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.     

Abstracts of papers to be presented at the Fiftieth Annual Meeting of the American Association of physical anthropologists Detroit,Michigan April 22–25, 1981

The critical problem confronting all allometric studies is the choice of an appropriate size variable, especially when body mass or some other measure of total size is unavailable. A method proposed by Jolicoeur (1963a,b) claims to generate an internal size variable by a principal components analysis of the covariance matrix of logarithmically transformed data, from which allometric coefficients can be computed. Despite the current popularity of this method, the precise relationship and degree of compatibility between such multivariate coefficients and the exponent of the bivariate power function (Y = βX^α) is unknown. This study evaluates the comparability and interpretability of allometric values computed by Jolicoeur's procedure and by standard bivariate regressions (least squares and major axis). Two primate data sets with known measures of size were utilized for these purposes: (1) longitudinal growth data from radiographs of two species of capuchin monkeys, Cebus apella and Cebus albifrons; and (2) interspecific osteometric data from a series of adult lemurs, species of prosimians from Madagascar. Consistent differences exist between multivariate and bivariate allometric coefficients for both ontogenetic and static data sets. Multivariate analysis underestimated the coefficients in the former and overestimated them in the latter. The internal size variable generated by principal components analysis is clearly not equivalent to, and hence not a suitable substitute for, known measures of size. Moreover, multivariate coefficients are very sensitive to the composition of variables in a given data set; α values of a variable changed appreciably depending on the other variables included in the analysis. The multivariate coefficients are also sample-specific, and provide misleading biological information when compared between samples (e.g., between species of capuchin monkey). For allometric investigations designed to evaluate scaling parameters relative to total size, alternative analytical solutions to the Jolicoeur method should be considered.     

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.     

Genetic mapping of allometric scaling laws

Many biological processes, from cellular metabolism to population dynamics, are characterized by particular allometric scaling relationships between rate and size (power laws). A statistical model for mapping specific quantitative trait loci (QTLs) that are responsible for allometric scaling laws has been developed. We present an improved model for allometric mapping of QTLs based on a more general allometry equation. This improved model includes two steps: (1) use model II regression analysis to estimate the parameters underlying universal allometric scaling laws, and (2) substitute the estimated allometric parameters in the mixture-based mapping model to obtain the estimation of QTL position and effects. This model has been validated by a real example for a mouse F2 progeny, in which two QTLs were detected on different chromosomes that determine the allometric relationship between growth rate and body weight.     

Ontogeny and phylogeny in papionin primates

This study investigates the developmental bases of size and shape variation in papionin primates (Macaca, Cercocebus, Mandrillus, Lophocebus, and Papio). The analysis tests hypotheses predicting that heterochronic changes in ontogeny, particularly in the degree of overall size growth, can account for cranial diversity and "allometric scaling" in this clade. Large developmental samples of extant papionin crania are examined to test heterochronic hypotheses using bivariate allometric methods. Analyses indicate that the crania of larger papionins (Mandrillus and Papio) are generally peramorphic, surpassing size and shape ranges of smaller, and probably less-derived, macaques and mangabeys. At least two heterochronic processes, including acceleration and hypermorphosis, can account for this pattern. Ontogenetic changes include decoupling of growth and development among cranial regions, along with simple shifts in size. Allometric scaling has complex developmental bases. Size change itself is not sufficient to explain all developmental differences among papionins, but these changes are extremely important in comparisons within cranial regions such as the face. Results imply that Papio exhibits strongly derived patterns of brain growth that impact postnatal patterns of size and shape transformation. Consideration of these results in the context of recent socioecological analyses suggests that derived patterns of cranial growth in Papio may be a response to selection during the early periods of ontogeny, resulting in a distinctive life history pattern.     

Phylogenetic analysis of the allometric scaling of therapeutic regimes for birds

The use of allometric scaling to estimate drug doses, regimes, and clearance rates (metabolic dosing) is based on the principle that the amount of drug to be administered is more closely related to daily energy use than to body mass (kg). Thus, by using the allometric estimations of minimal energy consumption (MEC) in kcal day⁻¹ based on the formula MEC= kM _b ^b , where b =3, it is thought to be possible to extrapolate appropriate drug dosage regimens to species for which direct MEC data are unavailable. However, the allometric equations for respiratory variables in birds were developed 30 years ago, and were based on a very small sample size, while the appropriate scaling exponent for the allometry of energy use is a matter of considerable debate. Hence, we revisit the issue of the scaling of therapeutic regimes in birds using the most current expanded database available (resting metabolic rate data for 296 species across 17 bird orders), taking account of the non-independence of species in this process using a phylogenetically independent approach. We show that the use of caloric values to estimate daily energy consumption introduces significant error into the formula, as there are a number of assumptions that are made when converting rate of oxygen consumption to a caloric value. We also show that there are significant differences in the proportionality or Hainsworth coefficients k across taxa when the data are examined in a phylogenetic context, although the allometric scaling exponent does not vary. We therefore recommend the use of only data based on oxygen consumption values, and not caloric values, and a multi-order phylogenetic model when calculating the appropriate drug dosage regime.     

Pitfalls and challenges of estimating population growth rate from empirical data: consequences for allometric scaling relations

The intrinsic rate of increase is a fundamental concept in population ecology, and a variety of problems require that estimates of population growth rate be obtained from empirical data. However, depending on the extent and type of data available (e.g. time series, life tables, life history traits), several alternative empirical estimators of population growth rate are possible. Because these estimators make different assumptions about the nature of age‐dependent mortality and density‐dependence of population dynamics, among other factors, these quantities capture fundamentally different aspects of population growth and are not interchangeable. Nevertheless, they have been routinely commingled in recent ecoinformatic analyses relating to allometry and conservation biology. Here we clarify some of the confusion regarding the empirical estimation of population growth rate and present separate analyses of the frequency distributions and allometric scaling of three alternative, non‐interchangeable measures of population growth. Studies of allometric scaling of population growth rate with body size are additionally sensitive to the statistical line fitting approach used, and we find that different approaches yield different allometric scaling slopes. Across the mix of population growth estimators and line fitting techniques, we find scattered and limited support for the key allometric prediction from the metabolic theory of ecology, namely that log₁₀(population growth rate) should scale as ?0.25 power of log₁₀(body mass). More importantly, we conclude that the question of allometric scaling of population growth rate with body size is highly sensitive to previously unexamined assumptions regarding both the appropriate population growth parameter to be compared and the line fitting approach used to examine the data. Finally, we suggest that the ultimate test of allometric scaling of maximum population growth rates with body size has not been done and, moreover, may require data that are not currently available.     

Patterns of cranial shape variation in the Papionini (Primates: Cercopithecinae)

Traditional classifications of the Old World monkey tribe Papionini (Primates: Cercopithecinae) recognized the mangabey genera Cercocebus and Lophocebus as sister taxa. However, molecular studies have consistently found the mangabeys to be diphyletic, with Cercocebus and Mandrillus forming a clade to the exclusion of all other papionins. Recent studies have identified cranial and postcranial features which distinguish the Cercocebus-Mandrillus clade, however the detailed similarities in cranial shape between the mangabey genera are more difficult to reconcile with the molecular evidence. Given the large size differential between members of the papionin molecular clades, it has frequently been suggested that allometric effects account for homoplasy in papionin cranial form. A combination of geometric morphometric, bivariate, and multivariate methods was used to evaluate the hypothesis that allometric scaling contributes to craniofacial similarities between like-sized papionin taxa. Patterns of allometric and size-independent cranial shape variation were subsequently described and related to known papionin phylogenetic relationships and patterns of development.Results confirm that allometric scaling of craniofacial shape characterized by positive facial allometry and negative neurocranial allometry is present across adult papionins. Pairwise comparisons of regression lines among genera revealed considerable homogeneity of scaling within the Papionini, however statistically significant differences in regression lines also were noted. In particular, Cercocebus and Lophocebus exhibit a shared slope and significant vertical displacement of their allometric lines relative to other papionins. These findings give no support to narrowly construed hypotheses of uniquely shared patterns of allometric scaling, either between sister taxa or across all papionins. However, more general allometric trends do appear to account for a substantial proportion of papionin cranial shape variation, most notably in those features which have influenced traditional morphological phylogenies. Examination of size-uncorrelated shape variation gives no clear support to molecular phylogenies, but underscores the absence of morphometric similarities between the mangabey genera when size effects are controlled. Patterns of allometric and size-uncorrelated shape variation indicate conservatism of cranial form in non- Theropithecus papionins, and suggest that Papio represents the primitive morphometric pattern for the African papionins. Lophocebus exhibits a divergent morphometric pattern, clearly distinguishable from other papionins, most notably Cercocebus. These results clarify patterns of cranial shape variation among the extant Papionini and lay the groundwork for studies of related fossil taxa.     

Allometric scaling in small colonies of the scleractinian coral Siderastrea siderea (Ellis and Solander)

Although most physiological traits scale allometrically in unitary organisms, it has been hypothesized that modularity allows for isometric scaling in colonial modular taxa. Isometry would allow increases in size without functional constraints, and is thought to be of central importance to the success of a modular design. Yet, despite its potential importance, scaling in these organisms has received little attention. To determine whether scleractinian corals are free of allometric constraints, we quantified metabolic scaling, measured as aerobic respiration, in small colonies (< or =40 mm in diam.) of the scleractinian Siderastrea siderea. We also quantified the scaling of colony surface area with biomass, since the proposed isometry is contingent upon maintaining a constant ratio of surface area to biomass (or volume) with size. Contrary to the predicted isometry, aerobic respiration scaled allometrically on biomass with a slope (b) of 0.176, and colony surface area scaled allometrically on biomass with a slope of 0.730. These findings indicate that small colonies of S. siderea have disproportionately high metabolic rates and SA:B ratios compared to their larger counterparts. The most probable explanations for the allometric scaling of aerobic respiration are (1) a decline in the SA:B ratio with size such that more surface area is available per unit of biomass for mass transfer in the smallest colonies, and (2) the small size, young age, and disproportionately high growth rates of the corals examined. This allometric scaling also demonstrates that modularity, alone, does not allow small colonies of S. siderea to overcome allometric constraints. Further studies are required to determine whether allometric scaling is characteristic of the full size range of colonies of S. siderea.     

Scaling in the animal kingdom

Several of the known scaling laws in the animal kingdom are based on a so-called allometric correlation in which some physical quantity is presumed to scale as some power of the mass of the animal. Such a simple correlation, when deduced purely as an empirical result, often hides the physical balances that fix the relevant scaling law. In particular, the emphasis on a simple allometric scaling has often masked the fundamental role played by time scales associated with the physical balances being struck. In this paper I have concentrated on three different attributes to which the use of dimensional analysis, scaling arguments and some judicious guesswork have led to new results and an understanding of some balances that occur in the animal kingdom. The running speed of animals is examined and a rationale deduced for the resolution of a conundrum first posed by A.V. Hill of why it is that many animals appear to have approximately the same maximum speed. A complete dimensional analysis for scaling the basal metabolic rate for a class of animals suggests that a detailed understanding of the physical balances that fix the metabolic rate could be quite subtle. However, the use of such an analysis has led to the discovery of a new correlation for mammals, relating the metabolic rate to the mass and the pulse rate of the animal. At the heart of many scaling laws for animal motion is the provision of an estimate of how the skeletal structure depends on the mass of the animal. It has been known for some time that the assumption of isometry between the builds of animals is too constrictive to describe the observed scaling laws. It is shown here how to relax the isometric assumption and deduce scaling laws in good agreement with observation. Thus, it appears that the skeletal dimensions of many animals with exoskeletons are fixed by the need to support static rather than dynamical loads. The scaling laws associated with endoskeletons are more complex, apparently, though the analysis does suggest that it is dynamical loading which is decisive for the skeletal design of land mammals.     

Projecting human pharmacokinetics of therapeutic antibodies from nonclinical data

The pharmacokinetics (PK) of therapeutic antibodies is determined by target and non-target mediated mechanisms. These antibody-specific factors need to be considered during prediction of human PK based upon preclinical information. Principles of allometric scaling established for small molecules using data from multiple animal species cannot be directly applied to antibodies. Here, different methods for projecting human clearance (CL) from animal PK data for 13 therapeutic monoclonal antibodies (mAbs) exhibiting linear PK over the tested dose ranges were examined: simple allometric scaling (CL versus body weight), allometric scaling with correction factors, allometric scaling based on rule of exponent and scaling from only cynomolgus monkey PK data. A better correlation was obtained between the observed human CL and the estimated human CL based on cynomolgus monkey PK data and an allometric scaling exponent of 0.85 for CL than other scaling approaches. Human concentration-time profiles were also reasonably predicted from the cynomolgus monkey data using species-invariant time method with a fixed exponent of 0.85 for CL and 1.0 for volume of distribution. In conclusion, we expanded our previous work and others and further confirmed that PK from cynomolgus monkey alone can be successfully scaled to project human PK profiles within linear range using simplify allometry and Dedrick plots with fixed exponent.     

Projecting human pharmacokinetics of therapeutic antibodies from nonclinical data: What have we learned?

The pharmacokinetics (PK) of therapeutic antibodies is determined by target and non-target mediated mechanisms. These antibody-specific factors need to be considered during prediction of human PK based upon preclinical information. Principles of allometric scaling established for small molecules using data from multiple animal species cannot be directly applied to antibodies. Here, different methods for projecting human clearance (CL) from animal PK data for 13 therapeutic monoclonal antibodies (mAbs) exhibiting linear PK over the tested dose ranges were examined: simple allometric scaling (CL versus body weight), allometric scaling with correction factors, allometric scaling based on rule of exponent and scaling from only cynomolgus monkey PK data. A better correlation was obtained between the observed human CL and the estimated human CL based on cynomolgus monkey PK data and an allometric scaling exponent of 0.85 for CL than other scaling approaches. Human concentration-time profiles were also reasonably predicted from the cynomolgus monkey data using species-invariant time method with a fixed exponent of 0.85 for CL and 1.0 for volume of distribution. In conclusion, we expanded our previous work and others and further confirmed that PK from cynomolgus monkey alone can be successfully scaled to project human PK profiles within linear range using simplify allometry and Dedrick plots with fixed exponent.Key words: monoclonal antibody, pharmacokinetics, clearance, allometric scaling, species-invariant time method     

Allometric growth and sperm competition in fishes

The allometric relationship between body mass and gonad mass in males of 23 fish species from 11 families was examined. There was no evidence of a single allometry for all fishes. A cross species analysis suggested a scaling coefficient of 1·04, which was significantly different from a previous study that reported a scaling coefficient of 0·904. A within species analysis generated scaling coefficients from 0·68 to 3·90. Furthermore, for those species characterized by alternative mating tactics, there was no correlation between the scaling coefficients of bourgeois‐type males and parasitic‐type males. These results are discussed in the context of the gonado‐somatic index and its use in testing sperm competition theory.     

Allometric Convergence in Savanna Trees and Implications for the Use of Plant Scaling Models in Variable Ecosystems

Theoretical models of allometric scaling provide frameworks for understanding and predicting how and why the morphology and function of organisms vary with scale. It remains unclear, however, if the predictions of ‘universal’ scaling models for vascular plants hold across diverse species in variable environments. Phenomena such as competition and disturbance may drive allometric scaling relationships away from theoretical predictions based on an optimized tree. Here, we use a hierarchical Bayesian approach to calculate tree-specific, species-specific, and ‘global’ (i.e. interspecific) scaling exponents for several allometric relationships using tree- and branch-level data harvested from three savanna sites across a rainfall gradient in Mali, West Africa. We use these exponents to provide a rigorous test of three plant scaling models (Metabolic Scaling Theory (MST), Geometric Similarity, and Stress Similarity) in savanna systems. For the allometric relationships we evaluated (diameter vs. length, aboveground mass, stem mass, and leaf mass) the empirically calculated exponents broadly overlapped among species from diverse environments, except for the scaling exponents for length, which increased with tree cover and density. When we compare empirical scaling exponents to the theoretical predictions from the three models we find MST predictions are most consistent with our observed allometries. In those situations where observations are inconsistent with MST we find that departure from theory corresponds with expected tradeoffs related to disturbance and competitive interactions. We hypothesize savanna trees have greater length-scaling exponents than predicted by MST due to an evolutionary tradeoff between fire escape and optimization of mechanical stability and internal resource transport. Future research on the drivers of systematic allometric variation could reconcile the differences between observed scaling relationships in variable ecosystems and those predicted by ideal models such as MST.     

Modeling the influence of body size on weightlifting and powerlifting performance

The purpose of this study was to examine 1) if lifting performance in both the weightlifting (WL) and powerlifting (PL) scale with body mass (M) in line with theory of geometric similarity, and 2) whether there are any gender differences in the allometric relationship between lifting performance and body size. This was performed by analyzing ten best WL and PL total results for each weight class, except for super heavyweight, achieved during 2000-2003. Data were analysed with the allometric and second-order polynomial model, and detailed regression diagnostics was applied in order to examine appropriateness of the models used. Results of the data analyses indicate that 1) women's WL and men's PL scale for M in line with theory of geometric similarity, 2) both WL and PL mass exponents are gender-specific, probably due to gender differences in body composition, 3) WL and PL results scale differently for M possibly due to their structural and functional differences. However, the obtained mass exponents does not provide size-independent indices of lifting performances since the allometric model exhibit a favourable bias toward middleweight lifters in most lifting data analyzed. Due to possible deviations from presumption of geometric similarity among lifters, future studies on scaling lifting performance should use fat-free mass and height as indices of body size.     

Allometric scaling of foraging rate with trail dimensions in leaf-cutting ants

Leaf-cutting ants (Atta spp.) create physical pathways to support the transport of resources on which colony growth and reproduction depend. We determined the scaling relationship between the rate of resource acquisition and the size of the trail system and foraging workforce for 18 colonies of Atta colombica and Atta cephalotes. We examined conventional power-law scaling patterns, but did so in a multivariate analysis that reveals the simultaneous effects of forager number, trail length and trail width. Foraging rate (number of resource-laden ants returning to the nest per unit time) scaled at the 0.93 power of worker numbers, the -1.02 power of total trail length and the 0.65 power of trail width. These scaling exponents indicate that individual performance declines only slightly as more foragers are recruited to the workforce, but that trail length imposes a severe penalty on the foraging rate. A model of mass traffic flow predicts the allometric patterns for workforce and trail length, although the effect of trail width is unexpected and points to the importance of the little-known mechanisms that regulate a colony's investment in trail clearance. These results provide a point of comparison for the role that resource flows may play in allometric scaling patterns in other transport-dependent entities, such as human cities.     

Active and resting metabolism in birds: allometry, phylogeny and ecology

Variation in resting metabolic rate is strongly correlated with differences in body weight among birds. The lowest taxonomic level at which most of the variance in resting metabolic rate and body weight is evident for the sample is among families within orders. The allometric exponent across family points is 0.67. This exponent accords with the surface area interpretation of metabolic scaling based on considerations of heat loss. Deviations of family points from this allometric line are used to examine how resting metabolic rates differ among taxa, and whether variation in resting metabolic rate is correlated with broad differences in ecology and behaviour. Despite the strong correlation between resting metabolic rate and body weight, there is evidence for adaptive departures from the allometric line, and possible selective forces are discussed.
The allometric scaling of active metabolic rate is compared with that of resting metabolic rate. The allometric exponents for the two levels of energy expenditure differ, demonstrating that active small-bodied birds require proportionately more energy per unit time above resting levels than do active large-bodied birds. No consistent evidence was found to indicate that the different methods used to estimate active metabolic rate result in systematic bias. Birds require more energy relative to body size when undertaking breeding activities than at other stages of the annual cycle.     

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.     

The effectiveness of scaling procedures for comparing ground reaction forces

Various scaling methods are used when attempting to remove the influence of anthropometric differences on ground reaction forces (GRF) when comparing groups. Though commonly used, ratio scaling often results in an over-correction. Allometric scaling has previously been suggested for kinetic variables but its effectiveness in partialing out the effect of anthropometrics is unknown due to a lack of consistent application. This study examined the effectiveness of allometric scaling vertical, braking and propulsive GRF and loading rate for 84 males and 47 females while running at 4.0 m/s. Raw, unfiltered data were ratio scaled by body mass (BM), height (HT), and BM multiplied by HT (BM1HT). Gender specific exponents for allometric scaling were determined by performing a log-linear (for BM and HT individually) or log-multilinear regression (BMHT). Pearson productmoment correlations were used to assess the effectiveness of each scaling method. Ratio scaling by BM, HT, or BM1HT resulted in an over-correction of the data for most variables and left a considerable portion of the variance still attributable to anthropometrics. Allometric scaling by BM successfully removed the effect of BM and HT for all variables except for braking GRF in males and vertical GRF in females. However, allometric scaling for BMHT successfully removed the effect of BM and HT for all reactionary forces in both genders. Based on these results, allometric scaling for BMHT was the most appropriate scaling method for partialing out the effect of BM and HT on kinetic variables to allow for effective comparisons between groups or individuals.