Outcome Regression

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Outcome Regression for Causal Inference

Outcome Regression (Covariate Adjustment): A method for estimating causal effects by modeling the outcome as a function of treatment and confounders, where the treatment coefficient represents the causal effect under specific conditions.

Outcome regression is perhaps the most widely used approach for causal inference in applied research. The basic idea is deceptively simple: model the outcome $Y$ as a function of treatment $A$ and confounders $L$ , then interpret the treatment coefficient causally. However, as Hernán and Robins emphasize in "What If" Chapter 15, this interpretation requires careful consideration of several conditions.

The standard linear outcome model takes the form:

$E[Y|A, L] = \beta_0 + \beta_1 A + \beta_2 L$

When does $\beta_1$ have a causal interpretation? Four key conditions must hold:

Condition	Requirement	Consequence if violated
Exchangeability	All confounders in $L$	Omitted variable bias
Correct functional form	Model is correctly specified	Specification bias
Effect modification	None, or correctly modeled	Biased average effect
Collapsibility	Effect measure is collapsible	Conditional $\neq$ marginal

Epidemiology Example: In the NHEFS analysis of smoking cessation and weight gain, researchers model: $$E[\text{WeightGain}|\text{Quit}, L] = \beta_0 + \beta_1 \text{Quit} + \beta_2^T L$$ where $L$ includes age, sex, race, education, smoking intensity, and years smoked. If all confounders are correctly measured and modeled, $\beta_1$ estimates the average causal effect of quitting on weight gain.

The elegance of outcome regression lies in its simplicity and familiarity to most researchers. However, this apparent simplicity often obscures the strong assumptions required for valid causal interpretation.

Reference: Hernán MA, Robins JM. "What If." Chapter 15: Outcome regression and propensity scores. Provides the theoretical foundation for understanding when regression coefficients have causal meaning.

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