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

Abstract

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.HighlightsThe 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.