Hi Kohei,

My goal is to examine correlations of relative abundance of each OTU between airway and fecal microbiota within subjects (ideally by using the R phyloseq package).

- I think you can directly use the cor() command in the base package of R after you've processed your data. Have you done any statistical tests and normalization in your data?

Hi Kohei and others following this question.

I am not a biologist working on microbiome data but a statistician dealing with compositional data.

My feeling is that frequencies of OTUs (or other larger units) are compositional data. The reasons for this consideration are:

1. Relevant information conveyed by the data is contained in the ratios of frequencies of different OTU's, and the total counts is mainly irrelevant for the analysis as it depends mostly on the measurement procedure and other experimental circumstances.

2. Always a limited number of OTU's are analysed, and the analyst assumes that the conclusions and results should be coherent with analyses performed using a larger number of OTU's. In the compositional framework, we call this assumption "subcompositional coherence".

From the times of Pearson, and specially after J. Aitchison works (in the 80's), it is well known that correlation analysis on compositional data (raw abundances, proportions, concentrations, probabilities,...) fail dramatically as these analyses violate the "subcompositional coherence" principle. Obviously any inference based on the correlation analysis can be wrong or, at least, methodologically biased. Fortunately, from time to time with no fixed rule, one can attain sound conclusions, but, in general, they can be misleading.

For these reasons, I can not agree with the opinion of Edward Alain suggesting the use of standard correlation or covariance on raw data. As a co-author of the paper recommended by Malte, I agree with his opinion. However, there is not a definitive compositional methodology to treat associations (I avoid the name "correlation") between OTU's. We are working on it.

Some references follow:

Aitchison, J.:The Statistical Analysis of Compositional Data. ondon, Chapman & Hall Ltd. (Reprinted in 2003 with additional material by The Blackburn Press), 1986.

Pearson, K. (1897) Mathematical contributions to the theory of evolution. On a form of spurious correlation which may arise when indices are used in the measurement of organs.Proceedings of the Royal Society of London, LX, 489–502.

Pawlowsky-Glahn, V., J. J. Egozcue, and R. Tolosana-Delgado (2015). Modeling and analysis of compositional data. Statistics in practice. John Wiley & Sons, Chichester UK. 272 pp.

Thank you very much, Edward, Malte, and Juan!  I'll try to analyze the methods you suggested. Some of them seem challenging!