Closed-Form Solution of the Unit Normal Loss Integral in 2 Dimensions, with Application in Value-of-Information Analysis

The unit normal loss integral (UNLI) has wide applicability in decision analysis and risk modeling, including as a solution for computation of various value-of-information (VoI) metrics. However, one limitation of the UNLI has been that its closed-form solution is available for only 1 dimension and thus can only be used for comparisons involving 2 strategies (where it is applied to the scalar incremental net benefit). We derived a closed-form solution for the 2-dimensional UNLI by the integration by parts technique. This enables the extension of the UNLI method to 3-comparison problems. We implemented this approach in R as part of the predtools package (https://github.com/resplab/predtools/) and verified the accuracy of this implementation via Monte Carlo simulations. A case study based on a 3-arm clinical trial was used as an example for VoI analysis. Methods based on the closed-form solutions for the UNLI can now be extended to 3-decision comparisons, taking a fraction of a second to compute and not being subject to Monte Carlo error. HIGHLIGHTS: The unit normal loss integral (UNLI) is widely used in decision analysis and risk modeling, including in the computation of various value-of-information metrics, but its closed-form solution is only applicable to comparisons of 2 strategies. We derive a closed-form solution for 2-dimensional UNLI, extending the applicability of the UNLI to 3-strategy comparisons. Such closed-form computation takes only a fraction of a second and is free from simulation errors that affect the hitherto available methods. In addition to the relevance in 3-strategy model-based and data-driven decision analyses, a particular application is in risk prediction modeling, where the net benefit of a classifier should always be compared with 2 default strategies of treating none and treating all.