Lloyd J. Edwards, PhD, Charles R. Katholi, PhD, Byron C. Jaeger, PhD

UAB Department of Biostatistics

Title: Properties of $R^2_\beta$ and Tests of Hypotheses for Fixed Effects in the Linear Mixed Model

Abstract:

Properties of the $R^2$ statistic used in linear regression with least squares estimation are well developed. Additionally, the $R^2$ statistic used in linear regression has a corresponding test of hypoyhesis associated with it. Similar properties are developed for $R^2_\beta$, an $R^2$ statistic for fixed effects in the linear mixed model. The central and non-central Beta distributions are used to approximate the distribution of $R^2_\beta$ under corresponding null and alternative test hypotheses. The asymptotic expectation and variance of $R^2_\beta$ are derived. Simulations are used to demonstrate the performance of the Beta approximation. Test statistics are proposed that are based on estimators of the derived expectation and variance. Simulations are used to compare the test statistics to the overall $F$ test for fixed affects in the linear mixed model. Using simulations, the Type I error rate of the proposed $R^2_\beta$ test statistics is shown to be equivalent to the Type I error rate for the overall $F$ test.

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- The following have been awarded Scholarships for 2017-2018:
**Simon Harris**won the Bond Memorial Scholarship.**Kevin Campbell**won the Kauffman Scholarship.**Alyssa Wilke**won the Severin Scholarship.**Ahmed Ghatasheh**won the Ward Memorial Scholarship.**Jonathan Kelleher**won the Warner Scholarship.**Kaitlyn Lee**won the Travis Wood Memorial Scholarship.**Kenneth Davis**,**Garrett Higginbotham**,**Harrison Mansour**,**Sakar Prasain**, and**Norman Schmitz**won Department of Mathematics Scholarships.
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