Skabbholmen data from: Concurrent ordination: Simultaneous unconstrained and constrained latent variable modeling

In community ecology, unconstrained ordination can be used to indirectly explore drivers of community composition, while constrained ordination can be used to directly relate predictors to an ecological community. However, existing constrained ordination methods do not explicitly account for community composition that cannot be explained by the predictors, so that they have the potential to misrepresent community composition if not all predictors are available in the data.
We propose and develop a set of new methods for ordination and Joint Species Distribution Modelling (JSDM) as part of the Generalized Linear Latent Variable Model (GLLVM) framework, that incorporate predictors directly into an ordination. This includes a new ordination method that we refer to as concurrent ordination, as it simultaneously constructs unconstrained and constrained latent variables. Both unmeasured residual covariation and predictors are incorporated into the ordination by simultaneously imposing reduced rank structures on the residual covariance matrix and on fixed-effects.
We evaluate the method with a simulation study, and show that the proposed developments outperform Canonical Correspondence Analysis (CCA) for Poisson and Bernoulli responses, and perform similar to Redundancy Analysis (RDA) for normally distributed responses, the two most popular methods for constrained ordination in community ecology. Two examples with real data further demonstrate the benefits of concurrent ordination, and the need to account for residual covariation in the analysis of multivariate data.
This article contextualizes the role of constrained ordination in the GLLVM and JSDM frameworks, while developing a new ordination method that incorporates the best of unconstrained and constrained ordination, and which overcomes some of the deficiencies of existing classical ordination methods.

Methods
This dataset was collected by Wolfgang and Hytteborn (1987) on the Swedish island Skabbholmen. It was provided from the Canoco example datasets by Prof. Cajo ter Braak (see also ter Braak 1987). It includes observations on an ordinal scale, of 70 species at 64 plots. Species composition at the plots was sampled in 1985 and 1987, except for two plots which were only sampled in one year. Additionally, elevation relative to the shoreline was recorded in centimeters.

Full species names were added by Bert van der Veen, and verified by Håkan Hytteborn and Bente Jessen Graae.