Choosing Between Mean and Failure-Time Structural Nested Models

Start With the Causal Quantity

A structural nested mean model (SNMM) and a structural nested failure-time model (SNFTM) can both describe longitudinal treatment effects, but they do not answer the same question by default. The first is organized around a contrast in a conditional mean on a chosen outcome scale. The second is organized around how treatment transforms failure time, often through an acceleration factor. Choosing between them therefore starts with the scientific interpretation required, not with which model is more familiar.

Let \(A_t\) denote treatment at decision time \(t\), and let \(H_t\) denote information measured before that decision, including relevant past treatment and covariates. Write \(Y^{\bar a}\) for the outcome that would be observed under treatment history \(\bar a\), and \(T^{\bar a}\) for the corresponding failure time. The target may be a change in mean symptom score, a change in mean survival time, or a multiplicative change in survival time. Those are distinct targets, even when they use the same study participants.

For either approach, causal interpretation requires a well-defined treatment strategy and consistency, no interference between individuals, sequential exchangeability of treatment given the measured pre-treatment history, and positivity for the treatment choices being compared within relevant histories. In brief, exchangeability rules out unmeasured confounding at each decision, while positivity requires that the compared choices actually occur with positive probability. A correct structural model and an adequate treatment and censoring analysis are additional requirements for estimation.

What the Two Blips Mean

In a mean SNMM, a blip at time \(t\) compares the conditional mean outcome under treatment \(a_t\) with the conditional mean under a reference choice, commonly \(0\), holding the relevant past history fixed and specifying a reference regime for subsequent treatment. A simple additive blip is

Here \(\psi\) is a difference in the units of \(Y\). If \(Y\) is a symptom score, \(\psi=-2\) means a two-point lower conditional mean score for the stated blip comparison. It is not automatically a population-average effect of assigning a whole longitudinal regime: later treatment decisions, effect modification, and the chosen blip structure affect that interpretation.

An SNFTM instead describes a transformation of failure time. In a simple one-treatment illustration, suppose

where \(T^0\) is the failure time under the reference treatment. Under this convention, \(r(a)\) is the treatment-to-reference time ratio: \(r(a)=0.8\) means failure time is 80% as long under treatment in this model. A positive \(\psi\) in this particular convention implies a ratio above one when \(a>0\), not a faster failure time. Signs depend on the model’s stated transformation convention, so always inspect the equation rather than infer direction from the word “acceleration.”

With repeated decisions, the failure-time model applies a sequence of blips to form a candidate treatment-free time \(U(\psi)\). The exact transformation depends on the model’s specification. At the true parameter, the transformed time should no longer show the treatment association that the model attributes to the blips, conditional on the relevant history. This is the basis for failure-time g-estimation.

Worked Examples of Scale Choice

Two Consequences That Help Choose the Model

The next result makes precise why an acceleration-factor model and a mean-time model are not interchangeable. It assumes finite means; without that condition, the stated mean contrast may not be defined.

Derivation. Since \(T^a=r(a)T^0\) and \(r(a)\) is a fixed finite number, linearity of expectation gives \(E[T^a]=E[r(a)T^0]=r(a)E[T^0]\). Subtracting \(E[T^0]\) yields \((r(a)-1)E[T^0]\). The finite-mean condition ensures both expectations exist. The examples above demonstrate the consequence: the same \(r(a)=0.8\) gives different differences when \(E[T^0]\) differs. \(\square\)

This calculation is not a claim that every SNFTM has a single common multiplier, or that every mean SNMM is additive. It shows why the model’s scale is part of the estimand. A mean SNMM with \(T\) as the outcome can target a mean-time contrast, but it does not acquire an acceleration-factor interpretation merely because the outcome is time.

A second useful derivation describes the role of blipping down in a simple failure-time model. Suppose the treatment relation is \(T^a=r(a)T^0\), with \(r(a)>0\), and define the candidate transformed time \(U(\psi)=T/r(A)\). At the true parameter \(\psi_0\), under consistency \(T=T^A\), so

Thus the correct transformation recovers the reference failure time in this simple model. If treatment is sequentially randomized given \(H_t\), the reference counterfactual time is independent of the treatment decision conditional on that history; the transformed time should consequently satisfy the corresponding treatment-independence restrictions. This reasoning depends on the model’s transformation being correct and on the treatment assignment assumptions. A failed association check can reflect misspecification, confounding, or inadequate support, not just an incorrect numerical estimate.

Data Requirements, Diagnostics, and Limitations

Both methods need the treatment and covariate history relevant to each decision, an outcome measured consistently, and adequate treatment overlap within histories. Their outcome requirements differ in emphasis:

For a mean SNMM, inspect whether the outcome scale and proposed blip form match the scientific question. Residual patterns, implausible predicted contrasts, or systematic differences across history strata can signal a misspecified blip function; such diagnostics do not prove identification. The earlier material on diagnostics for blip-function misspecification and treatment-model misspecification is relevant when interpreting these checks.

For an SNFTM, first establish that failure times and treatment timing are recorded well enough to define the proposed transformation. Right-censored observations do not reveal the failure time itself. Treating a censoring time as if it were a failure time can distort the transformed-time analysis; censoring must be handled under appropriate assumptions and with the needed censoring information. Check whether estimates or transformed-time associations are sensitive to the censoring specification, and whether few observations support some treatment choices or histories.

The main alternative is not always an either-or choice. If the scientific question concerns mean survival time, a mean SNMM using time as its outcome may be appropriate when the outcome and assumptions support it. If the question concerns a multiplicative time change, an SNFTM provides that interpretation directly. A distributional or rank-preserving specification makes further assumptions; do not infer those assumptions from a generic acceleration-factor model. Neither approach repairs unmeasured confounding, positivity violations, an ill-defined treatment regime, or poor measurement.

• Choose mean scale when the target is a conditional mean difference in a meaningful outcome unit and the outcome is observed or appropriately handled.
• Choose failure-time scale when the target is a time transformation or acceleration factor and failure-time, treatment-timing, and censoring data support that analysis.
• Pause before fitting if the target is described only as “the treatment effect.” Specify whether it means points, days, a ratio of times, or another explicit contrast.

Common Pitfall

A related warning is to avoid comparing coefficients across scales as if they were estimates of the same quantity. A mean difference in days and a log acceleration factor have different units and interpretations. A useful report states the outcome scale, reference treatment, blip definition, treatment-history comparison, and censoring assumptions alongside the estimate.