Regression Adjustment in Experiments

Regression can express the unadjusted difference in means and can use pretreatment covariates to improve precision.

Basic treatment regression

Consider

With an intercept and a binary treatment indicator, OLS gives the exact algebraic identity

Random assignment gives this coefficient its causal interpretation.

Additive covariate adjustment

A common adjusted model is

where $X_i$ contains pretreatment covariates. Adjustment can absorb outcome variation explained by $X$ and improve precision.

Treatment interactions

A fully interacted model is

Interactions allow the relationship between covariates and outcomes to differ by treatment arm. If covariates are centered at the full-sample mean, $\tau$ has a direct interpretation at that reference point and the fitted arm-specific means can be averaged to estimate an average effect.

Practical rules

• Use only pretreatment covariates for routine precision adjustment.
• Choose the adjustment specification before examining treatment effects when possible.
• Respect blocking, clustering, or unequal assignment probabilities in the analysis.
• Use standard errors appropriate to the design; heteroskedasticity-robust standard errors are common for individual randomization.
• Report the unadjusted difference alongside adjusted estimates so the role of modeling is visible.

What adjustment does not repair

Regression adjustment does not recreate randomization after a compromised design, and post-treatment controls can introduce bias by conditioning on variables affected by treatment.

See

• Indicator Variables and Interactions
• Linear Regression for Experimental Research
• Neyman Repeated-Sampling Inference
• Unconfoundedness and Overlap