What is an absorbing state?

A state with no outgoing transition in the specified model, such as permanent retirement within the modeled horizon.

Can a unit return to an earlier state?

Yes, if the allowed transition structure includes that return.

What is the difference between Markov and semi-Markov assumptions?

A semi-Markov formulation can allow the next transition to depend on time spent in the current state, beyond the current state alone.

AI Glossary / Statistics and probability / Advanced

Multistate model

Definition

A multistate model describes how a unit moves among a defined set of states over time. States might represent stages of a process, operating conditions or statuses in a longitudinal study. The model specifies which transitions are possible and how their probabilities or rates depend on time and other information. It extends a single-event perspective by representing intermediate states, competing transitions and sometimes returns to earlier states. Multistate models can be formulated in discrete or continuous time, and their assumptions should be matched to how the process is observed.

What it means

A unit's current state summarizes where it is in the modeled process. Some states allow further transitions; an absorbing state has no exit within the model. A Markov formulation assumes that, conditional on the current state and specified information, the future does not additionally depend on the earlier state history. Other formulations can depend on time already spent in a state or a richer history. The state definitions determine what the model can express, so merging distinct situations can conceal important differences in transition behavior.

How it works

Define mutually exclusive states, allowed transitions and the time scale. Record when units enter observation and when states or transitions are observed, accounting for censoring and incomplete observation. In discrete time, use transition probabilities for each interval. In continuous time, model transition intensities and derive probabilities over intervals; rates are not themselves probabilities. Include covariates and time dependence where justified. Estimate quantities such as transition probabilities, state-occupation probabilities or expected time in states. Assess fit, observation assumptions and sensitivity to the chosen state structure before interpreting the results.

Three practical examples

Equipment management: represent transitions among operating, repair and retired states.
Customer-process analysis: model movement among active, inactive and permanently closed statuses when those definitions are appropriate.
Workflow research: study intermediate stages rather than record only the final completion time. Each application requires a clear account of which transitions are reversible and which states are considered absorbing.

Think of it this way

Think of a map of rooms connected by doors. A multistate model specifies which rooms can be entered next and how movement unfolds over time. Some rooms have no exit in the map. The map also needs timing information: knowing that a door exists is different from knowing the chance of passing through it within the next hour.

Worked example

Consider an artificial monthly model with states operating, repair and retired. From operating, the next-month probabilities are 0.80, 0.15 and 0.05. From repair, they are 0.40, 0.50 and 0.10. Retired is absorbing, with probabilities 0, 0 and 1. Starting operating, the one-month state probabilities are (0.80,0.15,0.05). After two months, operating probability is 0.80 times 0.80 + 0.15 times 0.40 = 0.70. Repair probability is 0.80 times 0.15 + 0.15 times 0.50 = 0.195. Retired probability is 0.80 times 0.05 + 0.15 times 0.10 + 0.05 times 1 = 0.105. The probabilities sum to one. This original matrix-propagation example assumes unchanged monthly transition probabilities and the Markov property. It does not treat the listed probabilities as continuous-time transition rates.

Illustrative case study

In a fictional case, Fern Equipment tracks machines through operation, repair and retirement. The first model treats all repairs as one state, but analysts discover that transition behavior depends strongly on how long a machine has already been under repair. They evaluate a model that represents duration dependence rather than silently relying on a memoryless assumption. Records are observed only at scheduled inspections, so the team also accounts for uncertainty about transitions that may occur between visits.

Common misconceptions and mistakes

A multistate model is not automatically a Markov model, although Markov assumptions are common. An observed change between two inspections does not necessarily reveal every intermediate transition. A transition intensity can exceed one because it is a rate per unit time; it should not be read directly as a probability. State-occupation probability differs from the probability of ever entering a state. Absorbing status is a modeling assumption tied to the observation horizon and definition. Ignoring informative censoring or misclassified states can distort estimates even when the transition equations are correctly implemented.

Business use

Businesses can use multistate models to plan resources, understand process bottlenecks and quantify paths through a system. Start with state definitions that decision makers can interpret and data systems can measure reliably. Report the time horizon for every probability and distinguish transition rates from expected counts or occupancy. Evaluate whether policy changes alter the transition structure before reusing historical estimates. Examine uncertainty and alternative assumptions, especially when few transitions occur. A useful model makes the process visible while acknowledging what the observation schedule cannot reveal.

Check your understanding

A continuous-time transition intensity is 2 per year. Is the one-year transition probability automatically 2?

Answer: No. An intensity is a rate; a probability must be derived from the full transition model over the specified interval.

Related terms

Censoring
State transition
Markov model
semi-Markov model
transition intensity
absorbing state
state occupancy
competing risks

Learn next

Draw a state diagram and calculate a small discrete-time transition matrix. Then study continuous-time intensities and how irregular observation times affect estimation of a multistate process.

Sources

Christopher Jackson, Multi-State Models for Panel Data: The msm Package for R: https://www.jstatsoft.org/article/view/v038i08 Christopher Jackson, msm documentation, continuous-time Markov and hidden Markov models: https://chjackson.github.io/msm/reference/msm.html

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