Common G-Computation Implementation Errors

Why Implementation Details Change the Estimand

A parametric g-computation analysis can use correctly fitted models and still answer the wrong question if its implementation does not follow the intervention being evaluated. The most consequential errors often occur after model fitting: observed treatment values are inadvertently retained, post-treatment covariates are not regenerated, predictions are averaged over the wrong population, or an event is assigned to the wrong interval.

The central audit question is simple: for every simulated person and time point, does the computation represent the specified intervention and the history that would occur under it? The formulas below make that question precise. They are not a substitute for the identification assumptions or model diagnostics; they help ensure that the calculation implements the intended g-formula.

The Computation Being Implemented

Let \(L_0\) denote baseline covariates. At times \(t=0,\ldots,K-1\), let \(A_t\) be treatment and \(L_{t+1}\) the covariate history measured after that treatment and before the next treatment decision. Let \(Y\) be an outcome measured after the final treatment. Write \(a=(a_0,\ldots,a_{K-1})\) for a fixed treatment regime. The target is the mean outcome had everyone followed that regime, averaged over a specified baseline population.

Here \(\bar l_t=(l_0,\ldots,l_t)\) and \(\bar a_t=(a_0,\ldots,a_t)\). For discrete covariates, the integrals are sums. The formula presumes consistency, sequential exchangeability, and positivity for the histories and treatment levels needed by the regime. Its parametric implementation also requires adequate models for the covariate transitions and outcome regression. These assumptions concern identification and model validity; a coding check cannot establish them.

The computational order follows the factorization: draw or enumerate baseline covariates from \(F_0^*\), set treatment to the regime value, generate each next covariate from its transition model conditional on the generated history, and finally predict the outcome. The treatment values are set, not resampled from the observed treatment mechanism. The covariates are generated, not frozen at their observed values.

Result 1: Why Covariates Must Be Propagated Under the Regime

Derivation. Under consistency, the observed outcome and covariates equal their counterfactual counterparts for people whose observed treatment history follows \(a\). Sequential exchangeability allows the conditional distribution of each next covariate, given the history and the treatment at that time, to identify its counterfactual transition under that treatment. Positivity ensures these conditional distributions are supported by the observed data. Applying the chain rule to the counterfactual covariate history under the fixed regime therefore gives

The counterfactual mean is the expectation of the outcome conditional mean over this history distribution. Integrating that conditional mean against the displayed factorization gives the g-formula. In particular, the distribution of \(L_{t+1}\) depends on the treatment and covariate history leading to it. Holding \(L_{t+1}\) at its observed value instead averages the outcome regression over a different distribution, so it does not in general compute \(\psi(a)\). \(\square\)

Baseline Averaging: Use the Intended Population

The baseline distribution is part of the estimand, not a harmless choice of averaging weights. It may be the empirical distribution in the analysis sample, a weighted distribution representing a target population, or a specified distribution for a subgroup. The outcome predictions must be averaged over that stated population. Changing the baseline weights changes the answer even when every conditional model and regime-specific prediction is unchanged.

Event Timing and Risk Accumulation

In longitudinal event analyses, a time index must specify when treatment is assigned, when covariates are measured, and which interval an event hazard describes. Let \(D_{t+1}=1\) indicate an event during the interval from \(t\) to \(t+1\), and let \(R_t=1\) indicate survival and being event-free through time \(t\). Define the interval hazard \(h_{t+1}\) conditional on being at risk at time \(t\) and on the history available before that interval.

Derivation. By the definition of conditional probability, \(P(R_{t+1}=1\mid R_t=1,\text{history at }t)=1-h_{t+1}\). The event-free indicator at \(K\) requires being event-free at every preceding time. Applying the chain rule successively gives

and therefore \(P(D\text{ by }K\mid\text{history})=1-\prod_{t=0}^{K-1}(1-h_{t+1})\). With changing covariate histories, the hazards and histories must be generated and accumulated in the correct time order; the formula is applied along each simulated history before averaging across histories. \(\square\)

Audit Checks, Alternatives, and Limitations

Several checks can expose implementation problems before interpreting the estimate:

• Treatment-setting check: In a deterministic regime, verify that every simulated \(A_t\) equals the specified \(a_t\), regardless of the person's observed treatment.
• History check: Confirm that each generated \(L_{t+1}\) is drawn or integrated from the transition model conditional on the generated past and the regime treatment. Compare simulated covariate distributions across regimes; they may differ when treatment affects covariates.
• Population check: Print the baseline weights or stratum proportions used in the final average. Check that they represent the declared target, not an unintended complete-case subset or a resampled distribution.
• Timing check: Write a time-line showing when each \(A_t\), \(L_t\), and \(D_{t+1}\) is defined. Verify that no prediction uses future information and that events remove individuals from later risk sets.
• Numerical check: For risks, verify that estimates lie in \([0,1]\), and compare analytic integration with Monte Carlo integration when both are feasible. Increasing simulation size should reduce Monte Carlo error, but cannot correct a wrong treatment assignment, transition, or target distribution.

Analytic integration and Monte Carlo integration are alternative ways to evaluate the same g-formula when they use the same models, regime, and target population. Inverse-probability weighting is a different g-method: it builds a weighted pseudo-population rather than integrating outcome predictions over generated covariate histories. Agreement across approaches can be reassuring, but disagreement may reflect different modeling assumptions, target populations, or implementation defects; it is not by itself proof that one approach is correct.

G-computation remains sensitive to misspecified transition and outcome models, and positivity problems can force extrapolation even when code is flawless. Diagnostics can reveal implausible predictions, unsupported histories, or implementation inconsistencies, but cannot establish untestable causal assumptions. Treating an audit as a guarantee of causal validity is itself a mistake.

Common Mistake: Treating Prediction as Intervention

Incorrect reasoning: “The fitted model includes treatment, so predicting each person's outcome from their observed row estimates the outcome if everyone received the intervention.” It does not: observed treatment values preserve the natural treatment mix. Likewise, retaining observed post-treatment covariates preserves histories generated under that mix.

Correction: For each regime, explicitly set treatment to its prescribed value, propagate covariates forward from the corresponding transition distributions, predict the outcome for those histories, and average over the intended baseline population. Each of these steps has a separate role, and a correct outcome regression alone cannot replace them.