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I have two matrices, A and B, of the same size. Each row of A corresponds to a row of B. Theory predicts that some rows of A will negatively correlate with corresponding rows of B.

I am interested in testing the general relationship between these matrices. That is to say, I want to know to what extent the theory is correct.

One way I thought of approaching this problem was as follows: calculate the row-wise correlation coefficients for the two matrices, and compare it with a similar vector of correlation coefficients calculated for randomly sampled rows from the two matrices. Compare the distributions with Kolmogorov-Smirnov test.

I have over 80000 correlation coefficients. K-S shows a significant difference between the real and sampled distribution, but the effect is small. On the other hand, I have clear outliers with almost perfect negative correlation in the former test, while not so in the sampled distribution.

What do you think?

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    $\begingroup$ Have you considered a permutation test? It sounds like you've done most of the work for a permutation test already. $\endgroup$ – Anthony Jan 14 '15 at 15:10
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    $\begingroup$ Have you considered comparing simple summaries (means, or even tallies) of the correlations for the real and random data. Is there something that you can think of that might be causing the perfect negative correlations? If so, is it operating to a lesser extent for other rows? Also, if you have mean and SD for the random data, you can identify coefficients that are unusually large (in either direction). Of course, making 80,000 such determinations argues for setting the alpha level quite low. $\endgroup$ – Joel W. Jan 15 '15 at 20:42
  • $\begingroup$ Are you interested in finding which rows have particularly strong ("significant") negative correlation, or do you only want to check/test how strong/significant the overall effect is? $\endgroup$ – amoeba Apr 6 '15 at 16:06

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