Abstract
Constrained ordination methods aims at finding an environmental gradient along which the species abundances are maximally separated. The species response functions, which describe the expected abundance as a function of the environmental score, are according to the ecological fundamental niche theory only meaningful if they are bell-shaped. Many classical model-based ordination methods, however, use quadratic regression models without imposing the bell-shape and thus allowing for meaningless U-shaped response functions. The analysis output (e.g. a biplot) may therefore be potentially misleading and the conclusions are prone to errors. In this paper we present a log-likelihood ratio criterion with a penalisation term to enforce more bell-shaped response shapes. We report the results of a simulation study and apply our method to metagenomics data from microbial ecology.
Highlights
Constrained or Canonical Correspondence Analysis (CCA) is a well established method among environmental ecologists to study the relation between species abundances and the environmental conditions
In a CCA and Constrained Ordination Analysis (COA) the species abundances are regressed on environmental scores that result from linear combinations of the environmental conditions of the sampling locations
We proceed with the following steps for the generation of the simulated data: (1) construct an environmental data matrix, X, with observations of p = 4 environmental variables measured on n = 44 sampling locations; (2) specify two environmental gradients (α1 and α2); (3) specify s = 20 bell-shaped and U-shaped species response functions along the environmental gradients; (4) simulate 44 abundances for each of the 20 species
Summary
Constrained or Canonical Correspondence Analysis (CCA) is a well established method among environmental ecologists to study the relation between species abundances and the environmental conditions. Finding the gradient along which the species display maximally separated response functions is obtained by maximising their likelihood ratio criterion Their method will be referred to as Flexible Constrained Ordination Analysis (FCOA). Zhu et al [15] proposed an iterative scheme for the joint estimation of α and the βk parameters: (1) for an initial α the environmental variables are transformed to the z scores; (2) estimate the βk and β from the corresponding Poisson regression models; (3) estimate α by maximising the log-likelihood ratio (LLR) criterion. T k ðb^0k; b^1k; b^2kÞ denoting the vector with the parameter estimates for species k, model (3) results in a U-shaped response function iff b^2k > 0. We suggest to estimate the β2k parameters of model (3) with a penalised maximum likelihood method that favors negative parameter estimates and results in bell-shaped response functions. More gradients can be found by repeating this procedure (regressing x on all environmental scores)
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