A methodological proposal to estimate the correlation coefficient under range restriction and clustering

We propose a methodology for estimating the correlation coefficient between paired observations in the presence of range restrictions and clusters. We use a Bayesian nonparametric mixture model with Gaussian kernels to estimate the bivariate density. Within this model, we assume that the parameters of the Gaussian kernel follow a Dirichlet process, resulting in a flexible model that incorporates all available information. The overall correlation coefficient for the observations is estimated posteriorly from the components of the posterior variance-covariance matrix. Through a Monte Carlo study, we demonstrate that our methodology achieves the lowest Root Mean Squared Error (RMSE) compared to classical alternatives such as Pearson, Thorndike, FIML, and MICE. Additionally, we apply our methodology to real data obtained from one of the principal state universities in Colombia. In this application, our main objective is to estimate the relationship between admission test scores and grade-point averages.