The Difference Method Approach for Sampling Order Constrained Parameters: An Improved Implementation and Important Limitations

Abstract

A health economic model may include a set of related inputs whose true values are uncertain, but that can be assumed to follow a logical order. Various approaches are available for performing probabilistic sensitivity analysis while preserving the order constraint, one such approach is known as the difference method. The difference method approach appears to have many of the required properties, has been endorsed by good practice guidelines, and is likely to prove a popular approach. However, the proposed implementation of the difference method approach is cumbersome, requiring numerical estimation, which might present a barrier to its adoption. Furthermore, it is unclear whether the method can always be applied to three or more model inputs and whether it is unbiased across all possible input values. This study has investigated these three issues for ordered inputs bounded between 0 and 1. An analytic solution is given that allows for more straightforward and compact implementation. The difference method approach cannot always be applied to a set of three or more model inputs, and this depends on the relative size of the variances of the logit-transformed Beta distributions fitted to each variable. The approach can also produce samples with biased means and variances under certain combinations of input means and variances. It is recommended that the difference method approach be used where appropriate; however, an understanding of its limitations is necessary to identify such cases.