Non-collapsibility
The concrete version
Take a treatment that has exactly the same odds ratio in men and in women, and suppose sex is unrelated to treatment so there is no confounding whatsoever. Pool the men and women and compute the odds ratio: it will be CLOSER TO 1 than the common within-sex value. Nothing biased it. The odds ratio simply does not average that way.
Why this matters in practice
- The change-in-estimate rule for confounder selection misfires: an odds ratio that moves on adjustment may indicate confounding, non-collapsibility, or both, and the two cannot be separated by looking at the movement.
- Adjusted odds ratios from different studies adjusting for different covariate sets are not directly comparable, even if each is unbiased.
- A meta-analysis pooling conditional odds ratios from differently adjusted models is pooling different estimands.
- A logistic model's coefficient is a conditional effect; the policy question usually wants a marginal one.
- The gap widens as the outcome becomes more common, which is why the odds ratio is least troublesome for rare outcomes.
It is not the same as confounding
Confounding is a bias, caused by a common cause of exposure and outcome, and it can be removed by appropriate adjustment. Non-collapsibility is an arithmetic property of the measure, present with no bias at all, and adjustment does not remove it because there is nothing there to remove. Both make an estimate move on adjustment, and conflating them is the usual error.
Getting a marginal effect
Fit the model with the covariates for confounding control, then STANDARDIZE: predict each person's outcome probability under treatment and under no treatment, average each over the population, and form the contrast. That marginal estimate is collapsible and is the one a policy question asks for. Reporting a risk difference or risk ratio alongside the odds ratio serves the same end.
What to watch out for
The routine practice of comparing an unadjusted and an adjusted odds ratio to judge how much confounding there was is invalid, because the two would differ even with none. The adjusted odds ratio is also not an estimate of the population-average effect: it is the effect within strata of the covariates, which is systematically further from 1 than the marginal one. Hazard ratios are non-collapsible in the same way. Risk ratios and risk differences are not.
Clinical example
A trial reports an unadjusted odds ratio of 0.70 and an odds ratio of 0.62 after adjusting for prespecified prognostic covariates. In a randomized trial there is no confounding to remove, so the movement is non-collapsibility, and the adjusted figure is the conditional effect rather than a correction of the first.
- Noncollapsibility, confounding, and sparse-data bias. Part 1: The oddities of odds (opens in a new tab) · J Clin Epidemiol · doi 10.1016/j.jclinepi.2021.06.007
- Noncollapsibility, confounding, and sparse-data bias. Part 2: What should researchers make of persistent controversies about the odds ratio? (opens in a new tab) · J Clin Epidemiol · doi 10.1016/j.jclinepi.2021.06.004