OBJECTIVES

Markov models, which specify transition probabilities between discrete health states at each time step, are widely used within health economic modelling. Decision problems are usually assessed by attaching utilities and costs to states and summing the total utility and cost of decision options over some time-frame. Frequently, a correction method (e.g. half-cycle correction (HCC)) is applied to naïve discrete-time outputs to yield a closer approximation to an underlying continuous-time Markov chain. In this study, we note that an underlying continuous-time Markov chain that corresponds to the proposed discrete time analogue may not exist, question the rationale for corrections in this case, and introduce a novel approximation method based on Gaussian Quadrature (GQ).

METHODS

We considered a simple model with three health states – well, unwell and dead. We exploited analytical results for time-homogeneous Markov chains in terms of matrix exponentials and inverses to compactly express calculations, and to introduce a new n-order GQ-based numerical integration method, which is applied to naïve discrete-time output. An n-order GQ method approximates the continuous-time Markov chain result by a weighted sum of function values at specified points within the range of integration and yields exact values for polynomials of degree up to 2n – 1. We conducted a simulation study to compare the GQ methodology and other existing cycle correction methods (HCC, trapezoidal, and Simpson 1/3 and 3/8 methods), to the exact continuous-time process outcomes (gold standard).

RESULTS

At first order, we found that the GQ method replicated HCC. In our simulation study, the third-order GQ method outperformed other existing methods (HCC, trapezoidal and Simpson 1/3 and 3/8) in approximating the gold standard results. Simpson 1/3 turned out to be the second-best method.

CONCLUSIONS

The third-order GQ method, applied to naïve discrete-time output, can outperform other cycle correction methods for homogeneous models. Defining models in continuous-time avoids potential inconsistencies.