Abstract
The main purpose of this paper is to perform linear regression analysis on a continuous aggregate outcome from a Bayesian perspective using a Markov chain Monte Carlo algorithm (Gibbs sampling). In many situations, data are partially available due to privacy and confidentiality of the subjects in the sample. So, in this study, the vector of outcomes, Y, is realistically assumed to be missing and is partially available through summary statistics, sum(Y), aggregated over groups of subjects, while the covariate values, X, are availablefor all subjects in the sample. The results of the simulation study highlight both the efficiency of the regression parameter estimates and the predictive power of the proposed model compared with classicalmethods. The proposed approach is fully implemented in an example regarding systolic blood pressure for illustrative purposes.
Highlights
The relation among two or more observable quantities is the main concern of many scientific studies
The error term e follows the multivariate normal distribution, MVN 0, σ2In, where In is the identity n × n matrix, and Y is a vector of n outcomes
The improvement of the proposed model in reducing the M SE β1 is attributed to the practical importance of the truncation of the underlying multivariate normal distribution
Summary
The relation among two or more observable quantities is the main concern of many scientific studies. How a response outcome Y varies as function of a p−vector of quantities X. Both X and Y are available for each individual in the sample, let DC be the complete data set, where DC = {(Xi, Yi) , 1 ≤ i ≤ n }. The error term e follows the multivariate normal distribution, MVN 0, σ2In , where In is the identity n × n matrix, and Y is a vector of n outcomes. Based on this classical scenario, the standard statistical regression models can be used to draw meaningful statistical inferences about the regression parameters β and σ2
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