On easily interpretable multivariate reference regions of rectangular shape

Till now, multivariate reference regions have played only a marginal role in the practice of clinical chemistry and laboratory medicine. The major reason for this fact is that such regions are traditionally determined by means of concentration ellipsoids of multidimensional Gaussian distributions yielding reference limits which do not allow statements about possible outlyingness of measurements taken in specific diagnostic tests from a given patient or subject. As a promising way around this difficulty we propose to construct multivariate reference regions as p-dimensional rectangles or (in the one-sided case) rectangular half-spaces whose edges determine univariate percentile ranges of the same probability content in each marginal distribution. In a first step, the corresponding notion of a quantile of a p-dimensional probability distribution of any type and shape is made mathematically precise. Subsequently, both parametric and nonparametric procedures of estimating such a quantile are described. Furthermore, results on sample-size calculation for reference-centile studies based on the proposed definition of multivariate quantiles are presented generalizing the approach of Jennen-Steinmetz and Wellek.