Ecology, 88(2), 2007, pp. 315–323
Ó 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
2
AND
TIM M. BLACKBURN1
1
School of Biosciences, The University of Birmingham, Edgbaston, Birmingham, B15 2TT UK
Department 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 rate ¼ a 3 massb) 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 attempting 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 quarterpower scaling (Riisgård 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
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: [email protected]
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 b ¼ 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 representation 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
b ¼ 3/4 and b ¼ 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 b ¼ 3/4 compared to b ¼ 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
315
316
CRAIG R. WHITE ET AL.
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 b ¼ 3/4, but only for BMR, only for endotherms, and
only including a small and fortuitous selection of
exponents from the literature. For example, FarrellGray 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 subsequently been shown to be different (McKechnie et al.
2006). This suggests that Farrell-Gray and Gotelli’s
(2005: 2083) statement that ‘‘allometric exponents support 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).
Ecology, Vol. 88, No. 2
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 ectotherms 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 metabolic levels (e.g., White and Seymour 2005) has led to
the development of models that allow for scaling
exponent heterogeneity (Darveau et al. 2002, Hochachka 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 manipulation 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
February 2007
EVIDENCE AGAINST UNIVERSAL ALLOMETRY
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), thermoregulation (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 regression, 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 observations according to their heterogeneous variance and the
possible nonindependence of studies that have calculated 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 observations (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
317
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 Scheffé procedure (Scheffé 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 endotherms (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 generally consistent with the theoretical value of 0.67 for
endotherms and 0.75 for ectotherms. For all models,
mean b values from phylogenetically controlled regression 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 significant 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,
318
CRAIG R. WHITE ET AL.
Ecology, Vol. 88, No. 2
FIG. 1. Bivariate funnel plots of the relationship between (A) ordinary least-squares regression (OLS) slope value (allometric
exponent, or b) and sample size, and (B) standardized effect size (Zi) and sample size (n) for estimates of metabolic allometry
calculated from OLS regression for endotherms and ectotherms.
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 thermoregulation 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 nonindependence 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 is tempting to consider our findings with regard to
the debate surrounding the mechanistic basis of
metabolic scaling (Dodds et al. 2001, Agutter and
Wheatley 2004, Kozlowski and Konarzewski 2004,
Brown et al. 2005, Suarez and Darveau 2005, Weibel
and Hoppeler 2005, West and Brown 2005). However,
February 2007
EVIDENCE AGAINST UNIVERSAL ALLOMETRY
319
TABLE 1. Mean allometric exponents (slope estimates) b calculated separately for ectotherms and
endotherms, for different regression methods: ordinary least squares (OLS), reduced major axis
(RMA), and phylogenetically correct (PC).
Method
OLS slope estimates
Observed means
Ectothermic
Endothermic
Weighted means
Ectothermic
Endothermic
Weighted means with random effect
Ectothermic
Endothermic
RMA slope estimates
Observed means
Ectothermic
Endothermic
Weighted means
Ectothermic
Endothermic
Weighted means with random effect
Ectothermic
Endothermic
PC slope estimates
Observed means
Ectothermic
Endothermic
Weighted means
Ectothermic
Endothermic
Weighted means with random effect
Ectothermic
Endothermic
Estimate, b
95% CL
Difference
SE
0.804
0.704
0.77, 0.83
0.68, 0.72
0.100
0.02
6.11***
0.809
0.694
0.78, 0.83
0.68, 0.71
0.114
0.01
7.53***
0.800 0.714 ,à
0.74, 0.86
0.66, 0.77
0.085
0.03
2.50*
0.843
0.722
0.81, 0.87
0.70, 0.74
0.120
0.02
6.27***
0.866
0.711
0.84, 0.89
0.70, 0.72
0.154
0.01
10.36***
0.860
0.715 ,à
0.81, 0.91
0.67, 0.76
0.144
0.03
4.93***
0.842
0.687à
0.77, 0.91
0.66, 0.71
0.154
0.04
4.32***
0.834 0.683à
0.72, 0.95
0.67, 0.70
0.151
0.06
2.60*
0.837 0.670à
0.73, 0.94
0.62, 0.72
0.167
0.06
2.97**
t
Notes: Means were calculated from raw b values (observed means), from b values weighted by an
estimate of their variance (weighted means), and from b values weighted by an estimate of their
variance and including a random effect for study independence (weighted means with random effects).
* P , 0.05; ** P , 0.01; *** P , 0.001.
Least-square mean 95% confidence limits of exponent encompass 0.75.
à Least-square mean 95% confidence limits of exponent encompass 0.67.
while our results are most congruent with those models
that accommodate scaling exponent heterogeneity (Darveau et al. 2002, Hochachka et al. 2003, Kozlowski et al.
2003), these models do not provide testable predictions
for all taxa and metabolic states considered here.
Additionally, the fractal geometry model predicts
limited deviations from b ¼ 3/4 at small masses (West
et al. 1997), and the supply–demand balance model
FIG. 2. Ordinary least-squares regression (OLS) slope estimates (scaling exponent or b values; means 6 SE) for different metabolic
states in endotherms and ectotherms. There are no scaling exponent estimates for thermogenic or daily metabolic rates for ectotherms.
320
CRAIG R. WHITE ET AL.
Ecology, Vol. 88, No. 2
TABLE 2. Final (reduced or minimum adequate) models for ordinary least-squares regression
(OLS), reduced major axis regression (RMA), and phylogenetically correct regression (PC)
estimates of b in terms of metabolic state, thermoregulation mode, and interaction.
Fixed effect
least-square means
Thermoregulation mode or metabolic state
OLS reduced model
Thermoregulation
Ectothermic
Endothermic
Metabolic state
Daily
Exercise
Field
Resting
Thermogenesis
Thermoregulation 3 metabolic state interaction
RMA reduced model
Thermoregulation
Ectothermic
Endothermic
Metabolic state
Daily
Exercise
Field
Resting
Thermogenesis
Thermoregulation 3 metabolic state interaction
PC reduced model
Thermoregulation
Ectothermic
Endothermic
Type III tests
of fixed effects
Estimate
95% CL
df
0.775
0.690
0.72, 0.83
0.66, 0.72
1, 102
12.78***
0.601
0.871
0.776
0.741
0.676
0.51,
0.81,
0.72,
0.71,
0.58,
4, 102
69.65***
2, 102
9.56***
0.69
0.93
0.83
0.78
0.77
F
0.846
0.711
0.79, 0.90
0.67, 0.75
1, 86
12.80***
0.648
0.889
0.835
0.777
0.745
0.54,
0.81,
0.77,
0.73,
0.64,
4, 86
5.08**
2, 86
6.44*
1, 20
8.83**
0.837
0.670
0.75
0.97
0.90
0.82
0.85
0.73, 0.94
0.62, 0.72
* P , 0.05; ** P , 0.01; *** P , 0.001.
(Banavar et al. 2002) accommodates values of b between
2/3 and 3/4. Future expansion of these models may yet
account for the heterogeneity described here. Thus, it is
not clear whether scaling exponent heterogeneity repre-
sents a falsification of the ‘‘single exponent’’ models,
suggests that refinement of these models is necessary to
accommodate the documented heterogeneity, or merely
indicates that these models sacrifice some detail for the
TABLE 3. Effect size statistics for different nested metabolic states, thermoregulatory modes, and taxa.
State
Thermoregulation
Taxon
n
Zþ
95% CL
Exercise
Daily
Rest
Thermogenic
Field
Rest
Rest
Exercise
Rest
Thermogenic
Exercise
Rest
Rest
Rest
Total
endothermic
endothermic
endothermic
endothermic
endothermic, ectothermic
ectothermic
endothermic
ectothermic
ectothermic
endothermic
endothermic
ectothermic
ectothermic
ectothermic
mammals
birds
birds
birds
birds, mammals, reptiles
reptiles
mammals
arthropods
arthropods
mammals
birds
mammals
amphibians
unicells
4
4
12
2
11
10
32
4
10
2
2
2
2
4
110
2.59
2.45
2.26
2.21
2.16
2.15
2.10
1.86
1.85
1.85
1.76
1.76
1.70
1.51
2.08
2.31,
2.31,
2.19,
1.92,
2.07,
2.04,
2.07,
1.71,
1.74,
1.67,
1.44,
1.54,
1.55,
1.38,
2.06,
2.87
2.58
2.33
2.50
2.25
2.26
2.13
2.01
1.96
2.03
2.08
1.97
1.85
1.65
2.11
v2
4.76
22.50***
34.74***
8.98**
11.44
31.43***
586.67***
19.59***
31.09***
1.09
2.56
0.58
1.14
4.75
1000.32***
Notes: We tested the model specification that effect sizes are homogenous within classes by manually partitioning the classes across
each of the nested classification dimensions in a stepwise manner to yield finer and finer groupings. The order in which classification
dimensions were chosen was based on which hypothesis (when compared with the other remaining hypotheses) explained the most
effect-size homogeneity within classes for each subsequent partition. In each case we tested whether significant heterogeneity exists
within classes (v2). Classes that are italicized are those for which partitioning does not significantly resolve their class heterogeneity.
Zþ is the mean effect size, which is always significantly different from zero; n is the number of relationships included in each class.
* P , 0.05; ** P , 0.01; *** P , 0.001.
February 2007
EVIDENCE AGAINST UNIVERSAL ALLOMETRY
sake of generality. The last possibility is not necessarily a
shortcoming of these models, as any allometric relationship potentially obscures important variation between
species (e.g., BMR variation associated with environmental variables; Lovegrove 2003, Rezende et al. 2004),
and the mechanistic models that attempt to explain these
relationships can be no different. However, while the
sacrifice of some detail seems an inevitable consequence
of attempting to explain the allometric scaling of
metabolism, in our view, the significant heterogeneity
identified here represents a challenge to any model that
predicts only a single value of b.
Conclusions about effect size and scaling exponent
heterogeneity are unaffected by the statistical method
used to quantify the allometry of metabolic rate, as
similar patterns are observed for OLS, RMA, and PC
regression (Table 1). However, the precise values of the
exponents do differ across regression approaches
(Table 1), and any given value of b reflects idiosyncrasies
of the analytical methods. Ordinary least squares
regression assumes that the independent variable is
measured without error, and since this assumption is
violated by body mass, it may be argued that RMA is
the correct technique to use to quantify the allometry of
metabolic rate. Reduced major axis regression assumes
that error variances in the dependent and independent
variables are equal to their true variances. However,
McArdle (1988) suggests that OLS is the better
technique to use as long as the error variance in the
independent variable is ,1/3 of that in the dependent
variable. Since to our knowledge the error variances in
metabolic rate and body mass have never been assessed
in any comparative study, the question of which of these
techniques may be most justified in these terms remains
open. Nevertheless, it is a question that is likely to be
moot. Both OLS and RMA assume that data are
independent, and this assumption is likely to be violated
in any comparative study of allometry: closely related
species are likely to have similar body masses and
metabolic rates and so do not provide independent
information when estimating the relationship between
these two variables. Indeed, phylogenetic correlation in
relationships between metabolic rate and body mass
seems to be the norm (Elgar and Harvey 1987,
Freckleton et al. 2002, Blomberg et al. 2003, but see
McKechnie et al. 2006). This violates the assumption of
independence and can lead to biased estimates of
regression coefficients (allometric exponents) and underestimated standard errors (for a discussion in the
context of physiology, see Halsey et al. [2006]).
Assuming that metabolic rate is an evolved characteristic, and phylogenetic correlation is thus likely to be
widespread in such data, studies of metabolic allometry
require a phylogenetic perspective.
Support for a multiple exponent model raises the
important question of how properly to account for body
mass effects in broad interspecific analyses. The results
of our analysis suggest that dogmatic acceptance of b ¼
321
2/3, b ¼ 3/4, or any other exponent, is likely to
undermine such studies, but the extent of the problem
will depend on the question, which will determine the
metabolic state and the taxa involved. For example,
examination of patterns of FMR variation in mammals
might reasonably make use of a 3/4 exponent, because
mammalian FMR scales with an exponent not significantly different from 0.75 (Nagy et al. 1999, Savage et al.
2004, Nagy 2005). However, avian BMR, FMR, flight
MR, MMRex, and cold-induced MMR all scale with
exponents significantly different from 3/4 (Bennett and
Harvey 1987, Norberg 1996, Nagy et al. 1999, Tieleman
and Williams 2000, Frappell et al. 2001, Rezende et al.
2002, McKechnie and Wolf 2004, Anderson and Jetz
2005, Bishop 2005, Nagy 2005). Studies of relationships
between energy expenditure and ecological variables for
birds are therefore likely to be compromised by
assumptions of 3/4-power metabolic rate scaling.
Regardless of the applicability of a given exponent, a
better approach to accounting for body mass effects is,
where possible, to include body mass in an ANCOVA
model (e.g., Packard and Boardman 1988, 1999).
Inclusion of body mass effects by dividing metabolic
rate by body mass, dividing by body massb, or making
use of body mass residuals introduces a number of
potential problems that are not evident when an
ANCOVA approach is taken (Atchley and Woodruff
1976, Packard and Boardman 1988, 1999, Albrecht et al.
1993, Hayes and Schonkwiler 1996, Berges 1997,
Garcı́a-Berthou 2001, Hayes 2001, Brett 2004). An
ANCOVA approach also has the advantage of obviating
the need to assume a value for b, while phylogenetic
information can readily be incorporated using modern
statistical packages (see Halsey et al. 2006).
In concluding their meta-analysis, Savage et al. (2004)
suggested that a century of science was distorted by
trying to fit observations to an unsatisfactory surface
law (b ¼ 2/3). Given the apparent widespread acceptance
and application of b ¼ 3/4, it seems history is in danger
of repeating. Our analysis of 127 exponents suggests that
there is no single true allometric exponent relating
metabolic rate to body mass and no universal metabolic
allometry.
ACKNOWLEDGMENTS
We thank Andrew Clarke and two anonymous referees for
their constructive comments about an earlier version of this
manuscript, Jim Brown and Charles Darveau for helpful
discussions, and the many colleagues who have worked with
us on scaling, especially Roger Seymour. C. R. White was
supported by Natural Environment Research Council grant
NER/A/2003/00542 to G. R. Martin, P. J. Butler, and A. J.
Woakes.
LITERATURE CITED
Agutter, P. S., and D. N. Wheatley. 2004. Metabolic scaling:
consensus or controversy? Theoretical Biology and Medical
Modelling 1. hhttp://www.tbiomed.com/content/1/1/13i
Albrecht, G. H., B. R. Gelvin, and S. E. Hartman. 1993. Ratios
as a size adjustment in morphometrics. American Journal of
Physical Anthropology 91:441–468.
322
CRAIG R. WHITE ET AL.
Anderson, K. J., and W. Jetz. 2005. The broad-scale ecology of
energy expenditure of endotherms. Ecology Letters 8:310–318.
Aschoff, J., and H. Pohl. 1970. Rhythmic variations in energy
metabolism. Federation Proceedings 29:1541–1552.
Atchley, W. R., and D. S. Woodruff. 1976. Statistical properties
of ratios. I. Empirical results. Systematic Zoology 25:137–148.
Banavar, J. R., J. Damuth, A. Maritan, and A. Rinaldo. 2002.
Supply–demand balance and metabolic scaling. Proceedings
of the National Academy of Sciences (USA) 99:10506–10509.
Banavar, J. R., A. Maritan, and A. Rinaldo. 1999. Size and
form in efficient transportation networks. Nature 399:130–
131.
Bartels, H. 1982. Metabolic rate of mammals equals the 0.75
power of their body weight. Experimental Biology and
Medicine 7:1–11.
Bennett, P. M., and P. H. Harvey. 1987. Active and resting
metabolism in birds: allometry, phylogeny and ecology.
Journal of Zoology 213:327–363.
Berges, J. A. 1997. Ratios, regression statistics, and ‘‘spurious’’
correlations. Limnological Oceanography 42:1006–1007.
Bishop, C. M. 2005. Circulatory variables and the flight
performance of birds. Journal of Experimental Biology 208:
1695–1708.
Blomberg, S. P., T. Garland, Jr., and A. R. Ives. 2003. Testing
for phylogenetic signal in comparative data: behavioral traits
are more labile. Evolution 57:717–745.
Bokma, F. 2004. Evidence against universal metabolic allometry. Functional Ecology 18:184–187.
Brett, M. T. 2004. When is a correlation between nonindependent variables ‘‘spurious’’? Oikos 105:647–656.
Brody, S. 1945. Bioenergetics and growth. Reinhold, New
York, New York, USA.
Brown, J. H., G. B. West, and B. J. Enquist. 2005. Yes, West,
Brown and Enquist’s model of allometric scaling is both
mathematically correct and biologically relevant. Functional
Ecology 19:735–738.
Darveau, C. A., R. K. Suarez, R. D. Andrews, and P. W.
Hochachka. 2002. Allometric cascade as a unifying principle
of body mass effects on metabolism. Nature 417:166–170.
Dodds, P. S., D. H. Rothman, and J. S. Weitz. 2001. Reexamination of the ‘‘3/4-law’’ of metabolism. Journal of
Theoretical Biology 209:9–27.
Elgar, M. A., and P. H. Harvey. 1987. Basal metabolic rates in
mammals: allometry, phylogeny and ecology. Functional
Ecology 1:25–36.
Farrell-Gray, C. C., and N. J. Gotelli. 2005. Allometric
exponents support a 3/4-power scaling law. Ecology 86:
2083–2087.
Felsenstein, J. 1985. Phylogenies and the comparative method.
American Naturalist 125:1–15.
Frappell, P. B., and P. J. Butler. 2004. Minimal metabolic rate,
what it is, its usefulness, and its relationship to the evolution
of endothermy: a brief synopsis. Physiological and Biochemical Zoology 77:865–868.
Frappell, P. B., D. S. Hinds, and D. F. Boggs. 2001. Scaling of
respiratory variables and the breathing pattern in birds: an
allometric and phylogenetic approach. Physiological and
Biochemical Zoology 74:75–89.
Freckleton, R. P., P. H. Harvey, and M. Pagel. 2002.
Phylogenetic analysis and comparative data: a test and
review of evidence. American Naturalist 160:712–726.
Garcı́a-Berthou, E. 2001. On the misuse of residuals in ecology:
testing regression residuals vs. the analysis of covariance.
Journal of Animal Ecology 70:708–711.
Garland, T., Jr., and A. R. Ives. 2000. Using the past to predict
the present: confidence intervals for regression equations in
phylogenetic comparative methods. American Naturalist 155:
346–364.
Gates, S. 2002. Review of methodology of quantitative reviews
using meta-analysis in ecology. Journal of Animal Ecology
80:1142–1149.
Ecology, Vol. 88, No. 2
Glazier, D. S. 2005. Beyond the ‘3/4-power law’: variation in
the intra- and interspecific scaling of metabolic rate in
animals. Biological Reviews 80:1–52.
Grafen, A. 1989. The phylogenetic regression. Philosophical
Transactions of the Royal Society B 326:119–157.
Gurevitch, J., P. S. Curtis, and M. H. Jones. 2001. Metaanalysis in ecology. Advances in Ecological Research 32:199–
247.
Halsey, L. G., P. J. Butler, and T. M. Blackburn. 2006. A
phylogenetic analysis of the allometry of diving. American
Naturalist 167:276–287.
Hayes, J. P. 2001. Mass-specific and whole-animal metabolism
are not the same concept. Physiological and Biochemical
Zoology 74:147–150.
Hayes, J. P., and J. S. Schonkwiler. 1996. Analyzing massindependent data. Physiological Zoology 69:974–980.
Hedges, L. V., and I. Olkin. 1985. Statistical methods for metaanalysis. Academic Press, San Diego, California, USA.
Hemmingsen, A. M. 1960. Energy metabolism as related to
body size and respiratory surfaces, and its evolution. Reports
of the Steno Memorial Hospital and the Nordisk Insulinlaboratorium 9:1–110.
Heusner, A. A. 1991. Size and power in mammals. Journal of
Experimental Biology 160:25–54.
Hochachka, P. W., C. A. Darveau, R. D. Andrews, and R. K.
Suarez. 2003. Allometric cascade: a model for resolving body
mass effects on metabolism. Comparative Biochemistry and
Physiology A 134:675–691.
Kleiber, M. 1932. Body size and metabolism. Hilgardia 6:315–
353.
Kozlowski, J., and M. Konarzewski. 2004. Is West, Brown and
Enquist’s model of allometric scaling mathematically correct
and biologically relevant? Functional Ecology 18:283–289.
Kozlowski, J., and M. Konarzewski. 2005. West, Brown and
Enquist’s model of allometric scaling again: the same
questions remain. Functional Ecology 19:739–743.
Kozlowski, J., M. Konarzewski, and A. T. Gawelczyk. 2003.
Intraspecific body size optimization produces intraspecific
allometries. Pages 299–320 in T. M. Blackburn and K. J.
Gaston, editors. Macroecology: concepts and consequences.
Blackwell Science, Malden, Massachusetts, USA.
Lasiewski, R. C., and W. R. Dawson. 1967. A re-examination
of the relation between standard metabolic rate and body
weight in birds. Condor 69:13–23.
Lovegrove, B. G. 2000. The zoogeography of mammalian basal
metabolic rate. American Naturalist 156:201–219.
Lovegrove, B. G. 2003. The influence of climate on the basal
metabolic rate of small mammals: a slow–fast metabolic
continuum. Journal of Comparative Physiology B 173:87–
112.
Martins, E. P., and T. F. Hansen. 1997. Phylogenies and the
comparative method: a general approach to incorporating
phylogenetic information into the analysis of interspecific
data. American Naturalist 149:646–667.
McArdle, B. H. 1988. The structural relationship: regression in
biology. Canadian Journal of Zoology 66:2329–2339.
McKechnie, A. E., R. P. Freckleton, and W. Jetz. 2006.
Phenotypic plasticity in the scaling of avian basal metabolic
rate. Proceedings of the Royal Society B 273:931–937.
McKechnie, A. E., and B. O. Wolf. 2004. The allometry of
avian basal metabolic rate: good predictions need good data.
Physiological and Biochemical Zoology 77:502–521.
McNab, B. K. 1997. On the utility of uniformity in the
definition of basal rate of metabolism. Physiological Zoology
70:718–720.
Møller, A. P., and M. D. Jennions. 2002. How much variance
can be explained by ecologists and evolutionary biologists?
Oecologia 132:492–500.
Nagy, K. A. 2005. Field metabolic rate and body size. Journal
of Experimental Biology 208:1621–1625.
February 2007
EVIDENCE AGAINST UNIVERSAL ALLOMETRY
Nagy, K. A., I. A. Girard, and T. K. Brown. 1999. Energetics of
free-ranging mammals, reptiles and birds. Annual Review of
Nutrition 19:247–277.
Norberg, U. M. 1996. The energetics of flight. Pages 199–249 in
C. Carey, editor. Avian energetics and nutritional ecology.
Chapman and Hall, New York, New York, USA.
Osenberg, C. W., O. Sarnelle, S. D. Cooper, and R. D. Holt.
1999. Resolving ecological questions through meta-analysis:
goals, metrics, and models. Ecology 80:1105–1117.
Packard, G. C., and T. J. Boardman. 1988. The misuse of
ratios, indices and percentages in ecophysiological research.
Physiological Zoology 61:1–9.
Packard, G. C., and T. J. Boardman. 1999. The use of percentages
and size-specific indices to normalize physiological data for
variation in body size: Wasted time, wasted effort? Comparative Biochemistry and Physiology A 122:37–44.
Quinn, G. P., and M. J. Keogh. 2002. Experimental design and
data analysis for biologists. Cambridge University Press,
Cambridge, UK.
Rezende, E. L., F. Bozinovic, and T. Garland, Jr. 2004.
Climatic adaptation and the evolution of basal and
maximum rates of metabolism in rodents. Evolution 58:
1361–1374.
Rezende, E. L., D. L. Swanson, F. F. Novoa, and F. Bozinovic.
2002. Passerines versus nonpasserines: so far, no statistical
differences in the scaling of avian energetics. Journal of
Experimental Biology 205:101–107.
Riisgård, H. U. 1998. No foundation of a ‘‘3/4 power scaling
law’’ for respiration in biology. Ecology Letters 1:71–73.
Savage, V. M., J. F. Gillooly, W. H. Woodruff, G. B. West,
A. P. Allen, B. J. Enquist, and A. C. Brown. 2004. The
predominance of quarter-power scaling in biology. Functional Ecology 18:257–282.
Scheffé, H. 1953. A method for judging all contrast in the
analysis of variance. Biometrika 40:87–103.
323
Scheffé, H. 1959. The analysis of variance. Wiley, New York,
New York, USA.
Suarez, R. K., and C. A. Darveau. 2005. Multi-level regulation
and metabolic scaling. Journal of Experimental Biology 208:
1627–1634.
Tieleman, B. I., and J. B. Williams. 2000. The adjustment of
avian metabolic rates and water fluxes to desert environments. Physiological and Biochemical Zoology 73:461–479.
Weibel, E. R., L. D. Bacigalupe, B. Schmidt, and H. Hoppeler.
2004. Allometric scaling of maximal metabolic rate in
mammals: muscle aerobic capacity as a determinant factor.
Respiration Physiology and Neurobiology 140:115–132.
Weibel, E. R., and H. Hoppeler. 2005. Exercise-induced
maximal metabolic rate scales with muscle aerobic capacity.
Journal of Experimental Biology 208:1635–1644.
West, G. B., and J. H. Brown. 2005. The origin of allometric
scaling laws in biology from genomes to ecosystems: towards
a quantitative unifying theory of biological structure and
organization. Journal of Experimental Biology 208:1575–
1592.
West, G. B., J. H. Brown, and B. J. Enquist. 1997. A general
model for the origin of allometric scaling laws in biology.
Science 276:122–126.
West, G. B., J. H. Brown, and B. J. Enquist. 1999. The fourth
dimension of life: fractal geometry and allometric scaling of
organisms. Science 284:1677–1679.
White, C. R., N. F. Phillips, and R. S. Seymour. 2006. The
scaling and temperature dependence of vertebrate metabolism. Biology Letters 2:125–127.
White, C. R., and R. S. Seymour. 2003. Mammalian basal
metabolic rate is proportional to body mass2/3. Proceedings
of the National Academy of Sciences (USA) 100:4046–4049.
White, C. R., and R. S. Seymour. 2005. Allometric scaling of
mammalian metabolism. Journal of Experimental Biology
208:1611–1619.
APPENDIX
A summary of data used in the meta-analysis (Ecological Archives E088-019-A1).