Bias From Misspecified Transition and Outcome Models

Where Model Misspecification Enters

The parametric g-formula estimates a regime-specific mean by combining models for the evolving covariates with a model for the outcome. Under a specified treatment regime, it simulates covariate histories and averages predicted outcomes over those histories. This makes two distinct sources of error consequential: an incorrect transition model can generate the wrong histories, while an incorrect outcome model can assign the wrong mean outcome to a history. When both are wrong, their effects can interact.

Let \(L_t\) denote covariates measured at time \(t\), \(A_t\) treatment, and \(Y\) a terminal outcome. Write \(\bar L_t=(L_0,\ldots,L_t)\) and \(\bar A_t=(A_0,\ldots,A_t)\). For a fixed regime \(\bar a\), let \(\mu(\bar a)=E[Y^{\bar a}]\) denote the mean counterfactual outcome. Under consistency, sequential exchangeability, and positivity for the regime of interest, the g-formula expresses this mean using the baseline covariate distribution, the conditional distributions of subsequent covariates, and the conditional mean outcome. These conditions identify the target from the observed-data law; they do not guarantee that a chosen parametric model correctly represents that law.

For clarity, consider a finite number of time points, a correctly represented baseline distribution, and a bounded outcome \(Y\in[0,1]\). Suppose fitted models converge to limiting conditional distributions \(Q_t\) and a limiting outcome mean \(\tilde m\), which need not be the truth. The limiting plug-in estimate is then the mean of \(\tilde m\) over histories generated recursively from the \(Q_t\)'s under \(\bar a\). Increasing the sample size or the number of simulated histories can make the estimate approach this limit precisely; neither operation makes the limit equal \(\mu(\bar a)\) when the limiting models are wrong.

How Errors Change the Regime Mean

A useful way to separate the two errors is to imagine evaluating the fitted outcome mean first over the fitted history distribution, then compare that quantity with the true mean. Let \(P\) denote the true regime-specific distribution of covariate histories and \(Q\) the distribution generated by the fitted transition models. With \(m(h)=E[Y\mid h,\bar a]\) and fitted mean \(\tilde m(h)\), the discrepancy decomposes exactly:

The first term is outcome-model error averaged over histories the fitted procedure generates. The second is history-distribution error, evaluated using the true outcome regression. This identity does not imply that the two errors can be separately estimated from the data without additional information. It does show why checking only the outcome regression at observed histories is insufficient: the g-formula may average predictions over a different distribution of histories.

A Bound on Propagated Error

The following finite-state bound makes the accumulation mechanism precise. For probability distributions \(R\) and \(S\) on the same finite state space, define total variation distance by \(\|R-S\|_{\mathrm{TV}}=\sup_B|R(B)-S(B)|\). For each time \(t\), compare the true and fitted transition kernels under the regime, uniformly over histories that can be supplied to the model. Assume the baseline distribution is correct.

Proof. Generate a history under the true kernels and one under the fitted kernels, starting from the same baseline value. At each transition, couple the two conditional draws so that, when their histories agree, the probability of different next covariate values is their conditional total variation distance, at most \(\epsilon_t\). (For finite distributions such a coupling is obtained by matching the common probability mass at each state.) Once histories disagree, the two paths may be generated in any way consistent with their respective conditional distributions. The probability that the paths have disagreed by the end is at most \(\sum_{t=1}^T\epsilon_t\), by the union bound on the first transition at which they separate.

For any function \(g\) taking values in \([0,1]\), the absolute difference between its expectations under two coupled histories is at most the probability that the histories differ: when they agree the function values agree, and otherwise their difference is at most one. Apply this to the true outcome mean \(m\). The discrepancy between its expectations over the true and fitted history laws is therefore at most \(\sum_t\epsilon_t\). Finally, under the fitted history law, the pointwise bound \(|m-\tilde m|\leq\epsilon_Y\) implies \(|E_Q[m(H)-\tilde m(H)]|\leq\epsilon_Y\). The triangle inequality gives the stated result. \(\square\)

The bound is deliberately conservative: errors at different times can cancel, and a transition difference in a covariate unrelated to later risks may have little effect. But without such additional structure, small errors at several transitions can accumulate. The bound also makes explicit that a small outcome-model error alone says nothing about history-distribution error.

Assumptions, Comparisons, and Failure Clues

Interpretation depends on the distinction between identification assumptions and modeling assumptions. Consistency, sequential exchangeability, and positivity support identification of the regime mean from the observed-data distribution. Correct specification of all conditional models is an additional condition for the parametric plug-in procedure to consistently estimate that identified mean, assuming appropriate estimation and regularity conditions. If a conditional law is misspecified, a fitted parametric model may converge to a best-fitting or pseudo-true parameter under its estimation criterion, but the resulting g-formula generally targets the mean under that fitted law—not necessarily the causal estimand.

This differs from Monte Carlo error. With a fixed set of fitted models, increasing the number of simulated histories reduces numerical integration noise, not the gap between the fitted-law mean and the true regime mean. It also differs from positivity failure: extrapolation into poorly supported histories can make model misspecification especially consequential, but the source of bias here is a wrong conditional model, not simply a lack of treatment support. A correctly specified model cannot be learned reliably where the data provide no relevant support.

Useful warning signs include simulated covariate distributions that conflict with observed distributions in comparable histories, implausible transitions, outcome predictions that fail to track observed risk patterns, and marked changes in the regime mean when reasonable functional forms are changed. These clues do not prove misspecification, and good agreement on observed histories does not guarantee good predictions in histories generated under an intervention. Model diagnostics require examining both conditional predictions and the histories that those predictions generate.

The main practical alternatives are to improve the conditional models using defensible nonlinearities and interactions, use more flexible or less parametric models where supported by data, or compare the g-formula estimate with another causal estimator based on different modeling assumptions. None is automatically protected against all misspecification. A saturated model can represent observed conditional means in a finite table but becomes unstable with sparse histories; a flexible model can reduce functional-form restrictions while still extrapolating poorly. The relevant limitation is not that simulation itself is defective, but that the simulation inherits the conditional laws supplied to it.