Estimation of Underlying Normal Distribution Parameters From Dichotomized Data

Abstract

Accurately estimating the parameters of a continuous distribution from dichotomized or aggregated data is a common problem in biomedical and environmental research. Many studies report only the proportion of subjects exceeding a threshold, without releasing individual‐level measurements. To address this limitation, we develop a hierarchical binomial‐probit modeling framework to reconstruct the parameters of an underlying normal distribution from threshold‐based data. The framework considers two principal settings. In the fixed‐mean model, all studies are assumed to share a common mean and variance, and the parameters μ$\mu$ and σ$\sigma$ are estimated using a maximum‐likelihood estimator (MLE) and a generalized linear model (GLM) approximation. In the random‐mean model, each study has its own mean μi$\mu _i$ drawn from a population distribution N(μ0,τ2)$N(\mu _0,\tau ^2)$ with a common within‐study variance σ2$\sigma ^2$ ; parameters μ0$\mu _0$ , σ$\sigma$ , and τ$\tau$ are estimated using MLE, a generalized linear mixed model (GLMM) approximation, and a fully Bayesian Markov chain Monte Carlo (MCMC) method. Extensive simulations varying the number of studies, sample size, and heterogeneity ratio r=τ2/(σ2+τ2)$r=\tau ^2/(\sigma ^2+\tau ^2)$ evaluate estimator bias, variance, mean squared error, and coverage probability. Results show that MLE performs efficiently under well‐identified conditions, whereas GLMM and Bayesian estimators are more robust with small samples or strong heterogeneity. The proposed framework provides a unified and practical approach for inferring latent distributions from aggregated or privacy‐restricted data, with applications in clinical trial design, biomarker analysis, environmental monitoring, and quality control.