Splines in constrained ordination analysis of ecological data

Abstract

Multivariate Poisson Log-Normal (mvPA) models are shown to be a special case of Generalised Linear Mixed Models. The inclusion of regression splines further extends this to include general non-linear relationships between responses and covariates. A simulation process is developed for creating multivariate Poisson Log-Normal data which, coupled with estimates from real data, gives realistic contrived ecological datasets for assessing the efficacy of nonlinear multivariate methods. mvPA models of varying complexity are fit to contrived and real multivariate count data; subsequently mvPA simulation data are generated. A method is described that allows the grouping of multiple responses into classes that are similar in their relationships with their common covariates. The modelling is via penalized regression splines with automated selection of smoothness, and the similarities of the response surfaces are assessed geometrically by their parameter vectors. Actual grouping of surfaces is by hierarchical clustering and the stability of solutions is assessed by resampling. The Generalized Additive Model (GAM) framework underlying the process means the usual variety of link functions and error distributions can be employed. Multispecies response and covariate data from �ecological� studies are analysed and the method reduces complex multispecies data down to a small number of representative species groups with similar response surfaces. Standard dendrogram plotting gives clear insight into the similarity of responses and extensions to more complex multidimensional surfaces are straightforward. General applications beyond multivariate ecological data are apparent - specifically the analysis of repeated measures data. The summarising of a large set of individual curves by a small set of representative curves is demonstrated. Standard linear Redundancy Analysis (RDA) is generalised to include Generalised Linear Model link functions. This allows for the reduced space plotting of data previously beyond RDA, for example presence-absence responses. Modified RDA statistics and diagnostics are proposed. The generalised RDA (gRDA) is expanded to include general nonlinear relationships between responses and covariates via penalised regression splines, with smoothing levels optimised via Generalised Cross Validation. Contrived and real datasets are analysed and specific comparisons are drawn with a polynomial competitor, poly-RDA (Makarenkov and Legendre, 2002)