var (\widehat {\RD }) = p_1(1 - p_1)/n_1 + p_0(1 - p_0)/n_0\) exactly.
Proof. \(\log \widehat {\RR } = \log \hat p_1 - \log \hat p_0\), the difference of two independent terms. With \(g = \log \), \(g'(p) = 1/p\), so \(\Var (\log \hat p_1) \approx p_1(1 - p_1)/(n_1 p_1^2) = (1 - p_1)/(n_1 p_1)\); substituting \(\hat p_1 = a/n_1\) gives \((1 - a/n_1)/a = 1/a - 1/n_1\). The same holds for \(\hat p_0\), and the variances add. The risk difference is a difference of independent binomial proportions. □
The \(95\%\) interval for the risk ratio is \(\widehat {\RR }\,\exp (\pm 1.96\,\text {SE})\), with SE the square root of the estimated variance (often called the Katz interval), and for the risk difference \(\widehat {\RD } \pm 1.96\,\text {SE}\). The interval for the ratio is asymmetric about the estimate, as it should be for a quantity bounded below by \(0\).
Example 4.2.3. In a cohort study \(1200\) exposed workers and \(2400\) unexposed workers are followed for five years.
| Exposed | Unexposed | Total | |
| Cases | \(60\) | \(48\) | \(108\) |
| Non-cases | \(1140\) | \(2352\) | \(3492\) |
| Total | \(1200\) | \(2400\) | \(3600\) |
The risks are \(0.05\) and \(0.02\), so \(\widehat {\RR } = 2.5\). Then \[ \text {SE}(\log \widehat {\RR }) = \sqrt {\tfrac {1}{60} - \tfrac {1}{1200} + \tfrac {1}{48} - \tfrac {1}{2400}} = 0.190 , \] and the interval is \(2.5\,e^{\pm 1.96 \times 0.190} = (1.72,\ 3.63)\). The risk difference is \(0.030\) with \(\text {SE} = \sqrt {0.05 \times 0.95/1200 + 0.02 \times 0.98/2400} = 0.0069\), giving \((0.016,\ 0.044)\): between \(16\) and \(44\) extra cases per \(1000\) workers over five years.
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