Residual-on-Residual Regression as a Tool for Effect Estimation in Observational Data

Epidemiologists increasingly use machine learning to adjust for high-dimensional confounding.

Augmented inverse probability weighting (AIPW) and targeted maximum likelihood estimation (TMLE) are most widely used but may yield different results and both can become unstable under weak positivity violations.

Residual-on-residual regression is a stable alternative that estimates an exposure effect encoded in a partially linear model by fitting confounder adjusted models for the outcome and exposure, then regressing outcome residuals against exposure residuals using ordinary least squares.

We illustrate the approach using data from the Nulliparous Pregnancy Outcomes Study: Monitoring Mothers-to-Be (nuMoM2b; $n = 7{,}923$), estimating the association between high vegetable intake density and preeclampsia.

Residual-on-residual regression, AIPW, and TMLE yielded concordant estimates, indicating a modest reduction in preeclampsia risk.

In simulations, residual-on-residual regression was unbiased with near-nominal confidence interval coverage, performing comparably to AIPW and TMLE and substantially better than a misspecified parametric model when the exposure effect is approximately constant.

However, in simulation settings with positivity violations, residual on residual regression outperformed AIPW and TMLE when the true effect was coded in a partially linear model.

When the exposure effect is approximately constant, residual-on-residual regression is interpretable, computationally simple, and provides a triangulation strategy for observational causal inference.