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I work with datasets in which protein abundances are reported across samples. I have some measurements of biological samples that should be more or less equal in protein abundance.

After getting the data as intensities, they are normalized by subtracting the log$_2$ median of each sample. If I plot these normalized and log transformed intensities, I get a Pearson correlation of around 0.98. But when I divide each row by the row median and log$_2$ transform of the ratio, I get a Pearson correlation of about 0.3. I need to process my data by row normalizing them.

So I am clearly doing something very wrong when performing the correlation analysis after row normalization. How do I perform a correlation analysis on these values?

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It seems to me you're wandering towards correspondence analysis, which is based on modeling row normalized (and column normalized) measures in a low dimensional space. That literature should provide some principled normalization strategies. Greenacre has a very readable book on the topic:

Greenacre, Michael. Correspondence Analysis in Practice. Chapman and Hall/CRC, 2007.

There is also a related sub-literature on 'log-ratio' analysis that might be helpful.

For explicitly correlational questions, an algebraically identical (so far as I can see) strategy is called canonical correlation analysis. It's described in Section 9.6 of

Agresti, Alan. Categorical Data Analysis. 2nd ed. New York: Wiley-Interscience, 2002.

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