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First published online April 26, 2005
Journal of Experimental Biology 208, 1731-1747 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01566

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Problems of allometric scaling analysis: examples from mammalian reproductive biology

Robert D. Martin^1,*, Michel Genoud² and Charlotte K. Hemelrijk³

¹ Academic Affairs, The Field Museum, 1400 S. Lake Shore Drive, Chicago, IL 60605-2496, USA
² Département d'Ecologie et d'Evolution, Université de Lausanne, Bâtiment de Biologie, CH-1015 Lausanne, Switzerland
³ Centre for Ecology and Evolutionary Studies, University of Groningen, Kerklaan 30, 9751 NN Haren, The Netherlands

View larger version (13K): [in a new window]   Fig. 1. Illustration of three fundamental statistical problems involved in allometric scaling analyses. Problem 1 (A): results can be affected by the choice of line-fitting technique. Even in this case, where the correlation coefficient is relatively high (r=0.96), the least-squares regression (darker line) and the reduced major axis (lighter line) yield different results. The point indicated by the arrow lies below the regression but above the reduced major axis. The least-squares regression minimizes only deviations along the y-axis (as shown by the distance V for one point), whereas the reduced major axis minimizes deviations along both axes (area of triangle T shown for one point). Problem 2 (B): subsets of species may be separated by grade shifts, following the same scaling principle (common slope value) but differing in intercept value. In this case, two separate lines can be fitted to the dataset whereas an overall best-fit line yields a quite different result. Problem 3 (C): individual taxa in the comparison may not be statistically independent because of phylogenetic relationships within the tree to which they belong. (After Martin, 1998.)

View larger version (32K): [in a new window]   Fig. 2. Histograms showing distributions of (A) gestation periods and adult body mass for mammals (first author's dataset) and (B) neonatal mass and adult body mass for primates (data from Smith and Leigh, 1998). In both cases, it can be seen that neither variable meets the criterion of normality of distribution, as indicated by the superimposed curves. (See text for tests of significance.) Body mass (Mb; g); gestation (days).

View larger version (18K): [in a new window]   Fig. 3. Scaling of gestation periods for a sample of 429 placental mammal species (first author's dataset). Regardless of the method used, determination of a single best-fit line for the data yields a value for the scaling exponent () close to 0.25 (least-squares regression, 0.240; major axis, 0.255; reduced major axis, 0.285; rotation line, 0.260). (After Martin, 1989.) Gestation (days); body mass (g).

View larger version (20K): [in a new window]   Fig. 4. Scaling of gestation periods for the same sample of 429 placental mammal species, following subdivision into species with altricial neonates and those with precocial neonates. Best-fit lines fitted separately to the two grades have approximately similar slopes that are both markedly lower than the overall slope for the single distribution shown in Fig. 3 and are generally close to 0.15. (For altricial mammals: least-squares regression, 0.158; major axis, 0.160; reduced major axis, 0.198; rotation line, 0.176. For precocial mammals: least-squares regression, 0.135; major axis, 0.136; reduced major axis, 0.168; rotation line, 0.133.) (After Martin, 1989.) Gestation (days); body mass (g).

View larger version (11K): [in a new window]   Fig. 5. Plots of D-values (degree of dependence of marginal values of X and Y) derived by applying the rotation method to scaling of gestation periods for a large sample of placental mammal species. Rotation of the data in rads is indicated on the abscissa. (A) For the entire sample (N=429), a global minimum (thick vertical line) is located at 0.255 rads, corresponding to =0.260. However, a local minimum value (thin vertical line) is located at about 0.15 rads, corresponding to =0.151. (B) For altricial mammals taken alone (N=227), there is a clear single global minimum located at 0.174 rads, corresponding to =0.176. (C) For precocial mammals taken alone (N=202), there is a clear single global minimum located at 0.132 rads, corresponding to =0.133.

View larger version (29K): [in a new window]   Fig. 6. Within the two major groups of altricial and precocial mammals, there are further grade distinctions in the scaling of gestation periods between taxonomic categories. (A) Among altricial mammals, lipotyphlan insectivores and carnivores generally have relatively longer gestation periods than myomorph rodents. (B) Among precocial mammals, primates generally have relatively longer gestation periods than artiodactyls, while the latter tend to have longer gestation periods than hystricomorph rodents overall. (Lines are least-squares regressions, used simply as a visual guide to differences between taxonomic groups.) Gestation (days); body mass (Mb in g).

View larger version (25K): [in a new window]   Fig. 7. Histograms showing residual values for gestation periods of placental mammals calculated using an exponent value of 0.15, subdivided into individual taxonomic groups. Although there is some variability within each taxon, it can be seen that there is a fairly clear separation between groups with altricial neonates (carnivores, lipotyphlan insectivores, lagomorphs, myomorph and sciuromorph rodents), which typically have negative residual values, and groups with precocial neonates (hystricomorph rodents, pinnipeds, artiodactyls, cetaceans, perissodactyls and primates), which typically have positive residual values.

View larger version (16K): [in a new window]   Fig. 8. Scaling of neonatal body mass for the sample of 109 primate species (data from Smith and Leigh, 1998). A single best-fit line fitted to the data is seemingly appropriate. Body mass (Mb in g).

View larger version (12K): [in a new window]   Fig. 9. Plots of D-values derived by applying the rotation method to scaling of neonatal body mass for primate species (data from Smith and Leigh, 1998). Rotation of the data in rads is indicated on the abscissa. (A) For the entire sample (N=109), a global minimum (thick vertical line) is located at 0.742 rads, corresponding to =0.916. However, an additional local minimum value (thin vertical line) is located at 0.558 rads, corresponding to =0.624. (B) For strepsirrhine primates taken alone (N=28), the curve is somewhat irregular because of the small sample size, but there is a global minimum located at 0.603 rads, corresponding to =0.688. (C) For haplorhine primates taken alone (N=81), the sample size is considerably larger and there is a clear single global minimum located at 0.711 rads, corresponding to =0.862.

View larger version (19K): [in a new window]   Fig. 10. Scaling of neonatal body mass for the sample of 109 primate species (data from Smith and Leigh, 1998), following subdivision into strepsirrhines (N=28) and haplorhines (N=81). Best-fit lines fitted separately to the two grades have approximately similar slopes that are both markedly lower than the slope of the single best-fit line in Fig. 8. Body mass (Mb in g).

View larger version (14K): [in a new window]   Fig. 11. Flow diagram showing partial correlations in a four-way analysis of body mass, BMR, gestation period and brain mass for a sample of 51 placental mammal species. Substantial positive partial correlations exist between all pairs of variables except for BMR and gestation period, where the partial correlation is negative.

View larger version (36K): [in a new window]   Fig. 12. Histograms showing logarithmic distributions of data for body mass within (A) orders of placental mammals, and (B) families of primates. It can be seen that variation in body mass is typically quite tightly constrained within each lower-level taxon. Body mass (kg).

View larger version (32K): [in a new window]   Fig. 12B

View larger version (21K): [in a new window]   Fig. 13. Results of nested analysis of variance conducted on (A) raw values for body mass, gestation period, brain mass and basal metabolic rate (BMR) in placental mammals, and (B) the residual values for gestation period, brain mass and basal metabolic rate calculated relative to body mass. With all analyses of raw values, relatively little variance is found at the level of comparisons between species or between genera, directly reflecting the limited variation of body mass at those levels. By contrast, with analyses of residual values a substantial proportion of the variance is found at the level of comparisons between species or between genera for brain mass and BMR, but not for gestation period (which is clearly subject to marked phylogenetic inertia). This analysis replicates one previously reported by Ross (1989).

View larger version (24K): [in a new window]   Fig. 14. Illustration of different kinds of phylogenetic inertia: (A) inertia in both X and Y values (global inertia; repeat values); (B) inertia restricted to Y values (scaling inertia); (C) inertia restricted to X values (body size inertia); and (D) constrained allometric scaling of X and Y (allometric inertia).

View larger version (27K): [in a new window]   Fig. 15. Results of allometric analysis of brain:body scaling in placental mammals with raw values, contrast values and contrast values forced through the origin, randomly selecting either (A) one species from each order or (B) two species per order.