Exact unbiased covariance estimation for modified least-squares regression with binary outcomes

Summary Logistic regression has long been widely used in clinical and epidemiologic studies with binary outcomes. However, because odds ratios can be difficult to interpret, recent publications have increasingly recommended alternative regression models that yield more directly interpretable effect measures. Modified least-squares regression is one such approach, providing covariate-adjusted estimates of risk differences. Its point estimator is the ordinary least-squares estimator, whereas statistical inference is typically based on heteroskedasticity-robust sandwich covariance estimators. Although these covariance estimators are asymptotically valid, they may be biased in small to moderate samples, particularly in the presence of high-leverage observations. In this article, we show that, under a correctly specified linear mean model with fixed covariates, the covariance matrix of the modified least-squares estimator admits an exactly unbiased finite-sample estimator. The proposed construction uses the Hadamard square of the residual-maker matrix to recover unbiased estimates of the individual Bernoulli variances from the expected squared residuals. The resulting covariance estimator is algebraically simple and can be computed using only standard least-squares quantities. We also discuss its limitations, including the possibility of negative individual variance estimates, failure of the estimated covariance matrix to be positive semidefinite, and numerical instability when the Hadamard matrix is ill-conditioned. Exact unbiasedness of the covariance estimator does not imply exact finite-sample Wald inference.