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Usually the expression data is transformed to log space using either RPKM, FPKM or CPM, this is required when looking for differential expression because the data is tested against the normal distribution(limma) or the negative bionimal distribution (DESeq2).

In weighted gene co-expression analysis (WGCNA) there is also the recommendation to normalize the data (see FAQ 4) but says "RPKM, FPKM, or simply normalized counts doesn't make a whole lot of difference for WGCNA" because it is based in correlations.

For PCAs the data should be on the same scale to avoid missing the scale as a factor. See this for more references. But should it be normalized too to look like a normal distribution using logarithms?

How does this extend to other methods like CCA? Should the data look like a normal distribution for CCA? As it is based on correlation I don't expect much changes, so I am using raw counts (RSEM output), but the examples I've seen use normalized data.

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I've given an answer about using how I normalise for PCA here. PCA works best if the data are normal-ish, but as models go it is fairly robust to fairly large deviations from normal. Transforming is not essential, but if data can be easily transformed into something that looks a bit more normal, then conclusions derived from looking at the PCA should be a bit more robust.

I'm not familiar with canonical-correlation analysis (CCA), but because I see sums and square roots in the formulas, I would assume it has similar data cleaning requirements to PCA. Unfortunately the R documentation for cancor doesn't have any information about any assumptions made on the data.

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