BUILDING CORRELATIONS AMONG MODEL PARAMETERS: A PRACTICAL APPROACH

Purpose:  Probabilistic sensitivity analysis requires specifying distributions to describe the uncertainty about model parameters.  However, these parameters are often correlated in real life and modelers lack a simple tool to specify correlations among non-normally distributed parameters.  We propose a simple algorithm to build correlation that is not limited by the number or type of the parameter distributions, and always preserves the parameters’ marginal distributions.

   Methods:   Defining a joint distribution for correlated parameters is challenging except for the case of multivariate normal distribution (MVN) where all the parameters must be normally distributed.  In practice, modelers often use non-normal distributions for parameters that are bounded (e.g., probabilities) or skewed (e.g., odds ratios).  In these cases, sampling from a MVN distribution can be problematic.  Instead of sampling the correlated parameters simultaneously, we propose to separate sampling from correlation.  Our approach consists of three steps: (1) sample the parameter values from their desired distributions individually (e.g., beta, gamma), (2) sample the same number of random normal distributions from a MVN distribution with the desired correlation, and (3) sort the values of each parameter so that they have a rank similar to the corresponding normal distribution from the MVN distribution.  We demonstrate our approach with a previously published simulation model that involves treating a suspected case of herpes simplex encephalopathy (HSE).  The model compares three strategies: treat everyone (Treat), biopsy then treat (Biopsy), and do not treat or biopsy (No Treat).  We assume that the probability of severe complication due to biopsy (pSevereBiopsy) and the sensitivity of biopsy in detecting HSE (sensBiopsy) are correlated. 

   Results:  When pSevereBiopsy and sensBiopsy are uncorrelated, Biopsy, Treat and No Treat are optimal in 48%, 47%, and 5% of the simulations, respectively.  With a 0.5 correlation, these strategies are optimal in 47%, 49% and 4% of the simulations, respectively.  These strategies were also sensitive to correlations among various other parameter sets in the model.  In addition, our approach preserved the parameters’ marginal distributions and produced the desired rank correlation among the model parameters. 

   Conclusion:  Correlated parameters can impact model outcomes.  We outline a practical approach to specify the desired correlations among model parameters.  Our approach guarantees that the parameters’ marginal distributions are preserved and that the desired rank correlations are met.