4.5 Comparing rates
With \(D_1\) cases in \(T_1\) person-years among the exposed and \(D_0\) in \(T_0\) among the unexposed, the incidence rate ratio and rate difference are estimated by \[ \widehat {\text {IRR}} = \frac {D_1/T_1}{D_0/T_0}, \qquad \widehat {\text {IRD}} = \frac {D_1}{T_1} - \frac {D_0}{T_0} . \] By Theorem 3.3.5 the counts behave as independent Poisson variables with means \(\lambda _1 T_1\) and \(\lambda _0 T_0\), so \(\text {SE}(\log \widehat {\text {IRR}}) = \sqrt {1/D_1 + 1/D_0}\) and \(\text {SE}(\widehat {\text {IRD}}) = \sqrt {D_1/T_1^2 + D_0/T_0^2}\).
Theorem 4.5.1. If \(D_1 \sim \text {Poisson}(\lambda _1 T_1)\) and \(D_0 \sim \text {Poisson}(\lambda _0 T_0)\) independently, then given \(D = D_1 + D_0\), \[ D_1 \sim \text {Bin}(D, \pi ), \qquad \pi = \frac {\text {IRR}\;T_1}{\text {IRR}\;T_1 + T_0} , \] which depends on the rates only through \(\text {IRR} = \lambda _1/\lambda _0\).
Proof. \(D \sim \text {Poisson}(\mu _1 + \mu _0)\) with \(\mu _i = \lambda _i T_i\), so \begin {align*} P(D_1 = k \mid D) &= \frac {e^{-\mu _1}\mu _1^k/k! \cdot e^{-\mu _0}\mu _0^{D - k}/(D - k)!}{e^{-(\mu _1 + \mu _0)}(\mu _1 + \mu _0)^D/D!} \\ &= \binom {D}{k}\Bigl (\frac {\mu _1}{\mu _1 + \mu _0}\Bigr )^k\Bigl (\frac {\mu _0}{\mu _1 + \mu _0}\Bigr )^{D - k} . \end {align*}
Dividing numerator and denominator of \(\mu _1/(\mu _1 + \mu _0)\) by \(\lambda _0\) gives \(\pi \). □
Under no association \(\pi = T_1/(T_1 + T_0)\), the share of person-time that is exposed, and the exact test is a binomial test. An exact interval for \(\pi \) (Clopper–Pearson) transforms into one for the rate ratio through \(\text {IRR} = \{\pi /(1 - \pi )\}\,T_0/T_1\), which is increasing in \(\pi \).
Example 4.5.2. Among exposed workers \(30\) cases occur in \(2500\) person-years; among unexposed workers, \(25\) cases occur in \(5000\) person-years. The rates are \(12\) and \(5\) per \(1000\) person-years, so \[ \widehat {\text {IRR}} = 2.4, \qquad \text {SE}(\log \widehat {\text {IRR}}) = \sqrt {1/30 + 1/25} = 0.271 , \] and the interval is \((1.41,\ 4.08)\). Conditionally on the \(55\) cases, under no association \(D_1 \sim \text {Bin}(55, 1/3)\). The observed \(30\), against an expected \(18.3\), gives a one-sided exact \(P\) of \(0.0010\). The exact interval for \(\pi \) is \((0.406,\ 0.680)\), which transforms to \((1.36,\ 4.26)\) for the rate ratio. The rate difference is \(7.0\) per \(1000\) person-years, with interval \((2.3,\ 11.7)\).
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