Box–Cox transformations in the analysis of compositional data

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Abstract
The statistical analysis of compositional data is of fundamental importance to practitioners in general and to chemists in particular. The existing methodology is principally due to Aitchison, who effectively uses two transformations, a ratio followed by the logarithmic, to create a useful, coherent theory that in principle allows the plethora of normal‐based multivariate techniques to be used on the transformed data. This paper suggests that the well‐known class of Box–Cox transformations can be employed in place of the logarithmic to significantly improve the existing methodology. This is supported in part by showing that one of the most basic problems that Aitchison managed to overcome, namely the specification of an interpretable covariance structure for compositional data, can be resolved, or nearly resolved, once the ratio transformation has been applied. Hence the resolution is not directly dependent on the logarithmic transformation. It is then verified that access to the general Box–Cox family will allow a more accurate use of the normal‐based multivariate techniques, simply because better fits to normality can be achieved. Finally, maximum likelihood estimation and some associated asymptotics are employed to construct confidence intervals for ratios of the true, unknown compositional constituents. Heretofore this had not been done even in the context of the logarithmic transformation. Applications to real data are presented.