13.1 Every summary is a weighted average
With stratum-specific rate ratios \(\theta _k = r_{1k}/r_{0k}\), where \(r_{ik} = D_{ik}/T_{ik}\), each method of Sections 8–10 is a weighted average of the \(\theta _k\), on the ratio or the log scale. The methods differ only in their weights.
Proposition 13.1.1. The following summaries are weighted averages of the \(\theta _k\) with weights proportional to the stated quantities:
- 1.
- the CMF with standard population \(W_k\): weights \(W_k r_{0k}\) (Proposition 8.1.2);
- 2.
- the SMR of the exposed with the unexposed rates as standard, \(\sum _k D_{1k}/\sum _k T_{1k}r_{0k}\): weights \(T_{1k}r_{0k}\), the deaths expected among the exposed;
- 3.
- the Mantel–Haenszel rate ratio: weights \(D_{0k}T_{1k}/T_k\);
- 4.
- the inverse-variance estimator, on the log scale: weights \((1/D_{1k} + 1/D_{0k})^{-1}\).
Proof. (2) \(\sum _k D_{1k} = \sum _k T_{1k}r_{1k} = \sum _k (T_{1k}r_{0k})\,\theta _k\); divide by \(\sum _k T_{1k}r_{0k}\). (3) \(D_{1k}T_{0k}/T_k = (D_{0k}T_{1k}/T_k)\,\theta _k\), since \(D_{1k}T_{0k} = r_{1k}T_{1k}T_{0k} = \theta _k\,r_{0k}T_{0k}T_{1k} = \theta _k D_{0k}T_{1k}\); sum and divide. (4) is Theorem 9.3.1 with Woolf’s variance \(1/D_{1k} + 1/D_{0k}\) for a log rate ratio. (1) is Proposition 8.1.2. □
Two consequences follow. If the \(\theta _k\) are all equal, every method estimates the common value, and they differ only in precision. If the \(\theta _k\) differ, each method estimates a different average, and the differences between methods are a symptom of effect modification, not of error in any one of them. Maximum likelihood in the Poisson model with stratum and exposure terms is not an explicit weighted average, but under homogeneity it is the most efficient estimator, and under heterogeneity it too estimates a weighted average determined by the design.
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.