Closed-Form Solution of the Unit Normal Loss Integral in Two-Dimensions

In Value of Information (VoI) analysis, the unit normal loss integral (UNLI) frequently emerges as a solution for computation of various VoI metrics for both model-based and data-driven economic evaluations. However, one limitation of the UNLI has been that its closed-form solution is available for only one dimension, and thus can be used for comparisons involving only two strategies (where it is applied to the scalar incremental net benefit). We derive a closed-form solution for the two-dimensional UNLI, enabling closed-form VoI calculations for three strategies. A case study based on a three-arm clinical trial is provided as an example. VoI methods based on the closed-form solutions for the UNLI can now be extended to three-decision comparisons, taking a fraction of a second to compute and not being subject to Monte Carlo error. This method is implemented in R and is available through an R package (https://github.com/resplab/predtools/).