Worked Illness-Death Regression Analysis

Why Fit a Separate Regression for Each Transition?

In an illness-death model, a person may move from an unoccupied state to an ill state, move directly from unoccupied to death, or die after becoming ill. These are distinct processes: the people at risk of becoming ill are not the same as the people at risk of dying after illness. A worked regression analysis must therefore make the state-specific risk sets and interpretations explicit.

Here, state 0 means unoccupied, state 1 means ill, and state 2 means death; state 2 is absorbing. We fit a Cox proportional-hazards regression for each allowed transition: \(0\to1\), \(0\to2\), and \(1\to2\). Age is baseline age measured in decades, and treatment is an indicator coded 1 for treated and 0 for untreated. Thus an age coefficient compares people whose baseline ages differ by 10 years, and a treatment coefficient compares treated with untreated people.

For this analysis, follow-up time \(t\) is measured from study entry for all three transitions. Baseline age and treatment are fixed covariates. For \(0\to1\) and \(0\to2\), a person is at risk while still in state 0. For \(1\to2\), a person is at risk only after entering state 1 and until death or censoring. A person who becomes ill leaves the state-0 risk sets and enters the state-1 risk set.

Model, Risk Sets, and Assumptions

Let \(Y_i^{rs}(t)=1\) if person \(i\) is under observation and in state \(r\) immediately before time \(t\), and 0 otherwise. Let \(N_i^{rs}(t)\) count that person's observed \(r\to s\) transitions by time \(t\). The model for an allowed transition is

The Cox partial likelihood compares the covariates of the person who makes a transition with those of people in that transition's risk set at the same event time. It does not require a parametric form for \(h_{0,rs}(t)\). Its usual interpretation relies on a correctly specified proportional-hazards model for that transition, suitable covariate form, correctly recorded transition and censoring times, and censoring that is independent of the future transition process conditional on modeled covariates and relevant history. Repeated transitions from the same person are not treated as independent observations; the state history determines which transition risk sets that person enters.

A key assumption is that each transition-specific covariate effect is proportional over the chosen time scale. For example, the treatment hazard ratio for \(1\to2\) is assumed constant over follow-up among people currently in state 1. This does not require the baseline hazards to be the same across transitions. It also does not imply that the transition probabilities are proportional or that a hazard ratio is a risk ratio.

Why the Partial Likelihood Has This Form

Suppose, for one transition \(r\to s\), that event times are distinct. At an event time \(t_k\), condition on exactly one \(r\to s\) transition occurring among the people in the risk set \(R_{rs}(t_k)=\{i:Y_i^{rs}(t_k)=1\}\). Under the Cox model, the event rate for person \(i\) is proportional to \(h_{0,rs}(t_k)\exp(\beta_{rs}^{\mathsf T}x_i)\). The conditional probability that the event is person \(i_k\)'s is therefore

The common baseline hazard cancels. Multiplying these conditional contributions over event times for this transition gives its partial likelihood:

The three models use different risk sets and estimate separate coefficient vectors. If event times are tied, the distinct-time expression is not directly applicable; a specified ties method, such as Efron's approximation or the Breslow approximation, is used. Results should report which method was used when ties are present.

Worked Illness-Death Regression Results

Suppose a hypothetical study fits the three Cox models separately, using the covariates and coding defined above. The table reports coefficient estimates and model-based standard errors. Hazard ratios are exponentiated coefficients. For illustration, the confidence intervals and p-values below use the large-sample Wald method; they are approximate.

Diagnostics, Alternatives, and Limitations

Diagnostics should be examined separately for each transition, because a satisfactory fit on one edge does not validate the others. Check that the recorded state histories produce the intended risk sets and that event counts are adequate for the number of fitted terms. Inspect whether age has a plausible functional form, assess proportional hazards over the specified time scale (for example, with transition-specific Schoenfeld residuals), and investigate influential observations. If treatment or age effects vary over time, a constant-coefficient Cox model can give a misleading average summary. Sparse events, especially for \(1\to2\), can lead to unstable estimates and wide intervals.

The standard Cox model here uses baseline age. If the scientific question requires attained age as a changing covariate, the analysis must specify its time-varying form and ensure that values are defined appropriately as people move between states. The time scale must also be chosen deliberately; changing from time since entry to time since illness onset changes the risk-set comparisons and the meaning of baseline hazards. These decisions should be made before interpreting coefficients.

A transition-specific Cox analysis is one option among several. A model with a shared coefficient across transitions imposes a common-effect constraint; a stratified Cox model with transition-specific covariate interactions can instead allow edge-specific effects. Parametric proportional-hazards models specify baseline-hazard forms, while piecewise-exponential regression represents hazards as constant within chosen time intervals. Additive-hazards regression models absolute hazard differences rather than multiplicative hazard ratios. These alternatives answer related but not identical questions; their assumptions and interpretive scales should guide selection.

Separate Cox regressions do not, by themselves, produce state-occupation probabilities or the probability of a particular path by a given time. Those quantities require combining transition information under an appropriate multi-state probability calculation. Nor do edge-specific associations automatically identify causal effects. Results can also be sensitive to model misspecification, informative censoring, measurement error, and sparse transition events.