9.1 Stratification
Within a stratum in which the confounder \(C\) takes a single value, \(C\) cannot be associated with exposure, so it cannot confound (condition 2 of the definition in Section 7.4). The stratum-specific estimates are free of confounding by \(C\), though not by other variables.
Example 9.1.1. A case-control study of an exposure and a disease records age in three groups. In the notation of the fourfold table, \(a\) and \(b\) are the exposed and unexposed cases and \(c\) and \(d\) the exposed and unexposed controls.
| Age | \(a\) | \(b\) | \(c\) | \(d\) | \(\widehat {\OR }_k\) | \(v_k\) |
| Under 45 | 9 | 41 | 40 | 320 | 1.76 | 0.164 |
| 45–59 | 30 | 45 | 60 | 180 | 2.00 | 0.078 |
| 60+ | 64 | 26 | 50 | 50 | 2.46 | 0.094 |
| All ages | 103 | 112 | 150 | 550 | 3.37 | |
Here \(v_k = 1/a + 1/b + 1/c + 1/d\) is the estimated variance of \(\log \widehat {\OR }_k\). The crude odds ratio is \(3.37\), the stratum odds ratios between \(1.76\) and \(2.46\). Older people are both more often exposed (the proportion of controls exposed rises from \(11\%\) to \(50\%\)) and more often cases, so age inflates the crude association. The stratum estimates differ from one another by no more than chance would suggest (Example 9.4.2), and a single adjusted odds ratio of about \(2\) summarises them.
The crude estimate can differ from the stratum-specific ones in either direction, and can even reverse direction. That extreme case is Simpson’s paradox (Problem 9.6.1). There is nothing paradoxical about it once confounding is understood: the crude comparison mixes groups with different distributions of the confounder.
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