Anna. Rev. Ecol. Syst. 1989. 20:97-117 Copyright © /9.S9 _>>• <4nnua/ Reviews Inc. All rights reserved c1^ °( a\\omd>- *y 4epfuK jum a wcvrsc 5\/VHuXVi|\A - COV\5Vav\C\J in ANALYZING BODY SIZE AS d ( \S0WM2i A FACTOR IN ECOLOGY AND EVOLUTION
Michael LaBarbera
Department of Anatomy, The University of Chicago, 1025 East 57 Street, Chicago, Illinois 60637
INTRODUCTION An organism's size is perhaps its most apparent characteristic. The recent outpouring of studies on the influence of body size on all aspects of biology and evolution and the extension of the focus of such studies from physiology and functional morphology to ecological characteristics has resulted in a spate of books on the subject (17, 82, 99). As a result, the field of scaling, the study of the influence of body size on form and function, has rather suddenly become a prominent focus in ecology and evolutionary biology. [I prefer Schmidt-Nielsen's (99) term of "scaling" for this field over Gould's (35) "allometry," since the latter refers also to departures from geometric sim ilarity; confusion of the two senses of "allometry" has further complicated an already confused literature.] Because the study of body size was long relegated to a minor position in biology, standards within the field have tended to be lax and the literature is replete with papers of dubious methodology, analysis, and conclusions. Inappropriate comparison of different types of data is probably the most common error in the scaling literature (e.g. 42, 73); data plots often treat individuals and species as equivalent entities. Less obvious but just as erroneous is the common tendency to equate intraspecific trends with ontogenetic or evolutionary phenomena. In scaling studies, both the level of analysis and the kinds of data included must be carefully chosen if the results are to be meaningful. See Cock (22) for a clear and careful introduction to this topic. 97 0066-4162/89/11/20-0097$02.00 98 LABARBERA BODY SIZE 99
Allometric Models The oldest of such models is Sarras' "surface law of metabolism." The history of the "surface law" is an object lesson in the seductive potential of scaling The study of scaling is more an empirical than a theoretical science, but a "laws." Despite the fact that its assumptions were never critically tested and number of attempts have been made to justify the assumption that size per se the vast majority of the data was at variance with its predictions, the "surface influences the form and function of organisms. Central to the concept of law" was considered a fact until Kleiber demonstrated that mammalian scaling is the idea of similarity—a null model describing the expected change metabolic rates scaled more nearly as M° 75 (99). Discrepancies between the of some variable with change in size. In general, three kinds of similarity can "surface law" and relevant data were glossed over, in large measure because be distinguished: of the satisfyingly "physical" nature of the theory. (Parallels with Lord Kelvin's rejection of Darwinism because physics "proved" the earth could not physical similarity Often scale alone determines the relevant physical be old enough for natural selection to have operated are probably not coinci variables; as Steven Vogel (personal communication) observed, "Reality is size dependent." Typically the similarity criteria are constancy-of-force ratios dental; see 14.) All theoretical models which purport to predict scaling such as the familiar Reynolds (inertial/viscous forces) or Froude (inertial/ relations should be subject to careful testing; empirical scaling relations are gravitational forces) numbers (see 113). The relative importance of gravity, "noisy," and the desire of biologists for laws of the same power as those in physics is remarkably strong. inertia, viscosity, and surface tension all change with an organism's size; Elsewhere (61)1 have discussed the general areas of evolutionary and charming qualitative treatments can be found in Boycott (11) and Went (117) and a more quantitative treatment in Vogel (113). ecological scaling. Here 1 focus on appropriate methodology for studies of biological scaling and offer three examples of topics drawn from the ecologi geometric similarity For any scries of objects whose linear dimensions cal and evolutionary literature to illustrate the problems common in many of differ only by a constant multiplier, surface areas are proportional to some the controversies on scaling. characteristic length squared (A « L2), and volumes are proportional to that length cubed (V a L3). In biology, constancy in shape with change in size is THE METHODOLOGY OF SCALING STUDIES termed isometric growth (or isometry), the usual null hypothesis in morpho logical scaling studies. Departure from geometric similarity is termed al Virtually all work in biological scaling assumes a power function of the form lometry, Measurement of volume is both inconvenient and imprecise; most Y = a Mh, Huxley's (47) "simple equation of allometry" where a is the scaling studies assume that the density of different organisms is approximate scaling coefficient and b is the scaling exponent. In morphometries, a power ly the same and so they substitute mass for volume. There is much confusion law relation is usually justified by noting that relative growth rates of different in the literature about what constitutes isometry, since the appropriate null components of the organism will follow such a relation (35, 47, 96, 107), but model depends on the geometric variables of interest. Isometry is specified by it has also been derived from dimensional analysis and fractal geometry (43). the ratio of the dimensions of the variables; i.e. the expected slope of a plot of When the variable of interest is an ecological or demographic parameter, such log area vs log volume (see below) is AIV = L2/L3 = 2/3. explanations do not seem tenable, but power law models are used in the ecological scaling literature despite the lack of a rationale for their FUNCTIONAL similarity Any pair of functionally related variables that appropriateness. The few attempts at a general justification for a power law depend on different aspects of the geometry of the object—lengths vs areas, relation (e.g. 2, 55) have been unsuccessful. Whether a power law reflects a lengths vs volumes, or areas vs volumes—must change as size changes. For basic biological truth, the underlying structure of the universe we are em example, surface-to-volume ratios (relevant in diffusive transfer, heat loss, bedded in, or whether it is simply fairly robust at approximating a variety of and settling velocities) are proportional to L2/L2 = L~' and thus will change, data relations is yet to be determined. given a change in size alone. Most theoretical models which purport to For most of this discussion it will be assumed that a simple power law explain empirical scaling results are based on the maintenance of some function applies to most biological scaling problems. This assumption under functionally relevant variable constant over a range of size—often the predic lies much of the literature on scaling, but for some problems, a curvilinear or tion of a regular distortion of the organisms' geometry to counter the function bimodal scaling relation may be more appropriate (9; see Figure 1). In al changes which would occur under geometric similarity. Functional similar extreme cases, much more complex models are required (see (29) for an ity models underlie most of the scaling literature in ecology and physiology. exemplary model). When scaling relations vary either during ontogeny or
BODY SIZE 103 02 LABARBERA across the range of log M, and (d) log M is an independent variable, i.e. one erection, values for (he scaling coefflcienr (the antilog of the intercept of the whose values are known without error and are set by the investigator (see °ZSon line) will consistently be biased. The nse of nonlinear or Herat ve 105). ^.squares techniques to fi. data directly to the allometnc equationandU> Values of M are never determined without error, and in many scaling ,void log transformation of the data have been recommended (32 33, 122 studies these values are derived indirectly (from other scaling equations or 23 but such techniques have been challenged (69). The simplicity and secondary sources); such estimates are likely to have a large error term (see 'onvenience of regression techniques with log transformed data have made 22; p. 144). Attempts to circumvent this limitation of OLS regression by hem the method of choice for most workers. Correcting for bras introduced asserting that "body size is a primary determinant of numerous biological ,y a log transformation of the data is simple and routinely applied „ fo e ry characteristics" (53; p. 151) or by recasting the definition of an independent where power law relationships are used to esttmate productivity), bu variable as an "assumption in regression analysis of precision in the de rarely used in the scaling literature. The correction factor can be conveniently termination of the independent variable" (76; p. 37, and similar statements in s I cd from the standard error of log a (6, .0, 96, 108); this stabs tc ts 27, 82, 95) are invalid; neither causal relations or precision of measurement dy reported, however, so the correction usually cannot be performed po, are at issue. The values of M are not set by the investigator; they are hoc Since the magnitude of the correction factor depends on the variance of determined by the choice of data to be included in the analysis. Because OLS e log ran formed8 data and thus is unique to each study, published values for fits a line by minimizing the sum of squares of the residuals of the dependent baling coefficients must be viewed as suspect. The magnitude of this van- variable (log Y) only, the "best fit" line will not be an efficient estimator of the ^ce will depend on whether base . or base 10 logs were used; the nature ol functional relationship (56); the problem lies not in the technique, but in the the Iou transform should be routinely reported. fact that the underlying assumptions have been violated. Empirical analyses of the influence of body size on various aspects o In Model II regressions such as major axis (MA) or reduced major axis morphology, ecology, or behavior arc all ultimately dependent on a limited (RMA) regressions, on the other hand, both variables are assumed to have an number of satis.ical techniques. Although a broad variety o. sta tsl.cal mod associated error term; there is no "independent" variable and the line is fit by els and methods have been used in the past to clarify these rela tons, no. all minimizing the sum of products of residuals of both log Y and log M. MA and a. equally powerful and some (despite continued popularity) arc s imply RMA techniques make different assumptions about the error structure and K critique of Donhoflcr (28) is invalid (37, 90, 99>-,f appropriate variance relations of the variables. If a2,,, is the intrinsic variance of log M, a2 eg sion techniques are applied, the results are independent of the um.s is the intrinsic variance of log Y, a\ is the error variance of log M, and al is Involved. For simplicity, bivariale techniques and multivariate techniques are the error variance of log Y, then MA regression techniques assume that discussed separately. 0"« + 0"a-2 _= o-y + ai, 1.
B.VARIATE TECHNIQUES FOR THE ANALYSIS OF SCALING Although the while RMA techniques assume that biases introduced by log transformation of the raw data have been largely ignored in scaling studies, appropriate regression techniques to derive un a2/a2„ — a\la\ = a constant. biased estimates of log a and the scaling exponent have been the subject o intense debate. The problem is to identify the "true or functional 56) MA techniques arc sensitive to the absolute scale in which the variables are elation between the variables in the face of errors in the variables introduced measured and are not robust to rotation of the coordinate axes; RMA tech by statistical noise and measurement errors. The different regression models niques are insensitive to scale and rotation (89). OLS, MA, and RMA in statistics differ in their assumptions on the nature of the "tamfarp regressions are all special cases of the "general structural relation" (56, 71, between variables, their variance structure, and associated errors (63, 71, XJ). 89), itself a version of canonical variates analysis reduced to two dimensions. in the present context, Model I regressions such as ordinary least squares The choice of regression model to use in scaling studies is not a trivial linear (abbreviated OLS regression for the remainder of this discussion) consideration, for the various techniques yield different results. OLS regres regression techniques assume that: (a) the error term (log e) is normally sion consistently yields the lowest values for the scaling exponent, while distributed with a mean of zero and constant variance, (b) the distribution of either RMA or MA may yield the highest value depending on the variance log Y is normal at each value of log M, (c) the variance of log Y is constant t/-> o M
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T3 3 •8 "° 3 ^ Q - x S *q -5 CJ ra ^ 3 y u 3 £ 3 < CO 3 > ^ -a __, 28 3*^C2 5 CJ CJ '_) ^3 « O CJ ra .2 "3 CJ 3 cfl 3 ^~. ra cA ~ C J — U 3 or: 3-2 U3 _> CJ __ " t - i c r a cj > Cfl § cj -3 >n « o c « A . • — d■ Oh C J " 3 fi C J P 0 0 O 5 0 0 ^ 3C J - 81^ < * < . ra fi P, t- o -E — 2 .2 cfl " _T g2 fi 3 2 g O. _D.il C < o **> fe P sB CO "ra _- O -S *5* 3 > « I N CJ cu cj 3 ■ = v: tfl 106 LABARBERA BODY SIZE 107 cients were very similar to those derived from bivariate regression analyses now beyond question . . .," while Harper (cited in 72, p. 234) referred to it as (24, 44). If the true exponent relating one variable to size can be determined, "the first basic law demonstrated for ecology." then the PCA coefficients may be appropriately scaled (101). Without inclu The theoretical basis and evidence for the self-thinning rule have been sion of a size variable in the analysis or appropriate scaling after the fact, critically reviewed (115, 124). Methodological problems abound. As Weller however, the scaling exponents derived from PCA will equal the traditional (115) notes, M = kN~U2 is the correct formulation if regression analysis is to bivariate-derived exponents only by accident; extreme care should be used be applied to the problem (as it nearly universally was) since plotting log m vs when results are taken from the older literature. If a measure of overall size is log N is equivalent to plotting (total plot biomass/number of plants) against available, it should be included in the PCA analysis; care should be taken to (number of plants/total area). Since the number of plants appears in both exclude measures which are not size dependent, since they will grossly distort variables, statistical properties are grossly altered and correlation coefficients the analysis. Note that PCA assumes that the variables are linear and multi artificially inflated. OLS regressions were the norm in these analyses despite variate normal; PCA will yield undistortcd results only when all covariances the absence of any statistically independent variable, and no correction was are approximately equal (106). An alternative PCA based on the correlation made for the log transform despite the importance attached to empirical values rather than covariancc matrix is available (106). It is claimed to remove size of the thinning coefficient k in supporting theoretical expositions (e.g. 31). from multivariate data more efficiently than Jolicoeur's method but is not so For a discussion of a (distressing) number of further methodological difficul thoroughly explored. ties, see (115, 124). Analyzing a total of 488 data sets using PCA to establish the slopes of the EXAMPLES log transformed data, Weller (115) concludes that, although there is a clear relationship between stand biomass density and plant density, there is little The —312 Self-thinning "Law" in Plants support for a single rule describing this relationship in all plants. The ex Yoda et al (121) proposed that competition in even-aged stands of plants ponent relating M and N is a function of the plant species and degree of shade resulted in stunting and mortality such that the average plant mass (m) and tolerance, and different relations appear to apply to interspecific and in- density of the stand (number of plants per unit area, N) were related by the traspecific situations. The exponent is not a constant but rather a variable, one power law function m = A_V_3/2 or M = kN~1'2, where M equals the total describing specific aspects of the plants involved and the biological situation. plant mass per unit area (i.e. mN). The lines described by these functions were Using different methods, Zeide (124) comes to the same conclusion; "the law proposed to represent an absolute limit; as stands grew, the relation between is neither precise nor accurate" (124, p. 532). But self-thinning is an empirical density and biomass might follow any number of trajectories, but on reaching observation; should one conclude that the effects of competition among plants the limit defined by the above equations, all plant stands were proposed to are totally random? In a separate analysis, Weller (114) demonstrates that track along the defined slopes. The "—3/2 law" (also known as the self- 50% of the variation in thinning exponents can be explained by empirical thinning rule) was soon applied by plant ccologists and foresters to situations scaling relations of shape and biomass allocation for the individual species beyond the original formulation, including mixed plant communities (34), involved; a significant proportion of the remainder is presumably due to other ramets of clonal plants (85), marine macroalgae (25), and even phytoplankton biological characteristics of the plants such as shade tolerance, moisture in laboratory culture (1). The theoretical basis of the hypothesis as presented requirements, etc. Although plant ecology has apparently lost a "law," it by Yoda ct al (121) involved trade-offs between light interception and me would seem to have acquired a tool in the process-the thinning exponent has chanical support. This explanation was soon realized to be flawed, but the the potential of being a useful metric to compare plants in competitive self-thinning rule remained popular, and new theoretical developments based situations, and those biological characteristics that influence its value (growth on the theory of elastic similarity (73, 74) were proposed (31, 118). [For a full form, scaling relations, physiological characteristics) can be evaluated to historical development, see (115, 124); for a critique of elastic similarity, see determine their relative importance and to clarify the strategies that plants use M. LaBarbera, J. E. A. Bertram (unpublished ms)]. The fact that these in competition. The present situation, if less simple, is more satisfying. theoretical explanations could not apply to, e.g. marine macroalgae, was glossed over, as were counterexamples to some of the more extreme applica Cope's Law tions of the "law" (e.g. 94, 98). White (118, p. 479) stated that "The Cope's Law, the axiom that species within a lineage trend toward larger body empirical generality of the rule in its original formulation (Yoda et al 1963) is size with evolutionary time, has been generally accepted in paleontology and BODY SIZE 108 LABARBERA 109 den (67) studied were characterized by dwarfing rather than an increase in evolutionary biology (see e.g. 36, 79, 111). Most of the evidence in support of Cope's Law has, however, been anecdotal. Newell (79) outlined evidence body size, and those lineages that did increase in average size did not do so that Cope's Law is applicable to some groups of forams, corals, bryozoans, consistently through their history (Figure 2). Cope's Law in its strictest form does not hold, although size increase is pervasive. It is probably generally true cchinoderms, brachiopods, and molluscs, but claimed that size increase in that maximum size in a group tends to increase after the group's origin, but arthropods is rare and virtually nonexistent in most bryozoans, brachiopods, and graptolites. Newell (79) viewed Cope's Law as a potential source of until further studies are published that follow the history of body size in all branches of well-defined clades, the proper form in which Cope's Law should evolutionary novelty and nonadaptive trends (via allometry)—in modem be cast will remain unclear. jargon, that Cope's Law might drive macroevolutionary trends. Such an association between small size and morphological novelties in salamanders Scaling of Home Range and Population Density in Mammals has been recently documented (39). home range McNab (75) was the first to point out that there is a regular Stanley's (111) influential paper argued that Cope's Law is better viewed as scaling of home range size with adult body size in mammals. The scaling evolution from small body size rather than toward large body size; most exponent he calculated using OLS (b = 0.64) was not significantly different higher taxa arise from relatively small, unspecialized ancestors (see Table 1 in from the scaling exponent of metabolic rate (« M075), and McNab concluded 111), while their larger descendants tend to be more specialized (through that home range size was determined by metabolic rate. The RMA scaling scaling constraints). That adaptive breakthroughs occur primarily at small exponent that I calculate for McNab's (75) data (b = 0.74) is even closer to body sizes is confirmed by the observed Cambrian faunas; the first bivalves the scaling exponent for metabolic rate than McNab's original value, but these (58, 97) are less than a few millimeters in maximum dimension and the exponents differ strongly from those in subsequent studies (see Table I); the Tommotian-age shelly fauna (8) is characterized by small size. quality of the data McNab used must be considered suspect. Harestad & The generality of Cope's Law or its inverse, dwarfing (evolutionary de Bunnell (40) found that the home range of herbivores scaled as M102 (RMA: crease in body size), are subjects of contention; for a discussion, see (61). 1.18), the home range of omnivores scaled as M° 92 (RMA: 0.97), and the Hallam (38) documents persistent trends toward larger size in 41 species or home range of carnivores scaled as M136 (RMA: 1.51). Their in lineages of Jurassic bivalves and 19 of ammonites; 70% of these lineages terpretation—that home range increases more rapidly with size than does double in size, and some increase by as much as four times. Chaffee & metabolic rate, reflecting a declining ability of the habitat to support a Lindberg (18) present data for size of 149 molluscan taxa in the Cambrian; population as body size increases—is consistent with the corrected (RMA) maximum size certainly increases (from about 10 to over 50 mm), but it is unclear whether this represents an increase in size within clades of molluscs or 1 1 I i i 1 J- 1 1 simply an increase in the variance in size for molluscs as a group. - • • The best evidence offered to date to test Cope's Law is MacFadden's (67) - - D • - restudy of the evolution of horses, one of the "classic cases" of Cope's Law. x . Body sizes of fossils were estimated from OLS regressions of log body mass ?2- c • vs the log of v..ious osteological characters in the five living species of Eqiius 0 ■ _> I • (W.nin = 27.7 kg; pWR = 1.28). Unlike the case of self-thinning in plants, the 0 . • use of OLS regression in this study is perfectly appropriate; MacFadden's <_ > 1- • • . o • • a . • o goal was to predict the body size of fossil equids from dimensions of the • • • 0 •• o • - o preserved bones. One might question whether MacFadden's confounding of 1 • t intraspecific and interspecific allometry (use of multiple strains of Equus 40 60 40 20 caballus) was appropriate, but the question in this context is not clear cut. Million years Million years BP [Parenthetically, MacFadden (67) devotes considerable space ir. this paper to Figure 2 (left) Estimated body mass as a function of age for fossil horses. Note that size a discussion of the higher coefficients of variation for body mass than for increase does not occur at a constant rate and does not affect all lineages, (right) Rate of evolution linear dimensions; this is exactly what would be expected (62) given the of body size change in darwins (animals per square kilometer proportional to correction of the slope (RMA: - 1.55), nor does their estimate for neotropical M~° 75 (RMA: -0.87). Multiplying the scaling relationship for population mammals in general (b = -0.61; RMA: -0.91). Damuth (27) used a density times that for individual metabolic rate (<* M~"75 by his calculations), considerably larger data base (n = 467) to address the question of the scaling Damuth concluded that secondary production was independent of body size of population density in terrestrial mammals. His scaling exponent for herbi and that no mammalian herbivore species had an energetic advantage over any vores, -0.73, is very close to the -0.75 reported in Damuth (26); both values other on the basis of size alone. Unfortunately RMA yields a higher exponent convert to -0.87 using RMA. Damuth's (27) scaling exponent for population for the functional relation of popultion density to body mass, thus negating density of all mammals (b = -0.78; RMA: -0.98) differs markedly from Robinson & Redford (95), but these differences become insignificant using Table I Scaling ol" home range area with adult body size in terrestrial mammals. RMA to calculate the slopes. All of these studies are summarized in Table 2. The scaling exponent, its standard error (SE), the correlation coefficient (/), the Using an RMA regression model, the population density of terrestrial herbi number ol" species (N) involved, the minimum body mass (W„,;„). and log weight vores appears to be approximately proportional to M~i0, while carnivores range ratio (pWR) are given for each regression. show an apparently higher value, about M~lA. To a first approximation, the Exponent w„„„ population density of mammals, regardless of trophic mode, scales as M~l- . (OLS) (RMA) SE r N (kg) pWR Source Again, the agreement between published studies when the appropriate regres Herbivores 1.02 1.18 .11 .87 28 .016 4.41 40 sion model is applied is remarkable. Carnivores 1.36 1.51 .16 .90 20 .005 4.17 40 A number of papers have addressed the question of the relation between 1.12 1.34 .20 .84 15 .07 2.90 65" population density and body size in birds (see 51 for references), but correla Omnivores 0.92 0.97 .13 .95 7 .016 4.10 40 tion coefficients for log-log regressions have been so low (r2 = .03 to .27) as All 0.65 0.74 .07 .87 26 .005 4.86 75" to make the exercise of questionable ecological significance. 1.08 1.39 .12 .77 55 .005 4.96 40 1.42 1.65 .18 .So 23 .030 3.37 112 A large body of literature exists on the scaling of other ecological paramet ers. For example, Garland (30) discusses the ecological cost of transport, 'Recalculated from the data presented, averaging multiple entires for each species daily movement distances, daily food consumption, etc; Calder (15, 16) I? LABARBERA BODY SIZE 113
Table 2 Scaling of population density with adult body size in terrestrial mammals. as failing to correct for the bias due to a log transform and use of OLS (Conventions as in Table I.) regressions) that we know are inappropriate, and that future studies publish enough statistical information (or better yet, raw data) so that studies do not Exponent w„,in r Source automatically become irrelevant as better techniques are developed. I freely (OLS) (RMA) SE N (kg) pWR admit that 1 have sinned in this regard, but redemption takes only a little Herbivores -.75 -0.87 .026 -.86 307 0.005 2.43 2d thought and a few more keystrokes at the computer. -.88 -1.08 .035 -.81 108 0.01 5.40 83 1.22 Scaling studies paint nature with a very broad brush; they are more akin to -1.33 -1.55 .39 -.86 6 2.4 95 the gas laws of physics than to Newton's laws. They do not afford much -.73 -0.87 .024 -.84 368 0.005 5.76 27 Carnivores & -1.15 -1.32 .060 -.87 66 0.004 4.57 83 precision in their predictions for individuals or perhaps even species, but the Omnivores -.94 -1.11 .117 -.84 28 0.045 3.18 95a broad view—whether in space or time—is of interest in itself. Properly Carnivores -1.03 -1.14 .052 -.90 92 0.004 5.01 27b conducted, scaling studies can be one of our most powerful tools for achiev -.79 174 0.004 5.80 83 All -.86 -1.09 .032 ing a broad overview of ecosystems and evolution. -.61 -0.91 .07 -.67 103 0.015 4.30 95 -.78 -0.98 .027 -.80 467 0.004 5.83 27 Acknowledgments "Calculated from ihe pooled "insectivore-omnivore" and "carnivore" categories in (95) This work was supported in part by NSF grant BSR 84-06731. I thank John ''Calculated from the pooled "mammals: insect-eaters" and "mammals: vertebratc-ftcsh-eaters" categories in (27) Bertram, Rick Chappell, Andy Biewener, and Bob Full for useful dis cussions. discusses foraging time, efficiency, growth rates, litter size, and r and K Literature Cited selection; Peterson et al (84) discuss the allometry of population cycles. All of these works suffer from inappropriate regression models, addition or subtrac 1. Agusti, S., Duartc, C. M.. Kalff, J. foundland. Can. J. Earth Sci. 20:525- tion of empirical scaling exponents without knowledge of the covariance of 1987. Algal cell size and the maximum 36 density and biomass of phyloplankton. 9. Bertram, J. E. A. 1988. The biomechan the variables, or both. Swihart et al (112) measure the scaling of "time to Limnol. Oceanogr. 32:983-86 ics of bending and its implications for 2. Apple, M. S., Korostyshcvskiy, M. A. terrestrial support. PhD Thesis, Univ. independence" in home range use and relate this scaling to "physiological 1980. Why many biological parameters Chicago, 111. 167 pp. time" (reported as <* A/_l/4), but their calculated slopes are all significantly arc connected by power dependence. J. 10. Bird, D. P., Prairie, Y. T. 1985. Prac elevated when recalculated as RMA slopes; their conclusions, and similar Theor. Biol. 85:569-73 tical guidelines for the use of zooplank 3. Atchley. W. R. 1983. Some genetic ton length-weight regression equations. ones by Lindstedt ct al (65) and Reiss (91), arc thus highly questionable. aspects of morphometric variation. In J. Plankt. Res. 7:955-60 Numerical Taxonomy, cd. J. Felscn- 11. Boycott, A. E. 1919. On the sizes of stein, pp. 346-63. New York: Springer- things, or the importance of being rather CONCLUSIONS Vcrlag. 644 pp. small. In Contributions to Medical and 4. Atchley, W. R., Gaskins, C. T., An Biological Research Dedicated to Sir derson, D. 1976. Statistical properties of William Osier. Vol. 1, pp. 226-34. New Some may argue that I have deliberately chosen examples from the literature ratios. I. Empirical results. Syst. Zool. York: Hoeber. 649 pp. where statistical conventions were weak and the conclusions thus vulnerable. 25:137-48 12. Brower, J. C, Veinus, J. 1981. Allom 5. Baskcrville, G. L. 1970. Testing the etry in pterosaurs. Univ. Kansas Let me assure the reader that this is not true; I could have made these points Uniformity of Variance in Arithmatic Paleont. Contr. 105:1-32 and Logarithmic Units of a Y-Vari 13. Bryant, E. H. 1986. On use of loga just as well by reanalyzing data from the literature on scaling of production to able for Classes of an X-Vartable. Oak rithms to accommodate scale. Syst. biomass ratios, intrinsic rate of increase, reproductive effort, skeletal pro Ridge Nat. Lab. Publ. ORNL-IBP-70-1. Zool. 35:552-59 38 pp. 14. Burchfield, J. D.1975. Lord Kelvin and portions, or (in a different journal) metabolic rates. Some may find it dis 6. Baskerville, G. L. 1972. Use of loga the Age of the Earth. New York: Sci. couraging that many of the controversies in the literature are artifacts of bad rithmic regression in estimation of plant Hist. Publ. 260 pp. statistics. I find it encouraging that reanalysis indicates that there are indeed biomass. Can. J. For. Res. 2:49-53 15. Calder, W. A. III. 1982. The pace of 7. Beauchamp, J. J., Olson, J. S. 1973. growth: an allometric approach to com robust generalities; in some ways this reanalysis is the ultimate in double blind Correction for bias in regression es parative embryonic and post-embryonic studies, a rarity in any form in the ecological literature. timates after logarithmic transformation. growth. J. Zool.. Lond. 198:215-25 Ecology 54:1403-7 16. Calder. W. A. III. 1983. Ecological I do not argue for hegemony in statistical methods—I have no doubt that 8. Bcngtson, S., Fletcher, T. P. 1983. The scaling: mammals and birds. Annu. Rev. there are more robust techniques than I have recommended waiting for oldest sequence of skeletal fossils in the Ecol. Syst. 14:213-30 Lower Cambrian of southeastern New 17. Calder, W. A. III. 1984. Size, Function. someone to exploit them. I do urge that all involved abandon practices (such 114 LABARBERA BODY SIZE 115
and Life History. Cambridge, Mass: and standing crop in relation to density and scaling in primate biology with from "Eohippus" (Hyracotherium) to Harvard Univ. Press. 431 pp. in monospecific plant stands. Nature special reference to the locomotor Equus: scaling. Cope's Law, and the 18. Chaffee, C, Lindberg, D. R. 1986. Lar 279:148-50 skeleton. Yearbk. Phys. Anthropol. 27: evolution of body size. Paleobiology 12: val biology of early Cambrian molluscs: 35. Gould, S.J. 1966. Allometry and size in 73-97 355-69 the implications of small body size. ontogeny and phylogeny. Biol. Rev. 53. Jungers, W. L. 1984. Scaling of the 68. Madansky. A. 1959. The fitting ol Bull. Mar. Sci. 39:536-49 41:587-640 hominoid locomotor skeleton with straight lines when both variables arc 19. Chappell. R. 1989. Fitting bent lines to 36. Gould. S. J. 1982. Change in develop special reference to lesser apes. In The subject to error. Am. Slatis. Ass. J. data, with applications to allometry. J. mental timing as a mechanism of Lesser Apes: Evolutionary and Be 54:173-205 Theor. Biol. In press macroevolution. In Evolution and De havioral Biology, ed. H. Preuschoft, D. 69. Manaster, B. J., Manaster, S. 1975. 20. Chevcrud, J. M. 1982. Relationships velopment, cd J. T. Bonner, pp. 333- J. Chivcrs, W. Y. Brockelman, N. Techniques for estimating allometric among ontogenetic, static, and evolu 46. Berlin: Springer-Verlag. 356 pp. Creel, pp. 146-69. Edinburgh: Edin equations. J. Morphol. 147:299-308 tionary allometry. Am. J. Phvs. An- 37. Gunther, B., Morgado, E. 1987. Body burgh Univ. Press 70. Marshall, L. G., Corruccini, R. S. thropol. 59:139-49 size and metabolic rate: the role of units. 54. Jungers, W. L., German, R. Z. 1981. 1978. Variability, evolutionary rates, 21. Clarke, M. R. B. 1980. The reduced J. Theor. Biol. 128:397-8 Ontogenetic and interspecific skeletal and allometry in dwarfing lineages. major axis of a bivariate ' sample. 38. Hallam, A. 1975. Evolutionary size in allometry in nonhuman primates: bivari Paleobiology 4:101-19 Biomctrika 67:441-46 crease and longevity in Jurassic bivalves ate versus multivariate analysis. Am. J. 71. McArdle, B. H. 1988. The structural 22. Cock, A. G. 1966. Genetical aspects ol" and ammonites. Nature 258:493-96 Phys. Anthropol. 55:195-202 relationship: regression in biology. Can. metrical growth and form in animals. Q. 39. Hanken, J. 1985. Morphological novelty 55. Katz, M. J. 1980. Allometry formula: a J. Zool. 66:2329-39 Rev. Biol. 41:131-90 in the limb skeleton accompanies cellular model. Growth 44:89-96 72. Mcintosh, R. P. 1980. The background 23. Corruccini, R. S. 1978. Primate skeletal miniturization in salamanders. Science 56. Kendall, M. G., Stuart, A. 1979. The and some current problems of theoretical allometry and hominoid evolution. 229:871-74 Advanced Theory of Statistics. Vol. 2. ecology. Svnthese 43:195-255 Evolution 32:752-58 40. Harcstad, A. S., Bunnell, F. L. 1979. Inference and Relationships. London: 73. McMahon.T. A. 1973. Size and shape 24. Corruccini, R. S. 1983. Principal com Home range and body weight—a reeval- Griffin in biology. Science 179:1201-4 ponents for allometric analysis. Am. J. uation. Ecology 60:389-402 57. Kidwell, J. F., Chase, H. B. 1967. Fit 74. McMahon, T. A., Kronauer. R. E. Phvs. Anthropol. 60:451-53 41. Harvey, P. H. 1982. On rethinking al ting the allometric equation—a compari 1976. Tree structures: deducing the prin 25. Cousens, R., Hutchings. M. J. 1983. lometry. J. Theor. Biol. 95:37-41 son of ten methods by computer simula ciple of mechanical design. J. Theor. The relationship between density and 42. Heusncr, A. A. 1982. Energy metabo tion. Growth 31:165-79 Biol. 59:443-66 mean frond weight in monospecific sea lism and body size. I. Is the 0.75 mass 58. Krasilova, I. N. 1987. The oldest repre 75. McNab, B. K. 1963. Biocnergetics and weed stands. Nature 301:240-41 exponent of Klciber's equation a statis sentatives of the bivalve mollusks. the determination of home range size. 26. Damuth. J. 1981. Population density tical artifact? Rcsp. Physiol. 48:1-12 Paleontol. Jour. 1987:21-26 Am. Nat. 97:133-40 and body size in mammals. Nature 43. Heusncr, A. A. 1987. What does the 59. Ku. II. II. 1966. Notes on the use of 76. McNab, B. K. 1988. Complications in 290:699-700 power function reveal about structure propagation of error formulas. J. Res. herent in scaling the basal metabolic rate 27. Damuth, J. 1987. Interspecific allometry and function in animals of different size? Natl. Bureau Stand. C. Eng. lustrum. in mammals. Q. Rev. Biol. 63:25-54 of population density in mammals and Annu. Rev. Physiol. 49:121-33 70C263-73 77. Mosimann, J. E. 1970. Size allometry: other animals: the independence of body 44. Hills, M. 1982. Bivariate versus multi 60. Kuhry, B.. Marcus, L. F. 1977. Bivari size and shape variables with character mass and population energy-use. Biol. J. variate allometry: a note on a paper by ate linear models in biometry. Syst. izations of the lognormal and general Linn. Soc. 31:193-246 Jungers and German. Am. J. Phvs. An Zool. 26:201-9 ized gamma distributions. J. Am. Stat. 28. Donhoffer, S. 1986. Body size and thropol. 59:321-2 61. LaBarbera. M. 1986. The evolution and Assoc. 65:930-45 metabolic rate: exponent and coefficient 45. Hofman, M. A. 1988. Allometric scal ecology of body size. In Patterns and 78. Mosimann, J. E., James, F. C. 1979. of the allometric equation. The role of ing in palaeontology: a critical survey. Processes in the History of Life, ed. D. New statistical methods for allometry units. J. Theor. Biol. 119:125-37 Human Evol. 3:177-88 M. Raup., D. Jablonski, pp. 69-98. with application to Florida red-winged 29. Fccner, D. H., Lighton, J. R. B., Bar- 46. Hofman, M. A., Laan, A. C, Uy lings, Berlin: Springer-Vcrlag. 447 pp. blackbirds. Evolution 33:444-59 tholomew, G. A. 1988. Curvilinear al H. B. M. 1986. Bivariate linear models 62. Lande, R. 1977. On comparing coeffi 79. Newell, N. D. 1949. Phylctic size in lometry, energetics and foraging ecolo in neurobiology: problems of concept cients of variation. Svst. Zool. 26:214- crease, an important trend illustrated by gy: a comparison of leaf-culling ants and and methodology. J. Neurosci. Meth. 17 fossil invertebrates. Evolution 3:103- army ants. Fund. Ecol. 2:509-20 18:103-14 63. Laws. E. A., Archie, J. W. 1981. 24 30. Garland, T. Jr. 1983. Scaling the eco- 47 Huxley, J. S. 1932. Problems of Rela Appropriate use of regression analysis 80. Packard, G. C, Boardman, T. J. 1987. logical cost of transport to body mass in tive Growth. London: Methuen. 276 pp. in marine biology. Mar. Biol. 65:13- The misuse of ratios to scale physiolog terrestrial mammals. Am. Nat. 121:571- 48 Jensen, A. L. 1986. Functional regres 16 ical data that vary allomctrically with 87 sion and correlation analysis. Can. J. 64. Lcmcn, C. A, Freeman, P. W. 1984. body size. In New Directions in Ecologi 31. Givnish, T. J. 1986. Biomechanical Fish. Aquat. Sci. 43:1742-45 The genus: a macroevolutionary prob cal Physiology, cd. M. E. Feder, A. F. constraints on self-thinning in plant 49 Jolicoeur, P. 1963. The multivariate lem. Evolution 38:1219-37 Bennett, W. W. Burggren, R. B. Hucy, populations. J. Theor. Biol. 119:139- generalization of the allometry equation. 65. Lindstedt. S. L., Miller. B. J., Buskirk, pp. 216-39. Cambridge: Cambridge 46 Biometrics 19:497-99 S. W. 1986. Home range, time, and Univ. Press. 364 pp. 32. Glass, N. R. 1967. A technique for fit- 50 Jolicoeur, P.. Mosimann, J. E. 1968. body size in mammals. Ecology 67:413- 81. Packard, G. C, Boardman, T. J. 1988. ting models with nonlinear parameters to Intcrvallcs de confiancc pour la pente dc 18 The misuse of ratios, indices, and per biological data. Ecology 48:1010-13 Paxe majeur d'unc distribution normalc 66. Mace. G. M., Harvey, P. H., Clutton- centages in ecophysiological research. 33. Glass, N. R. 1969. Discussion of bidimensioncllc. Biomet.-Praximet. 9: Brock, T. H. 1983. Vertebrate home- Physiol. Zool. 61:1-9 calculation of power functions with 121-40 range size and energetic requirements. 82. Peters, R. H 1983. The Ecological Im special reference to respiratory mctabo- 51 Juanes, F. 1986. Population density and In The Ecology of Animal Movement, lism in fish. J. Fish. Res. Bd. Can. plications of Body Size. Cambridge: body size in birds. Am. Nat. 128:921- ed. I. R. Swingland, P. J. Greenwood, Cambridge Univ. Press. 329 pp. 26:2643-50 29 pp. 32-53. Oxford: Clarendon. 311 pp. 83. Peters, R. H., Raelson. J. V. 1984. Re 34. Gorham, E. 1979. Shoot height, weight 52 Jungers, W. L. 1984. Aspects of size 67. MacFadden, B. J 1986. Fossil horses lations between individual size and __ ■ *.«' 8 ».2 a u. CQ £ o S .-g.£BO 8'£ £ *3 .__• g ^H.**:- o t-.o u
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