Birge ratio method for modeling dark uncertainty in multivariate meta-analyses and inter-laboratory studies

In the paper, we introduce a new approach for combining multivariate measurements obtained in individual studies. The procedure extends the Birge ratio method, a commonly used approach in physics in the univariate case, such as for the determination of physical constants, to multivariate observations. Statistical inference procedures are derived for the parameters of the multivariate location-scale model, which is related to the multivariate Birge ratio method. The new approach provides an alternative to the methods based on the application of the multivariate random effects model, which is commonly used for multivariate meta-analyses and inter-laboratory comparisons. In two empirical illustrations, we show that the introduced multivariate Birge ratio approach yields confidence intervals for the elements of the overall mean vector that are considerably narrower than those obtained by the methods derived under the multivariate random effects model.