New expectation–maximization-type algorithms via stochastic representation for the analysis of truncated normal data with applications in biomedicine

Abstract

To analyze univariate truncated normal data, in this paper, we stochastically represent the normal random variable as a mixture of a truncated normal random variable and its complementary random variable. This stochastic representation is a new idea and it is the first time to appear in literature. According to this stochastic representation, we derive important distributional properties for the truncated normal distribution and develop two new expectation–maximization algorithms to calculate the maximum likelihood estimates of parameters of interest for Type I data (without and with covariates) and Type II/III data. Bootstrap confidence intervals of parameters for small sample sizes are provided. To evaluate the performance of the proposed methods for the truncated normal distribution, in simulation studies, we first focus on the comparison of estimation results between including the unobserved data counts and excluding the unobserved data counts, and we next investigate the impact of the number of unobserved data on the estimation results. The plasma ferritin concentration data collected by Australian Institute of Sport and the blood fat content data are used to illustrate the proposed methods and to compare the truncated normal distribution with the half normal, the folded normal, and the folded normal slash distributions based on Akaike information criterion and Bayesian information criterion.