Generalized inference for mixed linear models

Abstract

Tsui and Weerahandi (1989) introduced the concept of a generalized p-value for a hypothesis test and subsequently Weerahandi (1993) introduced the concept of a generalized pivotal quantity for a scalar parameter. Generalized pivotal quantities allow one to construct confidence intervals for parameters where standard frequentist confidence intervals are not available. Generalized confidence intervals are not guaranteed to be exact in the frequentist sense. Also, Weerahandi did not provide a systematic way for constructing generalized pivotal quantities. In this work we discuss three techniques for constructing generalized pivotal quantities - a simple recipe, a two stage procedure and a general structural method - all motivated by Fraser's structural method. We apply the simple recipe to obtain generalized pivotal quantities for the variance components of a general balanced mixed linear model and, using a theorem in Hannig et al. (JASA, March 2006), verify the asymptotic exact coverage of the associated confidence intervals. In addition we consider several commonly occurring problems and for each one derive a generalized pivotal quantity and evaluate the performance of the associated confidence interval using a Monte Carlo simulation. The list of problems we consider is (a) tolerance intervals for the distributions of population characteristics of the unbalanced one-way random model (b) tolerance intervals for population characteristics of the two-way random effects model (c) the proportion of conformance for population characteristics of a single normal population (d) proportion of conformance for populations characteristics of the one-way random model and (e) the Common Mean problem. We verify that all the generalized confidence intervals in (a)-(e) have asymptotically exact frequentist coverage.