Stratified Cox Regression Across Transitions

Why Stratify the Baseline by Transition?

In a multi-state process, a subject can be at risk for different transitions at different times. The underlying event rates may vary greatly by transition: moving from a healthy state to illness need not occur at the same rate as moving from illness to death. Yet a researcher may have a substantive reason to expect a particular covariate to act in the same direction and by the same multiplicative amount on each transition. Stratified Cox regression represents this combination: a separate baseline hazard for every transition, but a common covariate coefficient.

This is not the same as assigning one baseline hazard to all observed transitions. Nor does it fit an unrelated covariate coefficient for each transition. The model keeps the transition-specific baseline hazards flexible while imposing a shared effect. That restriction can simplify a model and make its estimated effect more precise when it is credible; it can also conceal important differences if the restriction is wrong.

Definition: Let $r\to s$ denote a specified transition, $N_{i,rs}(t)$ subject $i$'s counting process for that transition, and $Y_{i,r}(t)$ an indicator that the subject is in the origin state $r$ and eligible for the transition at time $t$. Let $Z_i(t)$ be a vector of covariates known just before $t$. A transition-stratified Cox model with a common coefficient vector $\beta$ specifies the conditional transition intensity as $$ \alpha_{i,rs}(t\mid\mathcal{F}_{t-})= Y_{i,r}(t)\alpha_{rs,0}(t)\exp\{\beta^{\mathsf T}Z_i(t)\}. $$ Here $\alpha_{rs,0}(t)$ is an otherwise unspecified baseline hazard for transition $r\to s$, and it may differ freely across transitions. The coefficient vector $\beta$, however, is the same for every transition in the model.

The intensity is zero when $Y_{i,r}(t)=0$, because the subject is not then at risk for a transition from state $r$. If the covariates are time-fixed, write $Z_i$ in place of $Z_i(t)$. The covariates must be defined consistently for the transitions being modeled; this formulation does not itself introduce transition-specific covariate coefficients.

What the Shared Coefficient Says

For a covariate vector $z$, the transition hazard at a time when a subject is at risk is $\alpha_{rs,0}(t)\exp(\beta^{\mathsf T}z)$. Compare two covariate values $z$ and $z+\delta$ for the same transition at the same time. Their hazard ratio is

\frac{\alpha_{rs,0}(t)\exp\{\beta^{\mathsf T}(z+\delta)\}} {\alpha_{rs,0}(t)\exp(\beta^{\mathsf T}z)} =\exp(\beta^{\mathsf T}\delta).

The baseline cancels, so this ratio does not depend on time or on which transition is considered. For a one-unit increase in covariate $j$, holding the other covariates fixed, the conditional hazard ratio is $\exp(\beta_j)$ on every modeled transition. “Common” refers to this relative covariate effect, not to the absolute hazard. Different baseline hazards can yield very different absolute transition hazards for subjects with the same covariates.

Worked Example: A Common Treatment Effect in an Illness-Death Model

Consider a hypothetical illness-death process with states 0 (healthy), 1 (ill), and 2 (dead), and modeled transitions $0\to1$, $0\to2$, and $1\to2$. Let $Z_i=1$ for treatment and $Z_i=0$ for control. Suppose the fitted common treatment coefficient is $\hat{\beta}=-0.4055$.

For each transition, the estimated treatment-to-control hazard ratio is $$ \exp(\hat{\beta})=\exp(-0.4055)\approx 0.667. $$

Under the model, at a given time and among subjects eligible for a given transition, the treated group has an estimated transition hazard about 0.667 times the control-group hazard. This same relative reduction applies to all three transitions, conditional on the model covariates and risk sets. It does not imply equal hazards for those transitions: their baselines $\alpha_{01,0}(t)$, $\alpha_{02,0}(t)$, and $\alpha_{12,0}(t)$ remain distinct. Nor does the result alone establish that treatment causes a reduction; causal interpretation requires an appropriate study design and additional assumptions.

Assumptions and Model Structure

The model is useful only when its assumptions match the analysis. In particular, the common coefficient is a substantive and statistical restriction, not a consequence of having multiple transitions.

Correct risk and transition definition. Subjects enter a transition’s risk set only when they are in its origin state and eligible for that transition. The state process and event records must identify the modeled transitions correctly.
Proportional hazards within each transition. For a fixed covariate contrast, the hazard ratio is constant over the chosen analysis time. Each transition has its own baseline, but the common $\beta$ does not vary with time.
Common covariate effects across transitions. The same $\beta$ applies to every transition included in the model. A coefficient that is genuinely different by transition violates this restriction.
Appropriate covariate timing and form. Time-varying covariates must be observed before the event time and represented in a way consistent with the model. The linear predictor $\beta^{\mathsf T}Z_i(t)$ must be an adequate form for the covariates used.
Independent subjects and suitable censoring. The usual inference requires independent subject-level observations (or a method that accounts for dependence) and censoring that is non-informative for the transition process, conditional on modeled information.
Estimable effects. There must be enough event information and covariate variation in the relevant transition risk sets to estimate the common coefficients. If a covariate barely varies, or events are too sparse, estimation may be unstable even though the model is defined.

A transition intensity conditional on the observed history need not, by itself, assert that the full process is Markov. The model describes the specified transition hazards given the conditioning information. If the process has duration or history dependence, that structure must be handled in the time scale, covariates, or model specification rather than assumed away.

Why the Baselines Must Remain Separate

The transition-specific baselines allow the underlying event rates to have different shapes and magnitudes over time. For example, a covariate can have a common hazard ratio for illness and death even when illness is much more common early in follow-up and death after illness becomes more common later. Constraining these baselines to be equal would add a different, usually unjustified, restriction.

A short algebraic check shows the model’s key separation. For transition $r\to s$, the hazard at covariate value $z$ is $\alpha_{rs,0}(t)\exp(\beta^{\mathsf T}z)$. For a second transition $u\to v$, it is $\alpha_{uv,0}(t)\exp(\beta^{\mathsf T}z)$. Their ratio at the same covariate value is

\frac{\alpha_{rs,0}(t)\exp(\beta^{\mathsf T}z)} {\alpha_{uv,0}(t)\exp(\beta^{\mathsf T}z)} = \frac{\alpha_{rs,0}(t)}{\alpha_{uv,0}(t)}.

Thus, the shared covariate term cancels when comparing transitions at the same $z$, but the baseline-hazard ratio need not equal one. The model shares a covariate multiplier; it does not make different transitions interchangeable.

Worked Example: Different Baselines, Same Relative Effect

Suppose a hypothetical model has one continuous covariate $Z$, measured in years, with common coefficient $\hat{\beta}=0.0800$ per year. At a particular time, the estimated baseline hazards are 0.0400 per year for transition A and 0.120 per year for transition B. Compare subjects with $Z=5$ and $Z=0$.

For either transition, the hazard ratio for a five-year increase is $$ \exp\{0.0800(5-0)\}=\exp(0.400)\approx 1.492. $$

For transition A, the estimated hazards are $0.0400\exp(0.0800\times0)=0.0400$ per year at $Z=0$, and $0.0400\exp(0.0800\times5)\approx0.0597$ per year at $Z=5$. For transition B, they are $0.120$ and $0.120\exp(0.400)\approx0.179$ per year, respectively. The absolute hazards differ between transitions, while the modeled five-year hazard ratio is the same.

When This Parameterization Is Useful

Transition stratification is most attractive when there is good reason to preserve distinct baseline event patterns but a defensible expectation that the modeled covariate effects are common. It can be a reasonable parsimonious starting point when some transitions have limited event information: instead of estimating a separate coefficient for every transition, information about the common coefficient can contribute across transition-specific risk sets. That potential gain depends on the common-effect assumption being adequate.

It is less appropriate when scientific knowledge or transition-specific evidence suggests materially different covariate effects. For example, a treatment might plausibly delay disease onset but have little effect on death after disease. A common coefficient would force one relative effect onto both transitions. A separate transition-specific Cox fit avoids that restriction but estimates more coefficients and can be less precise when events are few. These are modeling choices to be justified, not automatic rankings of one approach over another.

Approach	Baseline hazards	Covariate coefficients	Key implication
One pooled Cox model	Common baseline structure	Common coefficients	Can impose more similarity across transitions than is warranted.
Transition-stratified Cox model	Separate for each transition	Common coefficients	Allows different baseline patterns but assumes common covariate effects.
Separate transition-specific Cox models	Separate for each transition	Estimated separately	Allows effects to differ, at the cost of more parameters and potentially less precision.

Worked Example: Recognizing When a Common Effect Is Dubious

In a hypothetical analysis, separate exploratory Cox fits estimate a treatment hazard ratio of 0.70 for transition $0\to1$ and 1.40 for transition $1\to2$. These are estimates from separate models, not proof that the true effects differ. However, their directions are opposite, so imposing one common treatment coefficient would require a compromise rather than representing both estimates closely.

On the log-hazard scale, the estimates are $\log(0.70)\approx-0.357$ and $\log(1.40)\approx0.336$. A common coefficient cannot equal both values. Before adopting a shared-effect model, the analyst should investigate uncertainty, event counts, transition-specific risk-set composition, and model adequacy. A common estimate near zero could indicate weak overall evidence, but it would not establish that treatment has no effect on either transition separately.

Fitting, Diagnostics, and Failure Modes

The model is fit as a Cox regression stratified by transition: each transition has its own risk-set comparisons and baseline hazard, while the coefficient vector is shared. A subject contributes to a transition’s risk set only while eligible for that transition. Events from one transition must not be treated as though they occurred in the risk set for another transition. The specific partial-likelihood construction depends on the transition and event structure; this tutorial’s purpose is the model parameterization, not a derivation for a particular competing-risks setting.

Diagnostics should examine whether the restrictions are plausible within each transition. Useful checks include inspecting transition-specific residual or time-varying-effect patterns for departures from proportional hazards, and comparing exploratory transition-specific estimates with their uncertainty. A visible time trend in a covariate effect challenges proportional hazards; materially different transition-specific effects challenge the shared-coefficient assumption. Sparse events, separation-like behavior, or very broad uncertainty indicate that the nominal common estimate may still be unstable.

Do not infer equal baselines from shared coefficients. A shared $\beta$ places no equality constraint on $\alpha_{rs,0}(t)$.
Do not pool transition risk sets indiscriminately. Eligibility is transition-specific, even when coefficients are shared.
Do not treat a common estimate as evidence of homogeneity. The equality of effects is built into the model; the model does not test that restriction merely by fitting it.
Check time and history structure. Apparent coefficient differences or proportional-hazards violations may reflect an inadequate time scale, omitted history, or covariate specification.

Key takeaway: Transition-stratified Cox regression allows every modeled transition its own unspecified baseline hazard while sharing a single covariate coefficient vector. Use it when the common relative-effect restriction is substantively credible; do not mistake baseline flexibility for flexibility in the covariate effects.

Check Your Understanding

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

In a transition-stratified Cox model, what is allowed to differ across transitions, and what is constrained to be the same?
A covariate coefficient is $\beta=0.12$ per unit. What hazard ratio does the model assign to a two-unit increase, on each modeled transition?
Why does a shared coefficient not imply that two transitions have the same absolute hazard?
What modeling restriction is questionable if exploratory transition-specific fits suggest hazard ratios of 0.6 and 1.5 for the same covariate?
Name one diagnostic pattern that would challenge the proportional-hazards assumption within a transition.

Solutions. (1) Each transition has its own baseline hazard; the covariate coefficient vector is common. (2) The ratio is $\exp(0.12\times2)=\exp(0.24)\approx1.271$, on every modeled transition. (3) The absolute hazards also depend on transition-specific baseline hazards, which need not be equal. (4) The shared-effect restriction is questionable because one common coefficient cannot represent both estimated log-hazard effects closely; uncertainty and event information should also be examined. (5) A covariate effect that changes systematically over time, such as a residual pattern or estimated time-varying effect, challenges proportional hazards.