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## Confidence Intervals on Variance Components in Mixed Linear Models

**EPA Grant Number:**U914815

**Title:**Confidence Intervals on Variance Components in Mixed Linear Models

**Investigators:**Purdy, Kathleen G.

**Institution:**

**Oregon State University**

**EPA Project Officer:**Broadway, Virginia

**Project Period:**January 1, 1995 through January 1, 1996

**Project Amount:**$102,000

**RFA:**STAR Graduate Fellowships (1995) Recipients Lists

**Research Category:**Fellowship - Mathematics , Environmental Statistics , Academic Fellowships

### Objective:

The objective of this research project is to determine optimal designs for obtaining efficient estimates of variance components.

### Approach:

This problem applies to environmental studies, particularly in developing a sampling protocol to minimize the cost of a study, while maximizing the precision of the estimates. Measuring the source and magnitude of components of variation has important applications in industrial, environmental, and biological studies. This project considers the problem of constructing confidence intervals for variance components in Gaussian mixed linear models. Several methods based on the usual ANOVA mean squares have been proposed for constructing confidence intervals for variance components in balanced mixed models. Some authors have suggested extending balanced model procedures to unbalanced models by replacing the ANOVA mean squares with mean squares from an unweighted means ANOVA. However, the unweighted means ANOVA is only defined for a few specific mixed models. We define a generalization of the unweighted means ANOVA for the three variance component mixed linear model, and illustrate how the mean squares from this ANOVA may be used to construct confidence intervals for variance components. Computer simulations indicate that the proposed procedure gives internal that are generally consistent with the stated confidence level, except in the case of extremely unbalanced designs.

### Supplemental Keywords:

*fellowship, mixed linear models, confidence intervals, variance components, balanced mixed models.*, RFA, Scientific Discipline, Ecosystem Protection/Environmental Exposure & Risk, Mathematics, Monitoring/Modeling, balanced mixed models, mixed linear models, confidence intervals, Gaussian mixed linear models