epsGG

The Greenhouse-Geisser epsilon value measures by how much the sphericity assumption is violated. Epsilon is then used to adjust for the potential bias in the F statistic. Epsilon can be 1, which means that the sphericity assumption is met perfectly. An epsilon smaller than 1 means that the sphericity assumption is violated. The further it deviates from 1, the worse the violation; it can be as low as epsilon = 1/(k - 1), which produces the lower bound of epsilon (the worst case scenario). The worst case scenario depends on k, the number of levels in the repeated measure factor. In real life epsilon is rarely exactly 1. If it is not much smaller than 1, then we feel comfortable with the results of repeated measure ANOVA.
The Greenhouse-Geisser epsilon is derived from the variance-covariance matrix of the data. For its evaluation we need to first calculate the variance-covariance matrix of the variables (S). The diagonal entries are the variances and the off diagonal entries are the covariances. From this variance-covariance matrix, the epsilon statistic can be estimated. Also we need the mean of the entries on the main diagonal of S, the mean of all entries, the mean of all entries in row i of S, and the individual entries in the variance-covariance matrix. There are three important values of epsilon. It can be 1 when the sphericity is met perfectly. This epsilon procedure was proposed by Greenhouse and Geisser (1959).

Syntax: function epsGG(X)

Inputs:
X - Input matrix can be a data matrix (size n-data x k-treatments)

Output:
x - Greenhouse-Geisser epsilon value.