Why can a logistic model's conditional odds ratio differ from the marginal odds ratio?

## Two different quantities **Marginal (population-averaged) effect**: compare the outcome distribution if the whole population were treated with the distribution if none were. This is what a randomised trial's headline number describes. **Conditional (covariate-specific) effect**: compare the same two worlds within a fixed covariate stratum — among users of the same tenure, the same plan history, and so on. Whether these two coincide depends on the effect measure, and that is the whole content of collapsibility. ## Collapsible and non-collapsible measures A measure is **collapsible** when the marginal value is a weighted average of the stratum-specific values (in the absence of confounding). The **risk difference** and the **risk ratio** have this property: if the treatment lowers risk by 4 points in every stratum, it lowers population risk by 4 points, whatever the stratum mix. The **odds ratio** does not have it. Suppose treatment produces a conditional odds ratio of 2 in every covariate stratum, with no confounding whatsoever. The population's marginal odds ratio will be *less than* 2 — nearer to the null. Nothing went wrong; averaging risks and then converting to odds is not the same operation as averaging odds ratios, because the odds transformation is nonlinear. The **hazard ratio** is non-collapsible in the same way. This is a mathematical fact about the measure, entirely separate from bias. ## The consequence people get wrong Run a randomised experiment. Compute the crude odds ratio. Then fit a logistic model adding a covariate that strongly predicts the outcome but, thanks to randomisation, is balanced across arms and therefore confounds nothing. The adjusted odds ratio typically comes out **further from 1** than the crude one. An analyst who expects adjustment to be a bias-removing operation now has a puzzle: the estimate moved, but there was no bias to remove. The resolution is that the two numbers estimate different estimands. The crude one estimates the marginal odds ratio; the adjusted one estimates the conditional odds ratio, which for a non-collapsible measure is a genuinely different number. Both can be perfectly unbiased for their own target. The mistake is treating the movement as evidence about confounding — with a non-collapsible measure, a coefficient that shifts on adjustment tells you nothing on its own. ## Which one should you report Start from the decision, not the model. - **Policy and planning questions** — what happens if we roll this out to everyone — are marginal questions. Report a marginal risk difference or risk ratio, which are also easier for non-specialists to reason about and are collapsible, so the awkwardness disappears. - **Individual-level questions** — what does this mean for a user like this one — are conditional, and a conditional estimate is the right object. If you want the marginal effect but still want the precision that adjustment buys, the two goals are not in conflict: fit the covariate-adjusted model, then standardise. Predict each unit's risk with treatment forced on and forced off, average both sets of predicted risks across the sample, and form the contrast from those two averages. You get a marginal estimate on the scale you chose, out of an adjusted model. ## Naming it correctly in an interview The short version: the odds ratio is non-collapsible, so a conditional odds ratio and a marginal one are different parameters even without confounding; adjusting for a prognostic covariate pushes the odds ratio away from the null; risk differences and risk ratios do not behave this way; and if I need a marginal effect I standardise rather than reading the coefficient. Two common errors to avoid stating: that the movement on adjustment proves the covariate was a confounder, and that taking logs somehow fixes it — the log odds ratio is a monotone transform of a non-collapsible measure and is non-collapsible too.