
A command to get the probability that some of the eigenvalues of the selected PCA object are equal. A low probability means that it is not very likely that that these numbers are equal.
We test the hypothesis H_{0}: λ_{from} = ... = λ_{to} that r (= to–from+1) of the eigenvalues λ of the covariance matrix are equal. The remaining eigenvalues are unrestricted as to their values and multiplicities. The alternative hypothesis to H_{0} is that some of the eigenvalues in the set are distinct.
The test statistic is:
χ^{2} = –n Σ_{j=from..to} ln eigenvalue[j] + n r ln (Σ_{j=from..to} eigenvalue[j] / r), 
with r(r+1)/2 –1 degrees of freedom. Here n = totalNumberOfCases – 1.
A special case occurs when the variation in the last r dimensions is spherical. In a slightly more conservative test we may replace n by n–from–(2r^{2}+r+2)/6r.
Also see Morrison (1990), page 336.
© djmw 19981102