In this work, we derive an alternative analytic expression for the covariance matrix of the regression coefficients in a multiple linear regression model. In contrast to the well-known expressions which make use of the cross-product matrix and hence require access to individual data, we express the covariance matrix of the regression coefficients directly in terms of covariance matrix of the explanatory variables. In particular, we show that the covariance matrix of the regression coefficients can be calculated using the matrix of the partial correlation coefficients of the explanatory variables, which in turn can be calculated easily from the correlation matrix of the explanatory variables. This is very important since the covariance matrix of the explanatory variables can be easily obtained, or imputed using data from the literature, without requiring access to individual data. Two important applications of the method are discussed, namely in the multivariate meta-analysis of regression coefficients and the so-called synthesis analysis, the aim of which is to combine in a single predictive model, information from different variables. The estimator proposed in this work can increase the usefulness of these methods providing better results, as seen by application in a real dataset. Source code is provided in the Appendix.
If you use this software, please cite:
Bagos PG, Adam M. On the Covariance of Regression Coefficients. Open Journal of Statistics. 2015, 5, 680-701 [HTML] [PDF]