Structural Nested Failure-Time Models and Acceleration Factors

Why Model Treatment Effects on a Failure-Time Scale?

A treatment may change not only whether failure occurs, but also the time remaining until failure. A structural nested failure-time model (SNFTM) expresses this effect as an acceleration or deceleration of counterfactual survival time. Its blip parameters answer a local question: at a particular treatment decision, by what factor would treatment multiply the time remaining until failure, compared with no treatment at that decision?

The word structural signals that the model relates counterfactual outcomes under different treatment choices. The word nested signals that treatment effects are specified decision by decision, conditional on the treatment and covariate history already accrued. This is a different outcome scale from the mean-scale structural nested models discussed in the earlier material on proximal and cumulative treatment effects in SNMMs.

Counterfactual Survival and the Failure-Time Blip

Suppose treatment decisions occur at fixed times $t_0<t_1<\cdots<t_K$. Write $\bar a=(a_0,\ldots,a_K)$ for a treatment regime, with each $a_j$ equal to 0 or 1, and $T^{\bar a}$ for the counterfactual failure time under that regime. Let $H_j$ denote information available just before the treatment decision at $t_j$, including prior treatment and measured covariates. A failure time is measured from the common study origin.

At decision $j$, the remaining time under a specified regime is $T^{\bar a}-t_j$. The time-specific blip compares this remaining time when treatment at $j$ is set to 1 versus 0, holding the past history and the subsequent assigned regime fixed. The comparison is made among individuals eligible and alive at that decision under the common past history. Thus both remaining times in the comparison are positive.

Definition: A multiplicative SNFTM specifies, for each decision time $t_j$, a finite real-valued blip parameter $\psi_j(H_j)$ such that $$ T^{(\bar a_{j-1},1,\bar a_{j+1:K})}-t_j = \exp\{\psi_j(H_j)\} \left[T^{(\bar a_{j-1},0,\bar a_{j+1:K})}-t_j\right]. $$ Here the prior regime $\bar a_{j-1}$ fixes the history before decision $j$, and the subsequent regime is held fixed in the comparison. The quantity $\exp\{\psi_j(H_j)\}$ is the acceleration factor. This is a structural model assumption, not a conclusion guaranteed by randomization or by the observed data.

The model is stated on the remaining-time scale, not by multiplying the total study time $T$ by the same factor at every decision. If the acceleration factor is 1, then $\psi_j(H_j)=0$ and the model assigns no treatment effect at that history and decision. A factor greater than 1 means longer remaining survival under treatment; a factor between 0 and 1 means shorter remaining survival. These interpretations follow from the convention in the displayed definition.

Worked Example: Interpreting One Blip

At a decision time of 4 years, suppose the modeled acceleration factor for treatment is 1.25 for a particular history. Under the two regimes being compared, the counterfactual remaining survival time without treatment is 12 years.

The model gives $\psi_j(H_j)=\log(1.25)$, approximately 0.223. With treatment, the remaining time is $1.25(12)=15$ years. The corresponding failure times from the study origin are 4 + 12 = 16 years without treatment and 4 + 15 = 19 years with treatment.

The treatment adds 3 years in this particular comparison, but 3 years is not the blip parameter: the parameter is the log acceleration factor, approximately 0.223. If the untreated remaining time had instead been 4 years, the same blip would give 5 years under treatment, a 1-year difference.

How Blips at Successive Decisions Combine

A time-specific blip describes one decision while the other assigned treatments are held fixed. When several decisions are involved, the model can be applied recursively. For clarity, consider fixed treatment regimes and histories, and suppose the relevant counterfactual failure times lie after the decision times so that each remaining-time comparison is defined.

Start with the reference failure time under the regime that sets all decisions to 0. Beginning at the last decision and proceeding backward, replace the reference outcome at each decision by its treated counterpart using that decision’s acceleration factor. The recursion makes clear why an earlier treatment acts on the remaining time that already reflects later treatment.

Result: Let $Q_{K+1}=T^{\bar 0}$, the reference failure time under no treatment, and define recursively, for $j=K,K-1,\ldots,0$, $$ Q_j=t_j+\exp\{a_j\psi_j(H_j)\}(Q_{j+1}-t_j). $$ When the decision-specific blip model holds for these regime comparisons, $Q_0$ is the failure time under the specified treatment regime. Each factor acts on time remaining after its own decision, not on the amount of survival time added by earlier decisions.

To verify the recursion, hold fixed all decisions other than $j$. When $a_j=0$, the time after decision $j$ is $Q_{j+1}$, so the remaining time is $Q_{j+1}-t_j$. When $a_j=1$, the blip definition multiplies this remainder by $\exp\{\psi_j(H_j)\}$, and adding $t_j$ returns the failure time from the study origin. For $a_j=0$, the multiplier is $\exp\{0\}=1$. Applying this argument from $K$ back to 0 establishes each step of the recursion and the stated regime outcome. $\square$

Worked Example: Two Treatment Decisions

Take decisions at years 0 and 5. Suppose the no-treatment reference failure time is 20 years. At year 5, the acceleration factor is 1.5; at year 0, it is 1.2. Assume the relevant histories have these values and the failure times remain beyond both decision times.

First apply the year-5 blip. The reference remaining time at year 5 is $20-5=15$ years. Treatment at year 5 changes it to $1.5(15)=22.5$ years, so the resulting failure time is $5+22.5=27.5$ years.

Next apply the year-0 blip to the outcome that already reflects the year-5 decision. The remaining time at year 0 is 27.5 years, so treatment at year 0 changes it to $1.2(27.5)=33$ years. Under both treatments, the model therefore gives a failure time of 33 years.

The two factors do not imply that treatment adds a fixed 0.5 or 0.2 years. They multiply the remaining times at their respective decisions. In particular, reversing the calculation without respecting the reference regime would not represent the stated nested comparisons.

Result: For a single decision at time $t_j$, if the untreated remaining time $R_0$ is positive and the treated remaining time $R_1$ satisfies the SNFTM, then $$ \psi_j(H_j)=\log(R_1/R_0). $$ Indeed, the model states $R_1=\exp\{\psi_j(H_j)\}R_0$. Dividing by $R_0>0$ gives $R_1/R_0=\exp\{\psi_j(H_j)\}$; taking logarithms gives the identity. Thus $\psi_j$ is a log ratio of remaining survival times, and a unit increase in $\psi_j$ multiplies that ratio by $e$. $\square$

Worked Example: A Negative Blip

For a given decision and history, suppose untreated remaining survival is 10 years and the model’s parameter is $\psi_j(H_j)=\log(0.8)$, approximately $-0.223$. Since $\exp\{\psi_j(H_j)\}=0.8$, treated remaining survival is $0.8(10)=8$ years.

The factor is below 1, so the model describes a 20% shorter remaining time under treatment for this contrast. If the decision is at year 3, the corresponding failure times are 13 years without treatment and 11 years with treatment. The negative sign does not mean a negative survival time; it indicates a factor less than 1 on the positive remaining-time scale.

Assumptions, Diagnostics, and Limits

The blip equation requires well-defined counterfactual failure times under the regimes being compared, consistency between observed outcomes and the potential outcome under the received treatment, and a correctly specified form for $\psi_j(H_j)$. The finite-parameter multiplicative model also requires positive remaining times in each comparison to which it is applied. If treatment is selected using measured history, identifying and estimating these causal contrasts additionally requires appropriate treatment-assignment assumptions, including sequential exchangeability and positivity. Those assumptions do not follow from the SNFTM itself.

The model’s substantive restriction is that, at a given decision and history, treatment changes remaining survival through a multiplicative factor. A constant acceleration factor across histories is a strong restriction; allowing $\psi_j(H_j)$ to vary with history makes the model more flexible but does not eliminate the need to choose a suitable functional form. The same parameter need not apply at different decision times.

In practice, warnings include poorly supported treatment choices in parts of the history space, unstable estimates of time ratios, and systematic disagreement between model-based predictions and observed survival patterns across treatment and history groups. A fitted model that predicts implausible remaining times, or whose estimated blips change sharply under modest changes in model specification, deserves scrutiny. Such checks can reveal problems, but agreement with observed data cannot prove the counterfactual structural equation.

A mean-scale structural nested mean model describes a blip in a conditional mean, whereas an SNFTM describes a transformation of remaining failure time. A conventional accelerated failure-time model also uses time acceleration, but commonly specifies a regression model for an observed or regime-specific survival-time distribution. An SNFTM instead nests treatment contrasts at successive decisions and makes their causal interpretation explicit. Neither a mean-scale model nor a conventional accelerated failure-time model is automatically a substitute for the other: the choice depends on the outcome scale and causal question.

An acceleration factor is not an absolute treatment benefit. A factor of 1.25 corresponds to a 25% increase in remaining time in the modeled contrast, but the difference in years depends on the untreated remaining time. Nor does this tutorial’s definition establish that individuals retain the same ordering of failure times across regimes. The parameterization alone does not guarantee identification, correct model form, or a reliable estimate.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

At a decision time, the blip parameter is $\log(0.7)$. What does its sign imply, and what does the acceleration factor equal?
Without treatment, remaining survival is 6 years. The acceleration factor is 1.4. Calculate treated remaining survival and the absolute difference.
Why is a blip parameter not generally an additive number of years?
For decisions at years 0 and 4, a no-treatment reference failure time is 14 years. The year-4 factor is 2, and the year-0 factor is 1.5. Apply the recursion to find the failure time under both treatments.

Solutions. (1) The parameter is negative because $\log(0.7)<0$; the factor is $\exp\{\log(0.7)\}=0.7$, so the model gives 30% shorter remaining survival under treatment. (2) Treated remaining survival is $1.4(6)=8.4$ years, a difference of $8.4-6=2.4$ years. (3) The parameter is the logarithm of a ratio. The absolute difference is $(\exp\{\psi_j\}-1)R_0$, which depends on untreated remaining time $R_0$. (4) At year 4, untreated remaining time is $14-4=10$, and the treated time becomes $4+2(10)=24$. At year 0, the remaining time is 24, so the final failure time is $0+1.5(24)=36$ years.

Structural Nested Failure-Time Models and Acceleration Factors

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