Central tolerance regions and reference regions for multivariate normal populations

Reference intervals and regions are widely used to identify the measurement range expected from a reference population. Such regions capture the central part of the population, and have potential applications in the field of laboratory medicine. Furthermore, the uncertainty in an estimated reference region can be assessed using a central tolerance region, namely, a region that will contain the population reference region, with a specified confidence level. The construction of a central tolerance region is investigated in this article for a multivariate normal population, and also for a multivariate normal linear regression model. A theoretical framework is developed that will facilitate the numerical computation of the tolerance factor. The performance of a prediction region is also evaluated, in terms of capturing the central part of the population, and the prediction region is found to be unsatisfactory. Some examples from laboratory medicine are used to illustrate the results.