Multi-State Models

How long does it take for someone who was unemployed to take a temporary job and then move into a permanent one? How do people who get a permanent job right away differ from those who get there by way of temporary work? Can we also analyze the process of losing a job again after being employed?

The method that analyzes how units move between several states over time and how long it takes to make each transition is the multi-state model. Instead of checking only whether someone got a job once, it handles changes among distinct states such as unemployed, temporary employment and permanent employment.

Standard survival analysis usually focuses on the time until a single event. A multi-state model, by contrast, can analyze each transition separately: from unemployment to temporary work, from temporary to permanent work, and back to unemployment after being employed. This lets you look not only at transition hazards but also at the probability of being in each state at a given time and the sojourn time in each state.

Multi-state models are useful for data in which states change several times, such as employment histories, disease progression in patients, hospital admissions and discharges, and the stages of customers' service use. In this article we use a job-seeker example to look at defining states and transitions, setting up the data, interpreting transition hazards and state occupation probabilities, how the model differs from other survival analyses, and what to watch for when reporting results.

A multi-state model tracks states that change over time and analyzes from which state to which state, and when, units move. States are defined so that they do not overlap, such as 'unemployed', 'temporary' and 'permanent'. One person may move several times during the study period.

A transition is defined as a combination of a particular starting state and a destination state. For example, unemployed → temporary and temporary → permanent are different transitions, so the effects of explanatory variables can also be estimated separately.

Analyzing the time from unemployment to a first job tells you how fast people get hired. But whether that job is temporary or permanent, whether someone moved from a temporary job to a permanent one, or whether they became unemployed again is hard to express with a single event time.

A multi-state model analyzes the speed of each transition and the distribution across states separately. For example, job training experience may be unrelated to first employment yet related to how quickly people move from temporary to permanent jobs.

This example uses three states: unemployed (U), temporary (T) and permanent (P). Which of U→T, U→P, T→P, T→U, P→T and P→U to allow is decided by how the actual data were observed and the purpose of the study. An arrow being drawn does not necessarily mean every transition can be estimated.

Suppose 300 job seekers were followed for 24 months and their employment status was checked every month. At the first survey everyone is unemployed, and participation in job training, education, age and so on are also recorded. What follows is a hypothetical example for learning purposes.

For example, suppose A001 took a temporary job at 3 months, moved to a permanent job at 9 months and stayed in permanent work through 24 months. This one person's record contains two transitions, U→T and T→P.

The table above is a simple record format for understanding the concept. When entering data into actual software, you may expand the data to match the risk set for each transition or define a separate transition ID. If you observed only the state at each monthly survey rather than the exact time of transition, you may need an approach that reflects the fact that the transition happened somewhere within the interval.

A transition hazard is the instantaneous rate of moving to a given destination state among people currently in the starting state. The transition hazards for unemployed → temporary and unemployed → permanent are estimated separately, even though both start from the same unemployed state.

For example, if the U→P transition hazard ratio (HR) for job training participants is 1.40, then under the conditions and assumptions of the model, compared with non-participants in the same unemployed state, their instantaneous hazard of moving to a permanent job is 40% higher. It does not mean the probability of getting hired is exactly 40% higher.

In the example above, the 24-month values are shown as 25% unemployed, 30% temporary and 45% permanent. These are the probabilities of being in each state at that time, which differ from the share of people who were ever in a permanent job during the 24 months. When there are reverse transitions such as losing a job again, the gap between the two can be large.

Sojourn time is the time from entering a state until leaving it. Even if the sojourn time in unemployment gets shorter, the quality of employment can be judged differently depending on whether the next state is temporary or permanent.

A transition probability is the probability of being in the destination state after a certain amount of time, given the current state. Because a person may have passed through another state and come back in between, do not confuse the transition probability between states over an interval with the number of direct transitions in that interval.

Suppose we estimated the effect of job training in hypothetical transition-specific Cox models. Because the effects on unemployed → permanent and temporary → permanent may differ, summarizing every transition with a single hazard ratio loses information.

The figures in the table are hypothetical values made up to illustrate how results are reported; they were not computed from the data on 300 people. A transition hazard ratio greater than 1 means the instantaneous risk of that transition is higher, but whether that is a good or bad outcome depends on the direction of the transition.

Observation interval and exact transition times

With a monthly survey, you may not know the exact date a state changed, or you may miss states that changed several times within a month. Treating such data as if every transition time were observed exactly can introduce bias.

The Markov assumption and sojourn time

A basic Markov model uses a structure in which future transitions are explained by the current state (and the time and covariates in the model) and do not further depend on the earlier path. If the length of time spent unemployed itself affects the chance of getting a job, a semi-Markov approach that reflects sojourn time, among others, may be more suitable.

Risk sets and number of transitions

A given transition is possible only for people in its starting state. If there are too few events for a transition, estimates become unstable, and when the same person makes repeated transitions, you may need to account for within-person correlation.

Censoring and time-varying covariates

Distinguish dropout, end of observation, the start of job training and so on. Treating time-varying information such as job training participation as if people had participated from the start of the study can introduce bias.

Multi-state models can encompass competing risks analysis, but they are especially useful when re-entry into or return between states matters. Not every problem needs the most complex model; choose the analysis based on the research question.

A report should include the people followed and the observation period, the starting state, the possible transitions, the actual transition frequencies, how censoring was handled, the time scale, the model and assumptions used, and hazard ratios with confidence intervals for each transition. If you present state occupation probabilities, state the estimation method and the time points.

In actual research, do not use this sentence as is; write it to reflect the number of events, estimates, model diagnostics and state occupation probabilities from your own data.

In the job-seeker example, checking only whether someone got a first job makes it hard to explain the difference between people who moved from unemployment into temporary work and those who went straight into permanent jobs. By distinguishing starting and destination states and looking at the hazard of each transition and the state occupation probabilities at particular times, a multi-state model helps you understand in more detail how employment status changes.

Before starting the analysis, however, you need to define states consistently and specify the possible transitions clearly. Also check whether a long observation interval makes transition times hard to know, whether the same person changes state repeatedly, and whether past paths and sojourn time matter in addition to the current state. Only by considering these conditions can you interpret transition hazards and probabilities in a way that fits the research question.