David I. Warton

Bivariate line-fitting methods for allometry

Fitting a line to a bivariate dataset can be a deceptively complex problem, and there has been much debate on this issue in the literature. In this review, we describe for the practitioner the essential features of line‐fitting methods for estimating the relationship between two variables: what methods are commonly used, which method should be used when, and how to make inferences from these lines to answer common research questions.

The arcsine is asinine: the analysis of proportions in ecology

The arcsine square root transformation has long been standard procedure when analyzing proportional data in ecology, with applications in data sets containing binomial and non-binomial response variables. Here, we argue that the arcsine transform should not be used in either circumstance. For binomial data, logistic regression has greater interpretability and higher power than analyses of transformed data. However, it is important to check the data for additional unexplained variation, i.e., overdispersion, and to account for it via the inclusion of random effects in the model if found. For non-binomial data, the arcsine transform is undesirable on the grounds of interpretability, and because it can produce nonsensical predictions. The logit transformation is proposed as an alternative approach to address these issues. Examples are presented in both cases to illustrate these advantages, comparing various methods of analyzing proportions including untransformed, arcsine- and logit-transformed linear models and logistic regression (with or without random effects). Simulations demonstrate that logistic regression usually provides a gain in power over other methods.

Global patterns of leaf mechanical properties

Leaf mechanical properties strongly influence leaf lifespan, plant-herbivore interactions, litter decomposition and nutrient cycling, but global patterns in their interspecific variation and underlying mechanisms remain poorly understood. We synthesize data across the three major measurement methods, permitting the first global analyses of leaf mechanics and associated traits, for 2819 species from 90 sites worldwide. Key measures of leaf mechanical resistance varied c. 500-800-fold among species. Contrary to a long-standing hypothesis, tropical leaves were not mechanically more resistant than temperate leaves. Leaf mechanical resistance was modestly related to rainfall and local light environment. By partitioning leaf mechanical resistance into three different components we discovered that toughness per density contributed a surprisingly large fraction to variation in mechanical resistance, larger than the fractions contributed by lamina thickness and tissue density. Higher toughness per density was associated with long leaf lifespan especially in forest understory. Seldom appreciated in the past, toughness per density is a key factor in leaf mechanical resistance, which itself influences plant-animal interactions and ecosystem functions across the globe.

Equivalence of MAXENT and Poisson Point Process Models for Species Distribution Modeling in Ecology

Modeling the spatial distribution of a species is a fundamental problem in ecology. A number of modeling methods have been developed, an extremely popular one being MAXENT, a maximum entropy modeling approach. In this article, we show that MAXENT is equivalent to a Poisson regression model and hence is related to a Poisson point process model, differing only in the intercept term, which is scale-dependent in MAXENT. We illustrate a number of improvements to MAXENT that follow from these relations. In particular, a point process model approach facilitates methods for choosing the appropriate spatial resolution, assessing model adequacy, and choosing the LASSO penalty parameter, all currently unavailable to MAXENT. The equivalence result represents a significant step in the unification of the species distribution modeling literature.

So Many Variables: Joint Modeling in Community Ecology.

Technological advances have enabled a new class of multivariate models for ecology, with the potential now to specify a statistical model for abundances jointly across many taxa, to simultaneously explore interactions across taxa and the response of abundance to environmental variables. Joint models can be used for several purposes of interest to ecologists, including estimating patterns of residual correlation across taxa, ordination, multivariate inference about environmental effects and environment-by-trait interactions, accounting for missing predictors, and improving predictions in situations where one can leverage knowledge of some species to predict others. We demonstrate this by example and discuss recent computation tools and future directions.

Poisson point process models solve the “pseudo-absence problem” for presence-only data in ecology

Presence-only data, point locations where a species has beenrecorded as being present, are often used in modeling the distribu-tion of a species as a function of a set of explanatory variables—whether to map species occurrence, to understand its associationwith the environment, or to predict its response to environmentalchange. Currently, ecologists most commonly analyze presence-onlydata by adding randomly chosen “pseudo-absences” to the data suchthat it can be analyzed using logistic regression, an approach whichhas weaknesses in model specification, in interpretation, and in imple-mentation. To address these issues, we propose Poisson point processmodeling of the intensity of presences. We also derive a link betweenthe proposed approach and logistic regression—specifically, we showthat as the number of pseudo-absences increases (in a regular or uni-form random arrangement), logistic regression slope parameters andtheir standard errors converge to those of the corresponding Poissonpoint process model. We discuss the practical implications of theseresults. In particular, point process modeling offers a framework forchoice of the number and location of pseudo-absences, both of whichare currently chosen by ad hoc and sometimes ineffective methods inecology, a point which we illustrate by example.

Putting plant resistance traits on the map: a test of the idea that plants are better defended at lower latitudes

• It has long been believed that plant species from the tropics have higher levels of traits associated with resistance to herbivores than do species from higher latitudes. A meta-analysis recently showed that the published literature does not support this theory. However, the idea has never been tested using data gathered with consistent methods from a wide range of latitudes. • We quantified the relationship between latitude and a broad range of chemical and physical traits across 301 species from 75 sites world-wide. • Six putative resistance traits, including tannins, the concentration of lipids (an indicator of oils, waxes and resins), and leaf toughness were greater in high-latitude species. Six traits, including cyanide production and the presence of spines, were unrelated to latitude. Only ash content (an indicator of inorganic substances such as calcium oxalates and phytoliths) and the properties of species with delayed greening were higher in the tropics. • Our results do not support the hypothesis that tropical plants have higher levels of resistance traits than do plants from higher latitudes. If anything, plants have higher resistance toward the poles. The greater resistance traits of high-latitude species might be explained by the greater cost of losing a given amount of leaf tissue in low-productivity environments.

The Time Value of Leaf Area

When a plant invests in construction of a leaf, the revenue‐stream that accrues is shaped by three variables: first, the light‐capture area per milligram dry mass invested, analogous to a potential rate of return on investment; second, the longevity of the leaf, analogous to the expected duration of the revenue stream; and third, a time‐discount rate, quantifying the fact that light‐capture area deployed in the immediate future is more valuable to the plant than the same area deployed at some later time. Recent comparative data make it possible to quantify the cross‐species trade‐off between the first variable and the second variable. Here we develop an approach through which the consequences of the third variable, the time‐discount rate, can be related to the trade‐off between the first variable and the second variable. The approach involves an equal‐benefit set, the cross‐species equivalent of a fitness set. A wide spread of strategies is actually observed to coexist in vegetation, from low to high light capture area per gram and, correspondingly, from high to low leaf longevity. The coexistence suggests that the different observed strategies do not have a clear‐cut advantage over the other. The equal‐benefit set can be used to investigate what levels of time discount would make it the case that neither the highest‐longevity nor the highest light‐capture area per milligram strategies would have a clear advantage over the other, with regard to the time‐discounted value of the revenue stream generated per milligram invested in leaf.

Point process models for presence-only analysis

Summary Presence-only data are widely used for species distribution modelling, and point process regression models are a flexible tool that has considerable potential for this problem, when data arise as point events. In this paper, we review point process models, some of their advantages and some common methods of fitting them to presence-only data. Advantages include (and are not limited to) clarification of what the response variable is that is modelled; a framework for choosing the number and location of quadrature points (commonly referred to as pseudo-absences or ‘background points’) objectively; clarity of model assumptions and tools for checking them; models to handle spatial dependence between points when it is present; and ways forward regarding difficult issues such as accounting for sampling bias. Point process models are related to some common approaches to presence-only species distribution modelling, which means that a variety of different software tools can be used to fit these models, including maxent or generalised linear modelling software.

Penalized Normal Likelihood and Ridge Regularization of Correlation and Covariance Matrices

High dimensionality causes problems in various areas of statistics. A particular situation that rarely has been considered is the testing of hypotheses about multivariate regression models in which the dimension of the multivariate response is large. In this article a ridge regularization approach is proposed in which either the covariance or the correlation matrix is regularized to ensure nonsingularity irrespective of the dimensionality of the data. It is shown that the proposed approach can be derived through a penalized likelihood approach, which suggests cross-validation of the likelihood function as a natural approach for estimation of the ridge parameter. Useful properties of this likelihood estimator are derived, discussed, and demonstrated by simulation. For a class of test statistics commonly used in multivariate analysis, the proposed regularization approach is compared with some obvious alternative regularization approaches using generalized inverse and data reduction through principal components analysis. Essentially, the approaches considered differ in how they shrink eigenvalues of sample covariance and correlation matrices. This leads to predictable differences in power properties when comparing the use of different regularization approaches, as demonstrated by simulation. The proposed ridge approach has relatively good power compared with the alternatives considered. In particular, a generalized inverse is shown to perform poorly and cannot be recommended in practice. Finally, the proposed approach is used in analysis of data on macroinvertebrate biomasses that have been classified to species.