A computer programme for multivariate statistical analysis of ecological data

Abstract

A programme is discussed which carries out a sequence of ‘classical’ multivariate statistical analyses on the composite matrix [ Y X ], in which the elements of Y are measures of species abundance, the elements of X are measures of environmental properties, rows refer to species or environmental properties and columns to samples ( e.g. quadrats). The basic model assumes that [ Y X ] has a multivariate normal distribution, although it is shown that, even with appropriate transformations, this ideal situation is seldom realized in natural data. The use of the symbols X and Y implies that the dependence of biota ( Y ) on environment ( X ) may be expressed in linear equations comparable to the simple linear regression equation: γ = β 0+ β 1 x+ ε. It is shown that, in practice, a rigorous solution to a realistic linear model imposes substantial difficulties: however, certain multivariate solutions appear to have interpretive value, even though the criteria by which they are judged have an element of subjectivity. The principal operations in the programme are as follows; principal component analysis of Y and of X individually, followed by correlation analysis between principal components, and canonical correlation analysis of [ Y X ]. The operations, which are illustrated by a numerical example, can be described by simple matrix operations, the only relatively complex subroutine being the solution for eigenvalues and eigenvectors. The programme follows an integrated sequence in which the principal components analyses are earlier steps in the canonical correlation computation. In general it is found that matrices of zero-order correlations are of greater interpretive value (in the descriptive sense) than are matrices of partial regression or canonical correlation vectors. Tests of significance originated by Bartlett are incorporated in the programme, and where appropriate are used to reduce the dimensionality of matrices or, when one or both of the data matrices contain no significant information, the programme may be terminated.