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A Comparison of Permutation Hotelling's T 2 Test and Log-Ratio Test for Analyzing Compositional Data

Compositional data can be identified as a vector X with non negative elements x 1, x 2,…, x D satisfying the constraint x 1 + x 2 + … + x D = 1. Such data play an important role in many disciplines such as medicine, geochemistry, and sedimentology and often display appreciable variability from vector to vector. It may be noted that because of the constraint, stringent range restrictions are imposed on the correlation structure. Thus, analyses of such data require special attention and an excellent account of various procedures can be found in Aitchison (2003) and Aitchision et al. (2005). In this article, we first present a motivating example and then conduct an empirical comparison of two methods namely the permutation Hotelling's T 2 and the analysis based on log-ratios. We find that the two methods give almost identical results but the permutation Hotelling's T 2 is preferred since one does not have to assume the underlying distribution to be multivariate normal and one does not need to worry about log-ratios blowing up if the divisor takes values close to zero. Further, using an example, we show that use of log-ratio can, at times, lead to misleading conclusions and one must proceed with caution.