Co-inertia analysis: an alternative method for studying species–environment relationships

• 1 Methods used for the study of species–environment relationships can be grouped into: (i) simple indirect and direct gradient analysis and multivariate direct gradient analysis (e.g. canonical correspondence analysis), all of which search for non-symmetric patterns between environmental data sets and species data sets; and (ii) analysis of juxtaposed tables, canonical correlation analysis, and intertable ordination, which examine species–environment relationships by considering each data set equally. Different analytical techniques are appropriate for fulfilling different objectives.
• 2 We propose a method, co-inertia analysis, that can synthesize various approaches encountered in the ecological literature. Co-inertia analysis is based on the mathematically coherent Euclidean model and can be universally reproduced (i.e. independently of software) because of its numerical stability. The method performs simultaneous analysis of two tables. The optimizing criterion in co-inertia analysis is that the resulting sample scores (environmental scores and faunistic scores) are the most covariant. Such analysis is particularly suitable for the simultaneous detection of faunistic and environmental features in studies of ecosystem structure.
• 3 The method was demonstrated using faunistic and environmental data from Friday (Freshwater Biology 18, 87-104, 1987). In this example, non-symmetric analyses is inappropriate because of the large number of variables (species and environmental variables) compared with the small number of samples.
• 4 Co-inertia analysis is an extension of the analysis of cross tables previously attempted by others. It serves as a general method to relate any kinds of data set, using any kinds of standard analysis (e.g. principal components analysis, correspondence analysis, multiple correspondence analysis) or between-class and within-class analyses.