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I am looking for relationships between mass-spectrometry results (566 intensities of peaks common to all samples) and sample metadata (24 variables with various chemical information). There are only 11 samples; furthermore, I'm interested in any relationships inside the groups that the samples divide into (5 and 6 samples respectively), if there are any.

Of course, I find a lot of correlations (the relationships I am looking for are probably non-linear, so I'm using Spearman's $ \rho $). Of course, once I correct for multiple testing (I use p.adjust(p.values, method = 'BH') in R), all of them (save a few) vanish.

Is there a way to find significant relationships between many variables (as is the case with high-resolution mass spectrometry) in a small number of samples (the samples are somewhat hard to obtain and I cannot do that myself)? This starts looking fundamentally impossible, especially since it doesn't take a lot of tries to get cor(runif(6), runif(6), method='spearman') as high as $ 0.89 $.

ETA: Part of the problem is that the mass-spectrometry values are correlated, and so are the chemical metadata. Should I look for correlations in the principal components of the data? On the one hand, this would reduce the number of comparisons I am going to make and might help highlight important patterns in the data. On the other hand, PCA itself works by looking at covariance (okay, not correlation) between a lot of variables - why shouldn't it be subject to the same kind of correction as multiple testing is?

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  • $\begingroup$ With 11 samples, there is really no hope of estimating covariances, nor whitening the data. BH sound like a sound way to go. If nothing is significant, you may simply have no power. $\endgroup$
    – JohnRos
    May 29, 2019 at 10:26
  • $\begingroup$ Thank you for confirming that I'm not missing anything and the dataset design is unsuitable for the task. One thing is curious: the 6 correlations that remain significant after correction have a p value of exactly 0. $\endgroup$
    – aitap
    May 29, 2019 at 10:45

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