Why Model Transitions at Visits?
In panel studies, a person's state may be observed only at scheduled visits. For example, a participant might be recorded as disease-free at one visit and as having disease at the next, while the exact event time is unknown. A discrete-time transition model describes the probability of the observed next state conditional on the current state. Its time unit is one step between specified visits, not an infinitesimal unit of calendar time.
The model is useful when the observation schedule is central to the data: it directly describes changes from one recorded visit to the next. It does not require the analyst to assign an event time within an interval. However, a probability of changing state between visits is not itself a transition intensity, and it generally does not reveal how many unobserved changes occurred in that interval.
Definition and Assumptions
A first-order Markov assumption on the visit sequence means that the conditional distribution of the next recorded state depends on the current recorded state, not additionally on the earlier recorded states: \(P(X_{k+1}=s\mid X_k,X_{k-1},\ldots)=P(X_{k+1}=s\mid X_k)\). This is an assumption about states at visits. It does not assert that the unobserved path within a visit interval has no intermediate transitions.
Estimation from state-pair counts also requires that the observations are classified consistently, the relevant visit-to-visit pairs are observed, and subjects are independent. Conditional on the states at the starts of the intervals, the Markov likelihood factors into contributions for the observed transitions. If each step has its own matrix, this gives separate probabilities for each \(k\). A time-homogeneous discrete model adds the stronger assumption that \(p_{rs,k}=p_{rs}\) for every visit step being pooled.
These assumptions are substantive. For example, if two participants both have state \(r\) at visit \(k\), but their transition chances differ because of an omitted history or covariate, a single \(p_{rs,k}\) may not describe both adequately. A conditional model can incorporate measured covariates, but that is a different specification from estimating an unrestricted transition matrix.
Likelihood and Maximum-Likelihood Estimates
Let \(N_{rs,k}\) be the number of observed intervals that start in state \(r\) at visit \(k\) and end in state \(s\) at visit \(k+1\), and let \(N_{r+,k}=\sum_s N_{rs,k}\) be the number starting in \(r\). Conditional on the starting states, the likelihood for visit-specific transition probabilities is
For a row with \(N_{r+,k}>0\), the likelihood contribution is a multinomial likelihood. Its maximum-likelihood estimate is the observed proportion ending in each destination:
This estimate applies to the transition probability for one visit step under the specified state and visit indexing. It is not the proportion who ever visited state \(s\), nor a rate per year unless the time step and model interpretation warrant that description.
Why the Row Proportions Maximize the Likelihood
For one origin-state row, write \(N_s\) for the destination counts, \(n=\sum_s N_s\), and \(p_s\) for the probabilities. The row likelihood is \(\prod_s p_s^{N_s}\), subject to \(p_s\geq0\) and \(\sum_s p_s=1\). Let \(S_+=\{s:N_s>0\}\). Assigning positive probability to every destination in \(S_+\) gives a positive likelihood. Therefore a maximizer must have positive probability on every destination in \(S_+\); otherwise its likelihood would be zero.
Now suppose a candidate maximizer assigns positive mass to any destination outside \(S_+\). Transfer some of that mass to a destination in \(S_+\). All factors for destinations outside \(S_+\) have exponent zero and do not reduce the likelihood, while the positive-probability factor for the receiving destination increases. Thus the likelihood strictly increases, contradicting maximality. A maximizer therefore assigns zero probability outside \(S_+\). On the remaining support, maximize the log likelihood \(\ell=\sum_{s\in S_+}N_s\log(p_s)\) subject to \(\sum_{s\in S_+}p_s=1\). The Lagrange equations give \(N_s/p_s=\lambda\); summing over \(s\) gives \(\lambda=n\), so \(p_s=N_s/n\). This also shows why a destination with zero observed count receives an estimate of zero, a boundary estimate, rather than requiring a separate positive-count formula.
If \(N_{r+,k}=0\), there is no observed transition from \(r\) at that visit step. The likelihood contains no information about that row, so its probabilities cannot be estimated from those data without additional assumptions or information. A zero estimated probability from a nonempty row is also not proof that the transition is impossible in the population; it may reflect limited data.
Worked Example: Estimating a Three-State Visit Matrix
Suppose a hypothetical panel study records participants in one of three states—well, mild disease, or severe disease—at two consecutive visits. The observed transitions are:
| State at visit \(k\) | Next: well | Next: mild | Next: severe | Row total |
|---|---|---|---|---|
| Well | 12 | 6 | 2 | 20 |
| Mild | 1 | 7 | 2 | 10 |
| Severe | 0 | 1 | 4 | 5 |
For participants recorded as well at visit \(k\), the estimated probabilities are \(12/20=0.600\) of being well, \(6/20=0.300\) of being mild, and \(2/20=0.100\) of being severe at the next visit. For the mild row they are \(1/10=0.100\), \(7/10=0.700\), and \(2/10=0.200\). For the severe row they are \(0/5=0\), \(1/5=0.200\), and \(4/5=0.800\).
The estimated matrix, with rows and columns ordered well, mild, severe, is
Each row sums to 1. Under this model, the estimated probability that a participant recorded as mild is recorded as severe at the next visit is 0.200. The estimate says nothing by itself about when, or how many times, that participant may have changed state between visits. The zero in the severe-to-well cell is the maximum-likelihood estimate for this sample, not evidence that recovery is impossible.
Worked Example: Competing Recorded Outcomes at the Next Visit
In a hypothetical study, 50 participants are recorded as event-free at a visit. At the next visit, 35 remain event-free, 9 are recorded with event type A, and 6 with event type B. Treat these as three mutually exclusive states at the next visit. The event-free row estimates are obtained by dividing each count by 50:
The estimates sum to \(0.700+0.180+0.120=1.000\). For this visit step, the estimated probability of being recorded with event type A at the next visit, conditional on being event-free at the current visit, is 0.180. Likewise, the corresponding probability for event type B is 0.120.
These endpoint categories are mutually exclusive as recorded states. If event type A occurred and then another state change happened before the next visit, the endpoint state alone may not identify that path. The estimated probabilities describe the recorded next state; they do not automatically identify the probability of a particular first event or the timing of an event within the interval.
Visit-Specific Versus Pooled Probabilities
Whether to estimate a separate matrix at each step or pool transitions is a modeling decision. Pooling reduces the number of parameters, but it assumes the same transition probabilities apply across the pooled steps. That assumption is not implied merely because the states have the same names at each visit.
Worked Example: Why Different Visit Steps May Need Separate Rows
Consider hypothetical observations of a binary state, 0 or 1. At the first visit step, 40 intervals start in state 0: 32 end in state 0 and 8 in state 1. At the second visit step, 30 intervals start in state 0: 18 end in state 0 and 12 in state 1.
The separate estimates for transition \(0\to1\) are
If a common probability is imposed across both steps, the pooled estimate is \(\widehat p_{01}=(8+12)/(40+30)=20/70\approx0.286\). This pooled value is a weighted average of the two step-specific proportions; it is appropriate as a common-probability estimate only if treating the steps as having the same transition probability is reasonable. The observed difference alone does not establish that the underlying probabilities differ, but it makes the homogeneity assumption worth examining in context.
Probabilities Are Not Continuous-Time Intensities
A discrete-time transition probability is indexed by a specified interval between visits. A continuous-time intensity \(q_{rs}(t)\), by contrast, describes an instantaneous transition rate from \(r\) to \(s\), conditional on being in \(r\) immediately before time \(t\). It has units of inverse time; a probability has no units. The continuous-time Markov likelihoods and intensity-recovery methods used for panel observations address a different parameterization.
Even under a continuous-time Markov model, the probability of an endpoint transition over an interval is generally not the intensity multiplied by the interval length. For a time-homogeneous model with generator \(Q\), the transition matrix over interval length \(\Delta\) is \(\exp(Q\Delta)\). Its entries account for possible paths through intermediate states. Thus, an observed transition probability can depend on all the intensities in the system and on the interval length. A visit-based estimate should not be reported as an intensity without a justified model connecting the two.
| Feature | Visit-indexed transition model | Continuous-time intensity model |
|---|---|---|
| Parameter | Probability of a next recorded state | Instantaneous transition rate |
| Time scale | A defined step between visits | Continuous calendar time |
| Within-interval path | Not specified by endpoint probabilities alone | Represented through the continuous-time model |
| Typical interpretation | Chance of a destination state at the next visit | Rate of an immediate transition out of a state |
Diagnostics, Limitations, and Common Pitfalls
Begin with simple checks: confirm that every row's counts add to its stated denominator, that every estimated row sums to one, and that the state definitions and visit indexing are consistent. Inspect row totals as well as proportions: a row estimated from a handful of transitions is much less stable than one estimated from many. A cell with zero count creates a boundary estimate, while a row with no transitions has no empirical estimate at all.
The Markov assumption is harder to verify than the arithmetic. If transition patterns plausibly differ by earlier history, time since entry, or measured participant characteristics, the current-state-only model may omit relevant structure. Likewise, a transition matrix for one visit schedule need not apply to another schedule: longer intervals allow more opportunity for change. These are model limitations, not defects that can be fixed by interpreting a probability as a rate.
- Mistake: interpreting \(p_{rs,k}\) as an instantaneous rate. A probability is the chance of an endpoint state over a specified visit step. It has no rate units and does not identify event timing.
- Mistake: assuming no observed transitions means a transition is impossible. A zero cell gives a boundary estimate for the sample. It may instead reflect sparse data; a structural zero requires scientific or design justification.
- Mistake: pooling all visit steps automatically. Pooling estimates a common probability and assumes homogeneity across those steps. If that assumption is inappropriate, report or model step-specific probabilities.
- Mistake: treating an empty origin-state row as an estimated row of zeros. With no observed starts in that state, the likelihood supplies no estimate of its destination probabilities.
Compared with a continuous-time intensity model, the visit-based model is direct and interpretable when the target is the state at the next scheduled visit and the visit schedule is meaningful. Its main limitation is that it does not reconstruct the path between observations. Conversely, a continuous-time model adds assumptions about transitions over time in exchange for a rate-based description. The choice should follow the scientific question and observation design, rather than a preference for one type of parameter.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In 25 hypothetical intervals starting in state A, 15 end in A, 7 in B, and 3 in C. What are the estimated transition probabilities from A?
- What does a zero count for a transition from a nonempty row imply about its maximum-likelihood estimate, and what does it not prove?
- Why is a transition probability over one visit step not automatically an intensity per year?
- At two visit steps, estimates of a transition probability are 0.10 and 0.35. What additional assumption is made if the counts are pooled to estimate one common probability?
Solutions. (1) Divide each destination count by 25: \(15/25=0.600\), \(7/25=0.280\), and \(3/25=0.120\); these sum to 1. (2) The maximum-likelihood estimate is zero for that destination, but the sample does not prove the population probability is zero. (3) The probability describes an endpoint over a specified interval, whereas an intensity is an instantaneous rate with inverse-time units; a model connecting them is needed. (4) Pooling assumes that the transition probability is common across both visit steps, rather than allowing it to vary by step.