Rectangular tolerance regions and multivariate normal reference regions in laboratory medicine.

Reference intervals are widely used in the interpretation of results of biochemical and physiological tests of patients. When there are multiple biochemical analytes measured from each subject, a multivariate reference region is needed. Because of their greater specificity against false positives, such reference regions are more desirable than separate univariate reference intervals that disregard the cross-correlations between variables. Traditionally, under multivariate normality, reference regions have been constructed as ellipsoidal regions. This approach suffers from a major drawback: it cannot detect component-wise extreme observations. In the present work, procedures are developed to construct rectangular reference regions in the multivariate normal setup. The construction is based on the criteria for tolerance intervals. The problems addressed include the computation of a rectangular tolerance region and simultaneous tolerance intervals. Also addressed is the computation of mixed reference intervals that include both two-sided and one-sided limits, simultaneously. A parametric bootstrap approach is used in the computations, and the accuracy of the proposed methodology is assessed using estimated coverage probabilities. The problem of sample size determination is also addressed, and the results are illustrated using examples that call for the computation of reference regions.

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.