Identification of distributions for risks based on the first moment and c-statistic

TL;DR

This paper demonstrates that for certain distribution families on [0,1], the first moment and c-statistic uniquely determine the distribution, enabling parameter estimation from summary risk prediction statistics.

Contribution

It establishes theoretical conditions under which the distribution can be identified from the first moment and c-statistic, and develops numerical algorithms for parameter estimation.

Findings

Algorithms accurately estimate distribution parameters in simulations.

Applicable to risk prediction models for sample size and value of information calculations.

Validated methods for common risk distributions like beta and normal-based models.

Abstract

We show that for any family of distributions with support on [0,1] with strictly monotonic cumulative distribution function that has no jumps and is quantile-identifiable (i.e., any two distinct quantiles identify the distribution), knowing the first moment and c-statistic is enough to identify the distribution. The derivations motivate numerical algorithms for mapping a given pair of expected value and c-statistic to the parameters of specified two-parameter distributions for probabilities. We implemented these algorithms in R and in a simulation study evaluated their numerical accuracy for common families of distributions for risks (beta, logit-normal, and probit-normal). An area of application for these developments is in risk prediction modeling (e.g., sample size calculations and Value of Information analysis), where one might need to estimate the parameters of the distribution of predicted risks from the reported summary statistics.