12.2 Logistic regression for case-control data
The logistic model is written for the probability of disease given exposure, a “prospective” quantity, yet a case-control study samples on disease. It is a remarkable fact that the model can be fitted to case-control data as if they were prospective.
Theorem 12.2.1 (Prentice and Pyke). Suppose that in the source population \(\operatorname {logit} P(D = 1 \mid x) = \alpha + \beta ^\top x\), and that cases are sampled with probability \(f_1\) and non-cases with probability \(f_0\), independently of \(x\). Then among sampled people \[ \operatorname {logit} P(D = 1 \mid x,\ \text {sampled}) = \alpha ^* + \beta ^\top x, \qquad \alpha ^* = \alpha + \log (f_1/f_0) . \] The coefficients \(\beta \), and hence all odds ratios, are unchanged; only the intercept is shifted.
Proof. Write \(S\) for the event of being sampled. By Bayes’ theorem, \[ \frac {P(D = 1 \mid x, S)}{P(D = 0 \mid x, S)} = \frac {P(S \mid D = 1, x)\,P(D = 1 \mid x)}{P(S \mid D = 0, x)\,P(D = 0 \mid x)} = \frac {f_1}{f_0}\,e^{\alpha + \beta ^\top x} , \] since sampling depends on disease status only. Taking logs gives the result. □
So ordinary logistic regression applied to case-control data estimates the population odds ratios correctly, with standard errors that are also valid (Prentice and Pyke, 1979, showed that the prospective likelihood gives the correct asymptotic variances). The intercept, which carries the baseline risk, cannot be interpreted unless the sampling fractions are known. This extends Theorem 1.4.2 from one binary exposure to any number of covariates.
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