Why the Choice Matters
In an illness-death process, people may move from a healthy state to an ill state, die before illness, or die after illness. Those transitions need not have the same baseline hazard. A treatment or prognostic factor may also have a different association with each transition. The modeling choice here is whether to allow that difference in covariate effects or to impose a common effect while retaining a separate baseline hazard for each transition.
The distinction is consequential. A shared-effect model can be more precise when the common-effect constraint is reasonable, but it can obscure genuine transition-specific associations. Separate fits are flexible, but use fewer events per estimate and can produce unstable results. The comparison should therefore be guided by the scientific question, the event information, and the adequacy of the model assumptions—not by which fit gives a smaller p-value.
Two Cox Model Specifications
Let the states be 1 (well), 2 (ill), and 3 (dead), with allowed transitions \(1\to2\), \(1\to3\), and \(2\to3\). For transition \(k\), let \(Y_k(t)\) indicate that a person is at risk for that transition at time \(t\), and let \(X(t)\) be the covariate vector. A transition-specific Cox model has hazard
Here \(\lambda_{0k}(t)\) is the baseline hazard for transition \(k\), and \(\beta_k\) is its covariate-effect vector. The stratified model keeps the baseline hazards distinct but constrains the coefficient vector to be the same for every transition.
Thus, stratifying by transition does not force the transition hazards to have the same shape or level. It is the effects of the included covariates—not the baseline hazards—that are shared. In a long-format data set, the shared-effect model can be fitted with a Cox model such as Surv(Tstart, Tstop, status) ~ treatment + age + strata(trans). Separate fits use the rows for one transition at a time and fit a Cox model for that transition. A pooled model with treatment-by-transition interactions can also estimate transition-specific treatment effects.
Assumptions and a Useful Justification
Both specifications require the transition structure and risk sets to be encoded correctly, and the chosen time scale and covariate definitions to be meaningful. Within each transition, the Cox proportional-hazards assumption says that a covariate’s hazard ratio is constant over the modeled time scale. The shared-effect model adds the substantive restriction that a given covariate has the same log hazard ratio on every transition. The separate-fit model does not add that equality restriction, but still assumes proportional hazards within each transition.
For ordinary right-censored follow-up, censoring must be noninformative conditional on the modeled covariates and observed history, in the sense needed for the transition-specific risk sets. Subjects should be independent of one another. Because one subject can contribute information to multiple transitions, a subject-clustered robust variance is often appropriate for inference; it changes uncertainty estimates, not the fitted coefficient estimates. Use a consistent tie-handling method, covariate coding, and time scale across the fits being compared.
The reason the stratified model retains separate baselines can be seen from its partial likelihood. For simplicity, suppose event times are untied. At an event time \(t_{kj}\) for transition \(k\), the contribution uses only people at risk for that transition. With shared effects, the partial likelihood is a product of transition-specific contributions evaluated at the same \(\beta\):
The risk set \(R_k(t)\) and event times can differ by transition, so each transition contributes its own comparison and baseline hazard cancels within that transition’s partial likelihood. Replacing \(\beta\) with \(\beta_k\) gives the separate-fit specification. The shared model is therefore the separate-effects model under the constraints \(\beta_1=\beta_2=\cdots\), rather than a model with one common baseline hazard.
Worked Example: Set Up Both Models for an Illness-Death Dataset
Hypothetical data. Suppose a follow-up dataset contains 240 people initially in state 1, with treatment assignment and age recorded. During follow-up, there are 58 transitions from state 1 to state 2, 44 direct transitions from state 1 to state 3, and 71 transitions from state 2 to state 3. Other people are censored before a transition of interest. The data have been expanded so each row represents a subject’s interval at risk for a particular transition, with transition identifier, interval start and stop, and event indicator.
A shared-effect model for treatment and age is Surv(Tstart, Tstop, status) ~ treatment + age + strata(trans). The model estimates one treatment coefficient and one age coefficient, but permits three different baseline hazards. If treatment is binary, its hazard ratio is \(\exp(\beta_{\text{treatment}})\) on each of the three transitions.
The separate alternative fits the same covariates within each transition: one Cox model for \(1\to2\), one for \(1\to3\), and one for \(2\to3\). This produces three treatment coefficients and three age coefficients. The comparison is fair only if the transition-specific risk sets, covariate definitions, time scale, and handling of tied event times are aligned with those in the stratified fit. The counts alone do not establish that either model is adequate; they do show why precision may differ, especially for the transition with fewer events.
Comparing the Effect Constraints
The shared-effect model is nested within the model with separate transition effects. For a single covariate and three transitions, the equality restriction is \(H_0:\beta_1=\beta_2=\beta_3\), which imposes two independent constraints. A joint Wald test can assess those constraints using the estimated coefficient vector and its covariance matrix. With repeated transition records per person, use a covariance estimator that accounts for clustering by person, and use that covariance for the test. A failure to reject the constraints does not prove that the effects are identical; the data may simply be too imprecise to distinguish them.
Worked Example: Test Whether Treatment Effects Differ by Transition
Hypothetical fitted results. Suppose separate fits give treatment log hazard ratios \(-0.55\), \(-0.10\), and \(-0.05\) for transitions \(1\to2\), \(1\to3\), and \(2\to3\), respectively. Suppose the subject-clustered covariance matrix reported for these estimates is diagonal, with variances \(0.0529\), \(0.0400\), and \(0.0625\). The standard errors are therefore \(0.23\), \(0.20\), and \(0.25\).
Test equality using two contrasts, \(C\beta=(\beta_1-\beta_2,\ \beta_1-\beta_3)^\mathsf{T}\). The observed contrasts are \((-0.45,-0.50)^\mathsf{T}\). From the stated covariance matrix,
The robust joint Wald statistic is
Compared with a chi-square distribution with 2 degrees of freedom, this gives \(p\approx0.237\). At the 0.05 significance level, these hypothetical data do not provide strong evidence against equal treatment effects across transitions. That is not evidence that the effects are exactly equal: for example, the separate estimates are uncertain, and their confidence intervals are wide. A decision to use the shared-effect model should also consider whether a common treatment effect is scientifically plausible and whether the sample can detect clinically important differences.
Worked Example: Interpret a Shared Treatment Effect
Suppose the shared-effect stratified fit gives a treatment coefficient of \(-0.30\) with standard error \(0.14\). The estimated hazard ratio is
Under the model’s shared-effect constraint, the estimated treatment-associated transition hazard is about 0.741 times the corresponding untreated hazard—about 25.9% lower—on each of the three transitions, conditional on the modeled covariates and being at risk for that transition. This is an association under the fitted model, not by itself a causal conclusion.
A Wald 95% confidence interval for the log hazard ratio is \(-0.30\pm1.96(0.14)=(-0.574,-0.026)\). Exponentiating the endpoints gives a treatment hazard-ratio interval of approximately \((0.563,0.975)\). Under the shared-effect model, this interval applies to the common hazard ratio, not to three independently estimated transition effects. Its interpretation depends on the common-effect and proportional-hazards assumptions being tenable.
If the scientific question is whether treatment has a different association with illness onset than with death, separate estimates or a transition-interaction model are more directly informative. In the hypothetical separate fits above, the treatment hazard ratios are \(\exp(-0.55)=0.577\), \(\exp(-0.10)=0.905\), and \(\exp(-0.05)=0.951\). Those point estimates differ, but their uncertainty and the joint test matter: point-estimate differences alone do not establish effect modification by transition.
Practical Choice, Diagnostics, and Limitations
| Question | Shared-effect stratified fit | Separate transition fits |
|---|---|---|
| Baseline hazards | Distinct and unspecified for each transition | Distinct and unspecified for each transition |
| Covariate effects | Constrained to be equal across transitions | Estimated separately by transition |
| Precision | Can gain precision if the shared-effect constraint is reasonable | May have less precision, especially with few events per transition |
| Primary concern | A common effect may conceal real transition differences | Many estimates may be unstable and difficult to compare |
Check the data expansion before interpreting either fit: verify that events are assigned to the correct transition, that a person enters the appropriate risk set at the correct time, and that no event is counted twice. Examine transition-specific event counts and covariate variation; a coefficient cannot be estimated reliably when a transition has very few events or little covariate overlap. Assess proportional hazards separately for each transition, for example by inspecting transition-specific Schoenfeld-residual patterns or testing time interactions. A shared coefficient does not repair a time-varying effect. If treatment effects vary over time, the issue is not solved merely by choosing separate transition fits with constant coefficients.
A common mistake is to reason that “stratified by transition” means the treatment effect is also different for every transition. It does not: strata permit different baseline hazards, while the ordinary treatment term in the model is shared. The correction is to add treatment-by-transition interactions or fit separate transition models when transition-specific effects are required. Conversely, fitting separate models does not automatically establish that the effects differ; compare the estimates with an appropriate joint test and their uncertainty.
Separate fits are not always the better alternative. With sparse transitions, they can yield very wide intervals or unstable estimates; a common-effect fit may be a useful parsimonious summary if its constraint is defensible. But a nonsignificant equality test is not a license to treat a clinically important difference as absent. Report which model was used, the reason for the constraint or its relaxation, transition-specific event information, and uncertainty. Neither specification alone addresses an incorrect transition structure, unmeasured confounding, or a poorly chosen time scale.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- In a model with \(\text{strata(trans)}\) and one treatment coefficient, what is shared and what is allowed to differ?
- Why should a test of transition-specific coefficients use a subject-clustered covariance estimate when people contribute to multiple transitions?
- A joint test of equal effects has \(p=0.24\). Does this show that the transition effects are identical?
- What should be checked if the treatment effect appears to change over time within one transition?
Solutions. (1) The treatment coefficient is shared; each transition has its own baseline hazard. (2) Transition records from the same person may be dependent, so a subject-clustered covariance estimate accounts for that dependence in uncertainty and the joint test. (3) No. The result does not provide strong evidence against equality at conventional levels, but it does not prove equality or rule out important differences. (4) Assess proportional hazards for that transition, including residual patterns or a treatment-by-time term; separate transition fits with constant effects would not address time variation within a transition.