Comparisons of the Satterthwaite Approaches for Fixed Effects in Linear Mixed Models

Abstract

Four approximate F- tests derived by Fai and Cornelious in 1996 to make inference for fixed effects in mixed linear models of rank greater than one. Two of these approaches derived by introducing a Wald-type statistic distributed approximately as an F distribution, and the denominator degrees of freedom computed by matching the approximated one moment of the Wald-type statistic with the exact one moment of the F distribution. The other two approaches were derived by introducing a scaled Wald-type statistic to be distributed approximately as an F distribution, and the denominator degrees of freedom and the scale factor computed by matching the two moments of the statistic with the moments of the F distribution. This paper proposes two more approximate F-tests analogous to the four approaches where an adjusted estimator of the variance of the estimate of fixed effects used. In addition, the paper evaluates and compares the performance of the six approaches analytically, and some useful results are presented. Also, a simulation study for block designs was run to assess and compare the performance of the approaches based on their observed test levels. The simulation study shows that the approaches usually perform reasonably based on their test levels, and in some cases some approaches found to more adequately than other approaches.