Central Tolerance Regions and Reference Regions for Some Normal Populations

Abstract

A reference interval represents a range of values that a physician can use in order to interpret a single test result from a patient. A 95% reference interval is simply the interval from the 2.5th to the 97.5th percentiles of the distribution of the test result. Since such an interval will typically depend on unknown parameters, we can use a random sample to compute an interval that will contain the reference interval with a specified confidence level. The interval so computed is referred to as a central tolerance interval. A central tolerance interval captures, with a given confidence level, a specified percentage of the central part of a univariate population. Reference regions and central tolerance regions are similarly defined for a multivariate population. This thesis considers the problem of deriving central tolerance intervals and regions for some normal populations, including multivariate normal distribution, multivariate linear regression model and the one-way and two-way random models. We also numerically investigate the computation of a simultaneous central tolerance interval for simple linear regression. Simultaneous hypotheses testing of two percentiles is a related problem of interest. The test is usually carried out using two one-sided tests which can be quite conservative. We make an attempt to derive a less conservative test by applying the bootstrap methodology. Numerical examples and real data applications are given to illustrate all of the proposed procedures.