Degree Type

Dissertation

Date of Award

2012

Degree Name

Doctor of Philosophy

Department

Statistics

First Advisor

Douglas G. Bonett

Abstract

Methods for constructing confidence intervals for the variance components from a random effects model have important applications in a variety of disciplines. A fundamental analysis with random effects models is confidence intervals for the variance components or functions of the variance components. Many methods for constructing confidence intervals are currently being used. These methods work well under normality, equal variance, and equal sample size, but are very sensitive to any violations of these assumptions. This dissertation addresses the problem of constructing confidence intervals for variance components when the random effects or the errors are not normally distributed. The focus is on balanced one way random effects models and four parameters - the between group variance, the ratio of between to within group variance components, the intra-class correlation, and the ``stepped-up" reliability - are examined. All of our proposed methods replace the usual estimate of the standard error calculated under the assumption of normality with an estimate calculated under non-normality. For the between group variance, this estimate includes an estimate of the kurtosis of the distribution of the random effect. For the other three parameters, the standard error estimate includes estimates of both the kurtosis of the distribution of the random effect and the kurtosis of the distribution of the errors. If the researcher does not have any information about the distribution of the random effects or the errors, a general kurtosis estimate is used which is based on Pearson's kurtosis estimator, but with adjustments suggested by Bonett and Shoemaker. If it seems reasonable to assume the random effect or the errors follow a Beta or Gamma distribution, the kurtosis is estimated by first estimating the parameters of these distributions and then using the parameter estimates to estimate the kurtosis. If a previous study has been conducted, kurtosis estimates from the previous study can be pooled with the kurtosis estimates from the current study. Finally, if the researcher can theoretically specify a kurtosis value based on expert knowledge about their field of study, this specified kurtosis value can be used in place of an estimate. Our findings indicate that the proposed methods, especially those that incorporate a researcher's knowledge about the distributions of the random effect and the errors, perform better than the current methods when the normality assumption is violated.

DOI

https://doi.org/10.31274/etd-180810-174

Copyright Owner

Kari Kraemer

Language

en

File Format

application/pdf

File Size

75 pages

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