A new technique for maximum-likelihood canonical Gaussian ordination

Canonical correspondence analysis (CCA) is probably the most popular ordination method in community ecology. However, it is only a heuristic approximation to maximum-likelihood estimated canonical Gaussian ordination (CGO), which is the ideal method. When proposed in the mid-1980s, CCA held two advantages over CGO: it was computationally cheaper, and its algorithm was, not complex. However, an exponential increase in computing speed over the last two decades has meant that computation cost is no longer such a compelling advantage. The computational complexity of CGO has always been its major difficulty, even though it is statistically more sound and Simpler to understand than CCA. For these reasons, no general computational framework or software has appeared until now. This article proposes a new class of statistical regression models called quadratic reduced-rank vector generalized linear models (QRR-VGLMs) for maximum-likelihood estimated CGO. This is achieved by extending a recently developed class of statistical models called RR-VGLMs. The extension is named QRR-VGLMs because of the addition of a quadratic form to each linear predictor, with the consequence that bell-shaped responses can be modeled as functions of latent environmental variables or gradients. QRR-VGLMs have several major positive features; for example, their framework is unifying and broad, so that canonical Gaussian ordination can potentially be performed on a wide range of data types. The two most important special cases of CGO in ecology, multispecies presence/absence and Poisson abundance data, are considered in this article. The methodology is illustrated with a real data set using a software implementation written by the author in the S statistical language. The code, called the VGAM package in R, is object-oriented and free, and it allows QRR-VGLMs to be fitted to moderate-sized data sets conforming reasonably closely to the Gaussian model.