Shared and Transition-Specific Covariate Effects

Why Specify Covariate Effects Across Transitions?

A multi-state process can have several possible transitions, and a covariate need not act in the same way on each one. For example, a treatment could be associated with a change in the intensity of moving from a healthy state to illness, while its association with the intensity of moving from illness to death could differ. A regression model must therefore specify whether the covariate coefficient is shared across transitions or allowed to vary by transition.

This is a choice about the linear predictors, not a declaration that transitions have identical baseline intensities or risk sets. The shared-coefficient model uses one coefficient vector in every transition-specific predictor; the alternative gives each transition its own vector. Both retain transition-specific baseline intensities.

Definition: For a permitted transition $g\to h$, let $Y_{igh}(t)$ indicate that subject $i$ is under observation and at risk for that transition at time $t$, and let $Z_i(t)$ be a $p$-dimensional covariate vector. A proportional transition-intensity model has the form $$ \lambda_{igh}(t\mid Z_i(t)) = Y_{igh}(t)\lambda_{gh,0}(t)\exp\{\eta_{igh}(t)\}. $$ Here $\lambda_{gh,0}(t)$ is the baseline intensity for transition $g\to h$, and $\eta_{igh}(t)$ is its covariate linear predictor. A shared-coefficient model sets $\eta_{igh}(t)=\beta^\mathsf{T}Z_i(t)$ for every modeled transition. A transition-specific model sets $\eta_{igh}(t)=\beta_{gh}^\mathsf{T}Z_i(t)$, with a separate vector for each transition.

What Is Shared—and What Is Not?

Suppose the modeled transitions are $0\to1$, $0\to2$, and $1\to2$. With two covariates, $Z_i(t)=(Z_{i1}(t),Z_{i2}(t))^\mathsf{T}$, the shared specification is

\begin{aligned} \eta_{i,01}(t)&=\beta_1 Z_{i1}(t)+\beta_2 Z_{i2}(t),\\ \eta_{i,02}(t)&=\beta_1 Z_{i1}(t)+\beta_2 Z_{i2}(t),\\ \eta_{i,12}(t)&=\beta_1 Z_{i1}(t)+\beta_2 Z_{i2}(t). \end{aligned}

The transition-specific version instead uses $\beta_{01}$, $\beta_{02}$, and $\beta_{12}$, each a two-element vector. In both models, $\lambda_{01,0}(t)$, $\lambda_{02,0}(t)$, and $\lambda_{12,0}(t)$ are distinct, unrestricted transition-specific baseline intensities. Sharing $\beta$ does not imply sharing the baselines.

The two specifications also leave the risk sets transition-specific. At a given time, the risk set for $0\to1$ contains observed subjects currently in state 0 and eligible for that transition. The risk set for $1\to2$ contains those currently in state 1. A shared coefficient does not combine these sets; it constrains the coefficient used in their respective transition models.

Worked Example: Writing Both Specifications

Consider a hypothetical illness-death process with transitions $0\to1$ (illness), $0\to2$ (death without recorded illness), and $1\to2$ (death after illness). Suppose $Z_i$ is a binary indicator for a hypothetical exposure. Write the shared model and a possible transition-specific model.

Shared model. With one coefficient $\beta$, $$ \eta_{i,01}=\beta Z_i,\qquad \eta_{i,02}=\beta Z_i,\qquad \eta_{i,12}=\beta Z_i. $$ The three baseline intensities remain $\lambda_{01,0}(t)$, $\lambda_{02,0}(t)$, and $\lambda_{12,0}(t)$.

Transition-specific model. Let the coefficients be $\beta_{01}=0.7$, $\beta_{02}=0.1$, and $\beta_{12}=-0.2$. For an exposed subject with $Z_i=1$, the three covariate contributions to the linear predictors are $0.7$, $0.1$, and $-0.2$. For an unexposed subject with $Z_i=0$, each contribution is 0.

These numbers specify different predictor contributions, not different baseline intensities. To obtain an intensity, each contribution is combined with the baseline for that particular transition. The example describes model specifications; it does not establish which specification is adequate for real data.

Partial Likelihood Under the Two Specifications

To use the Cox partial likelihood, assume transition-specific proportional intensities as defined above, correctly constructed risk sets, and covariates known just before each event time. Assume also that censoring is non-informative for the transition process conditional on the modeled covariate and observation history. For the simple product below, assume there are no tied transition-event times. The baseline intensity for a transition is left unspecified and cancels from its event-time contribution.

Let $R_{gh}(t)$ be the risk set for transition $g\to h$ at time $t$. If subject $i$ experiences that transition at time $t$, the partial-likelihood contribution under shared coefficients is

\frac{\exp\{\beta^\mathsf{T}Z_i(t)\}} {\displaystyle\sum_{j\in R_{gh}(t)} \exp\{\beta^\mathsf{T}Z_j(t)\}}.

The transition label determines the risk set, but the same $\beta$ appears in every factor. Under transition-specific coefficients, the corresponding factor is

\frac{\exp\{\beta_{gh}^\mathsf{T}Z_i(t)\}} {\displaystyle\sum_{j\in R_{gh}(t)} \exp\{\beta_{gh}^\mathsf{T}Z_j(t)\}}.

The full partial likelihood is the product of the relevant factors over observed transitions and event times. In the shared model, all factors depend on the same coefficient vector. In the transition-specific model, each factor depends on the vector for its own transition. For tied events, a suitable tie-handling likelihood approximation or exact method must replace the single-event factor.

Why the Transition-Specific Baseline Cancels

At an event time $t$ for transition $g\to h$, the proportional-intensity model assigns subject $i$ an instantaneous transition intensity proportional to $\lambda_{gh,0}(t)\exp\{\eta_{igh}(t)\}$. Conditional on one event of this transition among the subjects in $R_{gh}(t)$, the probability that subject $i$ is the one experiencing it is its intensity divided by the sum of intensities in that risk set. Under the shared model, this gives

\frac{\lambda_{gh,0}(t)\exp\{\beta^\mathsf{T}Z_i(t)\}} {\displaystyle\sum_{j\in R_{gh}(t)} \lambda_{gh,0}(t)\exp\{\beta^\mathsf{T}Z_j(t)\}} = \frac{\exp\{\beta^\mathsf{T}Z_i(t)\}} {\displaystyle\sum_{j\in R_{gh}(t)} \exp\{\beta^\mathsf{T}Z_j(t)\}}.

The common baseline factor for this transition and event time cancels. The same cancellation holds in the transition-specific model, using $\beta_{gh}$ in place of $\beta$. It does not make the baseline intensities equal across transitions: each transition has its own event-time contributions and risk sets.

A shared model with $K$ permitted transitions and $p$ covariates has $p$ covariate coefficients. A fully transition-specific model has $Kp$ such coefficients, assuming the same covariates enter every transition. The shared model is obtained by imposing the restrictions $\beta_{gh}=\beta$ for all modeled transitions. These restrictions change the predictor specification; they do not remove transition-specific risk sets or baselines.

Worked Example: Computing a Shared-Coefficient Partial Likelihood

Suppose a hypothetical study has three distinct event times. At each time, exactly two subjects are in the risk set for the event’s transition: one has binary covariate $Z=1$, and one has $Z=0$. The observed event subject has $Z=1$ at two of the times and $Z=0$ at the third. The events may be on different transitions, but use the same shared coefficient $\beta$. Assume the risk sets are correctly defined, covariate values are available just before each event, censoring is conditionally non-informative, and there are no ties.

At either time when the event subject has $Z=1$, the shared-model factor is $e^\beta/(1+e^\beta)$. At the time when the event subject has $Z=0$, it is $1/(1+e^\beta)$. Thus, writing $L(\beta)$ for this partial likelihood, $$ L(\beta)=\left(\frac{e^\beta}{1+e^\beta}\right)^2 \left(\frac{1}{1+e^\beta}\right) =\frac{e^{2\beta}}{(1+e^\beta)^3}. $$

Set $p_\beta=e^\beta/(1+e^\beta)$. Then $L(\beta)=p_\beta^2(1-p_\beta)$. Differentiating the log likelihood gives $$ \frac{d\log L(\beta)}{d\beta} =2-3\frac{e^\beta}{1+e^\beta}. $$ Setting this derivative to zero yields $e^\beta/(1+e^\beta)=2/3$, so $e^\beta=2$ and $\hat{\beta}=\log(2)\approx0.6931$. The second derivative is $-3p_\beta(1-p_\beta)$, which is negative at this estimate, confirming a local maximum. This calculation estimates one coefficient using contributions from all three transition-event risk sets; it does not impose a common baseline intensity.

Comparing the Specifications in Practice

Feature	Shared coefficients	Transition-specific coefficients
Linear predictor	One coefficient vector appears in every transition predictor	Each transition has its own coefficient vector
Baseline intensity	Separate for each transition	Separate for each transition
Risk sets	Separate by transition	Separate by transition
Coefficient restrictions	Constrains transition coefficient vectors to be equal	Allows coefficient vectors to differ
Main practical concern	May impose an overly restrictive common predictor	Uses more coefficients and needs adequate information for each transition

A shared specification can be a substantively motivated simplification when the same covariate contribution across transitions is plausible. A transition-specific specification allows the association to vary, but estimates more parameters. These are alternative modeling assumptions, not interchangeable descriptions of the same model. Neither one is justified merely because it is easier to fit or produces a more convenient summary.

Diagnostics, Assumptions, and Common Pitfalls

Before fitting either model, check that each transition risk set reflects who is genuinely eligible and observed at each event time. Verify that a subject contributes to the risk set for the correct transition given the subject’s current state and the observation scheme. Also check that covariates are available at the relevant time and that the chosen transition-intensity time scale and proportional-intensity assumption are defensible. A coefficient constraint cannot repair incorrect risk sets, unrecorded covariate history, or an unsuitable time structure.

After fitting, inspect whether the covariate specification appears plausible for each transition rather than relying only on an overall impression. Transition-specific event counts and covariate distributions can reveal weak support: for example, a covariate may barely vary among those at risk for one transition. Sparse support can make estimates unstable, especially when many transition-specific coefficients are fitted. Residual or proportionality checks should likewise respect the transition structure; a pattern confined to one transition can be obscured by a pooled summary.

Wrong reasoning: “The coefficients are shared, so all transitions have the same intensity.” Correction: the covariate coefficient vector is shared, but transition-specific baseline intensities and risk sets remain distinct.
Wrong reasoning: “All subjects in the study form one risk set because the coefficient is common.” Correction: each event-time likelihood factor uses the risk set for the transition that occurred.
Wrong reasoning: “Separate coefficients automatically fit better, so they should always be used.” Correction: flexibility is useful only when supported by the data and scientifically appropriate; additional coefficients may be weakly determined.
Wrong reasoning: “A common coefficient can be assumed because the same covariates were measured.” Correction: measuring a covariate on everyone does not establish that its association is the same for every transition.

The central limitation of the shared model is its equality constraint: it can conceal transition-specific differences if that constraint is not reasonable. The transition-specific model avoids that particular constraint but may be imprecise where transitions have few events or little covariate variation. Both specifications also rely on the underlying intensity model and observation assumptions; neither alone establishes a causal effect.

Key takeaway: “Shared” refers to the coefficient vector in the transition-indexed linear predictors. It does not mean shared baseline intensities, pooled risk sets, or identical transition processes.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

In a three-transition model, what remains transition-specific if all transitions use one shared coefficient vector?
For two covariates and four transitions, how many covariate coefficients are in a shared model and in a fully transition-specific model?
Why does the baseline intensity cancel from an event-time partial-likelihood contribution?
At an event on transition $1\to2$, which risk set is used in the partial-likelihood denominator?
Give one reason a transition-specific model may be difficult to estimate reliably.

Solutions. (1) The baseline intensities and risk sets remain transition-specific. (2) The shared model has 2 coefficients; the fully transition-specific model has $4\times2=8$. (3) At that event time, the baseline intensity is the same multiplicative factor for every subject in the risk set, so it cancels between numerator and denominator. (4) The risk set of observed subjects eligible for transition $1\to2$ at that time. (5) One transition may have few events or little covariate variation, leaving its separate coefficients weakly supported.