I think a few topics get mixed up here.

Of course, a sample can be too small to be representative (as in Andrea's 
example), and one should think carefully about the measures to take. It is 
also clear that an increase in sample size reduces standard errors of 
statistical estimates, including that of a covariance matrix and its 
eigenvalues. But, as mentioned by Dean, the standard errors of the 
eigenvalues are of secondary interest in PCA.

If one has a clear expectation about the signal in the data - and if one 
does not aim at new discoveries - a few specific measurements may suffice, 
perhaps even a few distance measurements. But effective exploratory 
analyses have always been a major strength of geometric morphometrics, 
enabled by the powerful visualization methods together with the large 
number of measured variables.

Andrea, I am actually curious what worries you if one "collects between 
2700 and 10 400 homologous landmarks from each rib" (whatever the term 
"homologous" is supposed to mean here)? 

Compared to many other disciplines in contemporary biology and biomedicine, 
a few thousand variables are not particularly many. Consider, for instance, 
2D and 3D image analysis, FEA, and all the "omics", with millions and 
billions of variables. In my opinion, the challenge with these "big data" 
is not statistical power in testing a signal, but finding the signal - the 
low-dimensional subspace of interest - in the fist place. But this applies 
to 50 or 100 variables as well, not only to thousands or millions. If no 
prior expectation about this signal existed (which the mere presence of so 
many variables usually implies), no hypothesis test should be performed at 
all. The ignorance of this rule is one of the main reasons why so many GWAS 
and voxel-based morphometry studies fail to be replicable.

Best wishes,


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