Title: A COMPARATIVE STUDY OF NOVEL METHODS FOR VARIANCES

Program: Mathematics MS

Committee Chair: Kyungduk Ko

Committee Co-Chair: Juna Goo

Committee: Kyungduk Ko, Juna Goo, Michael Perlmutter

Abstract: Understanding variability is fundamental to knowledge advancement across various
disciplines, including manufacturing, clinical research, biology, and genetics. Popu-
lation variances play a crucial role in processes such as quality control, treatment
evaluation, and the interpretation of biological mechanisms. However, statistical pro-
cedures for comparing variances across multiple populations remain less developed
than those for comparing means. Classical omnibus tests, such as Bartlett’s and Lev-
ene’s, can detect overall variance heterogeneity but fail to identify which populations
differ, thereby limiting their usefulness in multi-population analyses and increasing
the risk of inflated family-wise error rates through repeated testing.
This thesis develops and evaluates three multiple comparison procedures designed
to overcome these limitations. The Levene–Tukey-type procedure integrates absolute
deviations from population means with Tukey’s multiple comparison framework. The
F-ratio-type procedure extends the conventional F-test to multi-population contexts,
enhancing sensitivity while maintaining control of type I error. The Hartley’s F-max-
type procedure employs a ratio-based and stepwise elimination approach to identify
groups contributing to variance heterogeneity.
The proposed methods are assessed through extensive simulation studies under
varying sample sizes, variance structures, and distributional assumptions, including
non-normal data. Their practical relevance is further demonstrated using real-world
data. Collectively, these procedures advance statistical methodology by providing
robust, interpretable, and accessible tools for analyzing variance across populations,
thereby improving the reliability of applied statistical inference.