Maths implications: log2 transformation before or after normalisation

Question

[gablenord]

Hi guys,

A (hopefully) quite straightforward question:

What are the different implications of log2 transforming variables before or after performing normalisation (for example quantile norm) on a dataset?

I have in mind microarray gene expression data but I guess the question would stand for any type of data as well.

I found contradictory sources aorund, from norm functions (in R packages) that even expect log2 data input to people stating log2 transformation MUST be done after normalisation, I would like to understand the implications better.

Thanks,

[Martombo]

Well it depends on the normalization, if it's quantile normalization it should not make a difference if you log transform before or after, provided you don't have negative numbers.

[ATpoint]

See Normalisation before log2 transformation or after in Microarray Gene expression data?

[gablenord]

I had already read that, but I found it more prescriptive than descriptive, I was interested more in the why, not in the how :)

[Kevin Blighe]

Given the diversity of microarray designs and detection systems for each, I'm not surprised that you have come across seemingly contradictory material online.

As an example, for two-colour arrays, the 'raw' signal intensities are log (base 2) ratios between the cDNA in the test and reference samples - these are then further normalised and kept on the log (base 2) scale. Agilent produces most if not all of these two-colour arrays, I believe.

For the Affymetrix and Illumina arrays, the raw data is just fluorescent signal intensity from whatever detection system that they are using, so, it's not yet logged.

Answer 1

[jomo018]

Log2 is monotonic but a non-linear transformation. The ratios between elements in a sample are not kept. Once performed, a downstream linear operation such as depth normalization is less appropriate. In such cases, it makes more sense to begin with the linear operations and end with non-linear ones. Quantile normalization is also non-linear. In this case, both workflow arrangements are reasonable noting that right from start, ranking is kept but not the original ratios.

[gablenord]

Thank you Jomo, that's what I wanted to know!

[Kevin Blighe]

I moved this to an answer. Just to 'sure it up', the two approaches do produce different end results:

mat <- matrix(c(5,2,3,4,4,1,4,2,3,4,6,8), ncol = 3)

log2(preprocessCore::normalize.quantiles(mat))
         [,1]     [,2]     [,3]
[1,] 2.502500 2.369234 1.000000
[2,] 1.000000 1.000000 1.584963
[3,] 1.584963 2.369234 2.222392
[4,] 2.222392 1.584963 2.502500

preprocessCore::normalize.quantiles(log2(mat))
          [,1]      [,2]      [,3]
[1,] 2.4406427 2.3178151 0.8616542
[2,] 0.8616542 0.8616542 1.5283208
[3,] 1.5283208 2.3178151 2.1949875
[4,] 2.1949875 1.5283208 2.4406427

[gablenord]

Thanks Kevin. I did some tests myself on my data and I saw that, while the actual values are indeed slightly different as you showed, the results coming from downstream analysis (unsupervised clustering, PCA, paired t-tests,...) are basically totally comparable, if not the same. I was therefore interested in understanding if, apart from the specific empirical experience, there was a "mathematical" reason why one way should be preferred over the other and why.

From @jomo018 answer, I got that the choice should be based (mostly?) on the linear/non-linear nature of the transformation methods involved, so that there would be no "wrongs" in going either way in case of log2 and quantile normalisation (being both non-linear), while there would be a problem in case of other linear normalisations for example.

Did I get it right?

Thanks