13.4 Sparse strata
The methods differ most when the strata are many and small, as in matched studies. Two simulation experiments show this. In each, a case-control study is generated with a true odds ratio of \(2\) and a matching or stratifying factor associated with exposure, and \(2000\) (first experiment) or \(500\) (second) replicate studies are analysed by each method.
Example 13.4.1. Matched pairs. \(200\) case-control pairs, with the control’s probability of exposure varying from pair to pair between \(0.1\) and \(0.7\) and the case’s odds of exposure twice the control’s.
Moderate strata. \(10\) strata, each with \(20\) cases and \(20\) controls, the control exposure probability rising from \(0.1\) to \(0.7\) across strata. The true odds ratio is \(2\) in both experiments; ML stands for maximum likelihood.
| Median estimate | ||
| Method | Pairs | \(10\) strata |
| Crude | 1.84 | 1.77 |
| Mantel–Haenszel | 2.00 | 2.00 |
| Conditional ML | 2.00 | – |
| Woolf, \(\tfrac 12\) added to zero cells | 1.39 | 1.96 |
| Unconditional ML | 4.00 | 2.03 |
With moderately large strata all the stratified methods are close to the truth. With pairs, only the Mantel–Haenszel and conditional estimators are. The unconditional maximum likelihood estimator, with one intercept per pair, converges to \(\psi ^2 = 4\) (Problem 13.6.2). Woolf’s estimator is dragged towards \(1\) by the correction, which dominates tables with only two people in them. The crude estimate is biased towards \(1\) because matching on a factor associated with exposure makes the controls resemble the cases (Section 6.3).
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