Comparing Markov Models Across Visit Schedules

Why Visit Schedules Matter When Comparing Models

In panel data, a subject’s state is observed at visits, but changes between visits may be unseen. A transition from state \(i\) at one visit to state \(j\) at the next therefore describes the states at two observation times; it does not establish that exactly one event occurred during the gap. A longer gap gives more opportunity for transitions, including unobserved changes followed by a return to the original state.

This matters when comparing candidate models. A model should be judged against the transition probabilities implied for the actual gaps in the data, not against a count of apparent one-step changes that ignores time between visits. Likelihood criteria can compare models fitted to the same observations, but a lower criterion does not repair a misspecified transition process or an inappropriate treatment of the visit schedule.

The Markov assumption says that, conditional on the current state and the model’s included covariates or time effects, the future evolution does not depend on the earlier path. A time-homogeneous model adds the assumption that the transition rates do not vary with calendar or follow-up time. Candidate models might differ in which transitions are allowed, whether rates are constrained to be equal, or whether covariates are included. The likelihood-based comparisons below require valid likelihoods for the observations being compared and appropriate treatment of censoring and visit timing.

Gap-Specific Transition Patterns

For a continuous-time model, \(P(\Delta)\) changes with the gap \(\Delta\). In a discrete-time model, the corresponding transition probabilities must likewise be interpreted at the time step for which they were defined. Therefore, the same underlying process can produce different observed proportions of endpoint changes under different schedules. Raw transition counts are useful summaries, but they are not by themselves a fair comparison when groups have different gap lengths or starting-state distributions.

A meaningful pattern check compares observed transitions with model-implied probabilities at the observed gaps. For example, one can group intervals by starting state and gap length, then compare observed endpoint counts with expected counts obtained from the fitted \(P(\Delta)\). If the model repeatedly underpredicts a particular endpoint state at comparable gaps, that pattern may indicate inadequate transition structure, time variation, or a violation of the Markov assumption. Sparse cells and uncertain state histories can make such comparisons noisy, so they should be interpreted alongside likelihood and uncertainty.

Likelihood Criteria for Candidate Models

Suppose competing models are fitted to the same panel observations, with the same outcome definition, included subjects, and treatment of observation times and censoring. Let \(\ell\) be a model’s maximized log likelihood, \(k\) its number of estimated parameters, and \(n\) the number of independent subjects for the BIC convention used here. The Akaike information criterion (AIC) and Bayesian information criterion (BIC) are

Both criteria trade off fit against model complexity; smaller values are preferred among the candidates being compared. AIC uses a penalty of \(2\) per parameter, while BIC’s penalty grows with sample size. Neither criterion is a hypothesis test, and neither provides an absolute verdict that a model is true. Differences should be considered alongside model assumptions, parameter uncertainty, and the scientific question.

The likelihood comparison must refer to the same observed data and likelihood target. In particular, likelihood values from separate cohorts with different numbers of subjects or different observed outcomes cannot be compared as if they were on a common scale. Nor should one choose a model just because its likelihood is larger: adding parameters to a nested model ordinarily cannot reduce the maximized likelihood. AIC and BIC account for complexity, but only when computed consistently.

Nested Models and Likelihood-Ratio Tests

When one candidate is a special case of another, a likelihood-ratio test can assess the restrictions defining the smaller model. For \(r\) restrictions, the test statistic is \(2(\ell_{\text{full}}-\ell_{\text{restricted}})\). Under the null restrictions and standard regularity conditions—including identifiable parameters and a null value in the interior of the parameter space—this statistic has an approximate chi-square distribution with \(r\) degrees of freedom. The approximation may be poor in small samples or when parameters lie on a boundary.

The likelihood-ratio test applies to nested models, not to any two models that happen to have different parameter counts. For nonnested alternatives, criteria such as AIC or BIC, and predictive checks, are more appropriate comparison tools. A formal test also depends on the reference distribution being credible; boundary parameters, weak identifiability, or a very small sample can invalidate the usual chi-square approximation.

A Short Justification: Why an Endpoint Is Not a Single Transition

The gap effect follows directly from the Markov transition rule. If a gap of length \(2\Delta\) is split into two intervals of length \(\Delta\), the Chapman–Kolmogorov equation gives

The matrix product sums over every possible state at the intermediate time. For example, its \(i,j\) entry is

The sum includes paths that leave state \(i\), enter another state \(m\), and later reach \(j\), as well as paths that remain in or return to a state. Consequently, an observed pair of endpoints does not reveal the intermediate path. A model comparison that treats every observed interval as one identical discrete-time step discards this gap-dependent information unless all intervals truly have the same duration and the model is defined at that step.

Diagnostics, Alternatives, and Limitations

A comparison is most informative when it is paired with checks of how each candidate behaves across the observed schedules. Relevant warning signs include systematic discrepancies between observed and predicted endpoint patterns by starting state or gap length, unstable estimates, parameters near constraints, and conclusions that change sharply after a modest alteration to the model. The earlier material on diagnostics for panel-data transition models covers broader diagnostic strategies; here the central comparison question is whether a candidate explains the observed patterns at their actual time gaps.

Differences in visit schedules deserve special care. If two groups are observed at different gaps, compare their patterns using probabilities appropriate to each gap, rather than comparing raw endpoint proportions as if exposure time were equal. If the observation schedule is related to unobserved state changes or health status, conditioning on recorded visit times may not be enough; the observation process may need to be addressed in the analysis. Model comparison cannot remove bias caused by an unsuitable observation-process assumption.

• Wrong reasoning: “The group with more observed state changes must have faster transition rates.”
• Correction: The number of endpoint changes depends on visit gaps, starting states, and the possibility of unobserved transitions between visits. Compare model-implied probabilities at the recorded gaps before interpreting differences as rate differences.
• Wrong reasoning: “The model with the largest likelihood is best.”
• Correction: A more flexible model often has a larger maximized likelihood. Use a complexity-aware criterion for relative selection, or a likelihood-ratio test when the models are nested and its conditions hold.

No criterion can select among candidates that all omit an important state, impose an inappropriate time-homogeneity assumption, or rely on an unsuitable Markov structure. With sparse panel observations, distinct transition processes may also yield similar endpoint probabilities, limiting what can be identified from the data. Selection should therefore be reported as conditional on the candidate set, likelihood assumptions, and observation schedule.