Rank-Preserving and Distributional Failure-Time Models

Two Different Claims About Failure Times

A structural failure-time model compares a person's failure time under treatment with that same person's failure time under a reference treatment. The key issue in this tutorial is whether the model makes a claim about each individual’s pair of potential failure times, or only about the distributions of those times in a population. Those claims can imply the same marginal pattern while making different statements about individual effects.

Consider one binary treatment decision, with \(A\in\{0,1\}\). Let \(T^0\) and \(T^1\) be the potential failure times under no treatment and treatment, respectively, and assume they are positive. A multiplicative acceleration factor \(c>0\) means that treatment scales a failure time by \(c\); equivalently, write \(c=\exp(\psi)\), where \(\psi\) is the log acceleration factor. If \(c>1\), the modeled treated time is longer.

In the rank-preserving claim, the same person’s two potential times are linked by an exact equation. In the distributional claim, the two marginal distributions are linked, but the joint distribution of \((T^0,T^1)\) is left unspecified. This is a distinction about assumptions, not merely about how the same assumption is written.

Quantiles: The Marginal Implication

For a random variable \(X\) with distribution function \(F_X\), define its generalized quantile by \(Q_X(p)=\inf\{t:F_X(t)\ge p\}\), for \(0<p<1\). This definition also applies to discrete distributions, where quantiles can have jumps.

To verify the forward direction, for any \(t\), positivity of \(c\) gives \(F_{cT^0}(t)=P(cT^0\le t)=P(T^0\le t/c)=F_{T^0}(t/c)\). Thus \(T^1\overset{d}{=}cT^0\) gives \(F_{T^1}(t)=F_{T^0}(t/c)\). The quantile identity follows directly from the definition:

Here the third equality uses \(c>0\), so that \(t=cs\) preserves order. Conversely, a random variable can be represented in distribution by applying its quantile function to a uniform random variable on \((0,1)\). If the quantile functions satisfy \(Q_{T^1}(p)=cQ_{T^0}(p)\) for all \(p\), these representations show that the marginal law of \(T^1\) is that of \(cT^0\). This proves the result. \(\square\)

A rank-preserving model with \(T^1=cT^0\) implies the distributional result, because applying the same positive scaling to every individual scales the entire marginal distribution. The reverse implication does not follow: a distributional equality leaves open many possible ways to pair the two sets of potential times.

Same Marginals, Different Individual Pairings

The quantile relation does not reveal the individual pairing. A simple finite population makes this concrete. Keep the untreated values \(2,4,6\) and treated values \(4,8,12\), each occurring once, but pair them differently. The untreated and treated marginal distributions—and therefore all their quantiles—are unchanged.

The same point can be stated as a general identification limitation. Observed data typically reveal at most one potential failure time per person. Even in a randomized study, randomization can identify the marginal distributions of \(T^0\) and \(T^1\), subject to complete outcome ascertainment and consistency, but it does not reveal their joint distribution for the same individuals. Rank preservation specifies that joint relationship; it is an additional, generally untestable cross-world assumption.

The following result makes the implication precise: if the marginal distributions obey the scaling model, then every coupling of those marginals is compatible with the distributional model. Some couplings may satisfy the individual equation, while others may not. For the finite example above, the two explicitly listed pairings are such couplings: both have the same marginals, but only the aligned pairing has \(T^1=2T^0\) for every type. Therefore marginal scaling alone cannot identify the individual-level relationship. \(\square\)

Assumptions, Identification, and Diagnostics

A distributional scaling parameter is identified from marginal data only when the design and assumptions connect observed outcomes to the potential-time marginals. In the simple setting, the relevant conditions include:

• Consistency: the observed failure time equals \(T^A\) for the treatment actually received.
• Exchangeability: treatment assignment is independent of \((T^0,T^1)\), as in a properly randomized trial. In an observational study, a suitable conditional exchangeability assumption would be needed instead.
• Positivity: each treatment level has positive probability in the population or covariate strata being studied.
• Outcome ascertainment: failure times are observed or censoring is handled under appropriate additional assumptions. The examples here treat times as fully observed.
• Model and parameter uniqueness: the marginal distributions must actually satisfy the proposed scaling relation, and distinct values of \(c\) must not yield the same pair of distributions. For example, when the untreated median is positive, the median ratio uniquely specifies \(c\) if the scaling model holds.

These conditions separate three questions. First, can the arm-specific marginal distributions be identified from the observed data? Second, do those distributions satisfy the proposed scaling model? Third, should the model be interpreted as an individual-level equation? The first depends on design and identification assumptions; the second is a marginal model check; the third requires rank preservation or another explicit assumption about the joint potential outcomes.

In a randomized study with complete follow-up, the marginal implication can be examined by comparing estimated distributions or several quantiles. A single matching median is not enough to establish a scale relation: other quantiles may fail to scale by the same factor. In observational data, apparent marginal scaling may also reflect confounding, so agreement with the model does not by itself establish a causal interpretation.

The individual coupling is different. Because no person is observed under both treatment levels, the data do not directly check whether that person’s pair obeys \(T^1=cT^0\), or whether the person retains the same rank in both potential-time distributions. Thus a good marginal fit cannot be used as evidence that rank preservation is true.

Interpretation and Common Pitfall

A distributional factor summarizes how one potential-time distribution is scaled relative to another. It can support population-level statements about survival-time distributions or quantiles under the model. It does not, on its own, say that every person’s failure time changes by that factor, identify who benefits, or determine the distribution of individual treatment effects. Those questions require information or assumptions about the joint distribution of \((T^0,T^1)\).

The main alternative is therefore not a different way to calculate the same individual effect. It is a different modeling commitment. A rank-preserving model is stronger and permits an individual-level interpretation conditional on its assumptions. A distributional model makes a weaker claim about marginals, avoids asserting a particular cross-world pairing, and cannot answer individual-effect questions from the marginal relation alone. Neither model is automatically preferable: the choice must match the scientific estimand and the plausibility of the assumptions.