Multi-State Models in Survival Analysis

Multi-state models offer a comprehensive framework for analysing time-to-event data by partitioning an individual’s progression through a sequence of discrete states. Unlike traditional survival methods that consider only a single transition to an absorbing endpoint, multi-state approaches accommodate intermediate events, recurrent episodes and competing pathways. The core elements comprise a defined state space, transition intensities or hazard functions governing movement between states, and transition probabilities that quantify the likelihood of occupying each state over time. Variants include Markov models, which assume memoryless transitions, semi-Markov formulations that incorporate sojourn times, and non-homogeneous processes accommodating time-varying transition rates. These models have been applied in diverse clinical and public health contexts, from chronic disease trajectories and treatment response to hospital epidemiology and surgical prognosis. By capturing the sequence and timing of multiple events, multi-state models yield richer insights into natural histories, enable dynamic prediction of future outcomes and support the evaluation of interventions in complex settings.

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Recent work has illustrated the utility of multi-state modelling in vascular surgery by re-analysing randomised trial data on peripheral arterial disease. Non-homogeneous Markov models were used to compare bypass surgery and stenting, revealing distinct transition probability curves for primary and secondary patency over four years. These findings highlight how multi-state methods can address questions regarding differential durability and inform personalised follow-up strategies.

In geriatric epidemiology, an illness-death model with Bayesian estimation has been applied to a large cohort of elderly patients following hip fracture. Using Weibull baseline hazards within a Cox framework, investigators estimated disease incidence rates and transition probabilities for refracture and death. The analysis demonstrated sex and age differences in transition risks, emphasising the added value of multi-state models over competing risks approaches by capturing non-terminal health transitions.

A methodological review has underscored the general accessibility of multi-state models beyond specialist circles. The report introduced key estimands such as state occupation probabilities, expected sojourn times and cumulative transition hazards. Case examples from oncology and cardiology illustrate how these models extend Kaplan–Meier and Cox approaches to elucidate complex disease pathways and support dynamic risk prediction in routine practice.

Multi-State Models in Survival Analysis publication trend

The graph below shows the total number of articles in multi-state models in survival analysis across all publications each year (not limited to Nature Index journals).

Technical terms

State: A distinct condition or phase in a process (for example, healthy, diseased, recovered, or dead).

Transition intensity (hazard): The instantaneous rate of moving from one state to another, often modelled by a hazard function.

Transition probability: The probability of being in a particular state at a given time, conditional on the initial state.

Markov property: An assumption that future transitions depend only on the current state and not on the past history or sojourn time.

Semi-Markov model: A formulation that relaxes the Markov property by allowing transition rates to depend on the time spent in the current state.

Absorbing state: A state from which no further transitions occur (for example, death).

References

1. Role of Multistate Models to Predict Patency, Limb Salvage, and Survival: New Concepts to Analyse Data in Peripheral Arterial Disease. European Journal of Vascular and Endovascular Surgery (2024).
2. Estimating disease incidence rates and transition probabilities in elderly patients using multi-state models: a case study in fragility fracture using a Bayesian approach. BMC Medical Research Methodology (2023).
3. The Utility of Multistate Models: A Flexible Framework for Time-to-Event Data. Current Epidemiology Reports (2022).