Vector Of Regression Coefficients Research Articles

Matrix Derivation of Multiple Correlation

IN MANY textbooks on statistics for students in the behavioral sciences, a rather full discussion can be found of the type of problem situation and as sumptions involved in multiple correlation. How ever, their authors usually then present without derivation the and finally dis cuss computational techniques for solving these equations to obtain constants for a regression equa tion (3:406; 4:178; 5:237). Thus, for students of be havioral science a formal derivation of the normal equations expressed in terms of m atr ix algebra will serve to fill this logical gap. Assume that N persons have been measured on m predictor variables and one criterion variable. We can define the following vectors and m atr ices: j^isanNxi column vector of observed standard scores on the criterion, XisanNxm matrix of observed standard scores on the predictors, J3 is an m x 1 column vector of regression coefficients to be determined, and e is an N x 1 column vector of errors of estimate.

On the Bayesian Estimation of Multivariate Regression

Summary In this paper, we use a Bayesian approach to analyse sets of regression equations with correlated error terms. In the case that the matrix of “independent variables” is the same for all equations, our model reduces to the traditional multivariate regression model. For this case, the marginal posterior distribution of the regression coefficient vector for any equation is shown to be of the multivariate-t form. Further, the variances and covariances of the error terms have an “inverted” Wishart distribution a posteriori. Some properties of this distribution are given. Finally, the joint posterior distribution of the regression coefficients in the general model is derived and discussed.