Folded normal regression models with applications in biomedicine

In biomedical studies, a difference or deviation is usually measured and only the magnitude is recorded but the algebraic sign of the data is irretrievably lost, the resulting observed variable no longer follows a normal distribution, rather it follows a folded normal (FN) distribution. More importantly, the FN distribution could be used to fit data sets with the following two characteristics: (i) The density curve is similar to the normal density but truncated somewhere; (ii) The density curve of the truncated side is significantly higher than that of the other side. There are several issues on the statistical inferences with the FN distribution which are not (well) addressed in the existing literature. In this paper, starting from the stochastic representation, we develop a new expectation–maximization (EM) algorithm to calculate the maximum likelihood estimates of parameters in both FN distribution and the FN regression models. The EM structure can also facilitate the Bayesian inferences about the FN distribution and the FN regression models. Extensions to the generalized FN distribution are provided. Simulation studies are conducted to assess the estimation performances for the FN distribution and the FN regression model. Two real data sets are analyzed to illustrate the proposed methods.