PCA: Get equality of eigenvalues...

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 H0: λfrom = ... = λto that r (= tofrom+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 H0 is that some of the eigenvalues in the set are distinct.

Settings

Eigenvalue range
define the range of eigenvalues to be tested for equality.
Conservative test
when on, a more conservative estimate for n is chosen (see below).

Details

The test statistic is:

χ2 = –n Σj=from..to ln eigenvalue[j] + n r ln (Σj=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 nfrom–(2r2+r+2)/6r.

Also see Morrison (1990), page 336.

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© djmw, November 2, 1998