Inconsistent treatment estimates from mis-specified logistic regression analyses of randomized trials

TL;DR

This paper derives precise formulas for bias in logistic regression treatment effect estimates in randomized trials when important covariates are omitted, highlighting potential biases in common analysis methods.

Contribution

It provides accurate closed-form approximations for asymptotic bias due to omitted covariates in logistic regression, extending existing results and offering practical insights.

Findings

Bias formulas are accurate for Normal covariates

Bias approximations are applicable beyond Normal distributions

Bias persists even with additional binary covariates

Abstract

When the difference between treatments in a clinical trial is estimated by a difference in means, then it is well known that randomization ensures unbiassed estimation, even if no account is taken of important baseline covariates. However, when the treatment effect is assessed by other summaries, e.g. by an odds ratio if the outcome is binary, then bias can arise if some covariates are omitted, regardless of the use of randomization for treatment allocation or the size of the trial. We present accurate closed-form approximations for this asymptotic bias when important Normally distributed covariates are omitted from a logistic regression. We compare this approximation with ones in the literature and derive more convenient forms for some of these existing results. The expressions give insight into the form of the bias, which simulations show is usable for distributions other than the Normal. The key result applies even when there are additional binary covariates in the model.