Empirical Bayes Analysis for Systems of Mixed Models with Linked Autocorrelated Random Effects

Abstract

Empirical Bayes concepts are implemented in a simultaneous analysis of a system of mixed linear models having linked and serially correlated random effects. Emphasis is placed on the estimation of the random effects and exploration of the relationships between them. Application is made to the investigation of several series of laboratory assay data that were observed during overlapping time intervals and were therefore subjected to common systematic errors, or “daily effects.” The motivation for this work was the need to investigate methods of adjustment for such daily effects, and to estimate the degree to which concurrently run series are impacted in common. Attention is given to the construction of confidence intervals for daily effects. Tractable methods are proposed that yield approximately correct coverage for large samples. Although derived within a Bayes-empirical Bayes framework, these intervals are somewhat similar to intervals constructed by the method of Kackar and Harville. Implementation of Type III bootstrap confidence intervals is also discussed. The expectation maximization (EM) algorithm provides a natural parameter estimation method because of its intimate relationship with the estimation of the posterior distribution of the unobservable effects. The E step is shown to be essentially equivalent to Kalman smoothing of daily sums of residuals within each of the linear models of the system, while the M step admits a complete decoupling of the system into its individual components, followed by standard least squares calculations.

Key Words:

• Bootstrap confidence intervals
• EM algorithm
• EM algorithm
• Empirical Bayes estimates of random effects
• Immunological assays
• Kalman filter
• Systems of mixed linear models