Repeated Treatment Decisions in Failure-Time Models

Why Repeated Decisions Need a Sequential Model

In a failure-time study, treatment is often not a single baseline choice. A person may be untreated at first, begin treatment later, stop, and resume. A model with one treatment indicator cannot distinguish these histories, even though treatment at different decision times may have different effects on the remaining failure time. A structural nested failure-time model (SNFTM) represents these decisions as a sequence of local acceleration effects. Its blip-down calculation removes the effects in reverse time order.

Consider fixed decision times $t_0<t_1<\cdots<t_{K-1}$. Let $A_j\in\{0,1\}$ denote treatment status during the interval beginning at $t_j$, for a person alive and uncensored at that decision. A history such as $(0,1,0,1)$ represents starting treatment, stopping, and resuming. Let $T$ be the failure time. The parameter $\psi_j$ describes the acceleration factor for treatment at decision $j$: under the model, treatment multiplies the remaining failure time, after later treatment effects have been removed, by $\exp(\psi_j)$. Positive $\psi_j$ corresponds to longer remaining failure time under treatment.

Definition: For candidate parameters $\psi=(\psi_0,\ldots,\psi_{K-1})$, define $T^{(K)}_\psi=T$. Recursively, for $j=K-1,\ldots,0$, define $$ T^{(j)}_\psi = \begin{cases} t_j+\{T^{(j+1)}_\psi-t_j\}\exp(-\psi_j A_j), & T^{(j+1)}_\psi>t_j,\\ T^{(j+1)}_\psi, & T^{(j+1)}_\psi\leq t_j. \end{cases} $$ The quantity $T^{(j)}_\psi$ is the failure time after blipping down treatment at decision $j$ and all later decisions. The final $T^{(0)}_\psi$ is the candidate time with all modeled treatment effects removed.

The branch in this definition matters: a decision cannot affect failure time for someone whose transformed failure time is already at or before that decision. In ordinary observed data, $A_j$ is only defined for people alive at $t_j$; the risk set for decision $j$ is therefore essential.

Assumptions and Estimating Equations

The derivation below concerns failure times observed without censoring. With right censoring, the analysis must incorporate appropriate censoring adjustments, as in right-censoring adjustments in failure-time g-estimation. The sequential treatment assumptions are also substantive; a correctly computed transformation alone does not establish a causal effect.

Consistency. The observed failure time agrees with the counterfactual failure time under the treatment history actually followed.
Sequential exchangeability. Among people alive and eligible at $t_j$, treatment choice $A_j$ is independent of the counterfactual failure time with treatment at $j$ and later decisions removed, conditional on the measured pre-decision history $H_j$. The history includes relevant past treatment and measured covariates.
Positivity. In histories used for estimation, both treatment options have positive probability: $0<\pi_j(H_j)<1$, where $\pi_j(H_j)=P(A_j=1\mid H_j,T\geq t_j)$.
Correct blip specification. The interval-specific acceleration model describes the relevant treatment effects, and the chosen estimating moments exist. For unbounded outcomes such as failure time, finite moments are needed if the identity score is used.

A common choice is a centered treatment-residual moment. Let $g$ be a specified function for which the expectations exist. Under the assumptions above, the true parameter satisfies

E\left[I(T\geq t_j)\{A_j-\pi_j(H_j)\}g(T^{(j)}_\psi,H_j)\right]=0.

Here $T^{(j)}_\psi$ depends on the candidate blips at decision $j$ and later decisions, but not on earlier blips. This makes reverse-time estimation natural: estimate the last interval's blip first, then move backward. The empirical version replaces the expectation and propensity by sample averages and estimated propensities. One may use bounded scores, such as indicators of transformed failure-time ranges, rather than rely on a finite mean for $T^{(j)}_\psi$.

Why the Blip-Down and Moment Conditions Work

Result: If the local SNFTM holds, the recursion removes later treatment effects before applying the current interval's inverse acceleration. At the true parameters, $T^{(j)}_\psi$ is the counterfactual failure time with treatment at decision $j$ and later decisions set to zero, among those alive at $t_j$.

To see the recursion, suppose the later blips have already been correctly removed, giving $T^{(j+1)}_\psi$. The local model says that, conditional on being at risk, the remaining time under $A_j=1$ is the remaining time under $A_j=0$ multiplied by $\exp(\psi_j)$. Thus, if the later-blipped time is $t_j+r\exp(\psi_j A_j)$, its value after setting $A_j=0$ is $t_j+r$. Multiplication of the remaining time by $\exp(-\psi_j A_j)$ gives exactly $t_j+r$. If the time is already at or before $t_j$, there is no remaining time at that decision to transform. Applying this argument successively from $K-1$ down to 0 proves the stated interpretation.

Result: Under sequential exchangeability and positivity, the stated risk-set moment has expectation zero at the true $\psi_j$, for any integrable $g(T^{(j)}_\psi,H_j)$ for which the conditional expectation is defined.

Proof. Write $R_j=I(T\geq t_j)$. Conditional on $R_j=1$ and $H_j$, sequential exchangeability makes the correctly blipped time $T^{(j)}_\psi$ independent of $A_j$. Therefore $$ E[\{A_j-\pi_j(H_j)\}g(T^{(j)}_\psi,H_j)\mid R_j=1,H_j] = E[A_j-\pi_j(H_j)\mid R_j=1,H_j]\, E[g(T^{(j)}_\psi,H_j)\mid R_j=1,H_j]. $$ The first factor is zero by the definition of $\pi_j(H_j)$. The conditional expectation is consequently zero, and multiplying by $R_j$ and taking expectations gives the claimed moment. $\square$

This result is a population identity, not a guarantee that every finite sample equation has a root or that its root is unique. Identification also requires enough treatment variation and enough information in the chosen moments to distinguish the interval-specific parameters.

Worked Transformations and Estimation

Worked Example: Starting, Stopping, and Resuming

Take decision times $t_0=0,t_1=1,t_2=2,t_3=3$, observed failure time $T=5$, and history $(A_0,A_1,A_2,A_3)=(0,1,0,1)$. Suppose the candidate blips are $\psi_1=\log(1.5)$ and $\psi_3=\log(2)$, with $\psi_0=\psi_2=0$. Calculate the fully blipped-down time.

Start at the last decision. Since $A_3=1$, the remaining time after $t_3$ is multiplied by $\exp\{-\log(2)\}=1/2$: $$ T^{(3)}_\psi=3+(5-3)(1/2)=4. $$ At $j=2$, $A_2=0$, so the time is unchanged: $T^{(2)}_\psi=4$. At $j=1$, $A_1=1$, so $$ T^{(1)}_\psi=1+(4-1)\exp\{-\log(1.5)\} =1+3(2/3)=3. $$ Finally $A_0=0$, giving $T^{(0)}_\psi=3$. The calculation uses each treatment interval's own blip; it does not collapse the history to “ever treated.”

Worked Example: A One-Decision G-Estimate

Suppose there is one decision at time 0, no covariates, and two uncensored observations: one treated person with $T=6$ and one untreated person with $T=4$. For illustration, use known $\pi=1/2$ and the identity score $g(u)=u$. Solve the sample moment for $\psi$.

For the treated person, $T_\psi=6\exp(-\psi)$; for the untreated person, $T_\psi=4$. The sample moment, using the average of the two residual contributions, is $$ \frac{1}{2}\left[(1-\tfrac12)6\exp(-\psi)+(0-\tfrac12)4\right] =\frac{1}{2}\{3\exp(-\psi)-2\}. $$ Setting it equal to zero gives $\exp(-\psi)=2/3$, hence $\hat\psi=\log(3/2)$. Substitution verifies the root: $3(2/3)-2=0$. This tiny sample illustrates the equation only; it cannot support reliable uncertainty assessment or establish the assumptions.

Worked Example: A Risk Set and a Positivity Failure

At $t_1=1$, suppose three people are alive, with histories $H_1=a,a,b$, and treatment assignments $1,0,1$, respectively. In history $a$, the estimated propensity is $1/2$; in history $b$, everyone receives treatment, so the propensity is 1. Which observations can inform the treatment contrast at this decision?

For history $a$, the residuals are $1-1/2=1/2$ and $0-1/2=-1/2$. These observations provide both treatment options within the same history, so their residual-weighted transformed outcomes can contribute to the moment. For history $b$, the residual is $1-1=0$, and there is no untreated comparison. More importantly, positivity fails because $P(A_1=0\mid H_1=b,T\geq1)=0$. No g-estimating equation can recover the treatment contrast for that history from these data without additional assumptions or extrapolation.

Estimation Sequence, Diagnostics, and Alternatives

For several decisions, compute candidate transformations backward. At step $j$, the later blips have already been incorporated into $T^{(j+1)}_\psi$; vary $\psi_j$ and solve a risk-set moment for the current decision. This sequential procedure depends on the chosen moment functions, treatment models, and blip specification. If multiple moments are used, their joint equations may overidentify the parameter; they should not be treated as independent evidence when they are built from the same observations.

Specify the history and risk sets.
Record treatment status at each decision and the pre-decision history for people eligible to receive treatment.

Model treatment assignment.
Estimate $\pi_j(H_j)$ within each risk set, checking overlap for both choices.

Blip down in reverse order.
For candidate parameters, transform failure time at the last decision and continue backward.

Solve and inspect.
Find roots of the interval-specific moments, check sensitivity to score and model choices, and apply censoring adjustments when needed.

Useful diagnostics include treatment counts by decision and history, propensity distributions near 0 or 1, and plots of the empirical moment against candidate $\psi_j$. A nearly flat moment, multiple roots, or no root indicates weak information or incompatibility between the model and the data. Large residual associations between treatment and transformed failure time within history strata can signal a misspecified treatment model, an inadequate history, or an incorrect blip model. These checks cannot verify sequential exchangeability, which concerns unmeasured determinants of treatment and failure.

A single cumulative exposure model can be more parsimonious, but it imposes restrictions such as equal blips across intervals or a fixed effect of total exposure. The interval-specific SNFTM allows different effects for different decision times and represents stopping and resumption directly, at the cost of more parameters and greater positivity demands. A conventional time-dependent regression is another alternative, but its treatment coefficients do not automatically have the structural counterfactual interpretation of these blips. The choice depends on the scientific estimand and assumptions, not only on which model is easiest to fit.

The method is limited by unmeasured confounding, treatment-model error, misspecified acceleration factors, and sparse histories. The transformation also requires a meaningful time origin and decision schedule. Censoring that depends on unobserved failure time can invalidate the uncensored moment unless handled under defensible censoring assumptions. Confidence intervals and tests must account for estimation of the treatment models and the sequential estimating procedure; the point estimate alone does not quantify uncertainty.

Common Pitfalls

Collapsing the history to ever-treated. This loses the timing of initiation, stopping, and resumption. Keep the $A_j$ sequence.
Blipping in forward order. The transformation at an earlier decision must act on the time after later effects have been removed. Calculate from the last decision backward.
Using future treatment for a person who already failed. Restrict each treatment moment to its decision-time risk set.
Reading a fitted moment as proof of causal identification. A root can exist even when exchangeability or positivity fails. Assess these assumptions separately.

Key takeaway: Repeated treatment decisions are represented by separate interval-specific acceleration blips. Correct sequential blip-down removes those effects from last decision to first, while risk-set g-estimation uses treatment residuals to identify each blip only under exchangeability, positivity, and a suitable structural model.

Check Your Understanding

Answer each question, then compare with the solutions below.

For history $(0,1,0,1)$, why is “ever treated” insufficient to retain the treatment pattern?
At one decision, $t_j=2$, $T^{(j+1)}_\psi=8$, $A_j=1$, and $\psi_j=\log(2)$. What is $T^{(j)}_\psi$?
Why must the g-estimating moment be restricted to people at risk at the decision time?
If treatment is always assigned in one measured history stratum, what assumption fails and what is the consequence?

Solutions. (1) It records only whether treatment ever occurred, not that treatment started, stopped, and resumed at particular decisions; distinct histories become indistinguishable. (2) $T^{(j)}_\psi=2+(8-2)\exp\{-\log(2)\}=2+6/2=5$. (3) Treatment at that decision is defined only for eligible people, and people who have already failed cannot be assigned that decision's treatment; including them would not represent the decision-specific treatment contrast. (4) Positivity fails because one treatment option has conditional probability zero. The data contain no within-stratum contrast to identify that treatment effect without extra assumptions.