Integrated Problems in Interval-Censored Multi-State Data

From Visit Records to Inference

When states are observed only at scheduled visits, the recorded data usually identify a participant’s state at particular times, not the exact time or path of every transition between visits. A participant found in state 1 at a visit after being in state 0 at the previous visit may have changed state at any time in that interval. A participant found in state 2 may have passed through one or more intermediate states. Inference must therefore use probabilities of the observed records, rather than treating visit times as event times.

An integrated analysis has a sequence of decisions: specify a transition model and its assumptions, translate each observed record into a probability under that model, combine those probabilities into a likelihood, estimate the model parameters, and then report uncertainty for quantities of interest. The introductory material on continuous-time Markov likelihoods for panel data provides the basic panel-likelihood setup; here the emphasis is on carrying the pieces through complete inference problems.

Definition: Let $X(t)$ denote the state occupied at time $t$, and let $P_{ij}(s,t)=P\{X(t)=j\mid X(s)=i\}$ be the transition probability from state $i$ at time $s$ to state $j$ at time $t$. For a participant observed in states $x_0,\ldots,x_m$ at times $t_0<\cdots<t_m$, a time-homogeneous Markov model gives the panel-record likelihood contribution $$ L=\prod_{r=1}^{m}P_{x_{r-1},x_r}(t_{r-1},t_r). $$ This contribution is conditional on the observed starting state. Independent participants contribute multiplicatively to the likelihood.

The formula depends on a correctly specified state process and observation scheme. In the examples below, observed states are assumed to be measured without error, participants are independent, and the visit schedule is treated as fixed and non-informative about the unobserved state path. A continuous-time Markov model additionally assumes that, conditional on the current state, future transition behavior does not depend on the earlier path, and that the transition intensities are constant over time. These assumptions are substantive: if risks change with time since entering a state, or with calendar time, a time-homogeneous model may misrepresent the data.

Worked Example: Estimating an Interval-Censored Exponential Event Rate

In a hypothetical study, 20 participants begin in event-free state 0 and are assessed at month 2. Eight are found to have entered event state 1 by that visit; 12 remain in state 0. The exact event times are not observed. Assume independent participants and a constant transition intensity $\lambda$ per month from state 0 to absorbing state 1. The observation at month 2 is complete, and the exponential model is appropriate.

Under this model, the probability of still being in state 0 at month 2 is $e^{-2\lambda}$, and the probability of having entered state 1 by then is $1-e^{-2\lambda}$. The likelihood for the observed counts is therefore

L(\lambda)=(1-e^{-2\lambda})^8(e^{-2\lambda})^{12}.

Taking logs and differentiating gives

\ell(\lambda)=8\log(1-e^{-2\lambda})-24\lambda,\qquad \frac{d\ell}{d\lambda}=\frac{16e^{-2\lambda}}{1-e^{-2\lambda}}-24.

Setting the derivative to zero yields $e^{-2\hat{\lambda}}=12/20=0.6$, so

\hat{\lambda}=-\frac{\log(0.6)}{2}=0.2554\ \text{per month}.

The observed event proportion is $\hat{p}=8/20=0.4$, with approximate standard error $\sqrt{0.4(0.6)/20}=0.1095$. Since $\lambda=-\log(1-p)/2$, the delta method gives $SE(\hat{\lambda})\approx SE(\hat{p})/[2(1-\hat{p})]=0.0913$ per month. A Wald 95% interval is $0.2554\pm1.96(0.0913)$, or approximately $(0.0765,0.4343)$ per month. This interval uses a large-sample normal approximation; with only 20 participants, a profile-likelihood interval may have better small-sample behavior.

The observations establish only that each event occurred by month 2, not that it occurred at month 2. The likelihood correctly uses the probability of the interval outcome. Treating all eight events as exact events at the visit would change the data model.

Likelihoods for Multi-State Panel Observations

For a finite-state continuous-time Markov model, let $q_{ij}$ be the transition intensity from state $i$ to state $j$, for $i\ne j$. The diagonal entry is $q_{ii}=-\sum_{j\ne i}q_{ij}$. The transition-probability matrix over an interval of length $t$ is $P(t)=\exp(Qt)$, where $Q=(q_{ij})$. Its $i,j$ entry is the probability of being in state $j$ after time $t$, given state $i$ at the start.

A vital feature is that a transition probability can include several unobserved paths. For example, an observed change from state 0 to state 2 over a visit interval does not, by itself, establish whether an intermediate state was occupied. The model sums the probabilities of all allowable paths that end in the observed state. The following derivation makes this concrete.

Justification: Suppose the only transitions are $0\to1$ at intensity $a$ and $1\to2$ at intensity $b$, with state 2 absorbing. To be in state 1 at time $T$, the transition from 0 must occur at some time $u$ between 0 and $T$, and the process must then remain in state 1 until $T$. The density of the first transition at $u$ is $ae^{-au}$, and the probability of remaining in state 1 for the remaining $T-u$ is $e^{-b(T-u)}$. Thus

P_{01}(T)=\int_0^T ae^{-au}e^{-b(T-u)}\,du =\frac{a}{b-a}\left(e^{-aT}-e^{-bT}\right),\quad a\ne b.

The other probabilities from state 0 are $P_{00}(T)=e^{-aT}$ and $P_{02}(T)=1-P_{00}(T)-P_{01}(T)$. When $a=b$, the integral instead gives $P_{01}(T)=aTe^{-aT}$. These probabilities account for the possible unobserved transitions between visits.

Worked Example: Building a Multi-State Panel Likelihood

Consider hypothetical visit data under the three-state progression model just defined. Ten participants start in state 0 and are observed again six months later: three are in state 0, five in state 1, and two in state 2. Three other participants start in state 1: two remain in state 1 and one is in state 2 at six months. Suppose a fitted model has $a=0.2$ per month and $b=0.1$ per month. Assume the specified transition structure, constant intensities, accurate state observations, and independent participants.

For participants starting in state 0, the probabilities are

P_{00}(6)=e^{-0.2(6)}=e^{-1.2}\approx0.3012,

P_{01}(6)=\frac{0.2}{0.1-0.2}\left(e^{-1.2}-e^{-0.6}\right) =-2(0.3011942-0.5488116)\approx0.4952,

and, using the unrounded probabilities in the subtraction,

P_{02}(6)=1-e^{-1.2}-P_{01}(6)\approx0.2036.

For participants starting in state 1, $P_{11}(6)=e^{-0.1(6)}\approx0.5488$ and $P_{12}(6)=1-e^{-0.6}\approx0.4512$. The likelihood at these fitted intensities is

$$ L=(0.3011942)^3(0.4952348)^5(0.2035709)^2(0.5488116)^2(0.4511884). $$

The corresponding log likelihood is approximately $-12.29$. Each factor is the probability of the observed endpoint state conditional on the starting state; the two observed 0-to-2 records are not assigned a direct-transition probability, because the model reaches state 2 through state 1. If the rates were unknown, maximizing this likelihood over $a$ and $b$ would give the maximum-likelihood estimates. In a real fit, all observed visit intervals and starting states would contribute their corresponding transition-probability factors.

A basic check is that each row of the transition-probability matrix sums to one. Here the three probabilities from state 0 sum to 1, as do the two probabilities from state 1. Small discrepancies can result from rounding, but substantial discrepancies indicate a formula, coding, or model-implementation error.

Uncertainty for Fitted State Probabilities

A fitted transition probability is an estimate, not a known population probability. Uncertainty in fitted parameters therefore carries through to predicted state occupancy. For a smooth function $g(\theta)$ of an estimated parameter vector $\hat{\theta}$, the delta method approximates

\operatorname{Var}\{g(\hat{\theta})\}\approx \nabla g(\hat{\theta})^\mathsf{T}\widehat{\operatorname{Var}}(\hat{\theta})\nabla g(\hat{\theta}).

The parameter covariance matrix is commonly estimated from the inverse observed information matrix, when the likelihood is sufficiently regular and the model is identifiable. If the target is a transition probability, its gradient must be taken with respect to every fitted parameter on which that probability depends. A probability that depends only on one transition intensity does not inherit uncertainty from unrelated intensities.

Worked Example: Uncertainty in a Predicted Occupation Probability

Use the hypothetical data and fitted exponential model from the first example. At month 3, the probability of occupying event-free state 0 is $g(\lambda)=e^{-3\lambda}$. At $\hat{\lambda}=0.2554$ per month, the fitted probability is

\hat{g}=e^{-3(0.2554)}\approx0.4648.

The derivative is $g'(\lambda)=-3e^{-3\lambda}$. Using $SE(\hat{\lambda})=0.0913$ per month, the delta-method standard error is

SE(\hat{g})\approx3e^{-3\hat{\lambda}}SE(\hat{\lambda}) =3(0.4648)(0.0913)\approx0.1273.

A Wald 95% interval is $0.4648\pm1.96(0.1273)$, approximately $(0.2153,0.7143)$. Under the assumed exponential model, the fitted probability of remaining in state 0 through month 3 is about 0.465, with an approximate 95% confidence interval from 0.215 to 0.714. The interval describes uncertainty in the population probability due to estimation; it is not a 95% probability that the fixed probability lies inside this particular interval. The broad interval also signals that the small sample provides limited precision.

Diagnostics, Alternatives, and Common Pitfalls

Before interpreting results, check that the model’s transition structure matches the states and transitions possible in the scientific setting. Inspect whether the visit schedule leaves enough information to distinguish parameters: with very sparse visits, different intensity combinations can yield similar endpoint probabilities. Check convergence, parameter standard errors, the observed information matrix, and sensitivity to starting values. Compare predicted state proportions with observed proportions at visit times, while recognizing that agreement alone cannot establish the Markov assumption. Strong residual patterns by time since entry or calendar time may indicate non-constant intensities or inadequate state definitions.

A discrete-time transition model estimates changes over the visit intervals directly and can be attractive when only visit-based transition probabilities are of interest. A continuous-time model supplies probabilities for unequal time gaps and supports a coherent intensity-based description, but requires assumptions about transitions between visits. Interval-censored proportional hazards and accelerated failure time models address event-time covariate effects using different parameterizations; they are alternatives for suitable event-time questions, not substitutes for a multi-state likelihood when the observed states and paths matter.

Common mistake: assign an event to the visit date. A state change detected at month 6 generally locates the transition only between visits. Use the probability of the observed interval outcome or endpoint state, not an invented exact event time.
Common mistake: treat observed endpoints as the full path. An observation from state 0 to state 2 does not prove a direct transition. Include all allowable latent paths in the transition probability under the stated model.
Common mistake: interpret a fitted occupancy probability as an individual certainty. A value such as 0.465 is a model-based population probability conditional on the specified starting state and time, not a guarantee about a particular participant.

The methods are limited by what the observation schedule can identify. Misspecified state definitions, measurement error, informative visit timing, or a time-inhomogeneous process can undermine the likelihood and its standard errors. When these concerns are plausible, report the assumptions, examine diagnostics, and assess whether a more flexible model or sensitivity analysis changes the substantive conclusion.

Check Your Understanding

Work through each question, then compare with the solutions.

In the first example, what is the probability of remaining in state 0 at month 2 under the fitted model?
For the three-state model in the second example, why is $P_{02}(6)$ not set to zero even though no direct $0\to2$ transition is allowed?
If $a=b$ in the three-state model, what is $P_{01}(T)$?
Give one assumption that supports multiplying the participant-level likelihood contributions, and one diagnostic or check that could reveal a problem with the fitted transition model.

Solutions. (1) $e^{-2\hat{\lambda}}=e^{-2(0.2554)}\approx0.6000$, the fitted probability of remaining in state 0 through month 2. (2) Participants can move from 0 to 1 and then from 1 to 2 between visits, so the endpoint probability includes that unobserved path. (3) Taking the limit of the integral as $b$ approaches $a$, or evaluating the integral directly, gives $P_{01}(T)=aTe^{-aT}$. (4) Independent participants justify multiplying their likelihood contributions. A useful check is whether predicted state proportions at visits show systematic disagreement with observed proportions; such disagreement can indicate a model or implementation problem.

Integrated Problems in Interval-Censored Multi-State Data

What You'll Learn