12.4 Poisson regression for rates
Definition 12.4.1. In the Poisson regression model for grouped follow-up data, cell \(j\) (a combination of exposure, age band and period) has \(D_j\) events in \(T_j\) person-years, with \(D_j \sim \text {Poisson}(\lambda _j T_j)\) independently and \[ \log \lambda _j = x_j^\top \theta , \qquad \text {equivalently}\qquad \log E(D_j) = \log T_j + x_j^\top \theta . \] The known term \(\log T_j\) is called the offset.
As in the logistic model, \(e^\beta \) for the exposure is the rate ratio at fixed values of the other covariates. The model with indicators for age bands and exposure is the “common rate ratio” model of Section 10.6. Theorem 3.3.5 justifies treating counts of events over person-time as Poisson even though each person’s follow-up was random: the likelihood is the same.
Proposition 12.4.2. The score equations of the Poisson model are \(\sum _j (D_j - T_j\hat \lambda _j)x_j = 0\). For any indicator variable in the model, the fitted number of events among the cells it indicates equals the observed number.
Proof. The log-likelihood is \(\sum _j \{D_j x_j^\top \theta - T_j e^{x_j^\top \theta }\}\) plus a constant. Its gradient is \(\sum _j (D_j - T_j e^{x_j^\top \theta })x_j\). The component for an indicator \(x_{jk} \in \{0, 1\}\) is \(\sum _{j: x_{jk} = 1}(D_j - T_j\hat \lambda _j) = 0\). □
With age indicators and the exposure in the model, the fitted deaths among the exposed equal the observed deaths. The rate ratio is therefore the factor by which the unexposed age-specific rates must be multiplied to make the expected deaths among the exposed equal the observed ones, an internal SMR with the rates estimated jointly.
Example 12.4.3. For the British doctors’ data of Example 10.6.1, the Poisson model with five age bands and smoking gives \(\hat \beta = 0.355\) with \(\text {SE} = 0.107\), a rate ratio of \(1.43\) with interval \((1.15,\ 1.76)\), almost identical to the Mantel–Haenszel estimate. The fitted number of deaths among smokers is \(630\), the observed number. But the residual deviance is \(12.1\) on \(4\) degrees of freedom (\(P = 0.016\)): the model of a common rate ratio does not fit. Adding a term for smoking times age band (coded \(0\) to \(4\)) gives \[ \log \lambda = \text {age effects} + 1.136\,E - 0.309\,E \times \text {band} , \] with \(\text {SE} = 0.097\) for the product term. The likelihood ratio statistic for it is \(10.6\) on \(1\) degree of freedom, and the residual deviance falls to \(1.5\) on \(3\). The fitted rate ratio is \(e^{1.136} = 3.1\) at ages \(35\)–\(44\) and falls by a factor of \(e^{-0.309} = 0.73\) per decade, to \(0.91\) at \(75\)–\(84\). The model describes the effect modification found in Section 10 with two parameters instead of five.
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.