Estimation of Underlying Normal Distribution Parameters From Dichotomized Data

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 $\mu$ and $\sigma$ 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 ${\mu }_i$ drawn from a population distribution $N({\mu }_0,{\tau }^2)$ with a common within-study variance ${\sigma }^2$ ; parameters ${\mu }_0$ , $\sigma$ , and $\tau$ 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 $r={\tau }^2/({\sigma }^2+{\tau }^2)$ 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.