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.     

A new procedure to derive weighting factors for nonlinear regression analysis applied to enzyme kinetic data.

The experimental variance of enzymic steady-state kinetic experiments depends on velocity as approximated by a power function (Var(v) = K1 . valpha (Askel?f, P., Korsfeldt, M. and Mannervik, B. (1976) Eur. J. Biochem. 69, 61--67). The values of the constants (K1, alpha) can be estimated by making replicate measurements of velocity, and the inverse of the function can then be used as a weighting factor. In order to avoid measurement of a large number of replicates to establish the error structure of a kinetic data set, a different approach was tested. After a preliminary regression using a 'good model', which satisfies reasonable goodness-of-fit criteria, the residuals were taken to represent the experimental error. The neighbouring residuals were grouped together and the sum of their mean squared values was used as a measure of the variance in the neighbourhood of the corresponding measurements. The values of the constants obtained in this way agreed with those obtained by replicates.     

Statistical model of variable allometric growth: otolith growth in Micropogonias furnieri(Actinopterygii, Sciaenidae)

The main objective of this study was to develop a statistical model for accurate estimates of relative growth. The method was based on identifying patterns of the residuals obtained from the Huxley's allometric equation. Three different approaches were applied: (1) growth with variable proportionality and constant allometry coefficient, (2) growth with constant proportionality and variable allometry coefficient and (3) distinct growth phases in which proportionality and allometry coefficients remained constant. The proposed statistical models were applied to the relationship of the otolith size and fish size of whitemouth croaker Micropogonias furnieri . The best fit was obtained when using approach (3). A change in the growth parameters was associated with the attainment of sexual maturity.     

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.     

Mathematical Model for the Contribution of Individual Organs to Non-Zero Y-Intercepts in Single and Multi-Compartment Linear Models of Whole-Body Energy Expenditure

Mathematical models for the dependence of energy expenditure (EE) on body mass and composition are essential tools in metabolic phenotyping. EE scales over broad ranges of body mass as a non-linear allometric function. When considered within restricted ranges of body mass, however, allometric EE curves exhibit ‘local linearity.’ Indeed, modern EE analysis makes extensive use of linear models. Such models typically involve one or two body mass compartments (e.g., fat free mass and fat mass). Importantly, linear EE models typically involve a non-zero (usually positive) y-intercept term of uncertain origin, a recurring theme in discussions of EE analysis and a source of confounding in traditional ratio-based EE normalization. Emerging linear model approaches quantify whole-body resting EE (REE) in terms of individual organ masses (e.g., liver, kidneys, heart, brain). Proponents of individual organ REE modeling hypothesize that multi-organ linear models may eliminate non-zero y-intercepts. This could have advantages in adjusting REE for body mass and composition. Studies reveal that individual organ REE is an allometric function of total body mass. I exploit first-order Taylor linearization of individual organ REEs to model the manner in which individual organs contribute to whole-body REE and to the non-zero y-intercept in linear REE models. The model predicts that REE analysis at the individual organ-tissue level will not eliminate intercept terms. I demonstrate that the parameters of a linear EE equation can be transformed into the parameters of the underlying ‘latent’ allometric equation. This permits estimates of the allometric scaling of EE in a diverse variety of physiological states that are not represented in the allometric EE literature but are well represented by published linear EE analyses.     

Non-linear fitting program for biological data

A simple microcomputer program written in Microsoft Basic estimates pharmacokinetic parameters using the coordinate search technique to minimize the sum of squared errors. The program developed for portable computers combines a plot of data and curve fitting so as to find rapidly the initial parameters with the subsequent optimization of the parameter estimate.     

On the analysis of fluorescence decay kinetics by the method of least-squares

Analysis of fluorescence decay kinetics aims at the determination of the analytic expression and the numerical values of the pertinent parameters which describe the decay process. In the well-known method of least-squares, one assumes a plausible functional form for the decay data and adjusts the values of the parameters until the statistically best fit is obtained between the data and the calculated decay function, i.e., until the sum of the weighted squares of the residuals is at a minimum. It is shown that proper weighting of the squares of the residuals may markedly improve the quality of the analysis. Such weighting requires information about the character of the experimental noise, which is often available, e.g., when the noise is due to counting error in photon-counting techniques. Furthermore, dramatic improvements in the accuracy of the analysis may often be achieved by use of auxiliary information available about the system studied. For example, the preexponents in a multiexponential fluorescence decay of a mixture of chromophores (such as tryptophan residues in a protein molecule) may sometimes be estimated independently; much higher accuracy can then be attained for the decay lifetimes by analysis of the decay kinetics. It is proposed that the shape of the autocorrelation function of the weighted residuals may serve as a convenient criterion for the quality of fit between the experimental data and the decay function obtained by analysis. The above conclusions were reached by analysis of computer-simulated experiments, and the usefulness of this approach is illustrated. The importance of stating the uncertainties in the estimated parameters inherent in the analysis of decay kinetics is stressed.     

Characterizing morphological (co)variation using structural equation models: Body size,allometric relationships and evolvability in a house sparrow metapopulation

Body size plays a key role in the ecology and evolution of all organisms. Therefore, quantifying the sources of morphological (co)variation, dependent and independent of body size, is of key importance when trying to understand and predict responses to selection. We combine structural equation modeling with quantitative genetics analyses to study morphological (co)variation in a meta‐population of house sparrows (Passer domesticus). As expected, we found evidence of a latent variable “body size,” causing genetic and environmental covariation between morphological traits. Estimates of conditional evolvability show that allometric relationships constrain the independent evolution of house sparrow morphology. We also found spatial differences in general body size and its allometric relationships. On islands where birds are more dispersive and mobile, individuals were smaller and had proportionally longer wings for their body size. Although on islands where sparrows are more sedentary and nest in dense colonies, individuals were larger and had proportionally longer tarsi for their body size. We corroborated these results using simulations and show that our analyses produce unbiased allometric slope estimates. This study highlights that in the short term allometric relationships may constrain phenotypic evolution, but that in the long term selection pressures can also shape allometric relationships.     

Adjoint multi-start-based estimation of cardiac hyperelastic material parameters using shear data

Cardiac muscle tissue during relaxation is commonly modeled as a hyperelastic material with strongly nonlinear and anisotropic stress response. Adapting the behavior of such a model to experimental or patient data gives rise to a parameter estimation problem which involves a significant number of parameters. Gradient-based optimization algorithms provide a way to solve such nonlinear parameter estimation problems with relatively few iterations, but require the gradient of the objective functional with respect to the model parameters. This gradient has traditionally been obtained using finite differences, the calculation of which scales linearly with the number of model parameters, and introduces a differencing error. By using an automatically derived adjoint equation, we are able to calculate this gradient more efficiently, and with minimal implementation effort. We test this adjoint framework on a least squares fitting problem involving data from simple shear tests on cardiac tissue samples. A second challenge which arises in gradient-based optimization is the dependency of the algorithm on a suitable initial guess. We show how a multi-start procedure can alleviate this dependency. Finally, we provide estimates for the material parameters of the Holzapfel and Ogden strain energy law using finite element models together with experimental shear data.     

Adding further power to the Haseman and Elston method for detecting linkage in larger sibships: weighting sums and differences

Haseman and Elston (H-E) proposed a robust test to detect linkage between a quantitative trait and a genetic marker. In their method the squared sib-pair trait difference is regressed on the estimated proportion of alleles at a locus shared identical by descent by sib pairs. This method has recently been improved by changing the dependent variable from the squared difference to the mean-corrected product of the sib-pair trait values, a significantly positive regression indicating linkage. Because situations arise in which the original test is more powerful, a further improvement of the H-E method occurs when the dependent variable is changed to a weighted average of the squared sib-pair trait difference and the squared sib-pair mean-corrected trait sum. Here we propose an optimal method of performing this weighting for larger sibships, allowing for the correlation between pairs within a sibship. The optimal weights are inversely proportional to the residual variances obtained from the two different regressions based on the squared sib-pair trait differences and the squared sib-pair mean-corrected trait sums, respectively, allowing for correlations among sib pairs. The proposed method is compared with the existing extension of the H-E approach for larger sibships. Control of the type I error probabilities for sibships of any size can be improved by using a generalized estimating equation approach and the robust sandwich estimate of the variance, or a Monte-Carlo permutation test.     

Comparative methods as a statistical fix: the dangers of ignoring an evolutionary model

Abstract Comparative methods are widely used in ecology and evolution. The most frequently used comparative methods are based on an explicit evolutionary model. However, recent approaches have been popularized that are without an evolutionary basis or an underlying null model. Here we highlight the limitations of such techniques in comparative analyses by using simulations to compare two commonly used comparative methods with and without evolutionary basis, respectively: generalized least squares (GLS) and phylogenetic eigenvector regression (PVR). We find that GLS methods are more efficient at estimating model parameters and produce lower variance in parameter estimates, lower phylogenetic signal in residuals, and lower Type I error rates than PVR methods. These results can very likely be generalized to eigenvector methods that control for space and both space and phylogeny. We highlight that GLS methods can be adapted in numerous ways and that the variance structure used in these models can be flexibly optimized to each data set.     

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.     

On the practical identifiability of microbial growth models incorporating Michaelis-Menten type nonlinearities

The reason for difficulties in obtaining unique estimates of the parameters μ_m and K_s of the Michaelis-Menten equation are analysed for a microbial batch growth process. With the aid of simulation studies in which the influences of different types of noise on the parameter estimates are compared, it is shown that, although theoretically identifiable in the deterministic case with ideal measurements, the parameters cannot in general be correctly determined from noisy measurements. The difficulties are further illuminated by estimation examples using real data. It certain situations, in which the value of the ratio K_s/s_o is high or in which only few and noisy measurements are available, the linear approximation of the Michaelis-Menten equation gives a better fit. The practical difficulties in obtaining correct values of the model parameters do not limit the applicability of the Michaelis-Menten model, which in most cases explains the bacterial growth behavior excellently. Rather, they underline the fact that care must be taken when utilizing parameter estimates for biological interpretations.     

Nonparametrically Weighted Least Squares Estimation in Heteroscedastic Linear Regression

The weights used in iterative weighted least squares (IWLS) regression are usually estimated parametrically using a working model for the error variance. When the variance function is misspecified, the IWLS estimates of the regression coefficients β are still asymptotically consistent but there is some loss in efficiency. Since second moments can be quite hard to model, it makes sense to estimate the error variances nonparametrically and to employ weights inversely proportional to the estimated variances in computing the WLS estimate for β. Surprisingly, this approach had not received much attention in the literature. The aim of this note is to demonstrate that such a procedure can be implemented easily in S-plus using standard functions with default options making it suitable for routine applications. The particular smoothing method that we use is local polynomial regression applied to the logarithm of the squared residuals but other smoothers can be tried as well. The proposed procedure is applied to data on the use of two different assay methods for a hormone. Efficiency calculations based on the estimated model show that the nonparametric IWLS estimates are more efficient than the parametric IWLS estimates based on three different plausible working models for the variance function. The proposed estimators also perform well in a simulation study using both parametric and nonparametric variance functions as well as normal and gamma errors.     

Model selection and logarithmic transformation in allometric analysis

The standard approach to most allometric research is to gather data on a biological function and a measure of body size, convert the data to logarithms, display the new values in a bivariate plot, and then fit a straight line to the transformations by the method of least squares. The slope of the fitted line provides an estimate for the allometric (or scaling) exponent, which often is interpreted in the context of underlying principles of structural and functional design. However, interpretations of this sort are based on the implicit assumption that the original data conform with a power function having an intercept of 0 on a plot with arithmetic coordinates. Whenever this assumption is not satisfied, the resulting estimate for the allometric exponent may be seriously biased and misleading. The problem of identifying an appropriate function is compounded by the logarithmic transformations, which alter the relationship between the original variables and frequently conceal the presence of outliers having an undue influence on properties of the fitted equation, including the estimate for the allometric exponent. Much of the current controversy in allometric research probably can be traced to substantive biases introduced by investigators who followed standard practice. We illustrate such biases with examples taken from the literature and outline a general methodology by which the biases can be minimized in future research.     

A goodness-of-fit test for multinomial logistic regression

This article presents a score test to check the fit of a logistic regression model with two or more outcome categories. The null hypothesis that the model fits well is tested against the alternative that residuals of samples close to each other in covariate space tend to deviate from the model in the same direction. We propose a test statistic that is a sum of squared smoothed residuals, and show that it can be interpreted as a score test in a random effects model. By specifying the distance metric in covariate space, users can choose the alternative against which the test is directed, making it either an omnibus goodness-of-fit test or a test for lack of fit of specific model variables or outcome categories.     

Tree-Structured Exponential Regression Modeling

A method for fitting piecewise exponential regression models to censored survival data is described. Stratification is performed recursively, using a combination of statistical tests and residual analysis. The splitting criterion employed in cross-validation is the average squared error of the residuals. The bootstrap is employed to keep the probability of a type I error (the error of discovering two or more strata when there is only one) of the method close to a predetermined value. The proposed method can thus also serve as a formal goodness-of-fit test for the exponential regression model. Real and simulated data are used for illustration.     

Life‐history constraints on maximum population growth for loggerhead turtles in the northwest Atlantic

Conservation planning for protected species often relies on estimates of life‐history parameters. A commonly used parameter is the instantaneous maximum population growth rate (r_max) that can be used to limit removals and design recovery targets. Estimation of r_max can be challenging because of limited availability of species‐ and population‐specific data and life‐history information. We applied a method proposed by Neil and Lebreton, originally developed for birds, to loggerhead turtles. The method uses age‐at‐first‐reproduction and adult survival to estimate r_max. We used a variety of datasets and matrix population models to confirm an allometric assumption required by the method, and to generate estimates of age‐at‐first‐reproduction and adult survival. A meta‐analysis was applied to parameters from reported growth curves, which were then combined with the size distribution of neophyte nesters to derive estimates of age‐at‐first‐reproduction. Adult survival rates were obtained from an existing matrix population model. Monte Carlo simulation was then used to combine the estimates of the allometric coefficients, age‐at‐first‐reproduction, and adult survival to obtain a probability distribution of approximate r_max values. Estimated annual maximum population growth rates averaged 0.024, with a mode of 0.017 and a 95% highest density interval of 0.006–0.047. These estimates were similar to values reported by others using different methods and captured the variability in positive, annual change estimates across nesting beach sites for the northwest Atlantic loggerhead population. The use of life‐history parameters has a long history in wildlife and fisheries management and conservation planning. Our estimates of r_max, while having some biases and uncertainty, encompassed values presently used in recovery planning for loggerhead turtles and offer additional information for the management of endangered and threatened species.     

A Penalized Multinomial Approach to Smoothed Estimation of Disease Incidence

Raw estimates of disease rates over a geographical region are frequently quite variable, even though one may reasonably expect adjacent communities to have similar true rates. Smoother estimates are obtained by incorporating a penalty into a multinomial likelihood estimation procedure. For each pair of locations, this penalty increases with the difference between the rates and decreases with the distance between the two sites. The resulting estimates have smaller mean squared error than the raw estimates. Expansions are developed which demonstrate the contributions of the smoothing constant, spatial configuration, risk population and raw estimates to the amount of smoothing. Simulations and an example involving gastric cancer data illustrate the proposed method.     

Cluster detection based on spatial associations and iterated residuals in generalized linear mixed models

Summary .  Spatial clustering is commonly modeled by a Bayesian method under the framework of generalized linear mixed effect models (GLMMs). Spatial clusters are commonly detected by a frequentist method through hypothesis testing. In this article, we provide a frequentist method for assessing spatial properties of GLMMs. We propose a strategy that detects spatial clusters through parameter estimates of spatial associations, and assesses spatial aspects of model improvement through iterated residuals. Simulations and a case study show that the proposed method is able to consistently and efficiently detect the locations and magnitudes of spatial clusters.