ABSTRACT
In the analysis of a balanced mixedmodel, it may not be possible to obtain an exact F-test concerning a certain hypothesis from the corresponding ANOVA table. This occurs when no single mean square exists that can serve as an “error term” in the denominator of the test’s F-ratio. A common approach to this problem is to concoct a “synthetic error term” which consists of a linear combination of mean squares of random effects. It is also possible to construct an alternative test statistic by synthesizing both the numerator and the denominator of the F-ratio, that is, by creating two linear combinations of mean squares, one for the numerator and the other for the denominator. The choice of these linear combinations is based on requiring the numerator and denominator to have the same expected value under the null hypothesis to be tested. Under the alternative hypothesis, the expected value of the numerator exceeds that of the denominator by a positive constant. Each linear combination of mean squares of random effects is usually approximately represented as a scalar multiple of a chi-squared random variable whose number of degrees of freedom is estimated using the so-called Satterthwaite’s formula. This yields an F-ratio which has an approximate F-distribution. The whole process leadingup to this approximation is referred to asSatterthwaite’s approximation.
