Deriving the Panel-Data Likelihood From Transition Matrices

From Visit Records to a Likelihood

When a process is observed only at scheduled visits, the data usually give a sequence of states rather than exact transition times. For example, a participant might be in state A at baseline, state B at the next visit, and state C at a later visit. A likelihood for these data must describe the probability of that observed sequence, not assign an exact time to an unobserved change of state.

The key construction is local: for each pair of successive visits, use the transition probability from the state recorded at the first visit to the state recorded at the second. Multiplying these conditional probabilities gives the likelihood contribution for the observed history, provided the state process satisfies the stated Markov assumption. The same construction works when different intervals have different transition matrices.

The matrices may be identical across visits when the transition law is unchanged and the intervals have the same relevant length. More generally, \(P_j\) can vary with the interval, such as when visits are separated by different amounts of time or when the model allows calendar-time variation. The definition is conditional on the interval information used by the model; it does not imply that all intervals share one matrix.

Deriving the Likelihood Contribution

For one subject with observed sequence \(x_0,x_1,\ldots,x_m\), begin by conditioning on the observed initial state \(x_0\). The chain rule expresses the probability of the whole sequence as a product of successive conditional probabilities. The Markov assumption then reduces each conditional probability to the appropriate transition-matrix entry.

The first equality is the chain rule and does not require a Markov model. The second equality uses the Markov assumption, which is the substantive modeling step. The final equality follows from the definition of \(P_j\). If the entries depend on an unknown parameter vector \(\theta\), this sequence probability is the subject’s likelihood contribution \(L(\theta)\), conditional on the observed initial state and the visit schedule.

If the initial state is not conditioned on and its distribution is part of the model, its probability must also be included. If \(\pi_a=P\{X(t_0)=a\}\), the probability of the full observed sequence is \(\pi_{x_0}\prod_{j=1}^{m}(P_j)_{x_{j-1},x_j}\). Omitting this initial-state factor is appropriate when the analysis conditions on the baseline state; it is not appropriate when the model claims to describe the joint distribution including that state.

Combining Intervals and Subjects

For several subjects whose histories are independent under the model, the joint likelihood is the product of their individual likelihood contributions. This independence is ordinarily understood conditional on modeled covariates and any shared parameters. If all subjects share one transition matrix, observed transitions can also be summarized by counts: each matrix entry is raised to the number of observed interval transitions of that type. When matrices differ by interval, the corresponding interval-specific entries must remain distinct.

A useful general form makes the role of transition counts explicit. Let \(N_{j,ab}\) be the number of subjects observed in state \(a\) at the start of interval \(j\) and state \(b\) at its end. For conditionally independent subjects with common interval-specific matrices, the likelihood conditional on their observed initial states is

This is simply the individual-history product rearranged by matrix entry. A zero count contributes a factor of one. A positive count paired with a zero transition probability makes the likelihood zero, because the model assigns probability zero to an observed transition.

Assumptions, Checks, and Alternatives

The product formula is simple, but its interpretation depends on several conditions. The states must be defined consistently at every visit, and each recorded transition must correspond to the correct pair of successive visits. The transition matrices must describe the conditional probabilities for the relevant intervals and covariates. The Markov assumption must be defensible at the observation times, and the subject-level likelihood product requires independence across subjects conditional on the model.

The visit schedule and observation process also matter. The likelihood above is conditional on the stated visit times and observed states. If visit attendance, dropout, or measurement depends on unobserved state history in a way related to the parameters of interest, simply conditioning on the observed schedule may not capture the full data-generating process. Such a setting may require modeling the observation process as well; the basic transition product alone does not make informative observation harmless.

Useful checks include verifying that every row of each proposed matrix sums to one, confirming that the row index is the state before the interval and the column index the state after it, and checking that the selected matrix corresponds to the correct interval. Compare observed transitions with fitted probabilities, especially for states or transition types with sparse observations. If the same current state is followed by systematically different next-state patterns depending on earlier observed states, that is evidence against the assumed Markov structure. A zero fitted probability for a transition that appears in the data is a direct warning that the model or parameter restrictions are incompatible with those observations.

This likelihood is not an event-time density. A method for interval-censored event times targets probabilities associated with event-time intervals; here the observed data are a sequence of states, and the factors are transition probabilities between visits. A direct model for state occupation at a particular visit can target the marginal distribution at that visit, but does not by itself represent the joint probability of a complete observed path. Continuous-time transition models can impose additional structure on how probabilities vary with interval length; the transition-matrix likelihood presented here does not require that particular construction.

Why the Construction Matters

Writing the likelihood as a product makes the data-model connection transparent. Each observed step contributes exactly one probability, while the assumptions determine which probability is appropriate. This structure supports estimation by maximizing the likelihood over the transition probabilities or their model parameters. It also makes it easier to identify where a discrepancy originates: a particular interval, a particular starting state, or a particular observed destination.

The same clarity prevents overinterpretation. A panel history shows that the state differed between two visits, but it does not reveal when within the interval the change occurred. The transition-matrix likelihood uses the information actually recorded, without silently treating visit times as transition times.