12.3 Conditional logistic regression for matched data
In a matched study each matched set has its own baseline odds, a separate intercept \(\alpha _s\). There are as many intercepts as sets, so ordinary maximum likelihood fails: with matched pairs it estimates \(\psi ^2\) instead of \(\psi \), as noted in Section 10. The cure is to condition on the number of cases in each set, which eliminates \(\alpha _s\).
Proposition 12.3.1. In a matched set of one case and \(M\) controls with covariates \(x_0\) (the case) and \(x_1, \dots , x_M\), given that exactly one member is a case, the probability that it is the one observed is \[ \frac {e^{\beta ^\top x_0}}{\sum _{j=0}^{M} e^{\beta ^\top x_j}} , \] which does not depend on the set’s intercept \(\alpha _s\).
Proof. Under the logistic model with intercept \(\alpha _s\), the probability that member \(j\) is the only case is proportional to \(e^{\alpha _s + \beta ^\top x_j}\prod _{l}\{1 + e^{\alpha _s + \beta ^\top x_l}\}^{-1}\): the product over all members of the probability of each one’s status, where \(j\) contributes its odds of being a case times its probability of not being one. The product and the factor \(e^{\alpha _s}\) are common to every \(j\), and cancel in the ratio of the probability for \(j = 0\) to the sum over \(j\). □
The conditional likelihood is the product of these terms over sets, maximised over \(\beta \). For matched pairs with a binary exposure it reduces to the binomial likelihood of Theorem 6.3.1: concordant pairs give \(\tfrac 12\) whatever \(\beta \) is, and discordant pairs give \(\psi /(1 + \psi )\) or \(1/(1 + \psi )\). Conditional logistic regression is to the matched study what the Mantel–Haenszel estimator is to the stratified one, with the advantage that it handles several exposures and continuous covariates.
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