  Time-homogeneous Markov chains are often used as disease progression models
in studies of cost-effectiveness and optimal decision-making. Maximum
likelihood estimation of these models can be challenging when data are
collected at a time interval longer than the model's transition cycle length.
For example, it may be necessary to estimate a monthly transition model from
data collected annually. The likelihood for a time-homogeneous Markov chain
with 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 this
estimate 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 optimization
problem when a valid root is unavailable. The optimization problem is not
convex. Local convergence is explored in several case studies through graphical
representations of a grid search. The example cases use disease progression
data from the literature as well as synthetic data. The global maximum
likelihood estimate is increasingly difficult to locate as the number of cycles
or the number of states increases. What seems like a straightforward estimation
problem can be challenging even for relatively simple models. Researchers
should consider alternatives to likelihood maximization or alternative models.
