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?
 A: 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.,


*

*http://en.wikipedia.org/wiki/Bartlett's_test to

*https://personality-project.org/r/html/cortest.bartlett.html.

A: In response to the question title.
Bartlett's test of sphericity$^1$, which is often done prior PCA or factor analysis, tests whether the data comes from multivariate normal distribution with zero covariances. (Note please, that the standard asymptotic version of the test is not at all robust to the departure from multivariate normality. One might use bootstrapping with nongaussian cloud.) To put it equivalently, the null hypothesis is that the population correlation matrix is identity matrix or that the covariance matrix is diagonal one.
Imagine now that multivariate cloud is perfectly spherical (i.e. its covariance matrix is proportional to the identity matrix). Then 1) any arbitrary dimensions can serve principal components, so PCA solution is not unique; 2) all the components have the same variances (eigenvalues), so PCA cannot help to reduce the data.
Imagine the second case where multivariate cloud is ellipsoid with oblongness strictly along the variables' axes (i.e. its covariance matrix is diagonal: all values are zero except the diagonal). Then the rotation implied by PCA transformation will be zero; principal components are the variables themselves, only reordered and potentionally sign-reverted. This is a trivial result: no PCA was needed to discard some weak dimensions to reduce the data.

$^1$ Several (at least three, to my awareness) tests in statistics are named after Bartlett. Here we are speaking of the Bartlett's sphericity test.
