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I have a population that I have quantitatively profiled using two different technologies. The result is two data matrices that have the exact same set of individuals as rows, but different features (both in type and number) as columns. I want to know if the two technologies are actually characterizing anything distinct about the samples, or if both approaches are basically giving us the same data. What is a statistic that will capture some measure of "similarity" or "mutual information" between two differently sized matrices? Implementation in R would be ideal.

I'm looking for something that would yield an extreme high score if one matrix was just a copy or subset of another, and an extreme low score if features in one matrix are independent or non-inferable from the other. Most similarity measures I've found can measure between vectors or between matrices of identical dimension, but not between an $N$x$M$ matrix and an $N$x$P$ matrix. I've considered looking at the distribution of pairwise correlations between features from one matrix to another, but most features will generally be uncorrelated even if one feature is perfectly copied from one matrix to another. Another thought was to look at the distribution of maximum feature-pairwise correlations to see if features from one dataset have any good correlate in the other (rather than on average). But what I'd really like would be a single number to quantify similarity of differently sized matrices ("similarity" of course open to interpretation).

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