ab

Ecology, 88(2), 2007, pp. 315–323 � 2007 by the Ecological Society of America ALLOMETRIC EXPONENTS DO NOT SUPPORT A UNIVERSAL METABOLIC ALLOMETRY CRAIG R. WHITE,1,3 PHILLIP CASSEY,1,2 AND TIM M. BLACKBURN1 1School of Biosciences, The University of Birmingham, Edgbaston, Birmingham, B15 2TT UK 2Department of Ecology, Evolution, and Natural Resources, 14 College Farm Road, Rutgers University, New Brunswick, New Jersey 08901 USA Abstract. The debate about the value of the allometric scaling exponent (b) relating metabolic rate to body mass (metabolic rate¼ a3massb) is ongoing, with published evidence both for and against a 3/4-power scaling law continuing to accumulate. However, this debate often revolves around a dichotomous distinction between the 3/4-power exponent predicted by recent models of nutrient distribution networks and a 2/3 exponent predicted by Euclidean surface-area-to-volume considerations. Such an approach does not allow for the possibility that there is no single ‘‘true’’ exponent. In the present study, we conduct a meta-analysis of 127 interspecific allometric exponents to determine whether there is a universal metabolic allometry or if there are systematic differences between taxa or between metabolic states. This analysis shows that the effect size of mass on metabolic rate is significantly heterogeneous and that, on average, the effect of mass on metabolic rate is stronger for endotherms than for ectotherms. Significant differences between scaling exponents were also identified between ectotherms and endotherms, as well as between metabolic states (e.g., rest, field, and exercise), a result that applies to b values estimated by ordinary least squares, reduced major axis, and phylogenetically correct regression models. The lack of support for a single exponent model suggests that there is no universal metabolic allometry and represents a significant challenge to any model that predicts only a single value of b. Key words: allometry; metabolic rate; quarter-power; scaling. INTRODUCTION Debate about the value of the allometric scaling exponent (b) relating metabolic rate to body mass (MR ¼ a 3 Mb) has recently been stimulated by the publication of a number of competing models attempt- ing to explain the widely held observation that biological rates and times scale with M raised to multiples of 1/4 (West et al. 1997, 1999, Banavar et al. 1999, 2002). This has generated an increasingly acrimonious debate within which the competing models have been intensely scrutinized (Dodds et al. 2001, Agutter and Wheatley 2004, Kozlowski and Konarzewski 2004, 2005, Brown et al. 2005, Suarez and Darveau 2005, Weibel and Hoppeler 2005, West and Brown 2005), but has also led to close scrutiny of the empirical support for quarter- power scaling (Riisga˚rd 1998, Dodds et al. 2001, White and Seymour 2003, Bokma 2004, Savage et al. 2004, Farrell-Gray and Gotelli 2005, Glazier 2005). In this latter regard, four meta-analyses of studies of the allometric scaling of metabolic rate have recently been conducted, with conflicting results. Dodds et al. (2001) reanalyzed bird and mammal basal metabolic rate (BMR) data sets published by Heusner (1991), Bennett and Harvey (1987), Bartels (1982), Hemmingsen (1960), Brody (1945), and Kleiber (1932) and found little evidence for rejecting b ¼ 2/3 in favor of b¼ 3/4. Savage et al. (2004) combined the BMR data sets of Heusner (1991), Lovegrove (2000), and White and Seymour (2003) and, using a ‘‘binning’’ approach designed to account for nonuniform represen- tation of species within different body size classes, examined the scaling of BMR, field metabolic rate (FMR), and exercise-induced maximum metabolic rate (MMRex). They concluded that BMR and FMR scaled with b ¼ 3/4, while MMRex scaled with b . 3/4. The finding of non-3/4 scaling for MMRex was suggested to be explained by selection of species or methodological differences in addition to small sample size, but they nevertheless noted that MMRex clearly does not scale as M2/3 (Savage et al. 2004). Farrell-Gray and Gotelli (2005) used a likelihood analysis approach to compare b¼ 3/4 and b¼ 2/3 for 22 published BMR and standard metabolic rate (SMR) exponents for birds, mammals, reptiles, and insects. Likelihood ratios quantifying the relative probability of b¼ 3/4 compared to b¼ 2/3 were 16 074 for all species, 105 for mammals, 7.08 for birds, and 2.20 for reptiles (Farrell-Gray and Gotelli 2005). Farrell-Gray and Gotelli (2005) concluded that their analyses supported the idea of a universal metabolic exponent for endotherms, but not for ectotherms. Finally, and most recently, Glazier (2005) conducted Manuscript received 29 November 2005; revised 19 April 2006; accepted 27 April 2006; final version received 13 June 2006. Corresponding Editor: T. D. Williams. 3 E-mail: c.r.white@bham.ac.uk 315 an extensive descriptive review of intra- and interspecific scaling exponents and concluded that the ‘‘3/4-power scaling law’’ of metabolic rate is not universal. While the meta-analytical approach represents a significant advance in the debate on metabolic scaling, each of these studies has limitations. For example, Dodds et al. (2001) convincingly argue against b ¼ 3/4, but only for BMR. Savage et al. (2004) provide strong support for general quarter-power scaling, but only two of their three metabolic scaling exponents are not significantly different from b ¼ 3/4. Glazier’s (2005) analysis is the most extensive compilation of scaling exponents yet undertaken and strongly argues against a universal 3/4 exponent, but is largely descriptive. Farrell-Gray and Gotelli (2005), on the other hand, use a meta-analytical approach and find strong support for b¼ 3/4, but only for BMR, only for endotherms, and only including a small and fortuitous selection of exponents from the literature. For example, Farrell- Gray and Gotelli (2005) use BMR scaling exponents of 0.723–0.734 for birds (Lasiewski and Dawson 1967, Aschoff and Pohl 1970), values derived from separate regressions for passerines and non-passerines. While clade-specific regressions may be justified (Garland and Ives 2000), BMR or RMR exponents of 0.67, 0.677, and 0.68 have since been reported (Bennett and Harvey 1987, Tieleman and Williams 2000, Frappell et al. 2001) but were not included in Farrell-Gray and Gotelli’s (2005) analysis. Most recently, McKechnie and Wolf (2004) rigorously reviewed the published BMR data available for birds and found that only 67 of 248 measurements from an earlier analysis unambiguously met the criteria for BMR, which are strictly defined (McNab 1997, Frappell and Butler 2004). The regression for these rigorously selected bird BMR data had an exponent of 0.677, but the scaling exponents for captive-raised (b ¼ 0.670) and wild-caught birds (b ¼ 0.744) have subse- quently been shown to be different (McKechnie et al. 2006). This suggests that Farrell-Gray and Gotelli’s (2005: 2083) statement that ‘‘allometric exponents sup- port a 3/4-power scaling law’’ is premature. McKechnie and Wolf’s (2004) emphasis on the importance of data selection criteria was echoed by White and Seymour (2005), who reported that the BMR scaling exponent for mammals was positively correlated with the proportion of large herbivores within a data set. Given that measurement of BMR requires that the animals tested are in a postabsorptive state and such a state is difficult or impossible to achieve in at least ruminants (McNab 1997), the decision to include such species in BMR data sets must be made carefully, because non-BMR measurements will tend to increase the scaling exponent (White and Seymour 2005). Thus, excluding lineages for which basal conditions are unlikely to be met produces an exponent that is close to 2/3 (White and Seymour 2003), while including all species for which data are available produces an exponent close to 3/4 (Savage et al. 2004). The similar exponents obtained for rigorously selected bird and mammal data sets suggests that the BMR of endotherms does not scale with an exponent of 3/4 and argues against this as a universal exponent. Similarly, the standard metabolic rate (SMR) scaling of ecto- therms also fails to support the idea of a universal 3/4 exponent (Farrell-Gray and Gotelli 2005, White et al. 2006), as does the scaling of MMRex (Weibel et al. 2004, Bishop 2005, Weibel and Hoppeler 2005). Indeed, the variation in scaling exponents between different meta- bolic levels (e.g., White and Seymour 2005) has led to the development of models that allow for scaling exponent heterogeneity (Darveau et al. 2002, Hochach- ka et al. 2003, Kozlowski et al. 2003). However, much of the debate about the scaling of metabolic rate does not allow for such heterogeneity and presupposes that there is a single ‘‘true’’ allometric exponent and that it is either 2/3 or 3/4. Such a dichotomous distinction excludes the possibility that b is neither 2/3 nor 3/4 and the possibility that b is consistently different between, for example, different taxa and metabolic states. Ongoing attachment to a single-exponent paradigm, without favorable support, potentially represents a substantial barrier to understanding the causes of the non-isometric scaling of metabolic rate. In this analysis, we examine 127 published allometric scaling exponents using a meta-analytical approach (e.g., Osenberg et al. 1999, Gurevitch et al. 2001, Gates 2002). We aim to advance the debate over the form of published allometric scaling relationships by applying rigorous quantitative meta-analytical methodology to as comprehensive a set of such relationships as possible, thus addressing the criticisms, laid out above, of previous such analyses. Typically, the main objective of an ecological meta-analysis is to summarize estimates of the standardized magnitude of a response (i.e., the ‘‘effect size’’) relative to a given correlation or manip- ulation variable. However, the objective of most studies that examine the scaling of metabolic rate is not to estimate the strength of the relationship between log(mass) and log(metabolic rate), but to estimate the slope of the relationship between these two variables. Nevertheless, if the influence of mass on metabolism is indeed universal, both the slope and strength of the relationship might reasonably be predicted to be similar between taxonomically diverse groups and between metabolic states. Thus, we examine the 127 published slopes to determine whether a single scaling exponent and a single effect size characterize the relationship between mass and metabolism. MATERIALS AND METHODS Allometric exponents for 127 data sets relating metabolic rate to body mass were compiled from the literature (see Appendix). Exponents estimated by ordinary least-squares (OLS) regression were available for all 127 data sets; exponents estimated by reduced major axis (RMA) regression were available or could be CRAIG R. WHITE ET AL.316 Ecology, Vol. 88, No. 2 calculated from OLS exponents and r or r 2 values for 103 data sets. Phylogenetically correct (PC) regressions calculated by the method of independent contrasts (Felsenstein 1985) or phylogenetic generalized least squares (Grafen 1989, Martins and Hansen 1997, Garland and Ives 2000) were available for 30 data sets. Exponents were assigned to three sets of categorical variables based on taxonomy (amphibians, arthropods, birds, fish, mammals, reptiles, unicells), thermoregula- tion (endothermic, ectothermic), and metabolic state (daily [mean daily metabolic rate], exercise [flight metabolic rate or exercise-induced maximum metabolic rate], field [field metabolic rate], rest [basal or resting metabolic rate], thermogenic [cold-induced maximum metabolic rate]). Because the Central Limit Theorem applies to the allometric exponent b when calculated through regres- sion, its sampling distribution is normal with an expected value (mean) B and the standard deviation of its sampling distribution equal to the square root of the residual mean square divided by the sum of squares of X (Quinn and Keogh 2002). We analyzed the relationship between allometric exponents (OLS, RMA, PC) and three class variables (taxonomy, thermoregulation, metabolic state) using a weighted generalized linear mixed model (GLMM) in SAS version 8.0 (Proc MIXED; SAS Institute, Cary, North Carolina, USA). This model accounted for both weighting the observa- tions according to their heterogeneous variance and the possible nonindependence of studies that have calculat- ed exponents from shared data sets. Each OLS exponent was weighted by the reciprocal of an estimate of its variance (where available), whereas RMA and PC exponents were weighted by their number of observa- tions (sample size, N ). Standardized effect sizes were calculated from the Pearson correlation coefficients (r) estimated from OLS regression. This metric has been widely used in studies synthesizing correlational and linear relationships in ecology and evolution (Møller and Jennions 2002) and is the best-known index based on the variance accounted for owing to the introduction of an explanatory variable (Hedges and Olkin 1985). We used a random effects model to estimate the variance of the population of correlations and tested whether this variance differs from zero, using the large sample test for homogeneity of correlations given by Hedges and Olkin (1985). Following Hedges and Olkin (1985), we used a categorical model fitting procedure coded in SAS version 8.0 to assess the effect of particular predictor classes (taxonomy, thermoregulation, metabolic state) in influencing variability in effect sizes with respect to the relationship between body mass and metabolism. Any one class had to contain at least two body mass– metabolism relationship effect sizes to be included in the analysis. First, we tested the homogeneity of effect sizes across each of the three classes separately. The test of the hypothesis that the mean effect size does not differ across classes is analogous to the F test in an analysis of variance to test that class means are the same. The test is based on a between-class goodness-of-fit statistic. If this test led to the conclusion that effect sizes were not homogenous across classes (e.g., if it suggested that effect sizes differed for ectotherms and endotherms), then we compared the mean effect sizes of different classes by means of multiple (orthogonal) linear contrasts. The Scheffe´ procedure (Scheffe´ 1953, 1959) gives a simultaneous significance level a for all l contrasts. Second, we tested the model specification that effect sizes are homogenous within classes by manually partitioning the classes across each of the classification dimensions (in a stepwise manner) to yield finer and finer groupings. The order in which classification dimensions were chosen was based on which class (when compared with the other remaining classes) explained the most effect size homogeneity within classes for each subsequent partition. RESULTS Plots of both bOLS and effect size against sample size (Fig. 1A) were typically ‘‘funnel’’-shaped and showed convergence with increasing sample size, suggesting that analysis of b should be weighted by a measure of variance. In both cases, ectotherms and endotherms appeared to converge on different ‘‘true’’ means. Indeed, b values estimated by OLS, RMA, and PC regression are significantly different between ectotherms and endo- therms (Table 1), whether or not analysis is weighted by sample size and whether or not the weighted analysis includes a random effect for independence of study. The b values estimated from OLS and RMA regression support neither 0.67 nor 0.75 as a general scaling exponent (Table 1). In all models, the ectotherm b is significantly greater than 0.75, while the endotherm b lies between 0.67 and 0.75. Only for the weighted mean values from OLS and RMA regression calculated with the random effect model does the endotherm b not differ from 0.75, but this b value also does not differ from 0.67 for OLS regression. In contrast, b values calculated using phylogenetically controlled regression are gener- ally consistent with the theoretical value of 0.67 for endotherms and 0.75 for ectotherms. For all models, mean b values from phylogenetically controlled regres- sion differ significantly from 0.75 but not 0.67 for endotherms (Table 1). The equivalent mean b values for ectotherms differ significantly from 0.67 but not 0.75 for the weighted model and for the weighted random effect model (Table 1). Ectotherms and endotherms continue to show signif- icant differences between b values when different metabolic states are distinguished: examples for OLS regression are shown in Fig. 2. Modelling OLS b values in terms of metabolic state, thermoregulation mode, and taxon, with a random effect for independence of study, yields a minimum adequate model with metabolic state, February 2007 317EVIDENCE AGAINST UNIVERSAL ALLOMETRY thermoregulation mode, and the interaction between these two variables as predictors (Table 2). Thus, b values differ significantly between ectotherms and endotherms and between different metabolic states, and these values also vary differently for metabolic states between ectotherms and endotherms (e.g., Fig. 2). Similar results pertain to b values from RMA regression (Table 2), although slightly higher b values are obtained. The minimum adequate model of PC b values includes only thermoregulatory type as a predictor: ectotherms have higher b values than endotherms, with the latter consistent with an exponent of 2/3 and the former consistent with an exponent of 3/4 (Table 2). Whether ectotherms and endotherms also converge on different ‘‘true’’ averages for effect size is less obvious from the plot of effect size vs. sample size (Fig. 1B). In fact, the overall model for effect sizes (final line in Table 3) shows considerable heterogeneity. To attempt to explain this heterogeneity, we used stepwise manual partitioning to assign relationships to finer and finer groupings by metabolic state, thermoregulation mode, and taxon. The results of this procedure are given in Table 3. Effect sizes for field metabolic rate are not significantly heterogeneous between birds, mammals, and reptiles. All other metabolic states show effect size heterogeneity. Some of this heterogeneity is removed when studies are further partitioned by thermoregula- tion mode and taxon. For example, effect sizes are homogenous in studies of mammalian exercise metabol- ic rate. However, in several cases this partitioning fails to remove heterogeneity in effect sizes. Notably, effect sizes are heterogeneous for studies of resting metabolic rate in mammals, birds, reptiles, and arthropods (Table 3). DISCUSSION Our analysis of 127 allometric relationships for birds, mammals, fish, reptiles, amphibians, arthropods, and unicells at metabolic states ranging from standard/basal to maximum aerobic reveals significant heterogeneity in both effect size and scaling exponent. Our results account for heterogeneity in variance by weighting estimates accordingly and also account for the nonin- dependence of studies that have calculated exponents from shared data sets. Thus, the strength of the influence of mass on metabolic rate is not universal, and no single value of b adequately characterizes variation in the metabolic scaling of animals. Acceptance of any single exponent as the true exponent relating metabolic rate to body mass will therefore obscure significant and potentially important variability and is unlikely to provide a robust foundation for models explaining the allometric scaling of metabolism. It i