Why G-Computation Does Not Model Treatment Assignment

Why the Treatment-Assignment Model Disappears

In observed data, treatment is assigned according to the clinical or social process that generated the study. Under an intervention, treatment is instead set according to a specified regime. These are different distributions. The parametric g-formula uses models for how covariates and outcomes evolve under specified treatment values; it does not need to simulate the observed treatment-assignment process.

That distinction is not a claim that treatment assignment is unimportant. Assignment can be strongly associated with prognosis, and its probabilities help characterize confounding and the support available in the data. The point is narrower: once the intervention fixes treatment, the observed assignment probability is not a factor in the intervention distribution being averaged.

For clarity, first consider finite-valued treatments and covariates. The observed-data distribution factors into a baseline distribution, treatment-assignment factors, covariate-transition factors, and a terminal outcome factor:

Under regime \(d\), the factors \(g_t\) are not used to generate treatment. The intervention sets \(a_t=d_t(C_t)\). The distribution of later covariates and the outcome still depends on those assigned values, so their conditional models must retain treatment history as appropriate.

where \(a_t^d=d_t(\bar l_t,\bar a_{t-1}^d)\) is the treatment specified by the regime along that covariate history. For continuous covariates, the sums are integrals with respect to the relevant distributions. The expression requires a model for the baseline distribution, the covariate transitions, and the conditional outcome mean. It contains no \(g_t\) factor.

Deriving the Intervention Distribution

The cancellation is a consequence of replacing the assignment mechanism, not an approximation that treats observed treatment as random. For each history, a deterministic intervention assigns probability one to its prescribed treatment and zero to other values. Thus the treatment factor under intervention is an indicator, not the observed conditional probability.

More formally, the intervention joint distribution is the observed baseline, covariate, and outcome kernels multiplied by \(\prod_{t=0}^{K}I\{a_t=d_t(C_t)\}\). To obtain the marginal distribution of covariates and \(Y\), sum this joint distribution over all treatment sequences. At each step only the sequence consistent with \(d\) contributes. The result is the product of the baseline distribution and the covariate and outcome kernels evaluated along that sequence. Taking the expectation of \(Y\) gives the displayed g-formula. This derivation proves the algebraic form of the intervention distribution; identification of its kernels from observed data requires the assumptions below. \(\square\)

In a parametric implementation, the corresponding conditional models must also be adequately specified for the target histories. A correctly specified treatment model cannot repair a misspecified covariate-transition or outcome model in the g-formula. Conversely, omitting a treatment model from the g-formula does not by itself establish exchangeability or positivity.

Worked Examples

When Treatment Probabilities Still Matter

Although assignment probabilities are absent from the g-formula, they remain relevant to the analysis in several ways. First, sequential exchangeability is a causal assumption about treatment selection given measured history. A fitted treatment model may help describe that selection, but good prediction or calibration does not prove that all confounders were measured. Second, positivity concerns whether the regime-prescribed treatment occurs with positive probability at relevant histories. Near-zero probabilities signal that the outcome or transition models may be forced to extrapolate. Third, treatment probabilities are central to inverse-probability weighting, where observed records are reweighted to represent a target intervention. That is a different estimating strategy from g-computation, not an extra factor to append to its formula.

The main practical limitation is therefore not that g-computation ignores treatment selection, but that it relies on correctly estimated outcome and covariate-transition distributions at regime-relevant histories. Extrapolation can be severe when the data contain little support for those histories. A treatment model can help reveal weak support, but cannot turn unsupported histories into observed information. Nor does fitting every conditional model guarantee validity: causal identification assumptions and model adequacy remain necessary.

Common Mistake: Multiplying by the Propensity Score

A common error is to take the g-formula and multiply each term by \(P(A_t=d_t(C_t)\mid C_t)\), reasoning that the probability of observing the regime must be part of its counterfactual mean. That inserts the observational assignment mechanism into a distribution where treatment has already been set by intervention. It would generally downweight histories precisely because the observed assignment process made the intervention uncommon, changing the target quantity. In inverse-probability weighting, treatment probabilities appear for a different reason: weights reweight observed treatment histories. Keep the two constructions distinct.