Multivariate Analytical Treatment of Quantitative Species Associations: An Example from Palaeoecology

Abstract

A common observation made by micropalaeontologists in analysing series of borehole samples is that the proportions of species may fluctuate widely from sample to sample. It would not seem unreasonable to assume that the oscillations in numbers of individuals were caused by ecological factors. Although it seems hardly likely that the palaeoecologist will ever be in a position to identify such factors specifically, the statistical methods of principal component analysis and factor analysis, applied to frequency data, would seem to be suitable for supplying a picture of the order of complexity of the environment in which these species lived, and their reactions to it. Both methods attempt to divide up the total variation into entities, which one tries to provide with a meaningful interpretation. Both were originally devised for application to psychometric problems; factor analysis has been used for some 30 years in the analysis of biological variation, while principal component analysis has only more recently been applied to such problems. Numerous examples of the taxonomic use of principal component analysis with respect to the ostracod carapace are given in Reyment (1963). A discussion of factor analysis as applied to an ecological problem in botany, comparable to that treated in the present paper, is to be found in Greig-Smith (1957, pp. 147-9, 159, 161-3). A general text on multivariate statistical analysis, which contains a detailed account of the statistical theory involved in the following, is that of Kendall (1957). Ostracods are useful indicators in palaeoecology as they are amongst the most common animal fossils in borehole samples, they are benthonic (in, on, or close to the substratum) and, as opposed to their greatest potential rivals, the Foraminifera, they occur in a wide range of aquatic environments. A note on the interpretation of the mathematical results might be in place. The correlation coefficients between pairs of variables (species), ordered in a correlation matrix, are subjected to the mathematical procedure known as a transformation (equation (1)). The new matrix D in equation (1) has all off-diagonal elements equal to zero and its diagonal elements add up to the same total as the diagonal elements of the original correlation matrix, in our example 17. This diagonalization process is of wide application in applied mathematics. It was used by Jacobi over a hundred years ago in his study of the orbits of the planets and his solution is the one used in most electronic computer programmes today. Many statistical procedures make use of the process, including the statistical treatment of certain taxonomical problems and a problem in population dynamics. The element di of matrix D is larger than any of the succeeding elements. We shall here refer to it as an eigenvalue; other terms in use for it are 'latent root' and 'characteristic root'. Each eigenvalue is associated with two eigenvectors (for an asymmetric * Publication No. 6 of the Department of Geology, University of Ibadan, Nigeria. t Present address: Department of Geology, University of Ibadan, Ibadan, Nigeria.