Estimating the body mass of extinct ungulates: a study on the use of multiple regression

The correlation between body mass and both skeletal and dental measures in living mammals has enabled paleontologists to obtain reliable estimates of body size for extinct species, usually using log-transformed bivariate least-squares regression equations. Multiple regression, however, has rarely been used for estimating the mass of extinct species, although this technique can clearly improve the predictive equations compared with those adjusted by simple regression. However, the use of multiple regression is problematical, because even those functions explaining a high percentage of the variance of the dependent variable (i.e. body mass) can show a rather limited predictive power. After analyzing which factors determine the predictive ability of multiple regression equations, we propose a new set of algorithms that allow the estimation of the body mass of extinct ungulates. These algorithms are finally applied to three Miocene ungulate species, Dinohippus leidyanus , Stenomylus hitchcocki and Aletomeryx scotti .     

Aspects of line-fitting in bivariate allometric analyses

One of the fundamental problems involved in analyses of the scaling effects of body size (allometric analysis is the choice of an appropriate best-fit line in bivariate logarithmic plots. Following a discussion of some basic aspects of allometric analysis, the tow mai procedures for the determination of a best-fit line - the least-squares regression and the major axis - are examined with respect to their different properties and underlying models. It is important to distinguish intraspecific from interspecific scaling and to recognize the distinction between use of a best-fit line to define a relationship and use of the line for prediction. An alternative model to the bivariate normal distribution, referred to as the 'extruded normal distribution', is presented and its implications are examined with respect to two test cases (scaling of basal metabolic rate in human males; scaling of population density in mammals).     

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.     

Evolution of brain–body allometry in Lake Tanganyika cichlids

Brain size is strongly associated with body size in all vertebrates. This relationship has been hypothesized to be an important constraint on adaptive brain size evolution. The essential assumption behind this idea is that static (i.e., within species) brain–body allometry has low ability to evolve. However, recent studies have reported mixed support for this view. Here, we examine brain–body static allometry in Lake Tanganyika cichlids using a phylogenetic comparative framework. We found considerable variation in the static allometric intercept, which explained the majority of variation in absolute and relative brain size. In contrast, the slope of the brain–body static allometry had relatively low variation, which explained less variation in absolute and relative brain size compared to the intercept and body size. Further examination of the tempo and mode of evolution of static allometric parameters confirmed these observations. Moreover, the estimated evolutionary parameters indicate that the limited observed variation in the static allometric slope could be a result of strong stabilizing selection. Overall, our findings suggest that the brain–body static allometric slope may represent an evolutionary constraint in Lake Tanganyika cichlids.     

Heritable allometric variation in bumble bees: opportunities for colony-level selection of foraging ability

Different characters of an organism may be correlated if genes control the allometric relationship between them. If genetic variation exists for such genes then the allometric relation itself is potentially subject to change by selection. In social insects allometric relations represent colony-level characters. If colonies differ in these relations and this variation leads to differential productivity among colonies, then selection on allometric relations can operate at the level of the colony. We assessed the extent of heritable, between-colony variation for the allometric coefficients relating proboscis ( = glossa) length to wing length for two bumble bee species (Bombus huntii and B. occidentalis). We found that in both species colonies did not differ significantly in slope (b) but did differ significantly in intercept (a) of the regression of glossa length on wing length. Within-colony variation of the intercept was estimated by randomly constituting groups of five workers from each colony and calculating the regression for each group. The intraclass correlation was then calculated from the between- and within-colony mean squares. We found significant intraclass correlations in both species, giving heritabilities of 0.5 ± 0.3 in B. hunti and 0.7 ± 0.3 in B. occidentalis. If this allometric relation affects colony foraging success and foraging environments vary geographically, then the intercept should exhibit corresponding geographic variation. We tested this prediction by comparing intercepts calculated using wild-caught B. vagans workers from Alberta, Ontario and Maine. We found that the intercepts did differ significantly between sites, with the bees from Alberta having a significantly smaller intercept than the bees from eastern North America. Our results illustrate the opportunity for selection on an allometric relation that directly affects the foraging success of individual bumble bee colonies.     

The ratio and allometric scaling of speed, power, and strength in elite male rugby union players

This study compared the effectiveness of ratio and allometric scaling for normalizing speed, power, and strength in elite male rugby union players. Thirty rugby players (body mass [BM] 107.1 ± 10.1 kg, body height [BH] 187.8 ± 7.1 cm) were assessed for sprinting speed, peak power during countermovement jumps and squat jumps, and horizontal jumping distance. One-repetition maximum strength was assessed during a bench press, chin-up, and back squat. Performance was normalized using ratio and allometric scaling (Y/X), where Y is the performance, X, the body size variable (i.e., BM or BH), and b is the power exponent. An exponent of 1.0 was used during ratio scaling. Allometric scaling was applied using proposed exponents and derived exponents for each data set. The BM and BH variables were significantly related, or close to, performance during the speed, power and/or strength tests (p < 0.001-0.066). Ratio scaling and allometric scaling using proposed exponents were effective in normalizing performance (i.e., no significant correlations) for some of these tests. Allometric scaling with derived exponents normalized performance across all the tests undertaken, thereby removing the confounding effects of BM and BH. In terms of practical applications, allometric scaling with derived exponents may be used to normalize performance between larger rugby forwards and smaller rugby backs, and could provide additional information on rugby players of similar body size. Ratio scaling may provide the best predictive measure of performance (i.e., strongest correlations).     

Testing the cranial evolutionary allometric ‘rule’ in Galliformes

Recent comparative studies have indicated the existence of a common cranial evolutionary allometric (CREA) pattern in mammals and birds, in which smaller species have relatively smaller faces and bigger braincases than larger species. In these studies, cranial allometry was tested using a multivariate regression between shape (described using landmarks coordinates) and size (i.e. centroid size), after accounting for phylogenetic relatedness. Alternatively, cranial allometry can be determined by comparing the sizes of two anatomical parts using a bivariate regression analysis. In this analysis, a slope higher or lower than one indicates the existence of positive or negative allometry, respectively. Thus, in those species that support the CREA ‘rule’, positive allometry is expected for the association between face size and braincase size, which would indicate that larger species have disproportionally larger faces. In this study, I applied these two approaches to explore cranial allometry in 83 Galliformes (Aves, Galloanserae), ranging in mean body weight from 30 g to 2.5 kg. The multivariate regression between shape and centroid size revealed the existence of a significant allometric pattern resembling CREA, whereas the second analysis revealed a negative allometry for beak size and braincase size (i.e. contrary to the CREA ‘rule’, larger galliform species have disproportionally shorter beaks than smaller galliform species). This study suggests that the presence of CREA may be overestimated when using cranium size as the standard measurement.     

Allometric growth and reproductive biology traits of pikeperch Sander lucioperca at the southern edge of its range

This study describes some biological traits (allometric relationship, reproductive biology and condition factor) of pikeperch Sander lucioperca at the southern edge of its range. Data were compiled from field studies between January 1997 and January 2009 in three Tunisian reservoirs. Overall, 1497 fish were caught, from which 1308 specimens were sampled and dissected. Sexually undetermined, male and female fish from the different sites exhibited similar allometric growth, expressed through the equation M(T) = 5 × 10(-6) L(T) (3·06) , where M(T) is the total mass and L(T) is the total length. The exponent value is slightly but significantly greater than 3, indicating positive allometric growth. Sex ratio (males:females) was 47:53 but remained balanced only up to 55 cm L(T) and changed with increasing size towards female dominance. Spawning occurred in March and early April according to the changes in female size-adjusted gonad mass or corresponding gonado-somatic index (I(G) ) values. Thermal conditions seemed favourable in February, but March corresponded to the period of equal day and night lengths and the switch towards longer days in terms of photoperiod. Changes in both size-adjusted M(T) or Fulton's condition factor and size-adjusted liver mass or hepato-somatic index (I(H) ) were clearly associated with reproduction. Reserves were consumed during spring in association with spawning efforts. Nevertheless, the delay before the reconstitution of reserves seemed relatively long (from early spring to late summer) although both thermal and trophic conditions could be considered favourable for active feeding of the fish.     

Hydraulic architecture and the evolution of shoot allometry in contrasting climates

We used pairs of congeneric shrub species from contrasting habitats to test for repeated evolutionary divergence in leaf-stem allometry and shoot hydraulic architecture in response to water availability. Allometric relationships and mean ratios between leaf size (individual and total area and mass per shoot) and stem cross-sectional area were compared between habitats using six species pairs representing three genera (Arctostaphylos, Baccharis, Ceanothus). We measured correlations among evolutionary changes in allometric, morphological, and physiological traits using phylogenetic independent contrasts. Allometric analysis revealed habitat differences: slopes were homogeneous among species (=1.46), but the more mesic-adapted species generally supported more leaf area at a common stem cross-sectional area. Reducing bivariate allometry to a ratio obscured this pattern because ratios varied with stem size, which was unrelated to habitat. Mean individual leaf size also was not correlated with either water availability or leaf-stem allometry. Stem hydraulic conductivity was generally lower in the xeric-adapted species of each pair, and its evolution mirrored changes in shoot allometry. This study provides evidence for repeated evolutionary divergence in shoot allometry and hydraulic architecture associated with water availability and demonstrates the importance of shoot allometry to water relations, independent of leaf size.     

Developmental Processes, Evolvability, and Dental Diversification of New World Monkeys

The developmental processes that contribute to variation of morphological traits are the subject of considerable interest when attempting to understand phenotypic evolution. It is well demonstrated that most characteristics of tooth pattern can be modified by tinkering conserved signal pathways involved in dental development. This effect can be evaluated by comparing developmental models with naturally occurring variation within explicit phylogenetic contexts. Here, we assess whether evolutionary changes in lower molar (M) ratios among platyrrhines were channelled by alterations in the balance of activators and inhibitors as predicted by the inhibitory cascade (IC) model (Kavanagh et al. in Nature 449:427–432, 2007). Ordinary linear regression adjusted to M2/M1 versus M3/M1 ratios of 38 species of platyrrhines indicated that the slope and intercept were significantly different from the IC model. Conversely, when the phylogeny was incorporated into the regression analyses (PGLS), variation in molar ratios did not differ from the developmental model. PGLS also showed that changes in molar proportions are not an allometric effect associated with body size. Discrepancies between phylogenetically corrected and non-corrected analyses are mainly due to the departure of Callitrichines from the predicted values. This subfamily displays agenesis of M3 with higher than expected M2/M1 ratios, indicating that M3 fails to develop even when the inhibition by M1 on the subsequent molars is not increased. Our results show that evolution in molar ratios is concordant with slight changes in the proportion of activators and inhibitors that regulate molar development; however, other processes are required to account for variation in the number of teeth.     

Modeling the influence of body size on V(O2) peak: effects of model choice and body composition.

This study examined the bivariate relationship between peak oxygen uptake (V(O2) peak); l/min) and body size in adult men (n = 1,314, age 17-66 yr), using both "simple" and "full" iterative nonlinear allometric models. The simple model was described by V(O2) peak = M(b) (or FFM(b)) exp(c SR-PA) exp(a + d age) epsilon (where M is body mass in kg; FFM is fat-free mass in kg; SR-PA is self-reported physical activity; epsilon is a multiplicative error term; and exp indicates natural antilogarithms). The full model was described by V(O2) peak = M(b) (or FFM(b)) exp(c SR-PA) exp(a + d age) + e (epsilon), where e is a permitted Y-intercept term. The M exponent obtained from simple allometry was 0.65 [95% confidence interval (CI), 0.59-0.71], suggestive of a curvilinear relationship constrained to pass through the origin. This "zero Y-intercept" assumption was examined via the full allometric model, which revealed an M exponent of 1.00 (95% CI, 0.7-1.31), together with a positive Y-intercept term (e) of 1.13 (95% CI, 0.54-1.73). The FFM exponents were not significantly different from unity in either the simple or full allometric models. It appears that the curvilinearity of the simple allometric model (using total M) is fictitious and is due to the inappropriate forcing of the regression line through the origin. Utilizing FFM as the body-size variable revealed a linear relationship between body size and V(O2) peak, irrespective of model choice. We conclude that the population mass exponent for V(O2) peak is close to unity.     

Allometric changes during growth and reproduction in Eleutheria dichotoma (hydrozoa,athecata) and the problem of estimating body size in a microscopic animal

Changes in body morphology during growth and reproduction in the hydromedusa Eleutheria dichotoma are described in terms of variations in eight different characters: umbrella diameter, total surface area, tentacle area, umbrella area, tentacle knob diameter, number of embryos, and diameter and area of buds. Sexually (sex) and vegetatively (veg) reproducing medusae differ significantly in their body morphometrics. Statistically significant allometric relations exist between umbrella diameter and (1) central area (sex and veg); (2) tentacle area (veg); (3) total area (veg); (4) tentacle knob diameter (veg); (5) bud diameter; and (6) number of embryos. A significant correlation between umbrella diameter and area is also found in undetached buds. During sexual reproduction, umbrella area shows positive allometry and loses its correlations to total area, tentacle area, and tentacle knob diameter. Linear and nonlinear bivariate allometric coefficients allow estimation of total body size from only one or two easily measurable attributes, e.g., umbrella and tentacle knob diameter. Curve fitting by the classic allometric equation (y = bx^c) is only negligibly worse than that obtained with a “full” equation (y = a + ^c), and statistical confidence is better. Chemical analyses for carbon and nitrogen content allow estimation of biomass from the projection area of the body surface. The relation factors are 1.06 μgC mm^?2 (sex) and 1.14 μgC mm^?2 (veg) for carbon and 0.293 μgN mm^?2 (sex) and 0.287 μgN mm^?2 (veg) for nitrogen. The C:N ratios are 3.6 and 4.0 for sexual and vegetative medusae, respectively. The use of allometric regression formulas to calculate surface areas and to relate these to carbon content provides quick estimations of body size in a microscopic animal.     

祁连山区青海云杉林冠层叶面积指数的反演方法

叶面积指数(Leaf area index, LAI)是陆地生态系统的一个十分重要的结构参数。随着空间精细化模型的发展和基于过程的分布式模拟技术的应用, 对LAI的区域估算显得越来越重要, 但目前尚缺乏有效的估算手段。该项研究以青海云杉(Picea crassifolia)林为研究对象, 利用LAI-2000冠层分析仪、鱼眼镜头法和经验公式法对林冠层LAI进行了测定, 观测值分别为1.03~3.70、0.48~2.26和2.27~8.20, 显然, 仪器测定值偏低。针对针叶的集聚效应导致仪器测定值偏低的现象, 利用跟踪辐射与冠层结构测量仪(TRAC)测定的青海云杉林聚集系数计算调整系数, 对鱼眼镜头法获取的LAI值进行订正。根据高分辨率的遥感数据反演青海云杉林的植被指数与LAI的关系, 最后获得了较合理的该地区林冠层LAI的空间分布图。     

Morphological variability and allometric relationships of the herb Panax notoginseng in Yunnan,China

A plant's morphology changes throughout its ontogeny. Investigating the allometric relationships between different morphological traits could provide useful information for cultivation of medicinal plants. Here we collected 698 individuals of Panax notoginseng for allometric analysis from seven populations cultivated in Yunnan, Southwest China. The slopes and intercepts of the allometric relationships were estimated by Standardized Major Axis regression. Significant differences (p < 0.05) were found in each morphological variable considered among populations. Allometric analysis showed that all of the log-log relationships had different slopes or shared a common slope but differed in intercept (p < 0.001). The morphological traits showed flexible allometric relationships. However, the root biomass that considered as a target trait showed the least allometric variability (slope = 1.068–1.378) when compared to other variables. This could be because of the hundreds of years of cultivation and artificial selection.     

Testing of Corner’s rules across woody plants in Tiantong region,Zhejiang Province: effects of individual density

Aims Corner’s rules reflect the architectural strategies of plants with respect to deployment of twig size and leaf size, as well as of the number of twigs and leaves. The objective of this study was to examine how Corner’s rules would vary among plants with different individual densities.
Methods The study site is located in the Tiantong National Forest Park (29.87° N, 121.65° E), Zhejiang Province. We measured twig cross-sectional area (twig size), total leaf area (leaf size per twig), and the number of twigs at a given twig size (branching intensity) in woody plants across 25 plots differing in stem density to examine the effects of individual competition on Corner’s rules. The standardized major axis (SMA) analysis was conducted to determine the quantitative relationships of twig size with leaf size and branching intensity.
Important findings Significant, positive allometric relationships between cross-sectional area and total leaf area were found in individual plants across all communities. There was no significant difference among communities of different density intervals in the slope of the linear regression between cross-sectional area and total leaf area of individual plants, and the common slope of the regressions was significantly greater than 1 (p < 0.001). The intercept was significantly greater for plants in communities with higher density than in those with lower density (p < 0.001), indicating that plants in a high density community support greater total leaf area than in a low density community for a given twig size. In contrast, a significant, negative allometric scaling relationship was found between branching intensity and cross-sectional area in individual plants across different communities. Also, nosignificant difference was found among plants in communities of different density intervals in the slope of the regression between branching intensity and cross-sectional area, and the common slope of the regressions was significantly less than –1 (p < 0.001). The intercept for the regression relationship between twig area and branching intensity was the same among plants in communities of different density intervals (p > 0.05), suggesting that plants in a high density community do not deploy more twigs per twig size than in a low density community. In summary, this study demonstrated that plants responded to changes in individual density by maintaining an invariant regression slope for the twig size-leaf size relationship and the twig size-branching intensity relationship, and that the Corner’s rules were not affected by individual density of the communities in the Tiantong region. However, changes in the intercept of the regression between twig size and leaf size indicate that deployment strategies between twig and leaf sizes could be adjusted with increasing individual plant competition, thus structuring species coexistence through niche differentiation.     

Rotational distortion in conventional allometric analyses

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

Morphometric analysis of the skull of Dermophis mexicanus (Amphibia: Gymnophiona)

The levels of integration, patterns of allometric growth and size-related differences in skull dimensions of the caecilian Dermophis mexicanus were investigated by means of univariate, bivariate and multivariate statistics. The analyses indicate that (1) most but not all measurements of the skull arc more variable in adults than in juveniles; (2) growth is allometric and results in changes in shape between juveniles and adults; (3) there is a high level of integration of all variables by general size; and (4) homologous paired measurements show concordant allometric trends and are integrated beyond the sole effect of general size. These analyses provide baseline data, a set of hypotheses and a methodological framework for comparative studies of patterns of variation and integration among caecilians and among other vertebrates.     

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.     

Functional assessment of subfamily variation in maxillomandibular morphology among Old World monkeys

Among Old World monkeys, subfamily variation in maxillomandibular form is commonly attributed to divergent dietary and social behaviors. However, our knowledge of any musculoskeletal adaptations for gape in cercopithecines, and folivory in colobines, is incomplete. Such data are requisite to a more informed perspective on the evolutionary morphology of these taxa. Structural analyses of gape and biomechanical efficiency were applied to a representative sample of adult cercopithecids. Factors pertaining to the biomechanical scaling of cranial structures were evaluated with least-squares bivariate regression techniques. To assess subfamily differences in masticatory efficiency, analyses of covariance were made between relevant factors. Cercopithecines achieve increased gape and relative canine size mainly with strong positive allometry of the facial skull, combined with a larger gonial angle. Colobines possess a relatively long masseter lever arm and short facial skull, as well as an enlargened masseter-medial pterygoid complex. Subfamily differences in temporalis lever arm scaling are negligible. Biomechanical comparisons within and between subfamilies suggest that the mechanical advantage of the temporalis is relatively greater than that of the masseter, while the mechanical advantage of both muscles increases with face length. Evidence is presented to stress the need for adequate consideration of the dependent variable in allometric investigations of skull form.