Multi-State Models

The foundational step where the disease process is decomposed into a finite set of mutually exclusive, clinically meaningful states. Unlike binary survival models that only recognize 'alive' and 'dead', multi-state models define transient states (e.g., healthy, local recurrence, distant metastasis) that a subject can enter and leave, and absorbing states (e.g., death) from which no further transitions occur.

• Irreversible models: States progress in one direction (e.g., healthy → diseased → dead)
• Reversible models: Allow recovery or improvement (e.g., active disease → remission)
• Competing risks are a special case with one transient state and multiple absorbing states

The instantaneous risk of moving from one state to another, denoted as α_{hj}(t), representing the hazard of a transition from state h to state j at time t. These intensities form the transition intensity matrix, the core mathematical object of the model.

• Estimated using Markov models (intensities depend only on current state) or semi-Markov models (intensities depend on time spent in current state)
• Covariates can be incorporated to model how patient characteristics affect each specific transition
• Allows clinicians to quantify not just if a patient will progress, but how they progress through intermediate states

The Aalen-Johansen estimator is the non-parametric generalization of the Kaplan-Meier estimator for multi-state models. It calculates the probability of occupying any state at time t, accounting for all possible paths a subject could have taken through the state network.

• Produces state occupation probabilities: P(X(t) = j), the chance of being in state j at time t
• Handles right-censoring and left-truncation naturally
• Provides a richer prognostic picture than a single survival curve, showing, for example, the probability of being alive with recurrence versus alive without recurrence

The standard Markov multi-state model assumes the future depends only on the current state, not on the history of how the subject arrived there. When this assumption is violated, semi-Markov models incorporate sojourn time dependence.

• Markov property: α_{hj}(t) depends only on t, not on entry time into state h
• Semi-Markov (clock-reset): Intensity depends on time since entering the current state, useful when risk of further progression changes the longer a patient remains in a state
• Clock-forward models: Intensity depends on time since study origin, appropriate when disease age matters more than state duration

Multi-state regression models allow covariates to have transition-specific effects, meaning a treatment might reduce the risk of recurrence but have no effect on mortality after recurrence. This granularity is lost in standard Cox models.

• Stratified Cox models fit separate baseline hazards for each transition type
• A single covariate can have different hazard ratios for different transitions (e.g., age may increase recurrence risk but not post-recurrence mortality)
• Enables identification of treatment effect heterogeneity across the disease pathway, informing stage-specific interventions

Multi-state models enable dynamic prediction, updating a patient's prognosis as they transition through states. A patient's predicted 5-year survival changes when they experience an intermediate event like recurrence.

• Transition probabilities can be combined to calculate the probability of any future pathway
• Landmarking at the time of an intermediate event provides updated, clinically relevant predictions
• Supports personalized surveillance schedules by identifying when a patient's risk profile shifts, triggering more frequent monitoring

A framework that accounts for events that preclude the primary event of interest. In multi-state modeling, competing risks represent absorbing states that prevent transition to the target state.

• Uses the Cumulative Incidence Function (CIF) rather than 1-Kaplan-Meier
• Essential when modeling death from other causes alongside disease progression
• Distinguishes between cause-specific hazards and subdistribution hazards

The mathematical backbone of many multi-state models, assuming that the future state depends only on the current state, not the history of how the patient arrived there.

• Defined by a transition intensity matrix specifying instantaneous risks between states
• Can be time-homogeneous (constant hazards) or time-inhomogeneous (hazards vary with time)
• Enables calculation of state occupation probabilities and sojourn times

An extension of Markov models where transition intensities depend on the duration spent in the current state rather than calendar time. Critical for modeling recovery or disease progression where waiting time matters.

• Captures duration-dependent transitions like hospital discharge probability increasing with length of stay
• Requires specifying clock-reset or clock-forward time scales
• Often fitted using phase-type distributions or piecewise constant hazards

The probability that a subject occupies a specific state at a given time, accounting for all possible transition paths. The primary estimand in multi-state analysis.

• Calculated via the Aalen-Johansen estimator for non-parametric settings
• Requires estimating the transition probability matrix from observed data
• Provides clinically interpretable metrics like 'probability of being alive and progression-free at 5 years'

The simplest and most widely used multi-state model with three states: Healthy → Disease → Death, plus the direct Healthy → Death transition. The canonical framework for disease progression modeling.

• Captures both incidence (Healthy → Disease) and mortality (Disease → Death)
• Allows estimation of prevalence as a function of time
• Extensible to include recovery (Disease → Healthy) for reversible conditions

A regression framework combining Cox proportional hazards with multi-state structures, allowing covariate effects on each transition intensity to be estimated simultaneously.

• Each transition receives its own baseline hazard and covariate effects
• Can incorporate time-varying covariates that change as patients move through states
• Stratified by transition type while sharing information across related transitions