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Time-homogeneous Markov chains are often used as disease progression modelsin studies of cost-effectiveness and optimal decision-making. Maximumlikelihood estimation of these models can be challenging when data arecollected at a time interval longer than the model's transition cycle length.For example, it may be necessary to estimate a monthly transition model fromdata collected annually. The likelihood for a time-homogeneous Markov chainwith transition matrix $\mathbf{P}$ and data observed at intervals of $T$cycles is a function of $\mathbf{P}^T.$ The maximum likelihood estimate of$\mathbf{P}^T$ is easily obtained from the data. The $T$th root of thisestimate would then be a maximum likelihood estimate for $\mathbf{P}.$ However,the $T$th root of $\mathbf{P}^T$ is not necessarily a valid transition matrix.Maximum likelihood estimation of $\mathbf{P}$ is a constrained optimizationproblem when a valid root is unavailable. The optimization problem is notconvex. Local convergence is explored in several case studies through graphicalrepresentations of a grid search. The example cases use disease progressiondata from the literature as well as synthetic data. The global maximumlikelihood estimate is increasingly difficult to locate as the number of cyclesor the number of states increases. What seems like a straightforward estimationproblem can be challenging even for relatively simple models. Researchersshould consider alternatives to likelihood maximization or alternative models.

Maximum likelihood estimation for nonembeddable Markov chains when the cycle length is shorter than the data observation interval