Nonparametric hyperrectangular tolerance and prediction regions for setting multivariate reference regions in laboratory medicine

Reference regions are widely used in clinical chemistry and laboratory medicine to interpret the results of biochemical or physiological tests of patients. There are well-established methods in the literature for reference limits for univariate measurements; however, limited methods are available for the construction of multivariate reference regions, since traditional multivariate statistical regions (e.g. confidence, prediction, and tolerance regions) are not constructed based on a hyperrectangular geometry. The present work addresses this problem by developing multivariate hyperrectangular nonparametric tolerance regions for setting the reference regions. The approach utilizes statistical data depth to determine which points to trim and then the extremes of the trimmed dataset are used as the faces of the hyperrectangular region. Also presented is a strategy for determining the number of points to trim based on previously established asymptotic results. An extensive coverage study shows the favorable performance of the proposed procedure for moderate to large sample sizes. The procedure is applied to obtain reference regions for addressing two important clinical problems: (1) assessing kidney function in adolescents and (2) characterizing insulin-like growth factor concentrations in the serum of adults.