Recursive Parameterization of Longitudinal Treatment Effects

Why Treatment Effects Need a Recursive Parameterization

With treatment at several times, a final outcome reflects a sequence of decisions, not a single treatment contrast. A useful way to describe that sequence is to start with a reference treatment history and add one treatment decision at a time. Each addition has a blip: the change in outcome associated with changing treatment at that time while later treatments are set to the reference level.

The key point is that these increments are evaluated along a treatment history. If an earlier treatment changes a later covariate, the blip at a later decision may differ from the blip that would have occurred under the reference history. The decomposition is recursive because each step adds the next treatment to the prefix already considered.

This tutorial develops the algebraic parameterization. It does not establish when the blips are identified from observed data or how to estimate them; those questions require additional assumptions and methods.

Nested Outcomes and Sequential Blips

Consider \(T\) treatment decisions, with \(A_t\) the treatment at time \(t\), and let \(0\) denote the reference treatment. Write \(\bar a_t=(a_1,\ldots,a_t)\) for a treatment prefix. For a fixed prefix, define \(Y^{(t)}(\bar a_t)\) as the outcome if treatment follows \(\bar a_t\) through time \(t\) and is set to the reference level at every later time. Define \(Y^{(0)}=Y(0,\ldots,0)\).

These are nested counterfactual outcomes for the same target population. In particular, \(Y^{(T)}(\bar a_T)=Y(\bar a_T)\), the outcome under the full treatment history. For a given treatment history \(\bar a_T\), the sequential blip at time \(t\) is the difference between two consecutive nested outcomes.

The definition requires the relevant nested potential outcomes to exist and have finite means for mean-based results. It does not by itself require exchangeability, positivity, or a treatment-assignment model: these are not needed for the algebraic decomposition. To interpret a factual observed outcome as \(Y(\bar a_T)\) for a person whose observed history is \(\bar a_T\), consistency is also required. The intermediate counterfactual outcomes and blips are generally not observed.

A structural nested mean model typically represents a conditional mean of the individual blip. Let \(L_t\) denote covariates measured before treatment \(A_t\), and let \(H_t^{\bar a_{t-1}}\) denote the covariate history that would arise under the preceding treatment prefix. A conditional blip function can be written \( \gamma_t(\bar a_t,h_t)=E[\Delta_t(\bar a_t)\mid H_t^{\bar a_{t-1}}=h_t] \). The exact contents of the history may vary with the study design; the essential feature is that it is the history under the prefix before the time-\(t\) decision.

The Recursive Decomposition and Its Mean Form

The fundamental identity follows by adding consecutive differences. It is exact at the individual counterfactual level, so it does not depend on a statistical model being correctly specified.

Proof. By the definition of each blip, \(\Delta_t(\bar a_t)=Y^{(t)}(\bar a_t)-Y^{(t-1)}(\bar a_{t-1})\). Summing from \(t=1\) to \(T\) cancels every intermediate nested outcome: \(Y^{(1)}\) appears once with a positive sign and once with a negative sign, as does each \(Y^{(t)}\) for \(1<t<T\). The sum is therefore \(Y^{(T)}(\bar a_T)-Y^{(0)}\). Since \(Y^{(T)}(\bar a_T)=Y(\bar a_T)\), rearrangement gives the first identity. Taking expectations and using linearity of expectation, which applies because the terms have finite expectations, gives the second. \(\square\)

The conditional blip function gives a useful version of the mean identity. For each \(t\), the relevant conditioning history is generated under the prefix \(\bar a_{t-1}\). By the law of iterated expectations, \(E[\Delta_t(\bar a_t)]=E[\gamma_t(\bar a_t,H_t^{\bar a_{t-1}})]\). Consequently, the regime mean is the reference-regime mean plus the expected blip at each time, averaged over the history distribution under the corresponding prefix.

Derivation. The theorem gives \(E[Y(\bar a_T)]=E[Y^{(0)}]+\sum_t E[\Delta_t(\bar a_t)]\). For each \(t\), the conditional-expectation definition and the law of iterated expectations give \(E[\Delta_t(\bar a_t)]=E\{E[\Delta_t(\bar a_t)\mid H_t^{\bar a_{t-1}}]\}=E[\gamma_t(\bar a_t,H_t^{\bar a_{t-1}})]\). Substitution for every term yields the displayed result. Integrability ensures these expectations are well-defined. \(\square\)

How Parametric Recursive Models Accumulate Effects

A parametric structural nested mean model specifies each conditional blip as a function of treatment, history, and parameters. The recursive decomposition then determines how those time-specific functions combine. The blips need not be constant, and they need not be additive in treatment history.

The same logic applies when the model includes covariate effect modification: evaluate each blip at the history under the preceding treatment prefix, then average over that prefix's history distribution to obtain a marginal mean. A model that omits relevant treatment-history or covariate dependence may still yield a computable sum, but the sum need not represent the true sequence of causal increments.

Common Pitfalls, Diagnostics, and Alternatives

• Evaluating every blip at the reference history. This discards changes in covariate or treatment history induced by earlier decisions. The correction is to evaluate the time-\(t\) blip at the history under the prefix \(\bar a_{t-1}\).
• Calling a blip the effect of treatment at time \(t\) in every possible future. Here the blip compares nested regimes with future treatment set to reference. It is not automatically the effect under an arbitrary later-treatment plan.
• Treating the decomposition as an identification result. Telescoping is an identity among counterfactual outcomes. It does not show that the blips or regime means can be recovered from observed data.
• Assuming the observed outcome reveals its components. Consistency may connect the factual outcome to the full observed treatment history, but the reference outcome and intermediate nested outcomes are usually counterfactual.

In practice, a useful diagnostic is to check that each modeled blip is zero when the current treatment is set to the reference level, and then verify numerically that the recursive increments reproduce the modeled full-regime mean for selected histories. Large changes in a later blip across histories, or poor support for histories needed to evaluate a blip, warn that the parameterization may be unstable or that the model is extrapolating.

The parametric g-formula instead models covariate and outcome distributions under treatment regimes and uses them to calculate regime outcomes. A marginal structural model represents marginal outcomes across regimes, often through weighting. A structural nested mean model parameterizes incremental effects relative to nested reference regimes. These approaches target related causal questions but organize the modeling differently; the recursive identity alone does not make their assumptions or estimates interchangeable.

The decomposition is most useful when incremental effects are substantively meaningful and their dependence on treatment and covariate history can be represented plausibly. Its limitations include reliance on a chosen reference treatment, sensitivity to blip-model misspecification, and potentially weak support for some treatment histories. For continuous outcomes, finite-mean conditions are natural; other outcome scales require an appropriate model and a carefully defined blip contrast rather than assuming the same additive-mean identity applies unchanged.