Cajo J. F. ter Braak is WRF researcher and personal professor at Biometris, Wageningen University & Research and the inventor of Canonical Correspondence Analysis, resulting from a Eureka moment in 1987. He is the senior author of the software package Canoco, now in version 5.10, for visualization and testing of multivariate data (by constrained and unconstrained correspondence analysis, principal component analysis and NMDS). A second Eureka moment led to adaptive MCMC sampler called Differential Evolution Markov Chain (DE-MC aka DREAM), which is popular in hydrology and astronomy; for example, DE-MC has been used in the analysis of the first gravitational waves ever detected.
TITLE OF PRESENTATION
L-shaped data: from GLM to fourth corner correlation and double constrained correspondence analysis
L-shaped data consists of a non-negative central matrix with associated constraining (predictor) matrices for rows and columns. Formally, it is (weighted) bigraph with node predictors. Examples are preference data of consumers for products with features of both consumers and products as predictors, supervisory boards of firms with features of supervisors and firms as predictors for the membership, and, in ecology, abundance data of species and environmental variables with traits and environmental variables as predictors. We will discuss the statistical issues of analysing such data starting with GLM and GLMM models. With a single trait and environmental variable, the Rao score test on the interaction in a loglinear model is shown to reduce to the squared fourth-corner correlation introduced by Legendre et al. (1997). With multiple traits and environmental variables, it leads to double constrained correspondence analysis (dc-CA), which encompasses both simple and multiple correspondence analysis and canonical correspondence analysis. Some ecological applications will be discussed, in which dc-CA based forward selection of traits and environmental variables is used to select the most important traits and environmental variables that structure the central abundance table. The method is available in R and in Canoco 5.10.