Improving cycle corrections in discrete time Markov models : a Gaussian quadrature approach

Introduction: Discrete-time Markov models are widely used within health economic modelling. Analyses usually associate costs and health outcomes with health states and calculate totals for each decision option over some timeframe. Frequently, a correction method (e.g. half-cycle correction) is applied to unadjusted model outputs to yield an approximation to an assumed underlying continuous-time Markov model. In this study, we introduce a novel approximation method based on Gaussian Quadrature (GQ).

Methods: We exploited analytical results for time-homogeneous Markov chains to derive a new GQ-based approximation, which is applied to an unadjusted discrete-time model output. The GQ method approximates a continuous-time Markov model result by approximating a correction matrix, formulated as an integral, using a weighted sum of integrand values at specified points. GQ approximations can be made arbitrarily accurate by increasing order of the approximation. We compared the first five orders of GQ approximation with four existing cycle correction methods (half-cycle correction, trapezoidal and Simpson 1/3 and 3/8 rules) across 100,000 randomly generated input parameter-sets.

Results: We show that first-order GQ method is identical to half-cycle correction method, which is itself equivalent to trapezoidal method. The second-order GQ is identical to Simpson 1/3 method. The third, fourth and fifth order GQ methods are novel in this context and provide increasingly accurate approximations to the output of the continuous-time model. In our simulation study, fifth-order GQ method outperformed other existing methods in over 99.8% of simulations. Of the existing methods, Simpson 1/3 rule performed the best.

Conclusion: Our novel GQ-based approximation outperforms other cycle correction methods for time-homogeneous models. The method is easy to implement, and R code and an Excel workbook are provided as supplementary materials.