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

Michael Daniel Lucagbo, Thomas Mathew, Derek S. Young

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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.

Original languageEnglish
Pages (from-to)191-209
Number of pages19
JournalJournal of Biopharmaceutical Statistics
Issue number2
StatePublished - 2023

Bibliographical note

Funding Information:
This work is made possible in part by the University of the Philippines Faculty, REPS, and Administrative Staff Development Program, under which the first author has been a recipient of a doctoral fellowship. The authors are also grateful to a reviewer for drawing our attention to the Yang and Kolassa () paper.

Publisher Copyright:
© 2022 Taylor & Francis Group, LLC.


  • Bonferroni correction
  • mixed reference intervals
  • multivariate reference region
  • parametric bootstrap
  • rectangular prediction region

ASJC Scopus subject areas

  • Statistics and Probability
  • Pharmacology
  • Pharmacology (medical)


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