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

TL;DR

This paper derives a closed-form solution for the two-dimensional unit normal loss integral, enabling fast and accurate value of information calculations for three strategies, extending previous one-dimensional solutions.

Contribution

The authors developed a closed-form solution for the two-dimensional UNLI, allowing efficient VoI analysis for three strategies, which was not possible before.

Findings

Validated accuracy through simulation studies.

Enabled rapid VoI calculations for three strategies.

Provided an R implementation in the predtools package.

Abstract

In Value of Information (VoI) analysis, the unit normal loss integral (UNLI) frequently emerges as a solution for the computation of various VoI metrics. 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 derived a closed-form solution for the two-dimensional UNLI, enabling closed-form VoI calculations for three strategies. We verified the accuracy of this method via simulation studies. A case study based on a three-arm clinical trial was used 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. An R implementation of this method is provided as part of the predtools package (https://github.com/resplab/predtools/).