Outcome regression identifies causal means by modeling conditional outcome distributions and standardizing them over the target covariate distribution. The g-formula extends this idea to time-varying interventions.
Point-treatment standardization
Theorem. Under consistency, conditional mean exchangeability, and positivity, \[ \mu_a=\mathbb EY(a)= \int m_a(x)\,P_X(\mathrm{d}x), \qquad m_a(x)=\mathbb E(Y\mid A=a,X=x). \]
Proof. Iterated expectation gives \(\mathbb EY(a)=\int\mathbb E\{Y(a)\mid X=x\}\,P_X(\mathrm{d}x)\). Exchangeability, positivity, and consistency replace the conditional counterfactual mean with \(m_a(x)\).
Definition. The plug-in g-computation estimator is \[ \widehat\mu_a= \frac1n\sum_{i=1}^n\widehat m_a(X_i), \qquad \widehat\tau_{\mathrm g}=\widehat\mu_1-\widehat\mu_0. \]
Regression coefficients versus standardized effects
Example. If \[ m(a,x)=\beta_0+\beta_Aa+\beta_X^\top x+ a\,\gamma^\top x, \] then \[ \tau_{\mathrm{ATE}}= \beta_A+\gamma^\top\mathbb EX. \] The coefficient \(\beta_A\) equals the ATE only when the interaction average is zero. Standardization correctly averages heterogeneous conditional effects.
Binary-outcome standardization
For logistic outcome regression, \[ \widehat m_a(x)= \operatorname{expit}\{\widehat\beta_0+\widehat\beta_Aa+ \widehat\beta_X^\top x+\cdots\}. \] The causal risk difference is the average of predicted probabilities under \(a=1\) minus the corresponding average under \(a=0\). It is not the logistic treatment coefficient. Risk ratios and marginal odds ratios are obtained by transforming the two standardized means.
The parametric g-formula algorithm
- Fit models for each time-varying covariate and the final outcome conditional on past history.
- Resample baseline covariates from the target population.
- Set treatment according to the intervention rule.
- Simulate subsequent covariates and outcomes recursively from fitted conditional laws.
- Average simulated outcomes and compare intervention regimes.
This is Monte Carlo evaluation of a factorized identifying functional, not an assumption-free causal simulation.
The longitudinal g-formula
Theorem. Let \(\overline L_K=(L_0,\ldots,L_K)\) be covariate history, \(\overline A_K=(A_0,\ldots,A_K)\), and \(Y\) the final outcome. Under consistency, sequential exchangeability \[ Y^{\overline a}\perp\!\!\!\perp A_t \mid \overline L_t,\overline A_{t-1} \] and sequential positivity, \[ p(y^{\overline a}) = \int p(y\mid\overline l_K,\overline a_K) \prod_{t=0}^K p(l_t\mid\overline l_{t-1},\overline a_{t-1}) \,\mathrm{d}\overline l_K. \]
Proof. Repeatedly condition on covariate history. At each time, sequential exchangeability removes dependence of future potential outcomes on the observed treatment choice, consistency replaces observed and intervention histories when they agree, and positivity makes the conditional law estimable. Iteration yields the product of covariate transition kernels and the terminal outcome kernel under the fixed treatment sequence.
Why ordinary time-dependent regression can be biased
Treatment-confounder feedback. Suppose \(A_0\to L_1\), \(L_1\to A_1\), and \(L_1\to Y\). Adjusting for \(L_1\) in a regression of \(Y\) on \(A_0,A_1\) blocks part of the \(A_0\) effect, while omitting \(L_1\) confounds the \(A_1\) effect. The longitudinal g-formula or marginal structural models handle this feedback by defining a joint intervention and using the full temporal factorization.
Dynamic treatment regimes
Definition. A dynamic regime \(d=(d_0,\ldots,d_K)\) sets \[ A_t=d_t(\overline L_t,\overline A_{t-1}). \] Its value is \(\mathbb EY^d\). In the g-formula, replace each fixed \(a_t\) by the rule evaluated at the simulated history.
Stochastic interventions
A stochastic policy \(g_t^*(a_t\mid h_t)\) replaces the observed treatment kernel by a user-specified kernel. The identifying density is \[ p^{g^*}(y,\overline l,\overline a) =p(y\mid\overline l,\overline a) \prod_t p(l_t\mid h_{t-1},a_{t-1}) g_t^*(a_t\mid h_t). \] Incremental propensity interventions, which tilt rather than overwrite the observed propensity, can remain identifiable under weaker practical positivity than deterministic rules.
Model misspecification and extrapolation
Outcome regression can be stable under weak overlap because it does not divide by small probabilities, but it then relies heavily on extrapolating \(m_a(x)\) into covariate regions with few observations in arm \(a\). Flexible regression reduces functional-form restrictions but cannot create information outside support. Overlap plots and arm-specific prediction diagnostics remain essential.
Uncertainty
For a smooth finite-dimensional outcome model, the delta method propagates coefficient uncertainty through standardization. A nonparametric bootstrap can incorporate model fitting and empirical covariate averaging when the full estimator is sufficiently smooth. With adaptive machine learning, asymptotically linear doubly robust estimators are usually preferable because the raw plug-in g-formula can inherit first-order nuisance bias.