Probability bound analysis: A novel approach for quantifying parameter uncertainty in decision-analytic modeling and cost-effectiveness analysis

Abstract

Decisions about health interventions are often made using limited evidence. Mathematical models used to inform such decisions often include uncertainty analysis to account for the effect of uncertainty in the current evidence base on decision-relevant quantities. However, current uncertainty quantification methodologies, including probabilistic sensitivity analysis (PSA), require modelers to specify a precise probability distribution to represent the uncertainty of a model parameter. This study introduces a novel approach for representing and propagating parameter uncertainty, probability bounds analysis (PBA), where the uncertainty about the unknown probability distribution of a model parameter is expressed in terms of an interval bounded by lower and upper bounds on the unknown cumulative distribution function (p-box) and without assuming a particular form of the distribution function. We give the formulas of the p-boxes for common situations (given combinations of data on minimum, maximum, median, mean, or standard deviation), describe an approach to propagate p-boxes into a black-box mathematical model, and introduce an approach for decision-making based on the results of PBA. We demonstrate the characteristics and utility of PBA vs PSA using two case studies. In sum, this study provides modelers with practical tools to conduct parameter uncertainty quantification given the constraints of available data and with the fewest assumptions.

Keywords: cost-effectiveness analysis; decision-analytic modeling; parameter uncertainty; probability bound analysis; probability box; uncertainty quantification.

FIGURE 1

P‐boxes (solid) with different $𝒟$s and a normal CDF (dashed). CDF: cumulative distribution function [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 2

Outer discretization approach for approximating the p‐box of the model parameter ${\theta }_i$. Sub‐figure A shows the sampling of an interval using the quasi‐inverse of the p‐box, given a particular sub‐interval in $[0,1]$. Sub‐figure B and C Show the accuracy of the approximation when using, $n_{{\theta }_i}=10$ and $n_{{\theta }_i}=50$, respectively [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 3

A state‐transition model diagram used in case study 1

FIGURE 4

Uncertainty around model outcome of 4‐state model using p‐boxes vs precise CDFs for $c_1$ and $c_6$. Sub‐figure A portrays the comparison of the uncertainties in the model outcome resulting from a p‐box and a gamma distribution. Each (.) corresponds to the different combination of available data on parameter (${\theta }_b$). As more information is available, the p‐box enclosing the unknown precise CDF becomes tighter. Sub‐figure B illustrates the comparison between a p‐box and a uniform distribution and demonstrates how p‐box is more honest in representing the uncertainty, given information only on the minimum and the maximum values of the model parameters. CDF: cumulative distribution function [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 5

The accuracies of the approximations of the p‐box of the model outcome as a function of the increasing numbers of sub‐intervals ($n_{{\theta }_i}$) (as indicated by the numbers in the parentheses) for each parameter in ${\theta }_b$, given data on a, b $\mu$, and $\sigma$. CDF: cumulative distribution function [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 6

Uncertainties around the incremental net monetary benefit of computer‐assisted vs conventional total knee replacement surgeries using (1) precise CDFs, (2) p‐boxes with published minimum and maximum values, and (3) p‐boxes with extreme minimum and maximum values. The dashed vertical line lies at the zero INMB. The ranges of plausible values are [$\pounds 42642,\pounds 226579]$, $[-\pounds 17025,$ $\pounds 290681]$, and $[-$ $\pounds 10509248,$ $\pounds 383278]$ for precise CDF, p‐box (published ranges), and p‐box (extreme ranges), respectively. CDF: cumulative distribution function [Colour figure can be viewed at wileyonlinelibrary.com]