Ordinary least squares regression is indicated for studies of allometry

When it comes to fitting simple allometric slopes through measurement data, evolutionary biologists have been torn between regression methods. On the one hand, there is the ordinary least squares (OLS) regression, which is commonly used across many disciplines of biology to fit lines through data, but which has a reputation for underestimating slopes when measurement error is present. On the other hand, there is the reduced major axis (RMA) regression, which is often recommended as a substitute for OLS regression in studies of allometry, but which has several weaknesses of its own. Here, we review statistical theory as it applies to evolutionary biology and studies of allometry. We point out that the concerns that arise from measurement error for OLS regression are small and straightforward to deal with, whereas RMA has several key properties that make it unfit for use in the field of allometry. The recommended approach for researchers interested in allometry is to use OLS regression on measurements taken with low (but realistically achievable) measurement error. If measurement error is unavoidable and relatively large, it is preferable to correct for slope attenuation rather than to turn to RMA regression, or to take the expected amount of attenuation into account when interpreting the data.     

COMPARATIVE METHODS FOR THE ANALYSIS OF CONTINUOUS VARIABLES: GEOMETRIC INTERPRETATIONS

Abstract This study is concerned with statistical methods used for the analysis of comparative data (in which observations are not expected to be independent because they are sampled across phylogenetically related species). The phylogenetically independent contrasts (PIC), phylogenetic generalized least‐squares (PGLS), and phylogenetic autocorrelation (PA) methods are compared. Although the independent contrasts are not orthogonal, they are independent if the data conform to the Brownian motion model of evolution on which they are based. It is shown that uncentered correlations and regressions through the origin using the PIC method are identical to those obtained using PGLS with an intercept included in the model. The PIC method is a special case of PGLS. Corrected standard errors are given for estimates of the ancestral states based on the PGLS approach. The treatment of trees with hard polytomies is discussed and is shown to be an algorithmic rather than a statistical problem. Some of the relationships among the methods are shown graphically using the multivariate space in which variables are represented as vectors with respect to OTUs used as coordinate axes. The maximum‐likelihood estimate of the autoregressive parameter, ρ, has not been computed correctly in previous studies (an appendix with MATLAB code provides a corrected algorithm). The importance of the eigenvalues and eigenvectors of the connection matrix, W, for the distribution of ρ is discussed. The PA method is shown to have several problems that limit its usefulness in comparative studies. Although the PA method is a generalized least‐squares procedure, it cannot be made equivalent to the PGLS method using a phylogenetic model.     

Sampling effort, regression method, and the shape and slope of size–abundance relations

1. Despite a substantial body of work there remains much disagreement about the form of the relationship between organism abundance and body size. In an attempt at resolving these disagreements the shape and slope of samples from simulated and real abundance–mass distributions were assessed by ordinary least squares regression (OLS) and the reduced major axis method (RMA).
2. It is suggested that the data gathered by ecologists to assess these relationships are usually truncated in respect of density. Under these conditions RMA gives slope estimates which are consistently closer to the true slopes than OLS regression.
3. The triangular relationships reported by some workers are found over smaller mass and abundance ranges than linear relations. Scatter in slope estimates is much greater and positive slopes more common at small sample sizes and sample ranges. These results support the notion that inadequate and truncated sampling is responsible for much of the disagreement reported in the literature.
4. The results strongly support the notion that density declines with increasing body mass in a broad, linear band with a slope around −1. However there is some evidence to suggest that this overall relation results from a series of component relations with slopes which differ from the overall slope.     

Life-history invariants with bounded variables cannot be distinguish from data generated by random processes using standard analyses

A dimensionless approach to the study of life-history evolution has been applied to a wide variety of variables in the search for life-history invariants. This approach usually employs ordinary least squares (OLS) regressions of log-transformed data. In several well-studied combinations of variables the range of values of one parameter is bounded or limited by the value of the other. In this situation, the null hypothesis normally applied to regression analysis is not appropriate. We generate the null expectations and confidence intervals (CI) for OLS and reduced major axis (RMA) regressions using random variables that are bounded in this way. Comparisons of these CI show that, for log-transformed data, the patterns generated by random data and those predicted by life history invariant theory often could not be distinguished because both predict a slope of 1. We recommend that tests based on the putative invariant ratios and not the correlations between the two variables be used in the exploration of life-history invariants using bounded data. Because empirical data are often not normally distributed randomization test may be more appropriate than standard statistical tests.     

Use and misuse of the reduced major axis for line‐fitting

Many investigators use the reduced major axis (RMA) instead of ordinary least squares (OLS) to define a line of best fit for a bivariate relationship when the variable represented on the X‐axis is measured with error. OLS frequently is described as requiring the assumption that X is measured without error while RMA incorporates an assumption that there is error in X. Although an RMA fit actually involves a very specific pattern of error variance, investigators have prioritized the presence versus the absence of error rather than the pattern of error in selecting between the two methods. Another difference between RMA and OLS is that RMA is symmetric, meaning that a single line defines the bivariate relationship, regardless of which variable is X and which is Y, while OLS is asymmetric, so that the slope and resulting interpretation of the data are changed when the variables assigned to X and Y are reversed. The concept of error is reviewed and expanded from previous discussions, and it is argued that the symmetry‐asymmetry issue should be the criterion by which investigators choose between RMA and OLS. This is a biological question about the relationship between variables. It is determined by the investigator, not dictated by the pattern of error in the data. If X is measured with error but OLS should be used because the biological question is asymmetric, there are several methods available for adjusting the OLS slope to reflect the bias due to error. RMA is being used in many analyses for which OLS would be more appropriate. Am J Phys Anthropol, 2009. © 2009 Wiley‐Liss, Inc.     

Independent contrasts and PGLS regression estimators are equivalent

We prove that the slope parameter of the ordinary least squares regression of phylogenetically independent contrasts (PICs) conducted through the origin is identical to the slope parameter of the method of generalized least squares (GLSs) regression under a Brownian motion model of evolution. This equivalence has several implications: 1. Understanding the structure of the linear model for GLS regression provides insight into when and why phylogeny is important in comparative studies. 2. The limitations of the PIC regression analysis are the same as the limitations of the GLS model. In particular, phylogenetic covariance applies only to the response variable in the regression and the explanatory variable should be regarded as fixed. Calculation of PICs for explanatory variables should be treated as a mathematical idiosyncrasy of the PIC regression algorithm. 3. Since the GLS estimator is the best linear unbiased estimator (BLUE), the slope parameter estimated using PICs is also BLUE. 4. If the slope is estimated using different branch lengths for the explanatory and response variables in the PIC algorithm, the estimator is no longer the BLUE, so this is not recommended. Finally, we discuss whether or not and how to accommodate phylogenetic covariance in regression analyses, particularly in relation to the problem of phylogenetic uncertainty. This discussion is from both frequentist and Bayesian perspectives.     

Analysis of determinants of mammalian species richness in South America using spatial autoregressive models

Classically, hypotheses concerning the distribution of species have been explored by evaluating the relationship between species richness and environmental variables using ordinary least squares (OLS) regression. However, environmental and ecological data generally show spatial autocorrelation, thus violating the assumption of independently distributed errors. When spatial autocorrelation exists, an alternative is to use autoregressive models that assume spatially autocorrelated errors. We examined the relationship between mammalian species richness in South America and environmental variables, thereby evaluating the relative importance of four competing hypotheses to explain mammalian species richness. Additionally, we compared the results of ordinary least squares (OLS) regression and spatial autoregressive models using Conditional and Simultaneous Autoregressive (CAR and SAR, respectively) models. Variables associated with productivity were the most important at determining mammalian species richness at the scale analyzed. Whereas OLS residuals between species richness and environmental variables were strongly autocorrelated, those from autoregressive models showed less spatial autocorrelation, particularly the SAR model, indicating its suitability for these data. Autoregressive models also fit the data better than the OLS model (increasing R² by 5–14%), and the relative importance of the explanatory variables shifted under CAR and SAR models. These analyses underscore the importance of controlling for spatial autocorrelation in biogeographical studies.     

Richness patterns, species distributions and the principle of extreme deconstruction

Aim To analyse the global patterns in species richness of Viperidae snakes through the deconstruction of richness into sets of species according to their distribution models, range size, body size and phylogenetic structure, and to test if environmental drivers explaining the geographical ranges of species are similar to those explaining richness patterns, something we called the extreme deconstruction principle. Location Global. Methods We generated a global dataset of 228 terrestrial viperid snakes, which included geographical ranges (mapped at 1° resolution, for a grid with 7331 cells world‐wide), body sizes and phylogenetic relationships among species. We used logistic regression (generalized linear model; GLM) to model species geographical ranges with five environmental predictors. Sets of species richness were also generated for large and small‐bodied species, for basal and derived species and for four classes of geographical range sizes. Richness patterns were also modelled against the five environmental variables through standard ordinary least squares (OLS) multiple regressions. These subsets are replications to test if environmental factors driving species geographical ranges can be directly associated with those explaining richness patterns. Results Around 48% of the total variance in viperid richness was explained by the environmental model, but richness sets revealed different patterns across the world. The similarity between OLS coefficients and the primacy of variables across species geographical range GLMs was equal to 0.645 when analysing all viperid snakes. Thus, in general, when an environmental predictor it is important to model species geographical ranges, this predictor is also important when modelling richness, so that the extreme deconstruction principle holds. However, replicating this correlation using subsets of species within different categories in body size, range size and phylogenetic structure gave more variable results, with correlations between GLM and OLS coefficients varying from –0.46 up to 0.83. Despite this, there is a relatively high correspondence (r = 0.73) between the similarity of GLM‐OLS coefficients and R² values of richness models, indicating that when richness is well explained by the environment, the relative importance of environmental drivers is similar in the richness OLS and its corresponding set of GLMs. Main conclusions The deconstruction of species richness based on macroecological traits revealed that, at least for range size and phylogenetic level, the causes underlying patterns in viperid richness differ for the various sets of species. On the other hand, our analyses of extreme deconstruction using GLM for species geographical range support the idea that, if environmental drivers determine the geographical distribution of species by establishing niche boundaries, it is expected, at least in theory, that the overlap among ranges (i.e. richness) will reveal similar effects of these environmental drivers. Richness patterns may be indeed viewed as macroecological consequences of population‐level processes acting on species geographical ranges.     

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.     

MacroCAIC: revealing correlates of species richness by comparative analysis

Abstract. Studies of species richness have been hampered by the use of methods that fail to account for phylogenetic non-independence of character states. MacroCAIC is a computer program that extends the method of phylogenetically independent contrasts to encompass species-richness data. It examines user-selected characters for correlation with species richness, thus allowing clearer identification of the factors driving large-scale patterns of diversity.     

The Applicability of Ordinary Least Squares to Consistently Short Distances between Taxa in Phylogenetic Tree Construction and the Normal Distribution Test Consequences

Short phylogenetic distances between taxa occur, for example, in studies on ribosomal RNA-genes with slow substitution rates. For consistently short distances, it is proved that in the completely singular limit of the covariance matrix ordinary least squares (OLS) estimates are minimum variance or best linear unbiased (BLU) estimates of phylogenetic tree branch lengths. Although OLS estimates are in this situation equal to generalized least squares (GLS) estimates, the GLS chi-square likelihood ratio test will be inapplicable as it is associated with zero degrees of freedom. Consequently, an OLS normal distribution test or an analogous bootstrap approach will provide optimal branch length tests of significance for consistently short phylogenetic distances. As the asymptotic covariances between branch lengths will be equal to zero, it follows that the product rule can be used in tree evaluation to calculate an approximate simultaneous confidence probability that all interior branches are positive.     

Scaling of respiratory variables and the breathing pattern in birds: an allometric and phylogenetic approach

Allometric equations can be useful in comparative physiology in a number of ways, not the least of which include assessing whether a particular species deviates from the norm for its size and phylogenetic group with respect to some specific physiological process or determining how differences in design among groups may be reflected in differences in function. The allometric equations for respiratory variables in birds were developed 30 yr ago by Lasiewski and Calder and presented as "preliminary" because they were based on a small number of species. With the expanded data base now available to reconstruct these allometries and the call for taking account of the nonindependence of species in this process through a phylogenetically independent contrasts (PIC) approach, we have developed new allometric equations for respiratory variables in birds using both the traditional and PIC approaches. On the whole, the new equations agree with the old ones with only minor changes in the coefficients, and the primary difference between the traditional and PIC approaches is in the broader confidence intervals given by the latter. We confirm the lower VE/VO2 ratio for birds compared to mammals and observe a common scaling of inspiratory flow and oxygen consumption for birds as has been reported for mammals. Use of allometrics and comparisons among avian groups are also discussed.     

Permutation tests for phylogenetic comparative analyses of high‐dimensional shape data: What you shuffle matters

Evaluating statistical trends in high‐dimensional phenotypes poses challenges for comparative biologists, because the high‐dimensionality of the trait data relative to the number of species can prohibit parametric tests from being computed. Recently, two comparative methods were proposed to circumvent this difficulty. One obtains phylogenetic independent contrasts for all variables, and statistically evaluates the linear model by permuting the phylogenetically independent contrasts (PICs) of the response data. The other uses a distance‐based approach to obtain coefficients for generalized least squares models (D‐PGLS), and subsequently permutes the original data to evaluate the model effects. Here, we show that permuting PICs is not equivalent to permuting the data prior to the analyses as in D‐PGLS. We further explain why PICs are not the correct exchangeable units under the null hypothesis, and demonstrate that this misspecification of permutable units leads to inflated type I error rates of statistical tests. We then show that simply shuffling the original data and recalculating the independent contrasts with each iteration yields significance levels that correspond to those found using D‐PGLS. Thus, while summary statistics from methods based on PICs and PGLS are the same, permuting PICs can lead to strikingly different inferential outcomes with respect to statistical and biological inferences.     

A METHOD FOR ASSESSING PHYLOGENETIC LEAST SQUARES MODELS FOR SHAPE AND OTHER HIGH‐DIMENSIONAL MULTIVARIATE DATA

Studies of evolutionary correlations commonly use phylogenetic regression (i.e., independent contrasts and phylogenetic generalized least squares) to assess trait covariation in a phylogenetic context. However, while this approach is appropriate for evaluating trends in one or a few traits, it is incapable of assessing patterns in highly multivariate data, as the large number of variables relative to sample size prohibits parametric test statistics from being computed. This poses serious limitations for comparative biologists, who must either simplify how they quantify phenotypic traits, or alter the biological hypotheses they wish to examine. In this article, I propose a new statistical procedure for performing ANOVA and regression models in a phylogenetic context that can accommodate high‐dimensional datasets. The approach is derived from the statistical equivalency between parametric methods using covariance matrices and methods based on distance matrices. Using simulations under Brownian motion, I show that the method displays appropriate Type I error rates and statistical power, whereas standard parametric procedures have decreasing power as data dimensionality increases. As such, the new procedure provides a useful means of assessing trait covariation across a set of taxa related by a phylogeny, enabling macroevolutionary biologists to test hypotheses of adaptation, and phenotypic change in high‐dimensional datasets.     

Genome size variation and evolution in Veronica

BACKGROUND AND AIMS: The amount of DNA per chromosome set is known to be a fairly constant characteristic of a species. Its interspecific variation is enormous, but the biological significance of this variation is little understood. Some of the characters believed to be correlated with DNA amount are alpine habitat, life history and breeding system. In the present study, the aim is to distinguish between direct causal connections and chance correlation of the amount of DNA in the genus Veronica. METHODS: Estimates of DNA amount were analysed for 42 members of Veroniceae in connection with results from a phylogenetic analysis of plastid trnL-F DNA sequences and tested correlations using standard statistical tests, phylogenetically independent contrasts and a model-based generalized least squares method to distinguish the phylogenetic effect on the results. KEY RESULTS: There appears to be a lower upper limit for DNA amount in annuals than in perennials. Most DNAC-values in Veroniceae are below the mean DNA C-value for annuals in angiosperms as a whole. However, the long-debated correlation of low genome size with annual life history is not significant (P = 0.12) using either standard statistical tests or independent contrasts, but it is significant with the generalized least squares method (P < 0.01). CONCLUSIONS: The correlation of annual life history and low genome size found in earlier studies could be due to the association of annual life history and selfing, which is significantly correlated with low genome size using any of the three tests applied. This correlation can be explained by models showing a reduction in transposable elements in selfers. A significant correlation of higher genome sizes with alpine habitats was also detected.     

Interpreting the evolutionary regression: the interplay between observational and biological errors in phylogenetic comparative studies

Regressions of biological variables across species are rarely perfect. Usually, there are residual deviations from the estimated model relationship, and such deviations commonly show a pattern of phylogenetic correlations indicating that they have biological causes. We discuss the origins and effects of phylogenetically correlated biological variation in regression studies. In particular, we discuss the interplay of biological deviations with deviations due to observational or measurement errors, which are also important in comparative studies based on estimated species means. We show how bias in estimated evolutionary regressions can arise from several sources, including phylogenetic inertia and either observational or biological error in the predictor variables. We show how all these biases can be estimated and corrected for in the presence of phylogenetic correlations. We present general formulas for incorporating measurement error in linear models with correlated data. We also show how alternative regression models, such as major axis and reduced major axis regression, which are often recommended when there is error in predictor variables, are strongly biased when there is biological variation in any part of the model. We argue that such methods should never be used to estimate evolutionary or allometric regression slopes.     

Red herrings revisited: spatial autocorrelation and parameter estimation in geographical ecology

There have been numerous claims in the ecological literature that spatial autocorrelation in the residuals of ordinary least squares (OLS) regression models results in shifts in the partial coefficients, which bias the interpretation of factors influencing geographical patterns. We evaluate the validity of these claims using gridded species richness data for the birds of North America, South America, Europe, Africa, the ex‐USSR, and Australia. We used richness in 110×110 km cells and environmental predictor variables to generate OLS and simultaneous autoregressive (SAR) multiple regression models for each region. Spatial correlograms of the residuals from each OLS model were then used to identify the minimum distance between cells necessary to avoid short‐distance residual spatial autocorrelation in each data set. This distance was used to subsample cells to generate spatially independent data. The partial OLS coefficients estimated with the full dataset were then compared to the distributions of coefficients created with the subsamples. We found that OLS coefficients generated from data containing residual spatial autocorrelation were statistically indistinguishable from coefficients generated from the same data sets in which short‐distance spatial autocorrelation was not present in all 22 coefficients tested. Consistent with the statistical literature on this subject, we conclude that coefficients estimated from OLS regression are not seriously affected by the presence of spatial autocorrelation in gridded geographical data. Further, shifts in coefficients that occurred when using SAR tended to be correlated with levels of uncertainty in the OLS coefficients. Thus, shifts in the relative importance of the predictors between OLS and SAR models are expected when small‐scale patterns for these predictors create weaker and more unstable broad‐scale coefficients. Our results indicate both that OLS regression is unbiased and that differences between spatial and nonspatial regression models should be interpreted with an explicit awareness of spatial scale.     

Locomotor mode, maximum running speed, and basal metabolic rate in placental mammals

The locomotor performance (absolute maximum running speed [MRS]) of 120 mammals was analyzed for four different locomotor modes (plantigrade, digitigrade, unguligrade, and lagomorph-like) in terms of body size and basal metabolic rate (BMR). Analyses of conventional species data showed that the MRS of plantigrade and digitigrade mammals and lagomorphs increases with body mass, whereas that of unguligrade mammals decreases with body mass. These trends were confirmed in plantigrade mammals and lagomorphs using phylogenetically independent contrasts. Multiple regression analyses of MRS contrasts (dependent variable) as a function of body mass and BMR contrasts (predictor variables) revealed that BMR was a significant predictor of MRS in the complete data set, as well as in plantigrade and nonplantigrade mammals. However, there was severe multicollinearity in the nonplantigrade model that may influence the interpretation of these models. Although these data show mass-independent correlation between BMR and MRS, they are not necessarily indicative of a cause-effect relationship. However, the analyses do identify a negligible role of body size associated with MRS once phylogenetic and BMR effects are controlled, suggesting that the body size increase in large mammals over time (i.e., Cope's rule) can probably rule out MRS as a driving variable.     

Examining the extinction risk of specialized folivores: a comparative study of Colobine monkeys

Species extinctions are nonrandom with some taxa appearing to possess traits that increase their extinction risk. In this study, eight predictors of extinction risk were used as independent variables to predict the IUCN category of a subfamily of specialized folivorous primates, the Colobinae. All data were transformed into phylogenetically independent contrasts and were analyzed using bivariate regressions, multiple regression, and a maximum likelihood approach using Akaike's Information Criterion to assess model performance. Once an outlier was removed from the data set, species that devote a smaller proportion of their diet to mature leaf consumption appear to be at a greater risk of extinction. Also, as female body mass increases, so does extinction risk. In contrast, as maximum latitude and the number of habitat types increase, extinction risk appears to decrease. These findings emphasize the importance of examining detailed dietary variation for predicting extinction risk at a relatively fine taxonomic scale and, consequently, may help improve conservation management.     

Spatial autocorrelation and red herrings in geographical ecology

Aim Spatial autocorrelation in ecological data can inflate Type I errors in statistical analyses. There has also been a recent claim that spatial autocorrelation generates ‘red herrings’, such that virtually all past analyses are flawed. We consider the origins of this phenomenon, the implications of spatial autocorrelation for macro‐scale patterns of species diversity and set out a clarification of the statistical problems generated by its presence. Location To illustrate the issues involved, we analyse the species richness of the birds of western/central Europe, north Africa and the Middle East. Methods Spatial correlograms for richness and five environmental variables were generated using Moran's I coefficients. Multiple regression, using both ordinary least‐squares (OLS) and generalized least squares (GLS) assuming a spatial structure in the residuals, were used to identify the strongest predictors of richness. Autocorrelation analyses of the residuals obtained after stepwise OLS regression were undertaken, and the ranks of variables in the full OLS and GLS models were compared. Results Bird richness is characterized by a quadratic north–south gradient. Spatial correlograms usually had positive autocorrelation up to c. 1600 km. Including the environmental variables successively in the OLS model reduced spatial autocorrelation in the residuals to non‐detectable levels, indicating that the variables explained all spatial structure in the data. In principle, if residuals are not autocorrelated then OLS is a special case of GLS. However, our comparison between OLS and GLS models including all environmental variables revealed that GLS de‐emphasized predictors with strong autocorrelation and long‐distance clinal structures, giving more importance to variables acting at smaller geographical scales. Conclusion Although spatial autocorrelation should always be investigated, it does not necessarily generate bias. Rather, it can be a useful tool to investigate mechanisms operating on richness at different spatial scales. Claims that analyses that do not take into account spatial autocorrelation are flawed are without foundation.