A comparison of methods for fitting allometric equations to field metabolic rates of animals

We re-examined data for field metabolic rates of varanid lizards and marsupial mammals to illustrate how different procedures for fitting the allometric equation can lead to very different estimates for the allometric coefficient and exponent. A two-parameter power function was obtained in each case by the traditional method of back-transformation from a straight line fitted to logarithms of the data. Another two-parameter power function was then generated for each data-set by non-linear regression on values in the original arithmetic scale. Allometric equations obtained by non-linear regression described the metabolic rates of all animals in the samples. Equations estimated by back-transformation from logarithms, on the other hand, described the metabolic rates of small species but not large ones. Thus, allometric equations estimated in the traditional way for field metabolic rates of varanids and marsupials do not have general importance because they do not characterize rates for species spanning the full range in body size. Logarithmic transformation of predictor and response variables creates new distributions that may enable investigators to perform statistical analyses in compliance with assumptions underlying the tests. However, statistical models fitted to transformations should not be used to estimate parameters of equations in the arithmetic domain because such equations may be seriously biased and misleading. Allometric analyses should be performed on values expressed in the original scale, if possible, because this is the scale of interest.     

On the use of log‐transformation versus nonlinear regression for analyzing biological power laws

Xiao and colleagues re‐examined 471 datasets from the literature in a major study comparing two common procedures for fitting the allometric equation y = ax^b to bivariate data (Xiao et al., 2011). One of the procedures was the traditional allometric method, whereby the model for a straight line fitted to logarithmic transformations of the original data is back‐transformed to form a two‐parameter power function with multiplicative, lognormal, heteroscedastic error on the arithmetic scale. The other procedure was standard nonlinear regression, whereby a two‐parameter power function with additive, normal, homoscedastic error is fitted directly to untransformed data by nonlinear least squares. Xiao and colleagues articulated a simple (but explicit) protocol for fitting and comparing the alternative models, and then used the protocol to examine each of the datasets in their compilation. The traditional method was said to provide a better fit in 69% of the cases and an equivalent fit in another 15%, so the investigation appeared to validate findings from a large majority of prior studies on allometric variation. However, focus for the investigation by Xiao and colleagues was overly narrow, and statistical models apparently were not validated graphically in the scale of measurement. The present study re‐examined a subset of the cases using a larger pool of candidate models and graphical validation, and discovered complexities that were overlooked in their investigation. Some datasets that appeared to be described better by the traditional method actually were unsuited for use in an allometric analysis, whereas other datasets were not described adequately by a two‐parameter power function, regardless of how the model was fitted. Thus, conclusions reached by Xiao and colleagues are not well supported and their paradigm for fitting allometric equations is unreliable. Future investigations of allometric variation should adopt a more holistic approach and incorporate graphical validation on the original arithmetic scale. © 2014 The Linnean Society of London, Biological Journal of the Linnean Society, 2014, 113 , 1167–1178.     

Allometric equations for predicting body mass of dinosaurs

We use data from the literature to compare two statistical procedures for estimating mass (or size) of quadrupedal dinosaurs and other extraordinarily large animals in extinct lineages. Both methods entail extrapolation from allometric equations fitted to data for a reference group of contemporary animals having a body form similar to that of the dinosaurs. The first method is the familiar one of fitting a straight line to logarithmic transformations, followed by back-transformation of the resulting equation to a two-parameter power function in the arithmetic scale. The second procedure entails fitting a two-parameter power function directly to arithmetic data for the extant forms by nonlinear regression. In the example presented here, the summed circumferences for humerus plus femur for 33 species of quadrupedal mammals was the predictor variable in the reference sample and body mass was the response variable. The allometric equation obtained by back-transformation from logarithms was not a good fit to the largest species in the reference sample and presumably led to grossly inaccurate estimates for body mass of several large dinosaurs. In contrast, the allometric equation obtained by nonlinear regression described data in the reference sample quite well, and it presumably resulted in better estimates for body mass of the dinosaurs. The problem with the traditional analysis can be traced to change in the relationship between predictor and response variables attending transformation, thereby causing measurements for large animals not to be weighted appropriately in fitting models by least squares regression. Extrapolations from statistical models obtained by back-transformation from lines fitted to logarithms are unlikely to yield reliable predictions for body size in extinct animals. Numerous reports on the biology of dinosaurs, including recent studies of growth, may need to be reconsidered in light of our findings.     

Is non‐loglinear allometry a statistical artifact?

The allometric equation, y = ax^b, is commonly fitted to data indirectly by transforming predictor (x) and response (y) variables to logarithms, fitting a straight line to the transformations, and then back‐transforming (exponentiating) the resulting equation to the original arithmetic scale. Sometimes, however, transformation fails to linearize the observations, thereby giving rise to what has come to be known as non‐loglinear allometry. A smooth curve for observations displayed on a log–log plot is usually interpreted to mean that the scaling exponent in the allometric equation is a continuously changing function of body size, whereas a breakpoint between two (or more) linear segments on a log–log plot is typically taken to mean that the exponent changes abruptly, coincident with some important milestone in development. I applied simple graphical and statistical procedures in re‐analyses of three well‐known examples of non‐loglinear allometry, and showed in every instance that the relationship between predictor and response can be described in the original scale by simple functions with constant values for the exponent b. In no instance does the allometric exponent change during the course of development. Transformation of data to logarithms created new distributions that actually obscured the relationships between predictor and response variables in these investigations, and led to erroneous perceptions of growth. Such confounding effects of transformation are not limited to non‐loglinear allometry but are common to all applications of the allometric method. © 2012 The Linnean Society of London, Biological Journal of the Linnean Society, 2012, ?? , ??–??.     

Determining best complete subsets of specimens and characters for multivariate morphometric studies in the presence of large amounts of missing data

Missing data are frequent in morphometric studies of both fossil and recent material. A common method of addressing the problem of missing data is to omit combinations of characters and specimens from subsequent analyses; however, omitting different subsets of characters and specimens can affect both the statistical robustness of the analyses and the resulting biological interpretations. We describe a method of examining all possible subsets of complete data and of scoring each subset by the 'condition' (ratio of first eigenvalue to second, or of second to first, depending on context) of the corresponding covariance or correlation matrix, and subsequently choosing the submatrix that either optimizes one of these criteria or matches the estimated condition of the original data matrix. We then describe an extension of this method that can be used to choose the 'best' characters and specimens for which some specified proportion of missing data can be estimated using standard imputation techniques such as the expectation-maximization algorithm or multiple imputation. The methods are illustrated with published and unpublished data sets on fossil and extant vertebrates. Although these problems and methods are discussed in the context of conventional morphometric data, they are applicable to many other kinds of data matrices.  © 2006 The Linnean Society of London, Biological Journal of the Linnean Society , 2006, 88 , 309–328.     

Rotational distortion in conventional allometric analyses

Three data sets from the recent literature were submitted to new analyses to illustrate the rotational distortion that commonly accompanies traditional allometric analyses and that often causes allometric equations to be inaccurate and misleading. The first investigation focused on the scaling of evaporative water loss to body mass in passerine birds; the second was concerned with the influence of body size on field metabolic rates of rodents; and the third addressed interspecific variation in kidney mass among primates. Straight lines were fitted to logarithmic transformations by Ordinary Least Squares and Generalized Linear Models, and the resulting equations then were re-expressed as two-parameter power functions in the original arithmetic scales. The re-expressed models were displayed on bivariate graphs together with tracings for equations fitted directly to untransformed data by nonlinear regression. In all instances, models estimated by back-transformation failed to describe major features of the arithmetic distribution whereas equations fitted by nonlinear regression performed quite well. The poor performance of equations based on models fitted to logarithms can be traced to the increased weight and leverage exerted in those analyses by observations for small species and to the decreased weight and leverage exerted by large ones. The problem of rotational distortion can be avoided by performing exploratory analysis on untransformed values and by validating fitted models in the scale of measurement.     

Logistic regression in comparative wood anatomy: tracheid types, wood anatomical terminology, and new inferences from the Carlquist and Hoekman southern Californian data set

Despite collecting copious amounts of data, wood anatomists rarely perform appropriate statistical analyses, especially in the case of categorical variables. Nevertheless, anatomists have succeeded in identifying strong ecological trends. We show that, with only a slightly more sophisticated analysis, the strength and significance of 'well-known' associations can be quantified, and new associations pinpointed. Using logistic regression to reanalyse the classic Carlquist and Hoekman data set for the southern Californian flora, we show strong support for the notion that true tracheid presence lowers vessel grouping; in contrast, vasicentric tracheids are associated with a diversity of vessel grouping strategies. We show that statistical models can refine anatomical interpretations by identifying unusual species. For example, Fremontodendron californicum and Baccharis salicifolia (=  B. glutinosa ) were identified as unusual in lacking vasicentric tracheids; a consultation of preparations revealed that they are indeed present. For purposes of ecological wood anatomy, anatomical terminology should reflect cell function; we suggest that terminological systems that yield better predictive power in statistical models such as ours are preferable. Finally, we make recommendations ranging from the statistical, e.g. the need to check assumptions and the need for the inclusion of phylogeny, to the biological, e.g. gathering data expressly designed to test functional hypotheses rather than all of the information in standardized lists.  © 2007 The Linnean Society of London, Botanical Journal of the Linnean Society , 2007, 154 , 331–351.     

Long-bone geometry in columnar-limbed animals: allometry of the proboscidean appendicular skeleton

Limb-bone allometry was investigated for 19 species of proboscideans, spanning almost the entire phylogenetic spectrum. More archaic proboscideans ('gompthotheres') have substantially thicker long-bone diaphyses relative to length than elephantids, as has been suggested previously, but contrary to previous suggestions it could not be confirmed that Mammuthus had more massive long-bone diaphyses on average than extant Elephas and Loxodonta . When correcting for phylogeny, the circumference of the limb bones to their length in proboscideans as a group generally scale with negative allometry, becoming stouter with increased length, as would be expected from limb mechanics. Few slopes were, however, statistically significantly negatively allometric. Correcting for phylogeny produced better correlations than traditional regression analyses, in contrast to most other studies where the reverse is the case. Intraspecific analyses of extant Elephas and Loxodonta , in addition to Mammuthus primigenius , Mammut americanum, and Gomphotherium productum , also resulted in negatively allometric regression slopes, frequently conforming to the theory of elastic similarity, as could be expected from the columnar posture of proboscideans. At present the reasons for the more massive limbs of gomphotheres s.l. are not fully understood. © 2007 The Linnean Society of London, Zoological Journal of the Linnean Society , 2007, 149 , 423–436.     

Polymorphism hides cryptic species in Oerstedia dorsalis (Nemertea, Hoplonemertea)

The marine hoplonemertean Oerstedia dorsalis is considered to be a highly polymorphic species with extensive geographic distribution. We show, based on mitochondrial cytochrome c oxidase subunit I, 16S and nuclear internal transcribed spacer sequences, that there are genetic subsets withing this species. Seventy-one specimens of various colours from different geographic localities (in Europe) were sequenced and analysed using statistical parsimony and Bayesian analysis. Both analyses supported nine major clades. We conclude that O. dorsalis hides different species with geographic resolution. These species, however, appear to be polymorphic as well, and we find no diagnostic features in pigmentation or external characters to separate species within this complex.  © 2009 The Linnean Society of London, Biological Journal of the Linnean Society , 2009, 98 , 556–567.     

Fitting statistical models in bivariate allometry

Several attempts have been made in recent years to formulate a general explanation for what appear to be recurring patterns of allometric variation in morphology, physiology, and ecology of both plants and animals (e.g. the Metabolic Theory of Ecology, the Allometric Cascade, the Metabolic‐Level Boundaries hypothesis). However, published estimates for parameters in allometric equations often are inaccurate, owing to undetected bias introduced by the traditional method for fitting lines to empirical data. The traditional method entails fitting a straight line to logarithmic transformations of the original data and then back‐transforming the resulting equation to the arithmetic scale. Because of fundamental changes in distributions attending transformation of predictor and response variables, the traditional practice may cause influential outliers to go undetected, and it may result in an underparameterized model being fitted to the data. Also, substantial bias may be introduced by the insidious rotational distortion that accompanies regression analyses performed on logarithms. Consequently, the aforementioned patterns of allometric variation may be illusions, and the theoretical explanations may be wide of the mark. Problems attending the traditional procedure can be largely avoided in future research simply by performing preliminary analyses on arithmetic values and by validating fitted equations in the arithmetic domain. The goal of most allometric research is to characterize relationships between biological variables and body size, and this is done most effectively with data expressed in the units of measurement. Back‐transforming from a straight line fitted to logarithms is not a generally reliable way to estimate an allometric equation in the original scale.     

Phenotypic and genetic variation in male genitalia in the seedbug, Lygaeus equestris (Heteroptera)

Male genitalia evolve through sexual selection and, in insects, tend to show negative static allometry, low phenotypic variation, and are usually relatively small. Much less is known about the genetic variation and heritability of male genitalia. Additionally, in instances where the intromittent organ is greatly elongated, it is unclear whether typical patterns of genital scaling and variation also apply. In the present study, we investigated the allometry, variation, and heritability of male genital length in the seedbug, Lygaeus equestris , a species with a greatly elongated intromittent organ (i.e. almost as long as male body size). We found that genital length was negatively allometric, in spite of its great length, and was no more variable than nongenital traits. Additionally, genital length was significantly heritable and had considerable evolvability.  © 2009 The Linnean Society of London, Biological Journal of the Linnean Society , 2009, 98 , 400–405.     

Genome size evolution and polyploidy in the Daphnia pulex complex (Cladocera: Daphniidae)

Genome size was estimated in 49 clones of the Daphnia pulex complex from temperate and subarctic locations using flow cytometry and microsatellite DNA analyses. Significant genome size differences were found in diploid species belonging to the two genetically distinct groups (the pulicaria and the tenebrosa groups), with clones from the tenebrosa group having genome sizes 22% larger than those in the pulicaria group. Combined flow cytometry and microsatellite DNA analyses revealed that nearly all polyploid clones in the D. pulex complex are triploid and not tetraploid, as was previously suggested. Sequencing analyses of the ND5 gene to position clones in their respective clades within the D. pulex complex have uncovered three triploid clones of Daphnia middendorffiana with a D. pulex maternal parent. This result was unexpected because Daphnia pulicaria has always been identified as the maternal parent of these hybrid polyploid clones. Triploid clones likely owe their origins to interactions between sexual and asexual populations. Further interactions in the tenebrosa group have generated tetraploid clones but these events have been rare.  © 2009 The Linnean Society of London, Biological Journal of the Linnean Society , 2009, 97 , 68–79.     

Variation in the digging apparatus of the subterranean silvery mole-rat, Heliophobius argenteocinereus (Rodentia, Bathyergidae): the role of ecology and geography

The skull of most subterranean tooth-digging rodents is markedly affected by their digging mode. In the present study, we investigated the cranial variation in a strictly subterranean, highly specialized Afrotropical tooth-digger, Heliophobius argenteocinereus (Bathyergidae, Rodentia), using a geometric morphometric approach and evaluated the effect of different factors on size and shape differences among four populations. No evidence for sexual dimorphism was found in skull size or shape. The cranial shape variation was large and influenced mainly by the type of habitat (miombo woodland versus farmland and grassland) and the latitudinal gradient. The dorsal side of the skull appears to be more plastic and adaptable to local environments, as well as more independent of size, than the ventral side. Only the shortening of the rostrum is presumably an adaptive process independent of size that leads to an increase of efficacy of the tooth-digging apparatus in Heliophobius , whereas the increase in the in-force and the more procumbent incisors both comprise size-related changes caused by ontogenetic allometric growth.  © 2009 The Linnean Society of London, Biological Journal of the Linnean Society , 2009, 97 , 822–831.     

Varying rates of diversification in the genus Melitaea (Lepidoptera: Nymphalidae) during the past 20 million years

The influence of Quarternary glacial cycles on the extant diversity of Holarctic species has been intensively studied. It has been hypothesized that palaeoclimatic changes are responsible for divergence events in lineages. A constant improvement in DNA sequencing and modeling methods, as well as palaeoclimatic reconstruction, permit a deeper exploration of general causes of speciation in geological time. In the present study, we sampled, as exhaustively as possible, the butterflies belonging to the genus Melitaea (Lepidoptera: Nymphalidae), which are widely spread in the Palaearctic region. We conducted analyses to assess the phylogeny of the genus and estimated the timing of divergence and the most likely distribution of ancestral populations. The results obtained indicate that the systematics of the genus is in need of revision and that the diversity of the genus has been profoundly shaped by palaeoenvironmental changes during its evolutionary history. The present study also emphasizes that, when employed with caveats, major palaeoenvironmental events could represent very powerful tools for the calibration of the dating of divergences using molecular data.  © 2009 The Linnean Society of London, Biological Journal of the Linnean Society , 2009, 97 , 346–361.     

Areas of endemism of Mexican mammals: reanalysis applying the optimality criterion

In order to test Mexican areas of endemism of mammals identified by previous parsimony analyses of endemicity (PAEs), we applied the optimality criterion to three data matrices (based on point records, potential distributional models and the fill option in software NDM). We modelled the ecological niches of 429 terrestrial mammal species using the genetic algorithm for rule-set prediction (GARP) and models were projected as potential distributional areas. We overlapped the point occurrence data and the individual maps of potential distributions to a grid of 1° latitude–longitude. Three matrices of 247 grid cells (areas) and 429 species were built: (1) a binary matrix with '0' for absence and '1' for presence of at least one record of the species inside the grid-cell; (2) a three-state matrix similar to (1) but assigning the state '2' to the assumed presence in the model of potential distribution; and (3) a three-state matrix similar to (2), but applying the fill option of software NDM instead of using a model. The optimality criterion was performed in NDM version 2.7 and results were examined with VNDM version 2.7. The first and second matrices showed 13 areas of endemism and the third identified 16 areas of endemism. NDM provided a better resolution than PAE, allowing us to identify several new areas of endemism, previously undetected. Ecological niche models, projected as potential distributional areas, and the optimality criterion are very useful to identify areas of endemism, although they should be used with caution because they may overpredict potential distributional areas. PAE seems to underestimate the areas of endemism identified.  © 2009 The Linnean Society of London, Biological Journal of the Linnean Society , 2009, 98 , 468–478.     

Population divergence in the amphicarpic species Amphicarpaea edgeworthii Benth. (Fabaceae): microsatellite markers and leaf morphology

Comparative analyses of the genetic differentiation in microsatellite markers ( F _ST) and leaf morphology characters ( Q _ST) of Amphicarpaea edgeworthii Benth. were conducted to gain insight into the roles of random processes and natural selection in the population divergence. Simple sequence repeat analyses on 498 individuals of 19 natural populations demonstrate that a significant genetic differentiation occurs among populations (mean F _ST = 0.578), and A. edgeworthii is a highly self-fertilized species (mean selfing rate s  = 0.989). The distribution pattern of genetic diversity in this species shows that central populations possess high genetic diversity (e.g. population WL with H _E = 0.673 and population JG with H _E = 0.663), whereas peripheral ones have a low H _E as in population JD (0.011). The morphological divergence of leaf shape was estimated by the elliptical Fourier analysis on the data from 11 natural and four common garden populations. Leaf morphology analyses indicate the morphological divergence does not show strong correlation with the genetic differentiation ( R  = 0.260, P  = 0.069). By comparing the 95% confidence interval of Q _ST with that of F _ST, Q _ST values for five out of 12 quantitative traits are significantly higher than the average F _ST value over eight microsatellite loci. The comparison of F _ST and Q _ST suggests that two kinds of traits can be driven by different evolutionary forces, and the population divergence in leaf morphology is shaped by local selections.  © 2009 The Linnean Society of London, Biological Journal of the Linnean Society , 2009, 96 , 505–516.     

Female-biased sexual size dimorphism in tinamous: a comparative test fails to support Rensch's rule

Rensch's rule states that degree of sexual dimorphism increases with body size in species with larger males, and decreases with body size in those with larger females. To test this rule, we assessed the pattern of sexual size dimorphism in tinamous using a comparative analysis of independent contrasts. Tinamous are a monophyletic group of primitive birds comprising at least 47 ground dwelling species with prominent or exclusive paternal care of eggs and offspring. Although the size of females exceeded that of males in most considered species, we found an isometric relationship between males and females, instead of the negative allometric one predicted by Rensch's rule. Previous studies in Strigiformes and Falconiformes found positive allometric and isometric relationships respectively, and, considering these findings with our results, we conclude that Rensch's rule is not supported by birds with exclusively female-biased sexual dimorphism in size.  © 2003 The Linnean Society of London, Biological Journal of the Linnean Society , 2003, 80 , 519–527     

Evolution of ontogeny in the hippopotamus skull: using allometry to dissect developmental change

Allometry describes the effect of size change on aspects of an organism's form and can be used to summarize the developmental history of growing parts of an animal. By comparing how allometric growth differs between species, it is possible to reveal differences in their pathways of development. The ability to compare and categorize developmental change between species is demonstrated here using morphometric methods. This involves the interspecific statistical comparison of a large number of bivariate relationships that summarize ontogenetic trajectories. These linear ontogenetic trajectories can be modified as they evolve in any of three ways: ontogenetic scaling indicative of change in the duration of growth, lateral shifts indicative of changes in prenatal development, and directional change indicative of novel modes of postnatal growth. I apply this analysis to skulls of the common hippopotamus ( Hippopotamus amphibius ) and the pygmy hippopotamus ( Hexaprotodon liberiensis ). The number of allometric changes falling into each category was statistically determined and Jolicoeur's multivariate generalization of simple allometry was used to provide an overview of cranial variation. For these skulls, directional change was not found to be statistically significant, but ontogenetic scaling and lateral shifts were both common. This indicates that conserved patterns of growth covariance (ontogenetic scaling) can be separated from novel or derived patterns (directional change and/or lateral shifts). This study demonstrates that He. liberiensis is not simply an ontogenetically scaled version of its larger relative. The evolutionary implications of allometric growth variation are discussed in the light of these findings and those of other studies.  © 2003 The Linnean Society of London, Biological Journal of the Linnean Society , 2003, 80 , 625–638.