Phylogeny, regression, and the allometry of physiological traits

Physiological and ecological allometries often pose linear regression problems characterized by (1) noncausal, phylogenetically autocorrelated independent (x) and dependent (y) variables (characters); (2) random variation in both variables; and (3) a focus on regression slopes (allometric exponents). Remedies for the phylogenetic autocorrelation of species values (phylogenetically independent contrasts) and variance structure of the data (reduced major axis [RMA] regression) have been developed, but most functional allometries are reported as ordinary least squares (OLS) regression without use of phylogenetically independent contrasts. We simulated Brownian diffusive evolution of functionally related characters and examined the importance of regression methodologies and phylogenetic contrasts in estimating regression slopes for phylogenetically constrained data. Simulations showed that both OLS and RMA regressions exhibit serious bias in estimated regression slopes under different circumstances but that a modified orthogonal (least squares variance-oriented residual [LSVOR]) regression was less biased than either OLS or RMA regressions. For strongly phylogenetically structured data, failure to use phylogenetic contrasts as regression data resulted in overestimation of the strength of the regression relationship and a significant increase in the variance of the slope estimate. Censoring of data sets by simulated extinction of taxa did not affect the importance of appropriate regression models or the use of phylogenetic contrasts.     

Relative size-at-sex-change in parrotfishes across the Caribbean: is there variance in a supposed life-history invariant?

Invariant life-history theory has been used to identify parallels in life histories across diverse taxa. One important invariant life-history model predicts that, given simple assumptions and conditions, size-at-sex-change relative to maximum attainable body size (relative size-at-sex-change, RSSC) will be invariant across populations and species in sequential hermaphrodites. Even if there are broad species-wide limits to RSSC, populations could fine-tune RSSC to local conditions and, consequently, exhibit subtle but important differences in timing of sex change. Previous analyses of the invariant sex-change model have not explicitly considered the potential for meaningful differences in RSSC within the confines of a broader ‘invariance’. Furthermore, these tests differ in their geographical and taxonomic scope, which could account for their conflicting conclusions. We test the model using several populations of three female-first sex-changing Caribbean parrotfish species. We first test for species-wide invariance using traditional log–log regressions and randomisation analyses of population-specific point estimates of RSSC. We then consider error around these point estimates, which is rarely incorporated into invariant analyses, to test for differences among populations in RSSC. Log–log regressions could not unequivocally diagnose invariance in RSSC across populations; randomisation tests identified an invariant RSSC in redband parrotfish only. Analyses that incorporated within-population variability in RSSC revealed differences among populations in timing of sex change, which were independent of geography for all species. While RSSC may be evolutionarily constrained (as in redband parrotfish), within these bounds the timing of sex change may vary among populations. This variability is overlooked by traditional invariant analyses and not predicted by the existing invariant model.     

Model selection and information theory in geographical ecology

Aim   Although parameter estimates are not as affected by spatial autocorrelation as Type I errors, the change from classical null hypothesis significance testing to model selection under an information theoretic approach does not completely avoid problems caused by spatial autocorrelation. Here we briefly review the model selection approach based on the Akaike information criterion (AIC) and present a new routine for Spatial Analysis in Macroecology (SAM) software that helps establishing minimum adequate models in the presence of spatial autocorrelation.
Innovation    We illustrate how a model selection approach based on the AIC can be used in geographical data by modelling patterns of mammal species in South America represented in a grid system ( n  = 383) with 2° of resolution, as a function of five environmental explanatory variables, performing an exhaustive search of minimum adequate models considering three regression methods: non-spatial ordinary least squares (OLS), spatial eigenvector mapping and the autoregressive (lagged-response) model. The models selected by spatial methods included a smaller number of explanatory variables than the one selected by OLS, and minimum adequate models contain different explanatory variables, although model averaging revealed a similar rank of explanatory variables.
Main conclusions    We stress that the AIC is sensitive to the presence of spatial autocorrelation, generating unstable and overfitted minimum adequate models to describe macroecological data based on non-spatial OLS regression. Alternative regression techniques provided different minimum adequate models and have different uncertainty levels. Despite this, the averaged model based on Akaike weights generates consistent and robust results across different methods and may be the best approach for understanding of macroecological patterns.     

Score Tests for Exploring Complex Models: Application to HIV Dynamics Models

In biostatistics, more and more complex models are being developed. This is particularly the case in system biology. Fitting complex models can be very time‐consuming, since many models often have to be explored. Among the possibilities are the introduction of explanatory variables and the determination of random effects. The particularity of this use of the score test is that the null hypothesis is not itself very simple; typically, some random effects may be present under the null hypothesis. Moreover, the information matrix cannot be computed, but only an approximation based on the score. This article examines this situation with the specific example of HIV dynamics models. We examine the score test statistics for testing the effect of explanatory variables and the variance of random effect in this complex situation. We study type I errors and the statistical powers of this score test statistics and we apply the score test approach to a real data set of HIV‐infected patients.     

Determinants of the steep species–area relationship of coral reef fishes

The increase in species richness with area is known as the species–area relationship (SPAR). Although several mutually non-exclusive processes may produce the SPAR, the null, often ignored, hypothesis states that a SPAR can be generated by random placement alone. The log–log-transformed SPAR of coral reef fishes on small patch-reefs revealed a steep slope of 0.55. However, this slope was dependent on the cumulative area of the reef examined and was therefore affected by random placement. After statistically removing the contribution of random placement from the SPAR, the slope was estimated to be 0.21. This is consistent with estimates from other, mostly terrestrial, systems. Furthermore, a randomization procedure, where the probability of fishes to reach a patch was proportional to reef area, showed that the field measured SPAR did not differ from random placement. In addition, fish assemblages on species poor reefs did not form subsets of species rich reefs (i.e., no nestedness) beyond that expected from random placement. Steep log–log-transformed SPARs can be formed by random placement alone, indicating that caution should be used when assigning an ecological meaning to SPARs generated from small spatial scales.     

Allometric estimation of metabolic rates in animals

The relationship between body mass (M) and metabolic rate (MR) typically accounts for most (>90%) of the inter-specific variation in MR. As such, when measurement of a species of interest is not possible, its MR can often be predicted using M. However, choosing an appropriate relationship to make such predictions is critical, and the choice is complicated by ongoing debate about the structure of the relationship between M and MR. The present study examines a range of methods including ordinary least squares (OLS), reduced major axis (RMA), and phylogenetically-informed (PI) approaches for estimating log(MR) from log(M), as well as non-linear approaches for estimating the relationship between MR and M without the need for log-transformation. Using data for the basal metabolic rates of mammals, it is shown that RMA regression overestimates the scaling exponent of MR (b, where MR=aM(b)), suggesting that OLS regression is appropriate for these data. PI approaches are preferred over non-PI ones, and the best estimates of log(MR) are obtained by including information on body temperature, climate, habitat, island endemism, and use of torpor in addition to log(M). However, the use of log-transformed data introduces bias into estimates of MR, while the use of non-linear regression underestimates MR for small mammals. This suggests that no single relationship is appropriate for describing the relationship between MR and M for all mammals, and that relationships for more narrow taxonomic groups or body mass ranges should be used when predicting MR from M.     

Research frontiers in null model analysis

Null models are pattern‐generating models that deliberately exclude a mechanism of interest, and allow for randomization tests of ecological and biogeographic data. Although they have had a controversial history, null models are widely used as statistical tools by ecologists and biogeographers. Three active research fronts in null model analysis include biodiversity measures, species co‐occurrence patterns, and macroecology. In the analysis of biodiversity, ecologists have used random sampling procedures such as rarefaction to adjust for differences in abundance and sampling effort. In the analysis of species co‐occurrence and assembly rules, null models have been used to detect the signature of species interactions. However, controversy persists over the details of computer algorithms used for randomizing presence–absence matrices. Finally, in the newly emerging discipline of macroecology, null models can be used to identify constraining boundaries in bivariate scatterplots of variables such as body size, range size, and population density. Null models provide specificity and flexibility in data analysis that is often not possible with conventional statistical tests.     

Nonparametric analysis of covariance for comparing change in randomized studies with baseline values subject to error

Change from baseline to a follow-up examination can be compared among two or more randomly assigned treatment groups by using analysis of variance on the change scores. However, a generally more sensitive (powerful) test can be performed using analysis of covariance (ANOVA) on the follow-up data with the baseline data as a covariate. This approach is not without potential problems, though. The assumption of ordinary ANCOVA of normally distributed errors is speculative for many variables employed in biomedical research. Furthermore, the baseline values are inevitably random variables and often are measured with error. This report investigates, in this situation, the validity and relative power of the ordinary ANCOVA test and two asymptotically distribution-free alternative tests, one based on the rank transformation and the other based on the normal scores transformation. The procedures are illustrated with data from a clinical trial. Normal and several nonnormal distributions, as well as varying degree of variable error, are studied by Monte Carlo methods. The normal scores test is generally recommended for statistical practice.     

Applying randomization tests to cluster analyses

Abstract. In applying randomization tests to hierarchical cluster analyses, we have noted a potentially misleading result: within a significant group, linkages are often identified as significant even when species are randomly distributed among the group's sites. We demonstrate this through a cluster analysis of a constructed matrix with two groups of 20 sites that share no species, and within each group species are randomly distributed among sites. A randomization test identified both of the groups and all linkages within them as significant, while the same test found all linkages non‐significant in the cluster analysis of a matrix containing just one of the two groups of 20 sites. In general, a non‐random distribution of species within a data set shortens linkages relative to distances in null distributions derived from randomized versions of the data. This confounds efforts to identify significant sub‐groups within a significant group. However, the significance of sub‐groups possibly could be tested by comparing linkage distances to a null distribution derived from the randomization and clustering of a sub‐matrix containing only the sites within the larger group. In essence, this comparison tests the null hypothesis that within the significant group, sites represent random assemblages of species. When applied to actual data sets, an approach involving sequential randomization tests could allow the evaluation of all nodes in a classification, increasing the utility of randomization tests and strengthening the interpretation of groups produced by cluster analysis.     

Optimal annual routines: behaviour in the context of physiology and ecology

Organisms in a seasonal environment often schedule activities in a regular way over the year. If we assume that such annual routines have been shaped by natural selection then life-history theory should provide a basis for explaining them. We argue that many life-history trade-offs are mediated by underlying physiological variables that act on various time scales. The dynamics of these variables often preclude considering one period of the year in isolation. In order to capture the essence of annual routines, and many life-history traits, a detailed model of changes in physiological state over the annual cycle is required. We outline a modelling approach based on suitable physiological and ecological state variables that can capture this underlying biology, and describe how models based on this approach can be used to generate a range of insights and predictions.     

Orthogonal Invariant Sets of the Diffusion Tensor and the Development of a Curvilinear Set Suitable for Low-Anisotropy Tissues

We develop a curvilinear invariant set of the diffusion tensor which may be applied to Diffusion Tensor Imaging measurements on tissues and porous media. This new set is an alternative to the more common invariants such as fractional anisotropy and the diffusion mode. The alternative invariant set possesses a different structure to the other known invariant sets; the second and third members of the curvilinear set measure the degree of orthotropy and oblateness/prolateness, respectively. The proposed advantage of these invariants is that they may work well in situations of low diffusion anisotropy and isotropy, as is often observed in tissues such as cartilage. We also explore the other orthogonal invariant sets in terms of their geometry in relation to eigenvalue space; a cylindrical set, a spherical set (including fractional anisotropy and the mode), and a log-Euclidean set. These three sets have a common structure. The first invariant measures the magnitude of the diffusion, the second and third invariants capture aspects of the anisotropy; the magnitude of the anisotropy and the shape of the diffusion ellipsoid (the manner in which the anisotropy is realised). We also show a simple method to prove the orthogonality of the invariants within a set.     

MPowering ecologists: community assembly tools for community assembly rules

Null model tests of presence–absence data (‘NMTPAs’) provide important tools for inferring effects of competition, facilitation, habitat filtering, and other ecological processes from observational data. Many NMTPAs have been developed, but they often yield conflicting conclusions when applied to the same data. Type I and II error rates, size, power, robustness and bias provide important criteria for assessing which tests are valid, but these criteria need to be evaluated contingent on the sample size, null hypothesis of interest, and assumptions that are appropriate for the data set that is being analyzed. In this paper, we confirm that this is the case using the software MPower, evaluating the validity of NMTPAs contingent on the null hypothesis being tested, assumptions that can be made, and sample size. Evaluating the validity of NMTPAs contingent on these factors is important towards ensuring that reliable inferences are drawn from observational data about the processes controlling community assembly.     

New perspectives for estimating body condition from mass/length data: the scaled mass index as an alternative method

Body condition is assumed to influence an animal's health and fitness. Various non‐destructive methods based on body mass and a measure of body length have been used as condition indices (CIs), but the dominant method amongst ecologists is currently the calculation of residuals from an ordinary least squares (OLS) regression of body mass against length. Recent studies of energy reserves in small mammals and starlings claimed to validate this method, although we argue that they did not include the most appropriate tests since they compared the CI with the absolute size of energy reserves. We present a novel CI (the ‘scaled mass index’) based on the central principle of scaling, with important methodological, biological and conceptual advantages. Through a reanalysis of data from small mammals, starlings and snakes, we show that the scaled mass index is a better indicator of the relative size of energy reserves and other body components than OLS residuals, performing better in all seven species and in 19 out of 20 analyses. We also present an empirical and theoretical comparison of the scaled mass index and OLS residuals as CIs. We argue that the scaled mass index is a useful new tool for ecologists.     

Testing Pairwise Association between Spatially Autocorrelated Variables: A New Approach Using Surrogate Lattice Data

Background

Independence between observations is a standard prerequisite of traditional statistical tests of association. This condition is, however, violated when autocorrelation is present within the data. In the case of variables that are regularly sampled in space (i.e. lattice data or images), such as those provided by remote-sensing or geographical databases, this problem is particularly acute. Because analytic derivation of the null probability distribution of the test statistic (e.g. Pearson''s r) is not always possible when autocorrelation is present, we propose instead the use of a Monte Carlo simulation with surrogate data.

Methodology/Principal Findings

The null hypothesis that two observed mapped variables are the result of independent pattern generating processes is tested here by generating sets of random image data while preserving the autocorrelation function of the original images. Surrogates are generated by matching the dual-tree complex wavelet spectra (and hence the autocorrelation functions) of white noise images with the spectra of the original images. The generated images can then be used to build the probability distribution function of any statistic of association under the null hypothesis. We demonstrate the validity of a statistical test of association based on these surrogates with both actual and synthetic data and compare it with a corrected parametric test and three existing methods that generate surrogates (randomization, random rotations and shifts, and iterative amplitude adjusted Fourier transform). Type I error control was excellent, even with strong and long-range autocorrelation, which is not the case for alternative methods.

Conclusions/Significance

The wavelet-based surrogates are particularly appropriate in cases where autocorrelation appears at all scales or is direction-dependent (anisotropy). We explore the potential of the method for association tests involving a lattice of binary data and discuss its potential for validation of species distribution models. An implementation of the method in Java for the generation of wavelet-based surrogates is available online as supporting material.     

Mammalian species richness and morphological complexity along an elevational gradient in the arid south-west

Aim We examined the relationship between species richness and morphological complexity of terrestrial mammal communities along an elevational gradient. Location The gradient is in the Sonoran Desert in Southern California and extends from a sand dune habitat near sea level to coniferous forest ending at >2600 m. Methods Morphological diversity, characterized by both size and shape of coexisting mammal species, was estimated within and between sites from projections of variables on principal components axes. Similarities among species were calculated as Euclidean distances. To tease apart size and shape, we constructed two principal component analyses: one based on log-transformed original measurements, the other on log-transformed proportional shape variables. To test whether species number accounted for the morphological diversity at each site we designed two null models. The models generated were random communities generated from the forty-two-species pool. Indices of morphological diversity for real communities were compared with the results of 500 simulations of each null model. Results Species richness varied along the gradient, peaking in the mid-elevation agave-ocotillo habitat. Morphological diversity of shapes and sizes correlated strongly with species richness. Locomotor, tooth, and skull traits were all important in distinguishing among species. Main conclusions Two important patterns emerged: (1) diversity of both sizes and of shapes of species within communities correlated positively with species number, and both sets of variables behaved similarly across this gradient; (2) the most species rich sites were not composed of specialists on these best places, but rather, a community of species derived from overlapping faunal groups.     

Flight Initiation Distance and Starting Distance: Biological Effect or Mathematical Artefact?

In many studies, flight initiation distance (FID, the distance at which a prey starts to flee at the approach of a walker) is positively related to starting distance (SD, the distance at which the walker begins to approach) and alert distance (AD, the distance at which the focal individual becomes alert to the threat). In spite of the fundamental differences between SD, a covariate that may not have any biological effect, and AD, a measure related to the behaviour of the animal, it is common to use SD as a proxy for AD when AD is hard to measure (e.g. in species that do not exhibit distinguishable alert postures). However, the relationship between SD and AD or FID may not have any biological reasons, but may instead simply result from a mathematical artefact because of the constraints SD ≥ AD ≥ FID. Under such constrains, the homoscedasticity assumption is violated, and thus, the classical null hypothesis of linear regression (slope = 0) is invalid. In this study, we first show that using SD as a proxy for AD can strongly affect the results on FID. Using data from FID tests on alpine marmots (Marmota marmota), a linear mixed model with AD as a covariate, suggested that the interaction between previous activity and AD had an effect on FID, while this effect was not detected when SD replaced AD as the covariate in the analysis. We then propose that the actual statistical test of the relationship between SD, AD and FID should be based on a null hypothesis that incorporates the constraint SD ≥ AD ≥ FID ≥ 0 and generate 95% CI of simulated slopes obtained from random values under this constraint. This null hypothesis can be rejected if the observed slope of the relationship between two of these variables is outside the 95% CI. We demonstrated that, for alpine marmots, the observed slope of the relationship between AD and SD was within the 95% CI of the simulated slopes. The absence of a statistically significant biological effect in the relationship between SD and AD raises important questions on the outcome of relationship between SD and FID. In Alpine marmot flight, decision should be studied separating the effect of SD on AD and the effect of AD on FID.     

Most Powerful Permutation Invariant Tests for Relatedness Hypotheses Using Genotypic Data

The problem of inferring kinship structure among a sample of individuals using genetic markers is considered with the objective of developing hypothesis tests for genetic relatedness with nearly optimal properties. The class of tests considered are those that are constrained to be permutation invariant, which in this context defines tests whose properties do not depend on the labeling of the individuals. This is appropriate when all individuals are to be treated identically from a statistical point of view. The approach taken is to derive tests that are probably most powerful for a permutation invariant alternative hypothesis that is, in some sense, close to a null hypothesis of mutual independence. This is analagous to the locally most powerful test commonly used in parametric inference. Although the resulting test statistic is a U-statistic, normal approximation theory is found to be inapplicable because of high skewness. As an alternative it is found that a conditional procedure based on the most powerful test statistic can calculate accurate significance levels without much loss in power. Examples are given in which this type of test proves to be more powerful than a number of alternatives considered in the literature, including Queller and Goodknight's (1989) estimate of genetic relatedness, the average number of shared alleles (Blouin, 1996), and the number of feasible sibling triples (Almudevar and Field, 1999).     

Bayesian inferences on umbrella orderings

In regression applications with categorical predictors, interest often focuses on comparing the null hypothesis of homogeneity to an ordered alternative. This article proposes a Bayesian approach for addressing this problem in the setting of normal linear and probit regression models. The regression coefficients are assigned a conditionally conjugate prior density consisting of mixtures of point masses at 0 and truncated normal densities, with a (possibly unknown) changepoint parameter included to accommodate umbrella ordering. Two strategies of prior elicitation are considered: (1) a Bayesian Bonferroni approach in which the probability of the global null hypothesis is specified and local hypotheses are considered independent; and (2) an approach which treats these probabilities as random. A single Gibbs sampling chain can be used to obtain posterior probabilities for the different hypotheses and to estimate regression coefficients and predictive quantities either by model averaging or under the preferred hypothesis. The methods are applied to data from a carcinogenesis study.     

Permutation tests for random effects in linear mixed models

Inference regarding the inclusion or exclusion of random effects in linear mixed models is challenging because the variance components are located on the boundary of their parameter space under the usual null hypothesis. As a result, the asymptotic null distribution of the Wald, score, and likelihood ratio tests will not have the typical χ(2) distribution. Although it has been proved that the correct asymptotic distribution is a mixture of χ(2) distributions, the appropriate mixture distribution is rather cumbersome and nonintuitive when the null and alternative hypotheses differ by more than one random effect. As alternatives, we present two permutation tests, one that is based on the best linear unbiased predictors and one that is based on the restricted likelihood ratio test statistic. Both methods involve weighted residuals, with the weights determined by the among- and within-subject variance components. The null permutation distributions of our statistics are computed by permuting the residuals both within and among subjects and are valid both asymptotically and in small samples. We examine the size and power of our tests via simulation under a variety of settings and apply our test to a published data set of chronic myelogenous leukemia patients.