## On the Equivalence Between the LRT, RLRT and F-test for Testing Variance Components in the Generalized Split-plot Models

Please use this identifier to cite or link to this item: http://hdl.handle.net/1928/25814

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Title

On the Equivalence Between the LRT, RLRT and F-test for Testing Variance Components in the Generalized Split-plot Models

Author(s)

Qeadan, Fares

Advisor(s)

Christensen, Ronald

Committee Member(s)

Lu, Yan

Sonksen, Michael

Kang, Huining

Sonksen, Michael

Kang, Huining

Department

University of New Mexico. Dept. of Mathematics and Statistics

Degree Level

Doctoral

Abstract

I study the equivalence between the Likelihood Ratio Test (LRT), Restricted Likelihood Ratio Test (RLRT) and the F-test when testing variance components within the class of generalized split-plot (GSP) models. In this work, I derive explicit expressions for both the maximum likelihood estimates (MLEs) and restricted maximum likelihood estimates (RMLEs) for the variance components of the GSP model and show the equivalence between the F-test, the LRT or the F-test and the RLRT when the level of the test, $\alpha$, is less or equal to one minus the probability, $p$, that the LRT or the RLRT statistic is zero. However, when $\alpha> 1-p$, I show that the F-test has a larger power than either the LRT or RLRT. Further, we derive the statistical distribution of these tests under both the null and alternative hypotheses $H_0$ and $H_1$ where $H_0$ is the hypothesis that the whole plot variance is zero.
To establish the power inequality for the case $\alpha> 1-p$, I developed a new stochastic inequality involving a class of distributions that includes, for example, the F and Gamma distributions. I call random variables (r.v.s.) that inherit this inequality to be quantile-stochastic. The stochastic representation of the new inequality involves $\alpha, p\in (0,1)$ such that if $p>\alpha$ and $k>1$ with $W$ being a random variable with an $F(\nu_1,\nu_2)$ or $Gamma(\tau,\theta)$ distribution then it's always true that
\[
\frac{1}{p}P\left(W<\frac{W_p}{k}\right)>\frac{1}{\alpha}P\left(W<\frac{W_{\alpha}}{k}\right),
\]
where $\gamma=P(W<W_\gamma)$. The inequality changes direction for $k\in [0,1)$ and becomes equality for $k=1$ and, trivially, for $k=\infty$.

Date

December 2014

Subject(s)

MLEs

REMLs

LRT

RLRT

F-Test

mixed model

variance components

stochastic inequality

quantile-stochastic

REMLs

LRT

RLRT

F-Test

mixed model

variance components

stochastic inequality

quantile-stochastic

##### Collections

- Statistics [16]