Why does sphericity diagnosed by Bartlett’s Test mean a PCA is inappropriate?

I understand that Bartlett’s Test is concerned with determining if your samples are from populations with equal variances.

If the samples are from populations with equal variances, then we fail to reject the null hypothesis of the test, and therefore a principal component analysis is inappropriate.

I’m not sure where the problem with this situation (having a homoskedastic data set) lies. What is the problem with having a data set where the underlying distribution of all your data is the same? I just don’t see the big deal if this condition exists. Why would this make a PCA inappropriate?

I can’t seem to find any good information anywhere online. Does anyone have any experience with interpreting why this test is relevant to a PCA?

Answer

It appears that there are two tests called Bartlett’s test. The one you referenced (1937) determines whether your samples are from populations with equal variances. Another appears to test whether the correlation matrix for a set of data is the identity matrix (1951). It makes more sense that you wouldn’t run PCA on data with an identity correlation matrix, since you will just get back your original variables as they are already uncorrelated. Compare, e.g.,

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Source : Link , Question Author : tumultous_rooster , Answer Author : amoeba

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