Rectangular multivariate normal prediction regions for setting reference regions in laboratory medicine

Reference intervals are among the most widely used medical decision-making tools and are invaluable in the interpretation of laboratory results of patients. Moreover, when multiple biochemical analytes are measured on each patient, a multivariate reference region (MRR) is needed. Such regions are more desirable than separate univariate reference intervals since the latter disregard the cross-correlations among variables. Traditionally, assuming multivariate normality, MRRs have been constructed as ellipsoidal regions, which cannot detect componentwise extreme values. Consequently, MRRs are rarely used in actual practice. In order to address the above drawback of ellipsoidal reference regions, we propose a procedure to construct rectangular MRRs under multivariate normality. The rectangular MRR is computed using a prediction region criterion. However, since the population correlations are unknown, a parametric bootstrap approach is employed for computing the required prediction factor. Also addressed in this study is the computation of mixed reference intervals, which include both two-sided and one-sided prediction limits, simultaneously. Numerical results show that the parametric bootstrap procedure is quite accurate, with estimated coverage probabilities very close to the nominal level. Moreover, the expected volumes of the proposed rectangular regions are substantially smaller than the expected volumes obtained from Bonferroni simultaneous prediction intervals. We also explore the computation of covariate-dependent MRRs in a multivariate regression setting. Finally, we discuss real-life applications of the proposed methods, including the computation of reference ranges for the assessment of kidney function and for components of the insulin-like growth factor system in adults.