Representing Disease Onset and Death
A regression coefficient for a multi-state process describes a transition hazard, not a single overall measure of disease risk. The first step is therefore to name the states and transitions. Consider a hypothetical cohort followed from a disease-free state through disease onset and, eventually, death. A person may also die before developing the disease.
Let state 0 denote alive and disease-free, state 1 alive after disease onset, and state 2 dead. State 2 is absorbing: a person who has died cannot leave it. The allowed transitions are \(0\to1\) (disease onset), \(0\to2\) (death before recorded disease), and \(1\to2\) (death after disease onset). The \(1\to2\) transition is the disease-related death transition in this model, meaning death after entering the disease state; the state structure alone does not establish that disease caused the death.
The three transitions have different origins and therefore different risk sets. Only people still alive and disease-free can experience \(0\to1\) or \(0\to2\). Only people who have developed the disease and remain alive can experience \(1\to2\). A death before disease onset removes a person from the risk set for disease onset; it is not treated as ordinary censoring for that transition.
Transition-Specific Regression Models
A common choice is a separate proportional-hazards model for each allowed transition. Let \(Z_i(t)\) be the covariate vector for individual \(i\), measured at or before time \(t\), and let \(\beta_{gh}\) be the coefficient vector for transition \(g\to h\). Then
Here \(\lambda_{0,gh}(t)\) is the transition-specific baseline hazard, and the coefficients are allowed to differ by transition. For example, a covariate might be associated with faster disease onset but not with death before disease, or it might have a different association with death after disease onset. A model that forces one coefficient to apply to all three transitions imposes an additional equality restriction; it is not the default implication of the three-state structure.
For a one-unit increase in covariate \(Z_j\), holding the other modeled covariates fixed, \(\exp(\beta_{gh,j})\) is the hazard ratio for transition \(g\to h\) at a given time, comparing otherwise equal covariate values. This interpretation requires proportional hazards: the hazard ratio does not vary with time. The coefficient is not a ratio of cumulative probabilities of being in a state.
The model requires correctly defined states and event times, a covariate specification appropriate to the scientific question, and adequate data at the covariate values being compared. Standard transition-specific partial-likelihood inference also assumes censoring is independent of the future transition process conditional on the modeled information. If the hazard depends on more history than current state, time, and the included covariates, that history must be represented appropriately; otherwise the stated model is misspecified. A Markov interpretation is appropriate only when the modeled transition rates depend on the current state and included covariate information, rather than unmodeled earlier history.
Worked Example: Defining the Three Risk Sets
Suppose six hypothetical participants, A through F, begin follow-up in state 0. A enters state 1 at time 2; B dies from state 0 at time 3; C enters state 1 at time 4; D is censored at time 4.5 while still in state 0; and A dies at time 7. Assume all event times are measured in years.
Immediately before time 2, everyone is alive and disease-free. Thus the risk set for the \(0\to1\) event at time 2 is \(\{A,B,C,D,E,F\}\). A moves out of state 0 at time 2, so A is no longer at risk for either \(0\to1\) or \(0\to2\) after that time.
Immediately before time 3, B, C, D, E, and F remain in state 0. They form the risk set for B's \(0\to2\) event. B then leaves state 0 and cannot later experience disease onset in this observed process. D remains eligible for a disease-onset or death event through time 4.5, then leaves follow-up by censoring.
At time 7, A and C are in state 1 and alive, so the risk set for \(1\to2\) is \(\{A,C\}\), assuming C remains alive and observed through time 7. A's death is an event for this risk set. E and F are not included: they are still in state 0, not state 1.
The transitions have distinct risk sets because they have distinct origin states. Treating all six participants as at risk for every transition would incorrectly include people who are in the wrong state or no longer observed.
Estimation and Coefficient Interpretation
For a Cox model on a particular transition, each event contributes a conditional probability: among individuals in that transition's risk set at the event time, the contribution is the modeled relative hazard of the person who experienced the event divided by the sum of relative hazards in that risk set. For an event \(i\) on transition \(g\to h\) at time \(t\), with no tied event at that time, the contribution is
where \(R_{gh}(t)\) is the set of individuals in origin state \(g\), observed and at risk immediately before \(t\). Multiplying the contributions over the relevant events gives the transition-specific partial likelihood. The baseline hazard cancels from these conditional comparisons. Tied event times require a specified tie-handling method; the simple expression above is for a single event at that time.
Worked Example: Interpreting Fitted Transition Coefficients
Suppose a hypothetical analysis uses age in 10-year units and reports the following transition-specific estimates. The standard errors are for the log hazard-ratio coefficients.
| Transition | Coefficient estimate | Standard error |
|---|---|---|
| \(0\to1\) | 0.50 | 0.20 |
| \(0\to2\) | -0.20 | 0.30 |
| \(1\to2\) | 0.70 | 0.25 |
For disease onset, the estimated hazard ratio per 10-year increase in age is \( \exp(0.50)=1.649 \), or about 1.65. Under the model, among people currently alive and disease-free with the same values of the other modeled covariates, a 10-year age increase is associated with an estimated 65% higher instantaneous disease-onset hazard.
A 95% Wald confidence interval on the log scale is \(0.50\pm1.96(0.20)\), giving endpoints \(0.108\) and \(0.892\). Exponentiating gives \( \big(\exp(0.108),\exp(0.892)\big)=(1.114,2.440) \). Thus the interval estimates the hazard ratio per 10 years of age for \(0\to1\); under repeated sampling and the model assumptions, the procedure has 95% coverage. This is not a 95% probability statement about the fixed parameter after observing this interval.
For death before disease, the fitted hazard ratio is \( \exp(-0.20)=0.819 \) per 10 years. For death after disease onset, it is \( \exp(0.70)=2.014 \) per 10 years. These are separate conditional transition-hazard comparisons. The different estimates do not imply that age has a particular effect on the overall probability of death by a given time, because movement between states also affects that probability.
The interpretation assumes the covariate is coded in 10-year units, the model has proportional hazards for each transition, and the estimates refer to adequately represented covariate values. In particular, the \(1\to2\) coefficient is informed by people who entered state 1, not by all cohort members.
Worked Example: A Partial-Likelihood Contribution
At a hypothetical disease-onset event time, suppose four people are in the \(0\to1\) risk set. One has an indicator \(Z=1\) and experiences disease onset; the other three have \(Z=0\). Let the fitted coefficient be \(\beta_{01}=\log(2)\). Assume there is one event at this time and no other covariates.
The event subject's relative hazard is \(\exp\{\log(2)\times1\}=2\). Each of the other three relative hazards is \(\exp\{\log(2)\times0\}=1\). Therefore the partial-likelihood contribution is
At these coefficient values, the model assigns a 0.400 conditional probability to this particular person being the one with the event, given one event in this risk set. The contribution is not the person's probability of developing disease over follow-up. Other event times contribute their own terms, using their own risk sets, to estimate the transition coefficient.
Justification, Diagnostics, and Alternatives
The risk-set construction and coefficient interpretation follow from the transition-specific hazard model. For two individuals \(i\) and \(j\) both at risk for the same transition at time \(t\), the model implies
The baseline hazard cancels, so the conditional relative hazard depends on the covariate contrast through the transition-specific coefficient. This cancellation also explains why each transition's likelihood contribution uses only people at risk from its origin state: a person in another state is not subject to that transition hazard at that instant.
Diagnostics should respect the transition structure. Check that state histories are chronologically possible, event types match the state immediately before each event, and each transition has enough events across the covariate range to support its estimates. Examine proportional-hazards diagnostics separately by transition; a satisfactory pattern for \(0\to1\) does not establish proportional hazards for \(1\to2\). Investigate influential observations and sparse risk sets, especially for the transition out of a less common state. Also verify that covariate updates occur at the correct times and that censoring assumptions are plausible given the recorded information.
The model is not a direct model for state-occupation probabilities. Transition-specific hazard ratios describe instantaneous rates among those currently in the relevant origin state; cumulative probabilities require combining the transitions over time. Separate transition-specific models are more flexible than a shared-coefficient model, while a shared effect can be more precise if equality across transitions is scientifically justified. Additive-hazards regression instead models covariate effects as hazard differences, and a piecewise-exponential model represents hazards as constant within specified time intervals. These approaches answer related but distinct modeling questions; none removes the need to define the origin-specific risk sets correctly.
A common mistake is to interpret an estimated \(1\to2\) hazard ratio as the ratio of death probabilities for all people who began follow-up. That reasoning ignores that only those who first enter state 1 can experience \(1\to2\), and that people may differ in whether and when they enter it. The correction is to state the comparison conditionally: it is a relative instantaneous death hazard among people currently alive in state 1, conditional on the modeled covariates. It is neither an overall mortality risk ratio nor evidence by itself that the disease caused death.
Check Your Understanding
Try each question, then compare with the brief solutions below.
- At a given time, who belongs to the risk set for \(0\to2\), and who belongs to the risk set for \(1\to2\)?
- A fitted coefficient for a binary covariate in the \(0\to1\) model is \(0.40\). Calculate and interpret the hazard ratio.
- Why is a person who died from state 0 not subsequently included in the \(0\to1\) risk set?
- What does a \(1\to2\) hazard ratio not tell you about overall mortality?
Solutions. (1) The \(0\to2\) risk set contains observed people alive and disease-free immediately before the time; the \(1\to2\) risk set contains observed people alive after disease onset. (2) \( \exp(0.40)=1.492 \), so the covariate-positive group has an estimated 1.49 times the instantaneous disease-onset hazard of the covariate-negative group, conditional on other modeled covariates and under proportional hazards. (3) Death removes the person from state 0 and is an absorbing outcome, so later disease onset is impossible in the observed process. (4) It does not give an overall mortality risk ratio for the original cohort; it compares instantaneous death hazards among those currently in state 1.