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Transition-Specific Cox Regression

Learn how to formulate and interpret a Cox regression for a single transition, including who is at risk, what the hazard ratio compares, and what assumptions to check.

Advanced 10 min read

What You'll Learn

  • Define the risk indicator for a specified transition and distinguish it from being under observation.
  • Write a transition-specific Cox intensity with a baseline hazard and covariate linear predictor.
  • Interpret a transition-specific hazard ratio for a covariate contrast.
  • Derive the partial-likelihood contribution for an untied event in a transition-specific risk set.
  • Check proportional-hazards, censoring, and risk-set assumptions for the modeled transition.
  • Distinguish a transition hazard ratio from an effect on cumulative incidence.

Why Model One Transition at a Time?

A multi-state process can move between several states, but a regression question can be specific: among individuals currently in state \(h\), how does a covariate relate to the instantaneous rate of moving to state \(k\)? Transition-specific Cox regression answers this question by modeling the hazard for the single transition \(h\to k\). The transition and its eligible risk set must be defined before fitting the regression: someone in another state is not at risk for this transition, even if they remain under observation.

The model describes a conditional transition rate, not the probability of being in a state at a future time. In particular, it does not by itself describe how a covariate changes the chance of eventually reaching \(k\), since other transitions may alter who remains able to make the transition of interest.

Model, Risk Indicator, and Assumptions

Definition: For subject \(i\), let \(N_i^{hk}(t)\) count observed transitions from state \(h\) to state \(k\) by time \(t\). Let \(Y_i^{hk}(t)\) be 1 when subject \(i\) is under observation and currently eligible to make the transition \(h\to k\) just before \(t\), and 0 otherwise. A transition-specific Cox model specifies the conditional intensity $$ E\!\left[dN_i^{hk}(t)\mid\mathcal F_{t-}\right] = Y_i^{hk}(t)\lambda_{0,hk}(t) \exp\!\left\{\beta_{hk}^{\mathsf T}Z_i(t)\right\}dt+o(dt). $$ Here \(\mathcal F_{t-}\) is the observed history immediately before \(t\), \(Z_i(t)\) is the covariate vector, \(\beta_{hk}\) is its coefficient vector for this transition, and \(\lambda_{0,hk}(t)\) is the unspecified baseline hazard for \(h\to k\).

When \(Y_i^{hk}(t)=1\), the hazard for the transition is \(\lambda_{0,hk}(t)\exp\{\beta_{hk}^{\mathsf T}Z_i(t)\}\). When the indicator is zero, the subject contributes no transition-specific risk at that instant. The risk indicator is determined by the state path and observation status: it is not simply an indicator that the subject is alive or enrolled. If a subject leaves \(h\) through a different transition, that subject leaves this transition's risk set at that time. If the process can later return to \(h\), the subject may again become eligible, depending on the model and the observation scheme.

The model is written on a chosen time scale, such as time since study entry or time since entering state \(h\). The baseline hazard is defined on that same scale. A covariate may vary over time, but \(Z_i(t)\) must be known just before \(t\) to serve as a predictable covariate in the intensity model. The linear predictor is \(\eta_{i,hk}(t)=\beta_{hk}^{\mathsf T}Z_i(t)\); it summarizes the covariate contribution on the log-hazard scale.

Interpretation and estimation require several conditions. Subjects should be independent, or the variance calculation must account for dependence. The risk indicator and transition times must be recorded accurately, and loss to follow-up must be independent of the future target transition conditional on the modeled information, or handled with an appropriate method. The proportional-hazards assumption is that each coefficient is constant over the chosen time scale: covariate contrasts multiply the target-transition hazard by a constant, rather than a time-varying factor. The baseline hazard can vary freely with time. No assumption that the whole multi-state process is Markov is required simply to specify this intensity; the conditioning history and time scale must, however, be appropriate for the scientific model.

Worked Example: Writing the Model for Illness-to-Death

Suppose a hypothetical study follows patients who may move from illness state \(h\) to death state \(k\). Let \(A_i=1\) indicate treatment and \(A_i=0\) indicate control. Let \(x_i=(\text{age}_i-50)/10\), so one unit is ten years of age. On the study-time scale, consider the model

$$ \lambda_{i,hk}(t\mid Y_i^{hk}(t)=1) = \lambda_{0,hk}(t)\exp\{-0.50A_i+0.18x_i\}. $$

For this model, \(\eta_{i,hk}(t)=-0.50A_i+0.18x_i\). Holding age fixed, the treatment-versus-control hazard ratio for illness-to-death is \(\exp(-0.50)=0.6065\). Thus, at a given time, among patients eligible for this transition and with the same age, the modeled instantaneous illness-to-death hazard in the treatment group is 0.6065 times the hazard in the control group. This is a 39.35% lower transition hazard, not a 39.35% lower probability of death by a specified follow-up time.

For a ten-year increase in age at fixed treatment, the hazard ratio is \(\exp(0.18)=1.197\), or about a 19.7% higher transition hazard. A twenty-year increase corresponds to \(\exp(2(0.18))=\exp(0.36)=1.433\). These are conditional comparisons for the illness-to-death transition, not comparisons of all-cause mortality or state occupancy.

The study description makes the source state and target transition explicit, but it does not establish that proportional hazards or conditional independent loss to follow-up hold. Those assumptions would need to be assessed using the observed transition-specific data. The numerical coefficients here are hypothetical, not estimates from a real study.

Transition-Specific Hazard Ratios

Definition: For two covariate vectors \(z_1\) and \(z_0\), with both individuals eligible for \(h\to k\) at time \(t\), the model-implied transition-specific hazard ratio is $$ \frac{\lambda_{0,hk}(t)\exp(\beta_{hk}^{\mathsf T}z_1)} {\lambda_{0,hk}(t)\exp(\beta_{hk}^{\mathsf T}z_0)} = \exp\!\left\{\beta_{hk}^{\mathsf T}(z_1-z_0)\right\}. $$

The baseline hazard cancels in this comparison, which is why the Cox model can estimate regression coefficients without specifying a parametric form for that baseline. For a one-unit increase in a continuous covariate \(Z_j\), with other covariates held fixed, the hazard ratio is \(\exp(\beta_{hk,j})\). For a change of \(c\) units, it is \(\exp(c\beta_{hk,j})\). A confidence interval for the coefficient can be exponentiated to give an interval for the hazard ratio; the interval is about the transition-specific relative hazard, not a probability.

The subscript \(hk\) matters. It identifies both the event being modeled and the coefficient vector. A coefficient for \(h\to k\) is not automatically the coefficient for another transition, even if the same covariate appears in both models. A hazard ratio compares instantaneous rates among individuals who are currently in the relevant risk set. Differences in who remains in that risk set over time can make the corresponding cumulative risks behave differently.

How the Cox Partial Likelihood Uses the Risk Set

For a single transition, consider distinct event times \(t_j\) at which an \(h\to k\) event occurs. Let \(R_{hk}(t_j)\) be the subjects with \(Y_i^{hk}(t_j)=1\), and let \(i_j\) be the subject making the target transition at \(t_j\). If exactly one target event occurs at that time, the partial-likelihood contribution is

$$ \frac{\exp\{\beta_{hk}^{\mathsf T}Z_{i_j}(t_j)\}} {\displaystyle\sum_{\ell\in R_{hk}(t_j)} \exp\{\beta_{hk}^{\mathsf T}Z_\ell(t_j)\}}. $$

The product of these contributions over target-transition event times is the partial likelihood. It contains no baseline-hazard term. This expression assumes correct risk sets and, as written, untied target-event times; tied events require a tie-handling method. Subjects who make a different transition out of \(h\) are no longer in \(R_{hk}\) after that exit. Their exit is part of the multi-state path, not a target \(h\to k\) event.

The cancellation behind the contribution is direct. Conditional on one target event occurring at \(t_j\) among the eligible subjects, the model assigns subject \(i\) a relative event rate proportional to \(\lambda_{0,hk}(t_j)\exp\{\beta_{hk}^{\mathsf T}Z_i(t_j)\}\). Dividing that rate by the sum over the risk set gives

$$ \frac{\lambda_{0,hk}(t_j)\exp\{\beta_{hk}^{\mathsf T}Z_i(t_j)\}} {\sum_{\ell\in R_{hk}(t_j)} \lambda_{0,hk}(t_j)\exp\{\beta_{hk}^{\mathsf T}Z_\ell(t_j)\}} = \frac{\exp\{\beta_{hk}^{\mathsf T}Z_i(t_j)\}} {\sum_{\ell\in R_{hk}(t_j)} \exp\{\beta_{hk}^{\mathsf T}Z_\ell(t_j)\}}. $$

The common baseline hazard cancels, leaving a likelihood for the regression coefficients. This also shows why an incorrect risk set is consequential: it changes the denominator and therefore changes the comparisons that estimate the coefficients.

Worked Example: A Target-Transition Risk Set

At one hypothetical event time, four patients are eligible for transition \(h\to k\): patients 2, 3, 5, and 6. Patients 2 and 5 are treated (\(A=1\)); patients 3 and 6 are controls (\(A=0\)). Patient 2 makes the target transition. Patients 1 and 4 are not in state \(h\) at that time, so they are not in the risk set. Suppose the fitted treatment coefficient is \(\hat\beta_{hk}=\log(0.70)\). There are no tied target events at this time.

The risk-set contribution for patient 2 is

$$ \frac{\exp\{\hat\beta_{hk}(1)\}} {\exp\{\hat\beta_{hk}(1)\}+\exp\{\hat\beta_{hk}(0)\} +\exp\{\hat\beta_{hk}(1)\}+\exp\{\hat\beta_{hk}(0)\}} = \frac{0.70}{0.70+1+0.70+1} = \frac{0.70}{3.40} = 0.2059. $$

This is the model-based conditional contribution associated with observing patient 2, rather than another eligible patient, as the target-transition event at that time. Including patients 1 and 4 merely because they were observed would be incorrect: neither was eligible for \(h\to k\). The risk set is determined by the transition and the state history, not by the total sample size.

Baseline Hazard and an Instantaneous Rate

The baseline hazard \(\lambda_{0,hk}(t)\) is the hazard for the reference covariate vector \(Z=0\), among individuals at risk for the specified transition. Its numerical meaning depends on the covariate coding and time units. The Cox model leaves its shape unspecified; it does not assume a constant hazard. The regression coefficients describe relative hazards, while estimating an absolute transition rate or a future state probability requires additional quantities and calculations.

Worked Example: Combining the Baseline With a Linear Predictor

In a hypothetical model, suppose the baseline hazard for \(h\to k\) at a particular time is \(0.040\) per month. Let \(A=1\) indicate treatment and \(x=(\text{age}-50)/10\), and suppose \(\hat\beta_A=-0.50\) and \(\hat\beta_x=0.08\). For a treated patient aged 60 who is eligible for the transition, \(A=1\) and \(x=1\), so the linear predictor is \(-0.50+0.08=-0.42\). The hazard is

$$ 0.040\exp(-0.42) = 0.040(0.6570) = 0.02628\text{ per month}. $$

For a control patient aged 50 in the same risk set, the linear predictor is zero and the hazard is \(0.040\) per month. Their hazard ratio is \(0.02628/0.040=0.6570\), agreeing with \(\exp(-0.42)\). These values are instantaneous transition rates at the stated time; \(0.02628\) per month is not a 2.628% monthly transition probability. The example assumes the stated baseline and coefficients apply at that time and uses the same time scale and covariate definitions for both patients.

Diagnostics, Alternatives, and Limitations

Diagnostics should be specific to the modeled transition. First check whether the recorded state histories produce plausible risk-set membership and whether target events occur in the stated risk sets. Then assess whether covariate effects are compatible with proportional hazards. For example, transition-specific residual patterns over time or estimated effects that vary systematically over time may signal nonproportionality. A time-varying coefficient or a different model may be needed if the constant hazard-ratio assumption is untenable. Sparse events, especially within covariate patterns, can also make estimates unstable or imprecise.

Check the observation process as well. Informative loss to follow-up can distort estimation if the chance of remaining observed depends on future transition experience after conditioning on modeled information. A competing transition is different: it moves a subject out of the risk set for \(h\to k\), because the subject is no longer in the source state. It does not need to be treated as if it were the target event. This risk-set construction defines a cause- or transition-specific hazard, but does not make the competing event irrelevant to future probabilities.

A parametric transition model is an alternative when a scientifically credible distributional form for the baseline hazard is appropriate; it can provide a more structured description of absolute rates but relies on that form being adequate. The Cox model is less restrictive about the baseline shape, but still assumes proportional covariate effects for the specified transition. A separate Cox model for one transition also does not test whether its coefficients equal coefficients for another transition. Its conclusions are limited to the modeled transition, time scale, covariates, and eligible population.

Common pitfall: “The treatment hazard ratio is 0.70, so treatment reduces the probability of the transition by 30%.” This is not justified. The ratio concerns instantaneous transition hazards among individuals currently eligible for that transition, conditional on the model covariates. It does not directly give a risk ratio or a difference in cumulative probability, especially when other transitions can remove individuals from the source state.

Check Your Understanding

Use the definitions and model for one specified transition to answer each question. Solutions follow.

  1. For the transition \(h\to k\), what happens to a subject's risk indicator immediately after the subject makes a different transition out of state \(h\)?
  2. A continuous covariate has coefficient \(0.12\) per five units in a Cox model for \(h\to k\). What is the hazard ratio for a ten-unit increase, holding other covariates fixed?
  3. Why does the baseline hazard not appear in an untied event's partial-likelihood contribution?
  4. State one reason why a transition-specific hazard ratio need not equal the corresponding ratio of cumulative transition probabilities.

Solutions. (1) The subject leaves the \(h\to k\) risk set because they are no longer in the source state \(h\), so \(Y_i^{hk}(t)\) becomes 0 unless the subject later returns to \(h\) and again becomes eligible. (2) Ten units are two five-unit increases, so the hazard ratio is \(\exp(2(0.12))=\exp(0.24)=1.271\). (3) At a target event time, the same baseline hazard multiplies the rate for every subject in the risk set; it cancels when the event subject's rate is divided by the sum of those rates. (4) Other transitions can remove people from the source-state risk set at different rates, changing who remains eligible over time; a hazard ratio is an instantaneous conditional comparison, not a cumulative-probability ratio.

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