What Is the Regression Target?
In a multi-state model, a subject can occupy different states over time and may make several kinds of transition. A regression effect therefore needs a precise target: it describes how a covariate relates to the rate of a particular transition, conditional on the subject being in the transition’s origin state and on the stated conditioning information. It is not automatically an effect on the probability of being in a state or on the probability of experiencing an event by a given time.
Let \(X(t)\) denote the state occupied at time \(t\), and let \(g\) and \(h\) be distinct states for which a direct transition from \(g\) to \(h\) is allowed. Write \(\mathcal H(t-)\) for the observed process history immediately before \(t\), and let \(Z(t)\) be a covariate vector known just before \(t\). The transition intensity is a conditional instantaneous rate:
The intensity has units of inverse time. Informally, over a sufficiently short interval \(\Delta t\), the conditional probability of the specified transition is approximately \(\alpha_{gh}(t\mid\cdot)\Delta t\). This definition presumes a jump-process setting in which transition probabilities over a short interval are governed by finite rates and simultaneous jumps have probability of smaller order than \(\Delta t\).
A widely used specification is a multiplicative model,
where \(\alpha_{gh,0}(t)\) is the baseline intensity for transition \(g\to h\), and \(\beta_{gh}\) is the transition-specific coefficient vector. The baseline intensity is allowed to vary with time. For a one-unit increase in covariate \(Z_k\), holding the other covariates fixed, the model’s intensity ratio is \(\exp(\beta_{gh,k})\), provided the same baseline and conditioning apply to the comparison.
This form asserts a constant multiplicative covariate effect over time for that transition. It does not assert that the intensity itself is constant over time. If covariates are time-varying, \(Z(t)\) must be defined using information available just before \(t\), so that the model does not use future information to explain a transition that has already occurred.
Conditional Effects and State Dependence
The transition index matters. A covariate can have one effect on moving from state \(g\) to state \(h\), and a different effect on moving from \(g\) to another state \(k\). Even when two transitions share an origin, their intensities describe different conditional rates. A coefficient for \(g\to h\) is not a general “effect of the covariate on the process.”
The model also specifies what history is conditioned on. If the intensity depends on the current state, time, and current covariates—but not on earlier history once these are given—then it has the Markov form, conditional on those covariates. An intensity may instead depend on time since entry to a state, prior visits, or other aspects of \(\mathcal H(t-)\). That is a valid history-dependent regression target, but it is not a Markov model merely because it is written as an intensity regression. The conditioning information must match the process being modeled.
The regression target is mathematical; estimation additionally relies on suitable observation and sampling conditions. For example, if transitions are not observed continuously, the observation scheme must contain enough information to identify the parameters under the chosen model. The definition of \(\alpha_{gh}\) alone does not guarantee that a particular data set can estimate it reliably.
Worked Example: Interpreting an Illness-Transition Coefficient
Consider a hypothetical illness-death process with state 0 = alive without illness, state 1 = alive with illness, and state 2 = dead. Suppose the transition \(0\to1\) follows the multiplicative model and the coefficient for a binary treatment indicator \(Z\) is \(\beta_{01}=0.40\). Assume the model’s proportional effect is appropriate for this transition, and compare otherwise identical subjects at the same time and under the same conditioning history.
The treatment-to-no-treatment intensity ratio for \(0\to1\) is
Thus, while subjects are in state 0, the model estimates a 1.492-fold instantaneous rate of transition to state 1 for treated versus untreated subjects, at the same time and conditional on the model’s other covariates and history. Equivalently, the modeled rate is about 49.2% higher. This is a rate comparison, not a statement that treatment raises the probability of becoming ill by 49.2% over a follow-up period. The transition intensity is measured per unit time; the ratio is dimensionless.
This interpretation is conditional on being in state 0. It does not describe the treatment effect on death from state 1, on direct death from state 0 if that transition is allowed, or on the probability of occupying state 1 at a later time.
Why an Intensity Effect Is Not an Occupation Effect
An intensity describes one possible next transition at an instant. State occupation and event probabilities accumulate the consequences of all possible paths over time. In a finite-state, time-inhomogeneous Markov model, let \(p_j(t)=P\{X(t)=j\}\). For \(j\) to be occupied at time \(t\), probability flows into \(j\) from other states and flows out through transitions from \(j\). The forward equation is
The first sum is the incoming flow; the second is the outgoing flow. A change to one intensity can therefore affect occupation probabilities both directly and through the probability of reaching other states. This equation is a Markov-model result: if intensities depend on additional history, the state probabilities alone may not contain enough information to express these flows.
A related distinction occurs in competing risks. A cause-specific intensity is the conditional instantaneous rate of one event type among subjects who remain event-free. Its cumulative incidence by time \(t\) is the probability of experiencing that event type by \(t\), accounting for the fact that a competing event can happen first. The latter depends on the intensities of all competing event types, not just the intensity for the event of interest.
Worked Example: Same Cause-Specific Intensity, Different Event Probability
Suppose, hypothetically, that two event types can occur from an event-free state. In a reference group, their constant intensities are \(\lambda_1=0.10\) per year and \(\lambda_2=0.10\) per year. In a comparison group, suppose the first intensity remains \(0.10\) per year, while the second is \(0.40\) per year. Assume these constant competing intensities fully describe event occurrence, with no other exit from the event-free state.
The cause-specific intensity ratio for event 1 is \(0.10/0.10=1\). Yet the probability of event 1 by five years differs between groups. With constant competing intensities, the probability of remaining event-free to time \(t\) is \(\exp\{-(\lambda_1+\lambda_2)t\}\). Multiplying the event-free probability at time \(u\) by the event-1 intensity and integrating gives
For the reference group at \(t=5\) years,
For the comparison group,
Both calculations use intensities in events per year, and the resulting probabilities are dimensionless. The event-1 intensity is unchanged, but the event-1 cumulative incidence is lower in the comparison group because the higher competing-event intensity removes more subjects from the event-free state first. Therefore, an intensity ratio of one for event 1 does not imply equal probabilities of that event by five years.
Specifying and Checking the Target
Before interpreting a coefficient, make the model’s target explicit. State the origin and destination states, the time scale, the covariates, and the history on which the intensity is conditioned. Then state the functional form. In the multiplicative model above, the transition-specific coefficient represents a log intensity ratio per unit covariate difference, conditional on the model’s other terms. This interpretation requires the proportional effect assumption: the coefficient does not vary with time or with omitted effect modifiers in the stated model.
Several checks can reveal that the proposed regression target or its functional form is inadequate:
- Check the transition definition. Confirm that the modeled transition is possible in the state graph and that state coding distinguishes the intended origins and destinations.
- Check the conditioning history. If duration in the current state or prior history affects the rate, a model conditioning only on current state and calendar time may be too restrictive.
- Check proportionality and covariate form. A covariate effect that changes over time, or a nonlinear effect represented as linear, undermines the stated coefficient interpretation.
- Check data support. Sparse transitions or covariate groups with little overlap can make transition-specific effects unstable or poorly identified.
- Compare model-implied paths with observed patterns. Systematic disagreement in transition timing or state occupancy can signal misspecification, though matching occupation probabilities alone does not prove that each transition intensity is correct.
These checks are not interchangeable. A model may describe one transition well but fail for another; good overall state-occupation predictions can also conceal offsetting errors in transition-specific intensities. Conversely, a well-defined intensity coefficient does not by itself answer every prediction question.
Choosing the Right Effect Scale
| Target | Question answered | What it depends on |
|---|---|---|
| Transition intensity | How does a covariate relate to the instantaneous rate of a specified transition? | The origin state and stated time, covariates, and history |
| State-occupation probability | How likely is a subject to occupy a particular state at a specified time? | All relevant pathways and transition behavior over time |
| Cumulative incidence | What is the probability of experiencing a specified event by a time? | The event intensity and the competing ways of leaving the event-free state |
Transition-intensity regression is useful when the scientific question concerns mechanisms or conditional rates of particular transitions, and when an intensity-based model is suitable for the process. State-occupation probabilities and cumulative incidences are more direct targets for questions about prognosis or absolute event probabilities. They can be derived from a specified transition model under its assumptions, but they are not interchangeable with its coefficients.
The distinction is especially important when comparing covariate effects across transitions. A larger intensity ratio for one transition does not necessarily imply a larger change in an occupation probability or absolute event risk. Those quantities reflect the full set of transitions and the time available for paths to unfold. The appropriate target depends on the scientific question, not on which regression coefficient is easiest to report.
Common Mistake and Limitations
Wrong reasoning: “The treatment has a transition hazard ratio of 1.5, so it increases the probability of that event by 50%.” The conclusion confuses a conditional instantaneous rate ratio with a probability ratio. It ignores follow-up duration, other transitions, and changes in who remains eligible to make the transition.
Correction: Report the ratio as a 1.5-fold conditional intensity for the specified transition, with the origin state and conditioning information stated. If the question concerns probability by a given time, calculate or estimate that probability using an appropriate model for the full set of relevant transitions and state the assumptions.
A further limitation is that intensity regression describes conditional associations, not necessarily causal effects. A causal interpretation requires additional design and identification assumptions, such as adequate control of confounding and a meaningful intervention definition; the multiplicative intensity formula alone does not provide them. Model form, covariate measurement, observation of transitions, and extrapolation beyond the observed covariate or time range can also limit interpretation.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A coefficient of \(-0.223\) is estimated for a one-unit increase in a covariate on a specified transition. What is the intensity ratio, approximately, and what does it mean?
- Why can the cumulative incidence of one event change even if its cause-specific intensity is unchanged?
- In one sentence, distinguish a Markov-form intensity from an intensity that depends on earlier history.
- Does a transition-intensity ratio of 2 imply that twice as many subjects will occupy the destination state at a fixed time? Explain briefly.
Solutions. (1) The ratio is \(\exp(-0.223)\approx0.800\). Under the model, a one-unit increase corresponds to an estimated 0.800-fold instantaneous rate—about 20.0% lower—for that transition, conditional on the stated covariates, time, origin state, and history. (2) A competing transition can remove subjects before the event of interest occurs, changing the probability of that event even when its own conditional intensity is fixed. (3) A Markov-form intensity depends on the current state and specified present-time variables but not on earlier history once those are given; a history-dependent intensity still depends on earlier process information. (4) No. Occupation probability depends on all relevant transitions and accumulated paths over time, not just one transition’s instantaneous intensity ratio.