The R2 statistic has become a widely used tool when analyzing data in the linear univariate setting. Many R2 statistics for the linear mixed model exist but their properties are not well established. The purpose of this dissertation is to examine the properties and performance of R2β for fixed effects in the linear mixed model.
Two approaches are considered in deriving approximations for the mean and variance of R2β under the null and alternative hypotheses which include using the Beta distribution and a Taylor series approximation. Test statistics based on these two approximations of the mean and variance are proposed and compared to the overall F test for fixed effects in the linear mixed model. Using simulations, the Type I error rate of the proposed R 2β test statistics derived from the Beta distribution was equivalent to the Type I error rate for the overall test. The Type I error rates for the test statistic based on the Taylor series approximation moments were slightly inflated.
The impact of covariance structure misspecification, estimation technique, and denominator degrees of freedom method on the asymptotic properties of R2β are explored. For the simulation studies examined, the estimation technique does not impact the values of R 2β. The values and asymptotic properties of R 2β using Kenward-Roger, containment and Satterthwaite methods are greatly impacted by covariance structure misspecification whereas R2β using the residual method is not. Simulations illustrate the impact of underspecification of the covariance structure as compound symmetric when the true structure is more complex. The asymptotic R2β's for the underspecified models using Kenward-Roger degrees of freedom are smaller than the true asymptotic R2 β's. Conversely, the asymptotic R2 β's for the underspecified models using residual methods are larger than the true asymptotic R2β.
The semi-parital R2β for the four denominator degrees of freedom are computed and compared to the corresponding model R 2β in both a real world example and simulation study. The semi-partial R2β using residual degrees of freedom never exceeded the model R2β, but the semi-partial R2β using the other three methods sometimes exceeded the model R2β. R2β is also evaluated as a fixed effects model selection tool. The performance of R2β is poor; so an adjusted R2β is created for purposes of fixed effects model selection. The adjusted R2β using residual degrees of freedom outperformed the adjusted R 2β defined using the other methods.
|Advisor:||Edwards, Lloyd J.|
|Commitee:||Bowling, James M., Preisser, John, Qaqish, Bahjat F., Sen, Pranab K.|
|School:||The University of North Carolina at Chapel Hill|
|School Location:||United States -- North Carolina|
|Source:||DAI-B 73/06, Dissertation Abstracts International|
|Keywords:||Linear mixed model, Linear univariate setting, Longitudinal data, R square statistic|
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