USE OF RICHARDSON'S EXTRAPOLATION TO REDUCE TIMING ERRORS IN MARKOV MODELS

Purpose: To explore the possibility of using Richardson's extrapolation in Markov models.

Method: For computational efficiency, it is desirable to use as large a cycle length as possible in a Markov model. However, large cycle lengths incur the risk of timing errors. Methods such as half-cycle correction and Simpson's rule can account for timing errors in calculating total costs and outcomes from Markov models, under the assumption that the state probabilities have been calculated correctly. However, it is also possible that timing errors may have been made in estimating the transition probabilities. Particular cases where there is scope for such errors are the presence of competing risks and the possibility of more than one event happening within a cycle, in particular for a progressive disease where some individuals may progress by more than one stage within a cycle. In principle these problems can be addressed by careful calculation of the probabilities: an alternative approach is called Richardson's extrapolation, in which the model is constructed and run using two or more different cycle lengths. Comparing the results from different cycle lengths allows an estimate of the timing errors to be made and hence an estimate of the results without timing errors. It is also possible to apply Richardson's extrapolation to appropriate powers of the transition matrices for models of different time cycles: the resulting transition matrix can then be used. Models have been built based on two different underlying continuous time models. The following two methods have been used for each model: (1) deriving transition probabilties directly from the continuous time models; (2) using simulated data sets sampled from the continuous time models. Richardson's extrapolation has been applied to each of these four cases, at both the results stage and the transition matrix stage.

Result: In all cases considered, Richardson's extrapolation gives an improvement in efficiency without any loss of accuracy in the model results.

Conclusion: Richardson's extrapolation is a promising approach for improving the efficiency of work with Markov models.