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Taylor’s power law (TPL) from empirical ecological theory has had many explanations proposed for its widespread observation in data. We show that the class of reduced-rank vector generalized linear models (RR-VGLMs) for coupling two parameters from a statistical distribution linearly together creates hybrid models that satisfy TPL or very similar. These include the RR-negative binomial, RR-inverse Gaussian and RR-generalized Poisson distributions. Some advantages of RR-VGLMs include the handling of covariates and an implementation exists in the form of the VGAM R package. The software is demonstrated to show how these models may be fitted conveniently.

One-parameter link functions play a fundamental role in regression via generalized linear modelling. This paper develops the general theory for two-parameter links in the very large class of vector generalized linear models by using total derivatives applied to a composite log-likelihood within the Fisher scoring/iteratively reweighted least squares algorithm. We solve a four-decade old problem with an interesting history as our first example: the canonical link for negative binomial regression. The remaining examples are fitting Weibull regression using both the mean and quantile directly compared to GAMLSS, and performing quantile regression based on the Gaussian distribution. Numerical examples based on real and simulated data are given. The methods described here are implemented by the VGAM and VGAMextraR packages, available on CRAN. Supplementary materials for this article are available online.

The euphotic–mesophotic transition is characterized by dramatic changes in environmental conditions, which can significantly alter the functioning of ecosystem engineers and the structure of their associated communities. However, the drivers of biodiversity change across the euphotic–mesophotic transition remain unclear. Here, we investigated the mechanisms affecting the biodiversity-supporting potential of free-living red coralline algae—globally important habitat creators—towards mesophotic depths. Across a 73 m depth gradient, we observed a general decline in macrofaunal biodiversity (fauna abundance, taxon richness and alphadiversity), but an increase in beta-diversity (i.e. variation between assemblages) at the deepest site (86 mdepth, where light levelswere less than 1% surface irradiance). We identified a gradient in abundance decline rather than distinct ecological shifts, driven by a complex interaction between declining light availability, declining size of the coralline algal host individuals and a changing host taxonomy. However, despite abundance declines, high betweenassemblage variability at deeper depths allowed biodiversity-supporting potential to be maintained, highlighting their importance as coastal refugia.

Over recent years parametric and nonparametric regression has slowly been adopted into extreme value data analysis. Its introduction has been characterized by piecemeal additions and embellishments, which has had a negative effect on software development and usage. The purpose of this article is to convey the classes of vector generalized linear and additive models (VGLMs and VGAMs) as offering significant advantages for extreme value data analysis, providing flexible smoothing within a unifying framework. In particular, VGLMs and VGAMs allow all parameters of extreme value distributions to be modelled as linear or smooth functions of covariates. We implement new auxiliary methodology by incorporating a quasi-Newton update for the working weight matrices within an iteratively reweighted least squares (IRLS) algorithm. A software implementation by the first author, called the vgam package for R, is used to illustrate the potential of VGLMs and VGAMs. [PUBLICATION ABSTRACT]

We propose a family of models called row–column interaction models (RCIMs) for two-way table responses. RCIMs apply some link function to a parameter (such as the cell mean) to equal a row effect plus a column effect plus an optional interaction modelled as a reduced-rank regression. What sets this work apart from others is that our framework incorporates a very wide range of statistical models, e.g., (1) log-link with Poisson counts is Goodman’s RC model, (2) identity-link with a double exponential distribution is median polish, (3) logit-link with Bernoulli responses is a Rasch model, (4) identity-link with normal errors is two-way ANOVA with one observation per cell but allowing semi-complex modelling of interactions of the form  A C T , (5) exponential-link with normal responses are quasi-variances. Proposed here also is a least significant difference plot augmentation of quasi-variances. Being a special case of RCIMs, quasi-variances are naturally extended from the M = 1 linear/additive predictor  η case (within the exponential family) to the M > 1 case (vector generalized linear model families). A rank-1 Goodman’s RC model is also shown to estimate the site scores and optimums of an equal-tolerances Poisson unconstrained quadratic ordination. New functions within the VGAM R package are described with examples. Altogether, RCIMs facilitate the analysis of matrix responses of many data types, therefore are potentially useful to many areas of applied statistics.

For several decades now, ecologists have sought to determine the shape of species' response curves and how they are distributed along unknown underlying gradients, environmental latent variables, or ordination axes. Its determination has important implications for both continuum theory and community analysis because many theories and models in community ecology assume that responses are symmetric and unimodal. This article proposes a major new technique called constrained additive ordination (CAO) that solves this problem by computing the optimal gradients and flexible response curves. It allows ecologists to see the response curves as they really are, against the dominant gradients. With one gradient, CAO is a generalization of constrained quadratic ordination (CQO; formerly called canonical Gaussian ordination or CGO). It supplants symmetric bell-shaped response curves in CQO with completely flexible smooth curves. The curves are estimated using smoothers such as the smoothing spline. Loosely speaking, CAO models are generalized additive models (GAMs) fitted to a very small number of latent variables. Being data driven rather than model driven, CAO allows the data to \"speak for itself\" and does not make any of the assumptions made by canonical correspondence analysis. The new methodology is illustrated with a hunting spider data set and a New Zealand tree species data set.

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.

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