Logarithmic transformation bias in allometry

It has been known for some time (DJ Finney, J. Roy. Stat. Soc. Suppl. 7:155–161, 1941) that transformation of an arithmetic data set to logarithms results in biased estimates when predicted values from a leastsquares regression are detransformed back to arithmetic units. Predicted values are estimates of the geometric mean of the dependent variable at that value of the independent variable, rather than the arithmetic mean. Since the geometric mean is always less than the arithmetic mean, detransformed predictions will underestimate the value in question. This bias affects the interpretations of allometric equations used for estimation, such as predicting fossil body mass from skeletal dimensions, and applications of allometry as a “criterion of subtraction,” in which residual variation is evaluated. A number of parametric and nonparametric corrections for transformation bias have been developed. Although this problem is relatively unexplored in mammalian morphometrics, it has received considerable attention in other disciplines that use power functions structurally identical to the allometric equation. Insights into transformation bias and the use of correction terms from economics, limnology, forestry, and hydrology are reviewed and interpreted for application to mammalian allometry. © 1993 Wiley-Liss, Inc.     

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.     

The allometry of CNS size and consequences of miniaturization in orb-weaving and cleptoparasitic spiders

Allometric studies of the gross neuroanatomy of adults from nine species of spiders from six web-weaving families (Orbicularia), and nymphs from six of these species, show that very small spiders resemble other small animals in having disproportionately larger central nervous systems (CNSs) relative to body mass when compared with large-bodied forms. Small spiderlings and minute adult spiders have similar relative CNS volumes. The relatively large CNS of a very small spider occupies up to 78% of the cephalothorax volume. The CNSs of very small spiders extend into their coxae, occupying as much as 26% of the profile area of the coxae of an Anapisona simoni spiderling (body mass < 0.005 mg). Such modifications occur both in species with minute adults, and in tiny spiderlings of species with large-bodied adults. In at least one such species, Leucauge mariana, the CNS of the spiderling extends into a prominent ventral bulge of the sternum. Tiny spiders also have reduced neuronal cell body diameters. The adults of nearly all orbicularian spiders weave prey capture webs, as do the spiderlings, beginning with second instar nymphs. Comparable allometric relations occur in adults of both orb-weaving and cleptoparasitic species, indicating that this behavioral difference is not reflected in differences in gross CNS allometry.     

Viscoelastic behavior of organic materials: consequences of a logarithmic dependence of force on strain rate

The viscoelastic and -plastic behavior of organic materials like bone, tendon or wood, as well as technical polymers, is amply documented. It is usually modeled using linear "Newtonian" friction, i.e., a viscous force proportional to the deformation rate. If the experimental results cannot be fitted with the resulting exponential "Debye" curves, a multitude of relaxation mechanisms or a spectrum of relaxation times is invoked. In this contribution experimental evidence is compiled which indicates that for polymers and organic materials a logarithmic dependence of the deformation force on the deformation rate is more appropriate. The corresponding equation of motion is solved in the quasi-static approximation and the solutions display just the typical deviations from the Debye behavior found experimentally, without any complications from multi-mechanism relaxation.     

Multivariate dental allometry in primates

Disagreement is current over the question of whether relatively large teeth in some large primates are a natural outcome of growth trends instead of an indication of intrinsic differences. A cross-primate survey of dental scaling relative to skull (and inferred body) size is given in this study, using a principal component technique to measure the multivariate growth relation between two sets of data: dental size and cranial size. Cheek teeth are strongly positively allometric in restriced taxonomic groups, especially in cercopithecoids. Conversely, the allometry drops to an almost linear proportional growth relation when variation in diet is controlled.     

Wing-bone length allometry in birds

A comparative analysis using both independent contrasts (CAIC) and a species level analysis was used to investigate the allometric scaling of avian wing-bone lengths. Total arm ( ta =humerus+ulna+manus) scaled with positive allometry as body mass ( M )^0.37–0.39. Similarly, and in accordance with previous studies, wing-span ( b ) was positively allometric, but CAIC suggested a lower allometric exponent ( M ^0.35) than found using species as independent data points ( M ^0.39). Contrary to previous studies, individual wing-bones appear to scale with similar exponents against M and scale isometrically with ta . In addition to a general trend for larger birds to have longer wings, wing-bones and ta , their ta was a larger proportion of their b . A detailed study of primary feather length and elbow joint angle across a wide range of bird species and bird size, however, is required before a conclusive explanation for this increase in ta relative to b in larger birds can be established. Scaling equations are presented that can be used to predict M , ta and b from individual wing-bone lengths, which may be of use to palaeontologists wishing to reconstruct whole animals from single bones.     

Biophysical and functional consequences of receptor-mediated nerve fiber transformation.

Stimulation of the nervous system by substance P, a G protein-coupled receptor, and subsequent receptor internalization causes dendrites to change their shape from homogeneous cylinders to a heterogeneous string of swollen varicosities (beads) connected by thin segments. In this paper we have analyzed this phenomenon and propose quantitative mechanisms to explain this type of physical shape transformation. We developed a mathematical solution to describe the relationship between the initial radius of a cylindrical nerve fiber and the average radii of the subsequently created varicosities and connecting segments, as well as the periodicity of the varicosities along the nerve fiber. Theoretical predictions are in good agreement with our own and published experimental data from dorsal root ganglion neurons, spinal cord, and brain. Modeling the electrical properties of these beaded fibers has led to an understanding of the functional biophysical consequences of nerve fiber transformation. Several hypotheses for how this shape transformation can be used to process information within the nervous system have been put forth.     

The allometry of patch selection in ruminants

An axiomatic feature of food consumption by animals is that intake rate and prey abundance are positively related. While this has been demonstrated rigorously for large herbivores, it is apparent from patch selection trials that grazers paradoxically tend to prefer short, sparse swards to tall, dense swards. Indeed, migratory herbivores often shift from areas of high to low sward biomass during the growing season. As nutritional quality is an inverse function of grass abundance, herbivores appear to sacrifice short-term intake for nutritional gains obtainable by eating sparse forage of higher quality. Explicit models of this trade-off suggest that individual ruminants maximize daily rates of energy gain by choosing immature swards of intermediate biomass. As body mass is related positively to both ruminant cropping rates and digestibility, there should be an allometric link between grass abundance and energy maximization, providing a tool for predicting patterns of herbivore habitat selection. We used previously published studies to develop a synthetic model of trade-offs between forage abundance and quality predicting that optimal sward biomass should scale allometrically with body size. The model predicts size-related variation in habitat selection observed in a guild of grazing ungulates in the Serengeti ecosystem.     

Biomechanical allometry in hominoid thoracic vertebrae

Considerable differences in spinal morphology have been noted between humans and other hominoids. Although comparative analyses of the external morphology of vertebrae have been performed, much less is known regarding variations in internal morphology (density) and biomechanical performance among humans and closely related non-human primates. In the current study we utilize density calibrated computed tomography images of thoracic vertebral bodies from hominoids (n = 8-15 per species, human specimens 20-40 years of age) to obtain estimates of vertebral bone strength in axial compression and anteroposterior bending and to determine how estimates of strength scale with animal body mass. Our biomechanical analysis suggests that the strength of thoracic vertebral bodies is related to body mass (M) through power law relationships (y ∝ M^b) in which the exponent b is 0.89 (reduced major axis) for prediction of axial compressive strength and is equal to 1.89 (reduced major axis) for prediction of bending strength. No differences in the relationship between body mass and strength were observed among hominoids. However, thoracic vertebrae from humans were found to be disproportionately larger in terms of vertebral length (distance between cranial and caudal endplates) and overall vertebral body volume (p < 0.05). Additionally, vertebral bodies from humans were significantly less dense than in other hominoids (p < 0.05). We suggest that reduced density in human vertebral bodies is a result of a systemic increase in porosity of cancellous bone in humans, while increased vertebral body volume and length are a result of functional adaptation during growth resulting in a vertebral bone structure that is just as strong, relative to body mass, as in other hominoids.     

The allometry of rodent intestines

This study examined the allometry of the small intestine, caecum, colon and large intestine of rodents (n = 51) using a phylogenetically informed approach. Strong phylogenetic signal was detected in the data for the caecum, colon and large intestine, but not for the small intestine. Most of the phylogenetic signal could be attributed to clade effects associated with herbivorous versus omnivorous rodents. The herbivorous rodents have longer caecums, colons and large intestines, but their small intestines, with the exception of the desert otomyine rodents, are no different to those of omnivorous rodents. Desert otomyine rodents have significantly shorter small intestines than all other rodents, reflecting a possible habitat effect and providing a partial explanation for the low basal metabolic rates of small desert mammals. However, the desert otomyines do not have shorter colons or large intestines, challenging claims for adaptation to water retention in arid environments. Data for the Arvicolidae revealed significantly larger caecums and colons, and hence longer large intestines, with no compensatory reduction in the length of the small intestine, which may explain how the smallest mammalian herbivores manage to meet the demands of a very high mass-specific metabolic rate. This study provides phylogenetically corrected allometries suitable for future prediction testing.     

Biogeochemical consequences of the transformation of native Cerrado into Pinus caribaea plantations in Brazil

Under the same climatic and edaphic conditions, native savanna vegetation in Brazil, the Cerrado, shows a lower stature and canopy cover than planted Pinus caribaea Morelet forests. To assess the differences in biogeochemical element cycling we compared the nutrient economy of Cerrado and Pinus on three replicate plots of each forest type. The mean nutrient storage in the soil organic layer under Pinus (N: 2630; P: 141; K: 103; Ca: 131; Mg: 20 kg ha^–1) was substantially higher than under Cerrado (N: 23; P: 1.2; K: 0.83; Ca: 5.8; Mg: 1.0 kg ha^–1) probably because the Pinus roots explored a larger soil volume. The Pinus trees had a higher nutrient-use efficiency as indicated by higher mean litter mass per unit nutrient in litter (N: 108; P: 2290; K: 729; Ca: 1360; Mg: 5420; S: 1190; Fe: 2960; Mn: 9990, Zn: 145000) than the Cerrado trees (N: 94; P: 1810; K: 619; Ca: 302; Mg: 938, S: 746; Fe: 1800; Mn: 7880; Zn: 63700). Mean annual small litterfall collected in 0.25-m² samplers between May 1997 and April 1999 was 2.1 Mg ha^–1 in Cerrado and 7.8 in Pinus. The litterfall rates of the 1–3 week collection intervals correlated negatively with the soil matric potential indicating that litterfall was partly related to water stress. The fluxes of N (73 kg ha^–1 year^–1), P (3.7), K (11), S (7.0), and Mn (0.83) to the soil with litterfall under Pinus were greater than the litterfall+turnover of the grass/herbs layer under Cerrado (N: 39, P: 2.8, K: 8.6, S: 5.4, Mn: 0.79 kg ha^–1 year^–1), those of Zn (0.06–0.07) were similar, and those of Ca (Pinus: 5.9/Cerrado: 10), Mg (1.5/4.4), and Fe (2.9/4.0) were smaller. Mean residence times of the organic matter and of all elements were longer in the soil organic layer under Pinus (3.7–26 years in the Oi horizon, 8.1–907 years in the whole organic layer) than under Cerrado (0.22–3.6 years in the Oi horizon, the only organic horizon under Cerrado). Our results demonstrate that the main differences in biogeochemical element cycling between the Pinus forest and the Cerrado consisted of a larger nutrient storage in the organic layer, a higher nutrient-use efficiency, and slower nutrient release rates from the organic layer in the Pinus forest than in the Cerrado. Nutrient cycling as assessed by the nutrient fluxes with litterfall was only partly faster in the Pinus forest than in the Cerrado.     

The allometry of saguaro height

The allometry of plant height H with respect to mean stem diameter D was determined based on 118 saguaro plants. The slope obtained for the reduced major axis regression analysis of the data was 2.36 ± 0.085, indicating that taller plants are disproportionately more slender than their shorter, presumably younger counterparts. The consequences of this positive, extremely anisometric relation on the elastic stability of stems were estimated by computing the critical buckling height H_crit for each of the 118 stems on the basis of the mean density-specific stiffness (i.e., the quotient of Young's elastic modulus E and bulk tissue density ρ) determined for a single section from a mature saguaro stem. E/ρ was nearly equivalent to that of tissue samples of sclerenchyma isolated from other plant species. Since the slope of H_crit vs. D equals ≈ 0.67 when E/ρ ≈ a constant, the safety-factor for saguaro stems (i.e., H_crit/H) appeared to be size-dependent such that it decreased with increasing plant height (i.e., H_crit/H ≈ D^-1.65). However, the mean safety-factor computed for the 118 saguaro specimens was 9.64, indicating that, on the average, plant height was well below H_crit. Additionally, circumstantial evidence suggests that saguaro stems become more stiff as they increase in size (and age) and that the rate of stem growth decelerates over time. The former would obtain a near size-independent safety-factor against elastic buckling while the latter protracts the time required to reach the critical buckling height. Comparisons among the allometries of H and H_crit for saguaro, other cacti, nonwoody, and highly branched tree species indicated that saguaro size overlaps with the lower size-range of the largest known dicot and gymnosperm tree specimens likely as a consequence of the high E/ρ of mature saguaro stems.     

The allometry of host-pathogen interactions

Background

Understanding the mechanisms that control rates of disease progression in humans and other species is an important area of research relevant to epidemiology and to translating studies in small laboratory animals to humans. Body size and metabolic rate influence a great number of biological rates and times. We hypothesize that body size and metabolic rate affect rates of pathogenesis, specifically the times between infection and first symptoms or death.

Methods and Principal Findings

We conducted a literature search to find estimates of the time from infection to first symptoms (t_S) and to death (t_D) for five pathogens infecting a variety of bird and mammal hosts. A broad sampling of diseases (1 bacterial, 1 prion, 3 viruses) indicates that pathogenesis is controlled by the scaling of host metabolism. We find that the time for symptoms to appear is a constant fraction of time to death in all but one disease. Our findings also predict that many population-level attributes of disease dynamics are likely to be expressed as dimensionless quantities that are independent of host body size.

Conclusions and Significance

Our results show that much variability in host pathogenesis can be described by simple power functions consistent with the scaling of host metabolic rate. Assessing how disease progression is controlled by geometric relationships will be important for future research. To our knowledge this is the first study to report the allometric scaling of host/pathogen interactions.