On the challenge of treating various types of variables: application for improving the measurement of functional diversity Sandrine Pavoine, Jeanne Vallet, Anne-Be ´atrice Dufour, Sophie Gachet and Herve ´ Daniel S. Pavoine (sandrine.pavoine@zoo.ox.ac.uk), Muse´um National d’Histoire Naturelle, De´pt Ecologie et Gestion de la Biodiversite´, UMR 5173 MNHN-CNRS-P6, 55 rue Buffon, FR 75005 Paris, France. Present address: Dept of Zoology, Univ. of Oxford, South Parks Road, Oxford OX1 3PS, UK.  J. Vallet and H. Daniel, Agrocampus Ouest, Centre d’Angers, Inst. National d’Horticulture et de Paysage, UP Paysage, 2 rue LeNoˆtre, FR49045 Angers Cedex 01, France.  A.-B. Dufour, Lab. de Biome´trie et Biologie Evolutive (UMR 5558), CNRS, Univ. de Lyon, Univ. Lyon 1, 43 Bd du 11 novembre 1918, FR69622 Villeurbanne Cedex, France.  S. Gachet, Muse´um National d’Histoire Naturelle, De´pt Ecologie et Gestion de la Biodiversite´, 57 rue Cuvier, FR75005 Paris, France, and IMEP, Univ. Paul Ce´zanne, FR 13397 Marseille, France. Functional diversity is at the heart of current research in the field of conservation biology. Most of the indices that measure diversity depend on variables that have various statistical types (e.g. circular, fuzzy, ordinal) and that go through a matrix of distances among species. We show how to compute such distances from a generalization of Gower’s distance, which is dedicated to the treatment of mixed data. We prove Gower’s distance can be extended to include new types of data. The impact of this generalization is illustrated on a real data set containing 80 plant species and 13 various traits. Gower’s distance allows an efficient treatment of missing data and the inclusion of variable weights. An evaluation of the real contribution of each variable to the mixed distance is proposed. We conclude that such a generalized index will be crucial for analyzing functional diversity at small and large scales. The measurement of distances or similarities among groups of organisms has become a critical step in studies of functional ecology. This increase in interest is largely due to the growth in the number of studies tackling the concept of functional diversity in the last decades (Petchey and Gaston 2006) and to the way that functional diversity is measured. Functional traits of organisms, which are phenotypic traits that enable species to function in their ecosystem, have become fundamental entities for under- standing ecosystem processes and for predicting the con- sequences of environmental modifications, especially on a large scale due to global changes. Here, we consider functional diversity as the variety of states that several functional traits possess in natural conditions. Various methods for measuring functional diversity exist in the literature (reviewed by Petchey and Gaston 2006). The first method distributes species into functional groups (Walker 1992), and measures functional diversity as the number of functional groups in a given community. The Shannon (1948) or Simpson (1949) index can also be applied to the relative abundances of the groups. Others have proposed the sum and the average of distances between species (Walker et al. 1999, Heemsbergen et al. 2004). Petchey and Gaston (2002) suggested the sum of the branches in a dendrogram (coefficient FD), which can be built using the distances between species. Another alter- native is Rao’s (1982) quadratic entropy, which includes phenotypic distances among species and an estimation of their abundance (Botta-Duka ´t 2005). A critical step of all of these indices is defining a general measure of distances based on mixed data. Indeed, phenotypic traits must be measured, and depending on the instruments or experts involved, the variables will be either nominal, ordinal, interval or ratio-scale (Anderberg 1973). Moreover, there may be special cases of scale variable types, such as binary, circular and fuzzy. A potentially high number of statistical types of variables must be integrated and a measure flexible enough to apply to any statistical types of variables must be identified. Several coefficients of distance or similarity have been developed to handle mixed data sets (Estabrook and Rogers 1966, Gower and Legendre 1986, Carranza et al. 1998). We focused on Gower’s (1971) general measure of distance because Gower defined the measure in a mathematical framework associated with interesting properties of Euclidean distances. Gower (1971) proposed measuring a general similarity among entities from the following types of variables: quantitative (variables measured on the interval and ratio scale), nominal, and ‘dichotomous’ (presence/ absence variables). Although his paper was directed towards taxonomists, it has impacted a much larger audience. His measure has been used in a variety of fields, including taxonomy, medicine (Kosaki et al. 1996), genetics (Mohammadi and Prasanna 2003), morphometry (Loo Oikos 118: 391402, 2009 doi: 10.1111/j.1600-0706.2008.16668.x, # 2009 The Authors. Journal compilation # 2009 Oikos Subject Editor: Owen Petchey. Accepted 30 September 2008 391