Conditional Models for Covariate Evolution

Why Model Covariate Evolution?

In longitudinal g-computation, a covariate measured after treatment may both predict later outcomes and be affected by earlier treatment. Under an intervention, the distribution of later covariates therefore needs to be generated from the history that would arise under that intervention. The relevant models are not unconditional models for each covariate in isolation: they are conditional transition models, indexed by the covariate and treatment history available at each time.

Use \(L_0\) for baseline covariates, \(A_t\) for treatment at time \(t\), and \(L_{t+1}\) for covariates measured after \(A_t\), for \(t=0,\ldots,K-1\). Write \(\bar L_t=(L_0,\ldots,L_t)\) and \(\bar A_t=(A_0,\ldots,A_t)\). The history immediately before \(A_t\) is \((\bar L_t,\bar A_{t-1})\); the history used to generate \(L_{t+1}\) includes \(A_t\) as well.

A regime specifies the treatment history to be used when generating a trajectory. It may be static, such as \(A_t=1\) at every time, or depend on past history. For a regime \(g\), insert its treatment value \(g_t(\bar l_t,\bar a_{t-1})\) into the transition model at time \(t\). The model still conditions on treatment history: it must describe how covariates evolve under the treatment values imposed by the regime.

The baseline distribution and the sequence of transition kernels determine the covariate-history distribution under the intervention. In particular, the models must preserve the temporal order: \(L_{t+1}\) is generated from history through \(A_t\), not from future treatment or future covariates. This conditional-model component is distinct from the later model for the outcome given the completed history.

Factorization Under an Intervention

Suppose the covariate history has a joint distribution and regular conditional distributions exist. The chain rule for conditional distributions gives its sequential factorization. Under a regime, the treatment values in each factor are replaced by those assigned by that regime. In density notation, for a static regime \(\bar a\), the resulting covariate-history law is

For a dynamic regime, the same expression uses the treatment values specified by \(g\) along the generated history. If the covariates at a time are a vector, \(q_{t+1}\) is their joint conditional distribution unless a further factorization is explicitly specified. Modeling vector components separately without conditioning on one another asserts a conditional-independence structure; it does not follow merely from writing several models.

Proof. For the first transition, the joint distribution of \((L_0,L_1)\) under the intervention factors as \(p(l_0)\) times the conditional distribution of \(L_1\) given \(L_0\) and the intervention’s \(A_0\). Sequential exchangeability and consistency identify that conditional distribution with \(q_1(l_1\mid l_0,a_0)\) at supported histories. Suppose the intervention-specific distribution through \(L_t\) is the product through time \(t-1\). By the chain rule, adjoining \(L_{t+1}\) multiplies that distribution by its conditional distribution given \((\bar L_t,\bar A_t)\). The same identification argument makes this factor \(q_{t+1}\) evaluated at the intervention’s treatment value. Induction through \(t=K-1\) gives the stated product. \(\square\)

This result explains what the covariate models must accomplish: together, they specify the distribution of whole histories under each intervention. Correctly modeling only the average of \(L_{t+1}\) is not enough if the g-formula requires its full conditional distribution. For instance, two distributions can have the same conditional mean but different probabilities of entering an important history.

Worked Examples

Model Accuracy and Trajectory Error

Errors in a transition model can propagate: an inaccurate distribution at one time changes the histories on which later transition models are evaluated. The following bound makes this accumulation explicit for discrete covariate histories. For probability distributions \(P,Q\), define total variation distance by \(\|P-Q\|_{\mathrm{TV}}=\sup_B|P(B)-Q(B)|\), where the supremum is over events \(B\).

Proof. Let \(P_t,Q_t\) denote the true and fitted distributions of histories through \(L_t\), and let \(d_t=\|P_t-Q_t\|_{\mathrm{TV}}\). Extending both distributions by the same fitted kernel cannot increase their distance: summing over the next covariate values shows that the absolute difference for any extended event is bounded by the total variation distance on the histories. Replacing the fitted kernel by the true kernel at a fixed history changes the conditional distribution by at most \(\epsilon_t\); averaging this bound over histories preserves it. The triangle inequality therefore gives \(d_{t+1}\leq d_t+\epsilon_t\). Since the baseline distributions agree, \(d_0=0\); induction yields \(d_K\leq\sum_{t=0}^{K-1}\epsilon_t\). \(\square\)

The bound is not a diagnostic that can be calculated from observed data without further assumptions: the true kernels are unknown. It does clarify why checking several transitions matters. Small errors at each step can add up, and a substantial error in one transition can alter many later histories. It also bounds errors in intervention-specific probabilities of trajectory events, since such probabilities differ by no more than the total variation distance.

Diagnostics, Alternatives, and Limitations

A useful review of fitted covariate models checks whether they are defined at the histories generated by the intervention, whether their predictions respect the covariate’s support, and whether they reproduce observed conditional distributions across important history strata. Large differences between observed and model-predicted transition frequencies, residual patterns over prior covariates or treatment, and sparse histories are warnings. Calibration in observed histories is informative but cannot establish correctness in unsupported histories or prove that an unobserved intervention-specific law is right.

The main modeling choice is between a single joint transition model for a covariate vector and a sequence of component-wise conditional models. A joint model can represent within-time dependence directly but may be demanding to estimate. Component-wise models can be convenient, but their factorization must be stated and justified; assuming independence merely for convenience can distort the generated history. Likewise, a model that omits a relevant prior covariate or treatment is not just more parsimonious: it may average over distinctions that the intervention makes consequential.

These models are parametric approximations. Their validity depends on the conditional distributions being adequately represented, and on the causal identification and support assumptions needed to interpret intervention-specific histories. Diagnostics can expose some problems but cannot eliminate model dependence. A model may fit common observed histories well while extrapolating poorly to histories emphasized by a particular intervention.