Box–Cox-chord transformations for community composition data prior to beta diversity analysis
Legendre Pierre, Borcard Daniel
In studies of spatial or temporal beta diversity, community composition data, often containing many zeros, must be transformed in some way before they are analysed by multivariate methods of data analysis. Data are transformed to reduce the skewness of species distributions and make dissimilarities double-zero asymmetrical. Criteria have recently been proposed to determine which dissimilarity functions (or the corresponding data transformations) can be used for beta diversity assessment. The chord transformation is often used as the preliminary transformation for frequency data. When the Euclidean distance is computed on chord-transformed data, a chord dissimilarity matrix D is produced, which obeys the proposed criteria. The Hellinger transformation, i.e. the chord transformation applied to square-root transformed frequencies, is also often used with community composition data prior to multivariate analyses; it leads to the Hellinger dissimilarity, which is another widely used D function in beta diversity studies. Among the data transformations often used in simple or multivariate data analysis, the Box–Cox method provides a useful series of transformations to make data distributions more symmetrical, where exponent 1 is the absence of a transformation, exponent 0.5 is the square-root, exponent 0.25 is the fourth-root, and the log transformation is the limit of the Box–Cox function corresponding to exponent 0. Combining the two previous ideas, this paper proposes to combine any transformation of the Box–Cox family with exponent in the [0,1] range with the chord transformation. In particular, one can compute the loge(y + 1) transformation of a community composition (or other frequency) data table and follow with a chord transformation. A D matrix can be computed from the doubly-transformed data. The transformations and D functions in that family inherit the properties of the chord dissimilarity, and this ensures that they all follow the necessary criteria for beta diversity assessment that have recently been proposed.