Greenhouse-Geisser Correction

When performing an analysis of variance with a one-factor, repeated-measurement design, the effect of the independent variable is tested by computing an F statistic, which is computed as the ratio of the mean square of effect by the mean square of the interaction between the subject factor and the independent variable. For a design with S subjects and A experimental treatments, when some assumptions are met, the sampling distribution of this F ratio is a Fisher distribution with ν1 = A − 1 and ν2 = (A − 1)(S − 1) degrees of freedom.

In addition to the usual assumptions of normality of the error and homogeneity of variance, the F test for repeated-measurement designs assumes a condition called sphericity. Intuitively, this condition indicates that the ranking of the subjects does not change across experimental treatments. This is equivalent to stating that the population correlation (computed from the subjects’ scores) between two treatments is the same for all pairs of treatments. This condition implies that there is no interaction between the subject factor and the treatment.

If the sphericity assumption is not valid, then the F test becomes too liberal (i.e., the proportion of rejections of the null hypothesis is larger than the a level when the null hypothesis is true). In order to minimize this problem, Seymour Greenhouse and Samuel Geisser, elaborating on early work by G. E. P. Box, suggested using an index of deviation to sphericity to correct the number of degrees of freedom of the F distribution. This entry first presents this index of nonsphericity (called the Box index, denoted?), and then it presents its estimation and its application, known as the Greenhouse-Geisser correction. This entry also presents the Huyhn-Feldt correction, which is a more efficient procedure. Finally, this entry explores tests for sphericity.

Index of Sphericity

Box has suggested a measure for sphericity, denoted ∊, which varies between 0 and 1 and reaches the value of 1 when the data are perfectly spherical. The computation of this index is illustrated with the fictitious example given in Table 1 with data collected from S = 5 subjects whose responses were measured for A = 4 different treatments. The [Page 545]standard analysis of variance of these data gives a value of

, which, with ν1 = 3 and ν2 = 12, has a p value of .014.

In order to evaluate the degree of sphericity, the first step is to create a table called a covariance matrix. This matrix is composed of the variances of all treatments and all the covariances between treatments. As an illustration, the covariance matrix for our example is given in Table 2.

Box defined an index of sphericity, denoted ∊, which applies to a population covariance matrix. If we call ζaa′ the entries of this A × A table, the Box index of nonsphericity is obtained as

Box also showed that when sphericity fails, the number of degrees of freedom of the FA ratio depends directly upon the degree of nonsphericity and is equal to · ν1 = ∊ (A-1)and ν2 = ∊(A-1)(S-1).

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