Multilevel and Quasi Monte Carlo methods for the calculation of the Expected Value of Partial Perfect Information

The expected value of partial perfect information (EVPPI) provides an upper bound on the value of collecting further evidence on a set of inputs to a cost-effectiveness decision model. Standard Monte Carlo (MC) estimation of EVPPI is computationally expensive as it requires nested simulation. Alternatives based on regression approximations to the model have been developed, but are not practicable when the number of uncertain parameters of interest is large and when parameter estimates are highly correlated. The error associated with the regression approximation is difficult to determine, while MC allows the bias and precision to be controlled. In this paper, we explore the potential of Quasi Monte-Carlo (QMC) and Multilevel Monte-Carlo (MLMC) estimation to reduce computational cost of estimating EVPPI by reducing the variance compared with MC, while preserving accuracy. In this paper, we develop methods to apply QMC and MLMC to EVPPI, addressing particular challenges that arise where Markov Chain Monte Carlo (MCMC) has been used to estimate input parameter distributions. We illustrate the methods using a two examples: a simplified decision tree model for treatments for depression, and a complex Markov model for treatments to prevent stroke in atrial fibrillation, both of which use MCMC inputs. We compare the performance of QMC and MLMC with MC and the approximation techniques of Generalised Additive Model regression (GAM), Gaussian process regression (GP), and Integrated Nested Laplace Approximations (INLA-GP). We found QMC and MLMC to offer substantial computational savings when parameter sets are large and correlated, and when the EVPPI is large. We also find GP and INLA-GP to be biased in those situations, while GAM cannot estimate EVPPI for large parameter sets