Baseline Effect Modification in Failure-Time Blips

Why Baseline Effect Modification Matters

A failure-time blip describes how treatment changes the counterfactual time remaining until failure at a treatment decision. A single acceleration factor assumes this change is the same for every baseline profile. That can be too restrictive: treatment may accelerate survival more for one baseline group than another. Baseline effect modification allows the blip to vary with covariates measured before treatment, while retaining a structural interpretation on counterfactual survival time.

The focus here is the model for that variation and its interpretation. The key distinction is between a covariate-dependent causal blip and a treatment-outcome association conditional on covariates. The former describes a contrast between counterfactual failure times and requires the assumptions that identify those counterfactuals.

Covariate-Dependent Acceleration Factors

Definition: At decision time $t_j$, let $A_j$ be a binary treatment and let $V$ be a vector of baseline covariates. A baseline-effect-modified failure-time blip has log acceleration parameter $\psi_j(V)$. For a treated decision, removing that treatment maps a failure time $U_j$ to $$ U_{j-1}=t_j+(U_j-t_j)\exp\{-\psi_j(V)\}. $$ Thus $\exp\{\psi_j(V)\}$ is the treatment-versus-no-treatment acceleration factor for residual failure time at that decision. The equation applies to a treated decision; if $A_j=0$, the blip transformation is the identity.

For example, a linear specification can be written $\psi_j(V)=\beta_{j0}+\beta_j^\mathsf{T}V$. The intercept and coefficients describe the log acceleration factor for the chosen covariate coding. A common binary-treatment model instead writes $\psi_j(A_j,V)=A_j(\beta_{j0}+\beta_j^\mathsf{T}V)$, so that untreated decisions have no blip. If $V$ is centered, the intercept refers to the covariate-centering profile; if it is not centered, it refers to $V=0$. The coding must therefore be reported when interpreting coefficients.

For repeated decisions, blip down treatments in reverse chronological order. At each step, the time origin is the corresponding decision time, and the covariate-dependent parameter applies to that decision. This transformation does not assert that the observed time is literally generated by a deterministic individual-level mechanism; it specifies a structural counterfactual relation.

Worked Example: Interpreting a Binary Baseline Modifier

Suppose $\psi(V)=0.20+0.30V$, where $V=0$ and $V=1$ identify two baseline groups. Treatment is given at time zero. Calculate each group’s acceleration factor and compare the counterfactual treated survival time if its no-treatment survival time is 8 years.

For $V=0$, the log acceleration factor is $0.20+0.30(0)=0.20$. The factor is $\exp(0.20)\approx1.2214$, so the treated counterfactual time is $8(1.2214)\approx9.771$ years.

For $V=1$, the log factor is $0.20+0.30(1)=0.50$. The factor is $\exp(0.50)\approx1.6487$, giving $8(1.6487)\approx13.190$ years.

The ratio of the two acceleration factors is $\exp(0.50)/\exp(0.20)=\exp(0.30)\approx1.3499$. Thus the factor for $V=1$ is about 35 percent larger than for $V=0$. This is a contrast between treatment acceleration factors, not a claim that the treated survival times themselves have a 35 percent ratio unless the two groups have the same no-treatment counterfactual time.

Worked Example: Blipping Down a Treated Failure Time

At a decision time $t=2$, a person with $V=1$ receives treatment and has observed failure time $T=10$. Let $\psi(V)=0.20+0.30V$. Compute the failure time blipped down to no treatment at this decision.

First, $\psi(1)=0.50$. The residual observed time after the decision is $10-2=8$. Therefore,

U=2+(10-2)\exp(-0.50) =2+8(0.6065307) \approx 6.852245.

The untreated counterfactual residual time is approximately $6.852245-2=4.852245$. Blipping it back up checks the calculation:

2+4.852245\exp(0.50) \approx 2+8 =10.

The transformation acts on time remaining after the treatment decision, not on the entire failure time when $t\ne0$. Treating the full time $T$ as the residual would use the wrong time origin.

Repeated Decisions and Group-Specific Blips

With two treatment decisions, suppose $t_0=0$, $t_1=5$, and the observed failure time is $T=12$. Let the decision-specific log acceleration parameters be $\psi_0(V)=0.10+0.20V$ and $\psi_1(V)=0.15+0.25V$. For $V=1$, both treatments are received. Blip down the later decision first:

U_1=5+(12-5)\exp(-0.40) =5+7(0.6703200) \approx9.692240.

Here $\psi_1(1)=0.40$. Next, remove the treatment at time zero, for which $\psi_0(1)=0.30$:

U_0=0+U_1\exp(-0.30) \approx9.692240(0.7408182) \approx7.180.

The resulting time is the sequentially blipped-down time under neither treatment, given this model and these parameter values. Reversing the order would apply the first decision’s transformation to a failure time that still contains the later treatment effect. As a check, transforming $U_0$ forward at time zero gives approximately $9.692240$; applying the time-5 treatment gives $5+(9.692240-5)\exp(0.40)\approx12$, up to rounding.

Identification and a G-Estimation Moment

A covariate interaction in the blip model is not identified merely because it can be fitted to observed data. The following result states the core logic for a decision $j$. Let $H_j$ denote the observed history just before that decision, including baseline $V$. Assume consistency, sequential exchangeability of treatment given $H_j$, positivity, and a correctly specified blip transformation. For clarity, first suppose failure times are fully observed.

Theorem: If, conditional on $H_j$, treatment assignment $A_j$ is independent of the relevant counterfactual failure times, then at the true blip parameter the correctly transformed failure time $U_{j-1}$ is conditionally independent of $A_j$ given $H_j$. Consequently, for any integrable function $q$ for which the expectation exists, $$ E\left[(A_j-e_j(H_j))q(U_{j-1},H_j)\right]=0, \qquad e_j(H_j)=P(A_j=1\mid H_j). $$

Proof. Under sequential exchangeability, the counterfactual failure time relevant to decision $j$ is independent of $A_j$ conditional on $H_j$. Under consistency and the correctly specified structural blip, applying the true blip transformation to the observed time recovers that counterfactual time for the treatment decision being removed, after later treatments have also been handled when needed. A measurable function of this recovered time and $H_j$ therefore remains conditionally independent of $A_j$. Hence

E[A_jq(U_{j-1},H_j)\mid H_j] =e_j(H_j)E[q(U_{j-1},H_j)\mid H_j].

Subtracting the right-hand side and taking expectations proves the stated moment, provided the displayed conditional expectations are finite. $\square$

In practice, $e_j(H_j)$ can be estimated, and $q$ can include functions of $V$ that make the moments informative about interaction coefficients. Identification additionally requires sufficient variation in treatment within relevant histories and a moment system that distinguishes the parameters. Positivity alone does not guarantee that a particular finite set of moments identifies every coefficient.

Worked Example: A Moment That Targets Effect Modification

Suppose the baseline modifier is binary, $V\in\{0,1\}$, and at a single decision the proposed model is $\psi(V)=\beta_0+\beta_1V$. Let $e(H)=P(A=1\mid H)$, and define $R=q(U,H)$ for an integrable function of the blipped-down time and history. The conditional-independence result gives two useful moments, one using $q$ and one using $Vq$.

The first is

$$ E[(A-e(H))R]=0. $$

The second is

$$ E[(A-e(H))VR]=0. $$

Because $V=0$ makes $V R=0$, the second moment receives its direct contribution from the $V=1$ group. It can therefore add information about the modifier coefficient $\beta_1$, while the first moment may primarily inform the overall blip. These equations do not guarantee identification by themselves: $R$ must respond to the candidate blip, treatment must vary in both groups, and the combined estimating equations must distinguish $\beta_0$ from $\beta_1$.

Assumptions, Diagnostics, and Alternatives

The structural interpretation relies on more than a well-fitting regression. In addition to consistency and the specified structural model, sequential exchangeability requires that, conditional on the measured history, treatment assignment carries no residual information about the relevant counterfactual failure times. Positivity requires a nonzero chance of each treatment option for the histories used to estimate the model. Correct specification requires the chosen covariate function to represent the way baseline profiles modify the log acceleration factor. With right-censoring, the analysis also needs suitable censoring assumptions and an appropriate adjustment; treating censoring times as failures does not meet this requirement.

Useful diagnostics include examining estimated treatment probabilities across baseline groups, checking whether treatment is rare in particular covariate strata, and assessing whether fitted blipped-down times retain systematic associations with treatment after conditioning on history. One can also compare plausible functional forms for $\psi_j(V)$, inspect residual or moment patterns by modifier strata, and check whether estimates change sharply when a small number of observations or strata are removed. Such checks can reveal instability or incompatibility with the model, but cannot establish exchangeability or prove the blip model true.

A common mistake is to read $\beta_1$ as a difference in mean survival time. In the linear log-acceleration model, $\beta_1$ is a difference in log acceleration factors for a one-unit change in $V$; exponentiating it gives a ratio of acceleration factors. Another mistake is to infer causal effect modification from a treatment-by-covariate term in an ordinary outcome regression. That term is an associational conditional-model feature unless its causal interpretation is supported by the relevant design and assumptions.

Approach	What it models	Key distinction
Failure-time blip with baseline modification	Covariate-dependent multiplicative changes in counterfactual residual failure time	Uses a structural contrast and g-estimation conditions
Conventional accelerated failure-time regression	A conditional distributional model for observed failure time	Does not, by itself, solve time-varying treatment confounding
Marginal structural model	A marginal counterfactual outcome contrast under treatment regimes	Targets a marginal parameter rather than this conditional blip parameter

The blip model is attractive when the scientific question concerns how baseline profiles alter the treatment acceleration factor. Its limitations include dependence on structural-model specification, possible weak identification in sparse strata, and sensitivity to unmeasured confounding. A single linear interaction may also miss thresholds or nonlinear effect modification. Flexible functions can reduce that restriction, but require enough data and careful parameter identification.

Check Your Understanding

Answer each question, then compare with the solutions below.

If $\psi(V)=0.10+0.20V$, what is the ratio of acceleration factors for $V=1$ versus $V=0$?
At $t=3$, a treated individual fails at $T=11$, and the applicable log acceleration factor is $0.25$. What is the blipped-down time?
Why must the residual time after a decision be transformed rather than the full failure time?
Name two assumptions needed for a causal g-estimation interpretation of a baseline-modified blip.

Solutions. (1) The factors are $\exp(0.30)$ and $\exp(0.10)$, so their ratio is $\exp(0.20)\approx1.2214$. (2) The residual time is $11-3=8$, so $U=3+8\exp(-0.25)\approx3+8(0.7788008)=9.2304$. (3) The blip concerns time remaining from the decision; transforming the full time would use the wrong origin when the decision occurs after time zero. (4) Examples include sequential exchangeability, positivity, consistency, and correct specification of the structural blip. Both exchangeability and positivity are needed for the usual causal identification argument.