Regression Techniques for Examining Land Use/Cover Change: A Case Study of a Mediterranean Landscape

In many areas of the northern Mediterranean Basin the abundance of forest and scrubland vegetation is increasing, commensurate with decreases in agricultural land use(s). Much of the land use/cover change (LUCC) in this region is associated with the marginalization of traditional agricultural practices due to ongoing socioeconomic shifts and subsequent ecological change. Regression-based models of LUCC have two purposes: (i) to aid explanation of the processes driving change and/or (ii) spatial projection of the changes themselves. The independent variables contained in the single ‘best’ regression model (that is, that which minimizes variation in the dependent variable) cannot be inferred as providing the strongest causal relationship with the dependent variable. Here, we examine the utility of hierarchical partitioning and multinomial regression models for, respectively, explanation and prediction of LUCC in EU Special Protection Area 56, ‘Encinares del río Alberche y Cofio’ (SPA 56) near Madrid, Spain. Hierarchical partitioning estimates the contribution of regression model variables, both independently and in conjunction with other variables in a model, to the total variance explained by that model and is a tool to isolate important causal variables. By using hierarchical partitioning we find that the combined effects of factors driving land cover transitions varies with land cover classification, with a coarser classification reducing explained variance in LUCC. We use multinomial logistic regression models solely for projecting change, finding that accuracies of maps produced vary by land cover classification and are influenced by differing spatial resolutions of socioeconomic and biophysical data. When examining LUCC in human-dominated landscapes such as those of the Mediterranean Basin, the availability and analysis of spatial data at scales that match causal processes is vital to the performance of the statistical modelling techniques used here.     

Poisson vs normal-errors regression in Mac Nally (1996)

Abstract Mac Nally (1996), in describing the application of ‘hierarchical partitioning’ in regression modelling of species richness of breeding passerine birds with response variable the species count, rejects the use of Poisson regression in favour of normal-errors regression on an incorrect basis. Mac Nally uses a function of the residual sum of squares, the root-mean square prediction error (RMSPE), calculated from predictions from each regression and rejects the Poisson regression because its RMSPE was 20% larger. This note points out that the RMSPE will always be larger for the Poisson regression, given the same link function and linear predictor is used, even if the response is truly Poisson. References to appropriate methods of determining the most suitable response distribution and link function in the context of generalized linear models are given.     

Testing the ability of species distribution models to infer variable importance

Models of species’ distributions and niches are frequently used to infer the importance of range- and niche-defining variables. However, the degree to which these models can reliably identify important variables and quantify their influence remains unknown. Here we use a series of simulations to explore how well models can 1) discriminate between variables with different influence and 2) calibrate the magnitude of influence relative to an ‘omniscient’ model. To quantify variable importance, we trained generalized additive models (GAMs), Maxent and boosted regression trees (BRTs) on simulated data and tested their sensitivity to permutations in each predictor. Importance was inferred by calculating the correlation between permuted and unpermuted predictions, and by comparing predictive accuracy of permuted and unpermuted predictions using AUC and the continuous Boyce index. In scenarios with one influential and one uninfluential variable, models failed to discriminate reliably between variables when training occurrences were < 8–64, prevalence was > 0.5, spatial extent was small, environmental data had coarse resolution and spatial autocorrelation was low, or when pairwise correlation between environmental variables was |r| > 0.7. When two variables influenced the distribution equally, importance was underestimated when species had narrow or intermediate niche breadth. Interactions between variables in how they shaped the niche did not affect inferences about their importance. When variables acted unequally, the effect of the stronger variable was overestimated. GAMs and Maxent discriminated between variables more reliably than BRTs, but no algorithm was consistently well-calibrated vis-à-vis the omniscient model. Algorithm-specific measures of importance like Maxent's change-in-gain metric were less robust than the permutation test. Overall, high predictive accuracy did not connote robust inferential capacity. As a result, requirements for reliably measuring variable importance are likely more stringent than for creating models with high predictive accuracy.     

Variation in reaction norms: Statistical considerations and biological interpretation

Analysis of reaction norms, the functions by which the phenotype produced by a given genotype depends on the environment, is critical to studying many aspects of phenotypic evolution. Different techniques are available for quantifying different aspects of reaction norm variation. We examine what biological inferences can be drawn from some of the more readily applicable analyses for studying reaction norms. We adopt a strongly biologically motivated view, but draw on statistical theory to highlight strengths and drawbacks of different techniques. In particular, consideration of some formal statistical theory leads to revision of some recently, and forcefully, advocated opinions on reaction norm analysis. We clarify what simple analysis of the slope between mean phenotype in two environments can tell us about reaction norms, explore the conditions under which polynomial regression can provide robust inferences about reaction norm shape, and explore how different existing approaches may be used to draw inferences about variation in reaction norm shape. We show how mixed model‐based approaches can provide more robust inferences than more commonly used multistep statistical approaches, and derive new metrics of the relative importance of variation in reaction norm intercepts, slopes, and curvatures.     

The effect of differences in methodology among some recent applications of shearing quotients

A shearing quotient (SQ) is a way of quantitatively representing the Phase I shearing edges on a molar tooth. Ordinary or phylogenetic least squares regression is fit to data on log molar length (independent variable) and log sum of measured shearing crests (dependent variable). The derived linear equation is used to generate an ‘expected’ shearing crest length from molar length of included individuals or taxa. Following conversion of all variables to real space, the expected value is subtracted from the observed value for each individual or taxon. The result is then divided by the expected value and multiplied by 100. SQs have long been the metric of choice for assessing dietary adaptations in fossil primates. Not all studies using SQ have used the same tooth position or crests, nor have all computed regression equations using the same approach. Here we focus on re‐analyzing the data of one recent study to investigate the magnitude of effects of variation in 1) shearing crest inclusion, and 2) details of the regression setup. We assess the significance of these effects by the degree to which they improve or degrade the association between computed SQs and diet categories. Though altering regression parameters for SQ calculation has a visible effect on plots, numerous iterations of statistical analyses vary surprisingly little in the success of the resulting variables for assigning taxa to dietary preference. This is promising for the comparability of patterns (if not casewise values) in SQ between studies. We suggest that differences in apparent dietary fidelity of recent studies are attributable principally to tooth position examined. Am J Phys Anthropol 156:166–178, 2015 © 2014 Wiley Periodicals, Inc.     

A new method for dealing with residual spatial autocorrelation in species distribution models

Species distribution modelling (SDM) is a widely used tool and has many applications in ecology and conservation biology. Spatial autocorrelation (SAC), a pattern in which observations are related to one another by their geographic distance, is common in georeferenced ecological data. SAC in the residuals of SDMs violates the ‘independent errors’ assumption required to justify the use of statistical models in modelling species’ distributions. The autologistic modelling approach accounts for SAC by including an additional term (the autocovariate) representing the similarity between the value of the response variable at a location and neighbouring locations. However, autologistic models have been found to introduce bias in the estimation of parameters describing the influence of explanatory variables on habitat occupancy. To address this problem we developed an extension to the autologistic approach by calculating the autocovariate on SAC in residuals (the RAC approach). Performance of the new approach was tested on simulated data with a known spatial structure and on strongly autocorrelated mangrove species’ distribution data collected in northern Australia. The RAC approach was implemented as generalized linear models (GLMs) and boosted regression tree (BRT) models. We found that the BRT models with only environmental explanatory variables can account for some SAC, but applying the standard autologistic or RAC approaches further reduced SAC in model residuals and substantially improved model predictive performance. The RAC approach showed stronger inferential performance than the standard autologistic approach, as parameter estimates were more accurate and statistically significant variables were accurately identified. The new RAC approach presented here has the potential to account for spatial autocorrelation while maintaining strong predictive and inferential performance, and can be implemented across a range of modelling approaches.     

Assessing the relative importance of neutral stochasticity in ecological communities

A central current debate in community ecology concerns the relative importance of deterministic versus stochastic processes underlying community structure. However, the concept of stochasticity presents several profound philosophical, theoretical and empirical challenges, which we address here. The philosophical argument that nothing in nature is truly stochastic can be met with the following operational concept of neutral stochasticity in community ecology: change in the composition of a community (i.e. community dynamics) is neutrally stochastic to the degree that individual demographic events – birth, death, immigration, emigration – which cause such changes occur at random with respect to species identities. Empirical methods for identifying the stochastic component of community dynamics or structure include null models and multivariate statistics on observational species‐by‐site data (with or without environmental or trait data), and experimental manipulations of ‘stochastic’ species colonization order or relative densities and frequencies of competing species. We identify the fundamental limitations of each method with respect to its ability to allow inferences about stochastic community processes. Critical future needs include greater precision in articulating the link between results and ecological inferences, a comprehensive theoretical assessment of the interpretation of statistical analyses of observational data, and experiments focusing on community size and on natural variation in species colonization order. Synthesis Community structure and dynamics have often been described as being underlain by ‘stochastic’ or ‘neutral’ processes, but there is great confusion as to what exactly this means. We attempt to provide conceptual clarity by specifying precisely what focal ecological variable (e.g. species distributions, community composition, demography) is considered to be stochastic with respect to what other variables (e.g. other species' distributions, traits, environment) when using different empirical methods. We clarify what inferences can be drawn by different observational and experimental approaches, and we suggest future avenues of research to better understand the role of neutral stochasticity in community ecology.     

Multiple regression and inference in ecology and conservation biology: further comments on identifying important predictor variables

Ecologists and conservation biologists frequently use multipleregression (MR) to try to identify factors influencing response variables suchas species richness or occurrence. Many frequently used regression methods maygenerate spurious results due to multicollinearity. argued that there are actually two kinds of MR modelling: (1)seeking the best predictive model; and (2) isolating amounts of varianceattributable to each predictor variable. The former has attracted most attentionwith a plethora of criteria (measures of model fit penalized for modelcomplexity – number of parameters) and Bayes-factor-based methods havingbeen proposed, while the latter has been little considered, althoughhierarchical methods seem promising (e.g. hierarchical partitioning). If the twoapproaches agree on which predictor variables to retain, then it is more likelythat meaningful predictor variables (of those considered) have been found. Therehas been a problem in that, while hierarchical partitioning allowed the rankingof predictor variables by amounts of independent explanatory power, there was no(statistical) way to decide which variables to retain. A solution usingrandomization of the data matrix coupled with hierarchical partitioning ispresented, as is an ecological example.     

Do we really need phytosociological classes to calibrate Ellenberg indicator values?

Abstract. Wamelink et al. (2002) calibrated Ellenberg indicator values for acidity and water availability against measured soil pH and measured mean spring groundwater level (MSL), respectively. Linear regression between indicator value and measured value of all the observations gave a poor fit. Regression lines per phytosociological vegetation class, on the other hand, generally described the observations well. In this article we demonstrate that this result is, at least partly, an artefact. First, because the data utilized are likely to contain systematic errors, and second, because a wrong regression model was applied. A sigmoid function for the relation between the indicator value for water availability and MSL gives a far better fit than a linear function does.‘Vegetation class’ is not an obvious choice as an extra explanatory variable for the regression, as it is only a convenient label for vegetation and should not be used as if it were a real independent environmental variable. In general, indicator values of plant species should be calibrated against environmental variables with great care. This implies that researchers should have knowledge about the ecological demands plants make on their environment, as well as about the spatial and temporal variability of this environment.     

Estimating and Modelling Bias of the Hierarchical Partitioning Public-Domain Software: Implications in Environmental Management and Conservation

Background

Hierarchical partitioning (HP) is an analytical method of multiple regression that identifies the most likely causal factors while alleviating multicollinearity problems. Its use is increasing in ecology and conservation by its usefulness for complementing multiple regression analysis. A public-domain software “hier.part package” has been developed for running HP in R software. Its authors highlight a “minor rounding error” for hierarchies constructed from >9 variables, however potential bias by using this module has not yet been examined. Knowing this bias is pivotal because, for example, the ranking obtained in HP is being used as a criterion for establishing priorities of conservation.

Methodology/Principal Findings

Using numerical simulations and two real examples, we assessed the robustness of this HP module in relation to the order the variables have in the analysis. Results indicated a considerable effect of the variable order on the amount of independent variance explained by predictors for models with >9 explanatory variables. For these models the nominal ranking of importance of the predictors changed with variable order, i.e. predictors declared important by its contribution in explaining the response variable frequently changed to be either most or less important with other variable orders. The probability of changing position of a variable was best explained by the difference in independent explanatory power between that variable and the previous one in the nominal ranking of importance. The lesser is this difference, the more likely is the change of position.

Conclusions/Significance

HP should be applied with caution when more than 9 explanatory variables are used to know ranking of covariate importance. The explained variance is not a useful parameter to use in models with more than 9 independent variables. The inconsistency in the results obtained by HP should be considered in future studies as well as in those already published. Some recommendations to improve the analysis with this HP module are given.     

New insights into butterfly-environment relationships using partitioning methods

Variation partitioning and hierarchical partitioning are novel statistical approaches that provide deeper understanding of the importance of different explanatory variables for biodiversity patterns than traditional regression methods. Using these methods, the variation in occupancy and abundance of the clouded apollo butterfly (Parnassius mnemosyne L.) was decomposed into independent and joint effects of larval and adult food resources, microclimate and habitat quantity. The independent effect of habitat quantity variables (habitat area and connectivity) captured the largest fraction of the variation in the clouded apollo patterns, but habitat connectivity had a major contribution only for occupancy data. The independent effects of resources and microclimate were higher on butterfly abundance than on occupancy. However, a considerable amount of variation in the butterfly patterns was accounted for by the joint effects of predictors and may thus be causally related to two or all three groups of variables. Abundance of the butterfly in the surroundings of the focal grid cell had a significant effect in all analyses, independently of the effects of other predictors. Our results encourage wider applications of partitioning methods in biodiversity studies.     

The sign of the bias of unmeasured confounding

Summary .   Unmeasured confounding variables are a common problem in drawing causal inferences in observational studies. A theorem is given which in certain circumstances allows the researcher to draw conclusions about the sign of the bias of unmeasured confounding. Specifically, it is possible to determine the sign of the bias when monotonicity relationships hold between the unmeasured confounding variable and the treatment, and between the unmeasured confounding variable and the outcome. Some discussion is given to the conditions under which the theorem applies and the strengths and limitations of using the theorem to assess the sign of the bias of unmeasured confounding.     

Red-shifts and red herrings in geographical ecology

I draw attention to the need for ecologists to take spatial structure into account more seriously in hypothesis testing. If spatial autocorrelation is ignored, as it usually is, then analyses of ecological patterns in terms of environmental factors can produce very misleading results. This is demonstrated using synthetic but realistic spatial patterns with known spatial properties which are subjected to classical correlation and multiple regression analyses. Correlation between an autocorrelated response variable and each of a set of explanatory variables is strongly biased in favour of those explanatory variables that are highly autocorrelated - the expected magnitude of the correlation coefficient increases with autocorrelation even if the spatial patterns are completely independent. Similarly, multiple regression analysis finds highly autocorrelated explanatory variables "significant" much more frequently than it should. The chances of mistakenly identifying a "significant" slope across an autocorrelated pattern is very high if classical regression is used. Consequently, under these circumstances strongly autocorrelated environmental factors reported in the literature as associated with ecological patterns may not actually be significant. It is likely that these factors wrongly described as important constitute a red-shifted subset of the set of potential explanations, and that more spatially discontinuous factors (those with bluer spectra) are actually relatively more important than their present status suggests. There is much that ecologists can do to improve on this situation. I discuss various approaches to the problem of spatial autocorrelation from the literature and present a randomisation test for the association of two spatial patterns which has advantages over currently available methods.     

基于光谱特征参量的‘温185’核桃叶片钾含量估测模型

基于田间人工定量施肥试验,采用Pearson相关分析筛选与果实不同生育时期‘温185’核桃叶片钾(K)含量呈极显著相关的光谱特征参量,并以筛选出的光谱特征参量为自变量,采用回归分析构建‘温185’核桃果实不同生育时期叶片K元素含量估测模型.结果表明: ‘温185’核桃果实不同生育时期均存在一到多个与叶片K含量呈极显著相关(P＜0.01)的光谱特征参量;分别以光谱特征参量绿色归一化差值指数、红边黄边面积比值、绿色比值指数和蓝边面积为自变量,采用三次函数建立的果实不同生育时期叶片K含量回归估测模型的拟合度(R²)均大于0.95,模型均方根误差小于0.8161 g·kg^-1,相对误差绝对值小于2.7%,模型估测值与实测值一致.基于光谱特征参量采用三次函数构建的‘温185’核桃叶片K含量估测模型具有较高的估测精度,光谱技术在核桃树体K元素营养信息探测方面有较大的应用潜力.     

An updated consumer’s guide to evenness and related indices

Ecologists widely agree that species diversity consists of two components, richness (the number of species) and evenness (a measure of the equitability of the proportional abundances of those species). However, no consensus on an exact definition of evenness (or equitability) has emerged. Instead, numerous equitability indices have been used in the ecological literature, as different researchers have preferred indices with different mathematical properties. In this paper, I show that the phrase ‘species diversity consists of two independent components, richness and evenness’ logically leads to one particular definition of evenness (Evenness = Diversity/Richness). To facilitate accurate communication, I propose that the term ‘evenness’ be used only to refer to this phenomenon, and that other terms be used for the equitability indices that measure other things. Here I provide a review of popular equitability indices, explain what each measures in practice, and show how they relate to each other and to evenness itself. I also explore how the partitioning of diversity into richness and evenness components is related to the partitioning of diversity into alpha and beta components. Dissecting the indices makes it easier to see the conceptual differences among them. Such understanding is necessary to ensure that an appropriate index is chosen for the questions at hand, as well as to interpret the index values correctly and to assess when index values can and when they cannot be considered comparable.     

Niche partitioning and the effect of interspecific competition on microhabitat use by two sympatric galaxiid stream fishes

1. Numerous interacting abiotic and biotic factors influence niche use and assemblage structure of freshwater fishes, but the strength of each factor changes with spatial scale. Few studies have examined the role of interspecific competition in structuring stream fish assemblages across spatial scales. We used field and laboratory approaches to examine microhabitat partitioning and the effect of interspecific competition on microhabitat use in two sympatric stream fishes (Galaxias‘southern’ and Galaxias gollumoides) at large (among streams and among sites within streams) and small (within artificial stream channels) spatial scales. 2. Diurnal microhabitat partitioning and interspecific competition at large spatial scales were analysed among three sympatry streams (streams with allotopic and syntopic sites; three separate catchments) and four allopatry streams (streams with only allotopic sites; two separate catchments). Electro‐fishing was used to sample habitat use of fishes at 30 random points within each site by quantifying four variables for each individual: water velocity, depth, distance to nearest cover and substratum size. Habitat availability was then quantified for each site by measuring those variables at each of 50 random points. Diet and stable isotope partitioning was analysed from syntopic sites only. Diel cycles of microhabitat use and interspecific competition at small spatial scales were examined by monitoring water velocity use over 48 h in artificial stream channels for three treatments: (i) allopatric G. ‘southern’ (10 G. ‘southern’); (ii) allopatric G. gollumoides (10 G. gollumoides) and (iii) sympatry (five individuals of each species). 3. One hundred and ninety‐four G. ‘southern’ and 239 G. gollumoides were sampled across all seven streams, and habitat availability between the two species was similar among all sites. Galaxias‘southern’ utilised faster water velocities than G. gollumoides in both the field and in channel experiments. Both species utilised faster water velocities in channels at night than during the day. Diet differences were observed and were supported by isotopic differences (two of three sites). No interspecific differences were observed for the other three microhabitat variables in the field, and multivariate habitat selection did not differ between species. Interspecific competition had no effect on microhabitat use of either species against any variable either in the field (large scale) or in channels (small scale). 4. The results suggest that niche partitioning occurs along a subset of microhabitat variables (water velocity use and diet). Interspecific competition does not appear to be a major biotic factor controlling microhabitat use by these sympatric taxa at any spatial scale. The results further suggest that stream fish assemblages are not primarily structured by biotic factors, reinforcing other studies de‐emphasising interspecific competition.     

A note on the use of multiple linear regression in molecular ecology

Multiple linear regression analyses (also often referred to as generalized linear models – GLMs, or generalized linear mixed models – GLMMs) are widely used in the analysis of data in molecular ecology, often to assess the relative effects of genetic characteristics on individual fitness or traits, or how environmental characteristics influence patterns of genetic differentiation. However, the coefficients resulting from multiple regression analyses are sometimes misinterpreted, which can lead to incorrect interpretations and conclusions within individual studies, and can propagate to wider‐spread errors in the general understanding of a topic. The primary issue revolves around the interpretation of coefficients for independent variables when interaction terms are also included in the analyses. In this scenario, the coefficients associated with each independent variable are often interpreted as the independent effect of each predictor variable on the predicted variable. However, this interpretation is incorrect. The correct interpretation is that these coefficients represent the effect of each predictor variable on the predicted variable when all other predictor variables are zero. This difference may sound subtle, but the ramifications cannot be overstated. Here, my goals are to raise awareness of this issue, to demonstrate and emphasize the problems that can result and to provide alternative approaches for obtaining the desired information.     

Rethinking patch size and isolation effects: the habitat amount hypothesis

I challenge (1) the assumption that habitat patches are natural units of measurement for species richness, and (2) the assumption of distinct effects of habitat patch size and isolation on species richness. I propose a simpler view of the relationship between habitat distribution and species richness, the ‘habitat amount hypothesis’, and I suggest ways of testing it. The habitat amount hypothesis posits that, for habitat patches in a matrix of non‐habitat, the patch size effect and the patch isolation effect are driven mainly by a single underlying process, the sample area effect. The hypothesis predicts that species richness in equal‐sized sample sites should increase with the total amount of habitat in the ‘local landscape’ of the sample site, where the local landscape is the area within an appropriate distance of the sample site. It also predicts that species richness in a sample site is independent of the area of the particular patch in which the sample site is located (its ‘local patch’), except insofar as the area of that patch contributes to the amount of habitat in the local landscape of the sample site. The habitat amount hypothesis replaces two predictor variables, patch size and isolation, with a single predictor variable, habitat amount, when species richness is analysed for equal‐sized sample sites rather than for unequal‐sized habitat patches. Studies to test the hypothesis should ensure that ‘habitat’ is correctly defined, and the spatial extent of the local landscape is appropriate, for the species group under consideration. If supported, the habitat amount hypothesis would mean that to predict the relationship between habitat distribution and species richness: (1) distinguishing between patch‐scale and landscape‐scale habitat effects is unnecessary; (2) distinguishing between patch size effects and patch isolation effects is unnecessary; (3) considering habitat configuration independent of habitat amount is unnecessary; and (4) delineating discrete habitat patches is unnecessary.     

Modeling Complex Biological Macromolecules: Reduction of Multibead Models

The shape of simple and complex biological macromolecules can be approximated by bead modeling procedures. Such approaches are required, for example, for the analysis of the scattering and hydrodynamic behavior of the models under analysis and the prediction of their molecular properties. Using the atomic coordinates of proteins for modeling inevitably leads to models composed of a multitude of beads. In particular, for hydrodynamic modeling, a drastic reduction of the bead number may become unavoidable to enable computation. A systematic investigation of different approaches and computation modes shows that the ‘running mean’, ‘cubic grid,’ and ‘hexagonal grid’ approaches are successful, provided that the extent of reduction does not exceed a factor of 100 and the grid approaches use beads of unequal size and the beads are located at the centers of gravity. Further precautions to be taken include usage of appropriate interaction tensors for overlapping beads of unequal size and appropriate volume corrections when calculating intrinsic viscosities. The applied procedures were tested with the small protein lysozyme in a case study and were then applied to the huge capsid of the phage fr and its trimeric building block. The appearance of the models and the agreement of molecular properties and distance distribution functions of unreduced and reduced models can be used as evaluation criteria.