Abstract
Bivariate continuous negatively correlated proportional data defined in the unit square (0,1)2 often appear in many different disciplines, such as medical studies, clinical trials and so on. To model this type of data, the paper proposes two new bivariate continuous distributions (i.e., negatively correlated proportional inverse Gaussian (NPIG) and negatively correlated proportional gamma (NPGA) distributions) for the first time and provides corresponding distributional properties. Two mean regression models are further developed for data with covariates. The normalized expectation–maximization (N-EM) algorithm and the gradient descent algorithm are combined to obtain the maximum likelihood estimates of parameters of interest. Simulations studies are conducted, and a data set of cortical thickness for schizophrenia is used to illustrate the proposed methods. According to our analysis between patients and controls of cortical thickness in typical mutual inhibitory brain regions, we verified the compensatory of cortical thickness in patients with schizophrenia and found its negative correlation with age.
Highlights
Accepted: 21 January 2022In many aspects, experimental results or measurements are reported in the form of ratios, scores, proportions or percentages, which is frequently encountered in sociology, psychology, epidemiology and clinical trials
Experimental results or measurements are reported in the form of ratios, scores, proportions or percentages, which is frequently encountered in sociology, psychology, epidemiology and clinical trials
The simplex distribution investigated by Zhang and Qiu [5] can be utilized to model such continuous proportional data, and they further pointed out the simplex regression model is more robust than the beta model
Summary
Experimental results or measurements are reported in the form of ratios, scores, proportions or percentages, which is frequently encountered in sociology, psychology, epidemiology and clinical trials. By mimicking the construction of beta distributions with gamma variates, Lijoiu et al [6] proposed a so-called normalized inverse Gaussian (IG) distribution by substituting the gamma variates with IG variates, as a new tool for modeling univariate proportional data. CepedaCuervo et al [14] defined a bivariate beta regression model from copulas and considered the Bayesian approach, in which the correlation could be positive or negative. PIG and PGA distributions, we will propose models to capture the negative correlation among components for multivariate proportional data. By combining the construction of multivariate PIG/PGA distributions and the negative correlation structure in beta/Dirichlet distributions, we define a new random vector x = ( X1 , X2 )> ∈ (0, 1) via the following SR: X1 =. Some technical details are put in the Appendices A and B, and others are shown in the Supplementary Material
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