Allometric Exponents Do Not Support a Universal Metabolic Allometry
Author(s): Craig R. White, Phillip Cassey, Tim M. Blackburn
Source: Ecology, Vol. 88, No. 2 (Feb., 2007), pp. 315-323
Published by: Ecological Society of America
Stable URL: http://www.jstor.org/stable/27651105
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Ecology, 88(2), 2007, pp. 315-323
? 2007 by the Ecological Society of America
ALLOMETRIC EXPONENTS DO NOT SUPPORT A UNIVERSAL
METABOLIC ALLOMETRY
Craig
1
School
R. White,1,3
of Biosciences,
of Ecology,
2Department
The
Abstract.
rate
metabolic
both
for and
often
revolves
recent
models
of
that
nutrient
or
shows
analysis
an
Such
to
we
conduct
a result
to b values
estimated
applies
correct
models.
regression
there is no universal
metabolic
that
any model
that
words:
Key
predicts
a single
only
metabolic
allometry;
about
the
value
of
the
a
X
has
recently
a number
of
Mh)
of
publication
allometric
been
stimulated
competing
and
times
et al.
(West
has generated
which
scale
with
M
al.
the
Kozlowski
and Konarzewski
and
Suarez
2005,
Hoppeler
2005, West
led to close
Darveau
Farrell-Gray
regard,
allometric
conducted,
2005,
and Gotelli
2004,
scaling
with
conflicting
support
and
between
exercise),
axis, and
scaling.
of
rate
et
Savage
have
al.
2004,
2005). In this
studies
recently
of
and
(FMR),
the
been
different
that
be
classes,
rate
rate
metabolic
BMR
and
FMR
scaled with b >
scaled
3/4. The
was
to
suggested
or methodological
to small
in addition
size, but
sample
they
noted
that MMRex
does not scale as
clearly
explained
(Savage
for MMRex
scaling
selection
by
nevertheless
M2/3
represen
metabolic
maximum
concluded
They
field
"binning"
size
body
BMR,
exercise-induced
of non-3/4
finding
et
al.
of
2004).
species
and
Farrell-Gray
Gotelli
(2005) used a likelihood analysis approach to compare
b = 3/4 and b= 2/3 for 22 published BMR and standard
rate
metabolic
and
exponents
(SMR)
insects.
Likelihood
for
birds,
ratios
mammals,
the
quantifying
relative probability of b = 3/4 compared to b = 2/3 were
Dodds et al. (2001) reanalyzed bird and mammal
rate (BMR) data sets published by
basal metabolic
16074
and
19 April
13 June
a
and,
using
for nonuniform
(2003)
to account
with b = 3/4, while MMRex
reptiles,
revised
received
Seymour
approach
designed
tation
of
within
species
examined
the
of
scaling
results.
29 November
received
2005;
Manuscript
27 April
2006; final version
2006; accepted
Editor: T. D. Williams.
2006. Corresponding
3 E-mail:
[email protected]
and
differences
for quarter
et al. 2001, White
2005, Glazier
four
meta-analyses
of metabolic
identified
rest, field, and
reduced
major
a single
model
exponent
a significant
to
challenge
represents
(MMRex).
et
Brown
Weibel
2005,
the empirical
Bokma
2003,
squares,
for
support
and
White
1/4
and Brown 2005), but has also
of
scrutiny
Seymour
latter
of
and Wheatley
2004,
power scaling (Riisg?rd 1998, Dodds
and
attempt
et al. 1999, 2002).
This
1997,
1999, Banavar
an increasingly
acrimonious
debate
within
models
have
been
competing
intensely
scrutinized (Dodds et al. 2001, Agutter
2004,
the
that biological
to multiples
raised
(MR
by
models
ing to explain the widely held observation
rates
also
(e.g.,
This
and
heterogeneous
endotherms
than for
(1991), Bennett and Harvey (1987), Bartels
(1982), Hemmingsen (1960), Brody (1945), and Kleiber
=
(1932) and found little evidence for rejecting b 2/3 in
favor of b= 3/4. Savage et al. (2004) combined the BMR
data sets of Heusner (1991), Lovegrove (2000), and
scaling
rate to body mass
exponent (b) relating metabolic
=
least
of
were
metabolic
states.
Heusner
Introduction
Debate
universal
metabolic
is significantly
is stronger
for
allometry
of b.
value
a
rate
by ordinary
The
lack
rate; quarter-power;
is
or between
on metabolic
rate
the effect
of mass
that, on average,
ectotherms.
differences
between
Significant
exponents
scaling
as well as between
ectotherms
and endotherms,
states
metabolic
phylogenetically
that
suggests
predicted
allow
for
there
taxa
differences
between
systematic
on metabolic
the effect
size of mass
that
University,
by Euclidean
the possibility
a meta-analysis
of 127
exponent
not
does
approach
In the present
study,
whether
determine
exponent.
exponents
a 2/3
and
networks
are
if there
B15 2TT UK
Rutgers
value
distribution
"true"
single
allometric
allometry
the
considerations.
is no
there
interspecific
Blackburn1
(metabolic
a 3/4-power
a dichotomous
surface-area-to-volume
Tim M.
of the allometric
exponent
scaling
(b) relating
rate = a X mass^)
is ongoing,
with
evidence
published
to accumulate.
law continuing
this debate
However,
scaling
distinction
between
the 3/4-power
exponent
by
predicted
about
mass
against
around
and
Cassey,1,2
The University
Edgbaston,
of Birmingham,
Birmingham,
14 College Farm Road,
and Natural
Resources,
Evolution,
New Brunswick,
New Jersey 08901 USA
debate
to body
Phillip
for
2.20
all
for
Farrell-Gray
analyses
exponent
Finally,
315
105
reptiles
(Farrell-Gray
and Gotelli
supported
for
for mammals,
species,
the
endotherms,
and most
7.08
and
(2005) concluded
idea
of
but
a universal
not
recently, Glazier
for
Gotelli
for
birds,
2005).
that their
metabolic
ectotherms.
(2005) conducted
R. WHITE
316
CRAIG
an extensive
review
scaling
concluded
descriptive
and
exponents
law" of metabolic
scaling
While
of
rate
intra-
and
that
is not
The
interspecific
the
universal.
meta-analytical
represents
approach
on metabolic
in the debate
advance
scaling,
these
et al.
Dodds
studies
(2001)
has
limitations.
For
argue
convincingly
example,
b = 3/4,
against
but only for BMR. Savage et al. (2004) provide strong
for
support
of
general
quarter-power
three metabolic
scaling
their
the most
is
analysis
exponents
a
small
the
0.723-0.734
for
Aschoff
and
0.68
have
and
been
exponents
reported
of
Farrell
1967,
from
BMR
an
earlier
analysis
met
unambiguously
the
and
0.670)
been
quently
bird
selected
2006).
This
(2005:
2083)
scaling
available
(b
subse
data
is premature.
(2004)
selection
on
emphasis
was
criteria
echoed
the
by
and Seymour (2005), who reported that the BMR
scaling
exponent
with
the proportion
Given
law"
scaling
of
importance
White
of
?
exponent
et al.
(McKechnie
that Farrell-Gray
and Gotelli's
that "allometric
sup
exponents
and Wolfs
that
animals
of
measurement
tested
was
for mammals
are
large herbivores
of BMR
positively
within
in a postabsorptive
a data
the
in
unlikely
sets
data
must
measurements
lineages
to be met
be
(White
for which
Seymour
exponent
species
for
which
close
to 3/4
data
are
(Savage
available
et al.
2004).
that
Thus,
are
is close
including all
produces
and
heterogeneity
allometric
much
rate
that
presupposes
and
exponent
of
not
does
there
it is either
that
for
example,
attachment
favorable
barrier
to
scaling
of
rate.
In this
analysis,
exponents
scaling
(e.g.,
Osenberg
aim
2002). We
we
examine
127 published
a meta-analytical
using
et al. 1999, Gurevitch
et al.
to advance
allometric
allometric
approach
Gates
2001,
over
the debate
the
scaling relationships
of
form
by applying
to as
quantitative
meta-analytical
methodology
a
set
as
of
such
comprehensive
relationships
possible,
thus
the
out
laid
of
criticisms,
above,
addressing
rigorous
such
previous
of an ecological
of
the
"effect
objective
estimates
of a response
the
(i.e.,
to a given
or manip
correlation
the objective
of most
studies
However,
relative
size")
variable.
ulation
that
the main
Typically,
is to summarize
analyses.
meta-analysis
standardized
magnitude
examine
estimate
the
the
and
log(mass)
of
the
slope
Nevertheless,
indeed
of metabolic
scaling
strength
of
rate
is not
to
the
between
relationship
to
but
estimate
the
rate),
log(metabolic
two variables.
between
these
relationship
on metabolism
if the influence
of mass
both
universal,
taxonomically
states.
Thus,
the
and
slope
an
diverse
we
strength
Materials
Allometric
rate
metabolic
literature
ordinary
for all
major
(see
exponents
to body
for
mass
Appendix).
least-squares
127 data
sets;
axis
and
(RMA)
(OLS)
and
groups
examine
the
to determine
a single
whether
slopes
a single
and
effect
size characterize
mass
between
and metabolism.
carefully,
increase
2005).
conditions
to 2/3 (White and Seymour 2003), while
exponent
to
scaling
Hochach
single-exponent
metabolic
basal
an
produces
a
to
that
the
requires
state and such a
tend
of metabolic
scaling
is consistently
different
between,
taxa and metabolic
states. Ongoing
b
between
made
"true"
for
allow
2/3 or 3/4. Such a dichotomous distinction excludes the
possibility that b is neither 2/3 nor 3/4 and the possibility
set.
will
and
exponent
scaling
excluding
BMR
non-BMR
such
is a single
that
et al. 2002,
(Darveau
et al. 2003). However,
is
of
the
relationship might reasonably be predicted to be similar
correlated
state is difficult or impossible to achieve in at least
ruminants (McNab 1997), the decision to include such
species
because
for
the
about
published
the criteria
captive-raised
have
0.744)
(b
different
to be
an
had
for
=
birds
suggests
statement
McKechnie
data
exponents
wild-caught
shown
a 3/4-power
port
BMR
models
exponent
heterogeneity
ka et al. 2003, Kozlowski
the debate
meta
different
and Seymour 2005) has led to
metabolic
for BMR, which are strictly defined (McNab 1997,
Frappell and Butler 2004). The regression for these
rigorously
but
0.677,
of
development
(Weibel et al. 2004,
2005). Indeed, the
and
for birds and found that only 67 of 248 measurements
from
the
3/4
et al.
White
2005,
between
exponents
Similarly,
ecto
support,
(2004)
data
in scaling
bolic levels (e.g., White
1987,
and Wolf
McKechnie
recently,
reviewed
the published
Gotelli
and
3/4
of
scaling
of a universal
and
While
and Harvey
(Bennett
support
and
of
exponent.
(SMR)
the idea
without
paradigm,
a substantial
potentially
represents
the causes
of the non-isometric
understanding
separate
justified
(Garland
of 0.67, 0.677,
rate
2006), as does the scaling of MMRex
Bishop 2005, Weibel and Hoppeler
that
Dawson
to
(Farrell-Gray
variation
2
selected
rigorously
exponent
a universal
metabolic
fails
an
scale with
as
this
also
for
88, No.
data sets suggests that the BMR of
not
does
exponent
different
Most
rigorously
therms
obtained
exponents
against
standard
and
Tieleman and Williams 2000, Frappell et al. 2001) but
were not included in Farrell-Gray and Gotelli's (2005)
analysis.
the
scaling exponents of
non-passerines.
be
may
and
selection
argues
allow
support
example,
derived
and
passerines
since
For
(Lasiewski
values
1970),
Pohl
strong
endotherms,
fortuitous
literature.
regressions
or RMR
BMR
clade-specific
Ives 2000),
find
for
birds
for
regressions
and
only
(2005) use BMR
Gray and Gotelli
(2005)
of scaling
compilation
a
argues
strongly
against
but
is largely
descriptive.
on
the other
hand,
(2005),
exponent,
3/4
and Gotelli
Farrell-Gray
use a meta-analytical
approach
for b = 3/4, but only for BMR,
including
from
exponents
not
and
universal
only
3/4. Glazier's
extensive
undertaken
yet
two
only
are
exponents
from b =
significantly different
but
scaling,
similar
endotherms
a
Vol.
Ecology,
bird and mammal
"3/4-power
the
significant
of
each
ET AL.
between
127 published
exponent
scaling
the relationship
Methods
127
data
sets
were
relating
the
from
compiled
estimated
Exponents
regression
were
estimated
exponents
were
available
regression
by
available
by reduced
or could
be
February 2007
calculated
EVIDENCE AGAINST UNIVERSAL ALLOMETRY 317
from
Garland
were
Exponents
variables
were
2000)
mammals,
or
rate
metabolic
allometric
expected
its sampling
mean
residual
(Quinn and Keogh
class
state)
resting
maximum
to
applies
the
deviation
standard
root
of
of
square
sum of squares
the
RMA,
(OLS,
(taxonomy,
a weighted
using
and
PC)
thermor?gulation,
accounted
model
for
to their
according
linear
generalized
weighted
variance
size, N).
effect
(sample
Standardized
Pearson
correlation
This
regression.
their
and
have
the
calculat
OLS
exponent
estimate
of
its
an
RMA
number
coefficients
metric
and
of
has
been
and
index
to the
for owing
and
(Hedges
model
to
calculated
PC
from
(r) estimated
used
widely
linear
Olkin
from
from
the
OLS
in studies
tested
the
using
of an explanatory
a random
We
used
of
whether
large
accounted
classes
8.0
(taxonomy,
influencing
relationship
one
class
metabolism
variability
between
to
had
the
sample
the
variable
body
contain
the three
that
(in
groupings.
were
dimensions
effect
subsequent
across
was
effect
effects
test
for
predictor
in
mass
at
effect
respect
and metabolism.
two
least
sizes
to be
the homogeneity
classes
the mean
separately.
effect
size
state)
to the
body
included
of
The
does
Any
mass
in the
effect
sizes
test of
not
all
finer
yield
on which
based
class
remaining
classes)
classes
within
size homogeneity
and
classification
(when
explained
for each
partition.
Results
Plots
of both
size against
size
?>ols an<i effect
sample
and
showed
1A) were
typically
"funnel"-shaped
with
that
convergence
size, suggesting
increasing
sample
a
measure
of b should
be weighted
of
analysis
by
In both
variance.
ectotherms
and
endotherms
cases,
the
differ
on different
to converge
appeared
b values
includes
a random
The
b values
values
from
the random
0.75,
OLS
for
neither
OLS
and
model
from OLS
nor
as a general
the ectotherm
0.75
the
endotherm
mean
the weighted
calculated
with
regression
the endotherm
b not differ
does
In
study.
and RMA
for
Only
RMA
this b value
regression.
of
independence
0.67
greater
0.67 and 0.75.
effect
is weighted
analysis
by
the weighted
analysis
1). In all models,
than 0.75, while
(Table
but
not
estimated
support
exponent
scaling
b is significantly
ectotherms
or not
effect
Indeed,
are
regression
and
endo
and PC
RMA,
by OLS,
different
between
or
means.
"true"
estimated
significantly
therms
(Table
1), whether
size and whether
sample
also
not
does
b
contrast,
differ
from
values
b values
sion differ
from
the weighted model
model (Table 1).
Ectotherms
and
endotherms
between
states
metabolic
regression
in terms
are
are
shown
of metabolic
with
yields
a minimum
effect
adequate
regres
continue
b
values
b values
for
not
for
to show
signif
different
when
thermor?gulation
for independence
model
with
0.75
random effect
distinguished:
examples
in Fig.
2. Modelling
OLS
state,
a random
taxon,
for
from 0.75 but not 0.67 for
and for the weighted
differences
gener
0.67
all models,
mean
(Table
1). The
equivalent
differ
from 0.67 but
significantly
endotherms
ectotherms
icant
of
controlled
phylogenetically
significantly
0.67
calculated
are
controlled
phylogenetically
regression
consistent
with
the theoretical
value
ally
and
endotherms
0.75
for ectotherms.
For
mean
homogeneity
particular
metabolic
for
using
this
of
a
level
that effect
specification
classes
by manually
each of the classification
within
other
linear
(Scheff? 1953, 1959)
to
manner)
stepwise
The
order
in which
the
endotherms),
of different
sizes
(orthogonal)
the model
chosen
with
compared
the most
not
that
suggested
and
significance
tested
a
it
effect
multiple
homogenous
the classes
finer
for
of
population
variance
differs
thermor?gulation,
in effect
sizes with
relationship
we tested
First,
analysis.
across
each of
hypothesis
assess
to
we
Second,
are
sizes
from
of correlations given by Hedges and Olkin (1985).
Following Hedges and Olkin (1985), we used a
categorical model fitting procedure coded in SAS
version
of
simultaneous
b lies between
in
relationships
the variance
1985).
the variance
and
zero,
on
based
introduction
estimate
correlations
a
gives
contrasts.
observa
ecology and evolution (M0ller and Jennions 2002) and is
the best-known
for
the mean
contrasts. The Scheff? procedure
regression
were
sizes
correlational
synthesizing
that
of
reciprocal
whereas
available),
by
observa
variance
the
by
the
weighting
sets. Each
data
(where
were
weighted
exponents
tions
both
heterogeneous
of studies
possible
nonindependence
ed exponents
from
shared
was
differed
if
(e.g.,
ectotherms
If this
were
(Fig.
model
(GLMM) in SAS version 8.0 (Proc
MIXED; SAS Institute, Cary, North Carolina, USA).
tions
classes
compared
by means
statistic.
sizes
of X
mixed
This
we
partitioning
dimensions
regres
through
an
normal
with
is
the
exponents
variables
metabolic
[basal
across
same.
goodness-of-fit
effect
that
conclusion
2002). We analyzed the relationship
allometric
between
or
to the
equal
divided
by
square
metabolic
calculated
distribution
sampling
B and the
value
(mean)
distribution
maximum
rest
Theorem
b when
exponent
its
Limit
[flight
the
homogenous
effect
sizes
in an analysis
of
test is
The
to the F test
is analogous
are the
that class means
a between-class
to
led
classes
arthropods,
[cold-induced
thermogenic
rate]).
the Central
Because
sion,
rate],
to test
on
test
then
categorical
rate], exercise
exercise-induced
rate],
metabolic
sets.
thermor?gula
state
metabolic
unicells),
and
classes
based
1997,
30 data
of
(amphibians,
[field metabolic
rate], field
metabolic
for
sets
three
[mean daily metabolic
(daily
Hansen
across
variance
least
generalized
available
ectothermic),
for
regressions
contrasts
independent
reptiles,
(endothermic,
r2 values
(PC)
and
to
assigned
on taxonomy
based
fish,
three
of
Martins
1989,
(Grafen
and Ives
birds,
correct
1985) or phylogenetic
(Felsenstein
squares
r or
arid
exponents
Phylogenetically
the method
by
calculated
tion
OLS
sets.
103 data
for
mode,
metabolic
OLS
b values
of
and
study,
state,
/
318
ET AL.
R. WHITE
CRAIG
2
88, No.
Vol.
Ecology,
Endotherms
a Ectotherms
CO
0.0
100
200
Fig.
Bivariate
or b) and
from OLS
1.
between
least-squares
plots of the relationship
(A) ordinary
effect size (Z,-) and sample size
sample size, and (B) standardized
for endotherms
and ectotherms.
regression
funnel
and
mode,
thermor?gulation
as
two variables
these
values
differ
and
values
these
states
between
Similar
also
vary
metabolic
states,
only
have
minimum
Our
regression
an
with
b values
includes
a predictor:
ectotherms
with
the latter
endotherms,
than
of
exponent
and
2/3
the
ectotherms
different
from
fact,
"true"
3)
to explain
considerable
heterogeneity,
assign
by metabolic
The
results
significantly
and reptiles.
heterogeneity.
when
studies
mode
homogenous
sizes
effect
size
sample
sizes
effect
on
converge
is less
size
state,
of
for
this
field
(Fig.
(final
To
obvious
thermor?gulation
are
procedure
rate
metabolic
Some
and
of
further
taxon.
in studies
this
IB).
line
partitioned
For
example,
of mammalian
by
value
in
given
are not
metabol
data
exponent
the
mass
important
scaling
Brown
variance
by
account
for
have
weighting
exponents
of
the strength
nonin
the
calculated
is not
influence
the
no
and
obscure
variability
(Dodds
our
the
et
2004, Kozlowski
Hoppeler
al.
to
is unlikely
the
explaining
of metabolism.
scaling
et
and
significant
and
for models
foundation
to consider
It is tempting
debate
surrounding
metabolic
and
that
therefore
will
a robust
Wheatley
are
standard/basal
b
metabolic
provide
allometric
in
also
and
rate
from
universal,
single
in the
characterizes
variation
adequately
of any
of animals.
single
scaling
Acceptance
as the true exponent
rate to
relating metabolic
of
body
scaling
sets. Thus,
on metabolic
and
arthropods,
significant
heterogeneity
accordingly
of studies
potentially
mode,
thermor?gula
effect
sizes
exercise
for
for birds,
relationships
in
heterogeneity
Our
results
exponent.
reveals
and
size
dependence
from shared
in
127 allometric
aerobic
effect
of mass
is removed
heterogeneity
to maximum
both
In
between
birds,
mammals,
heterogeneous
states
All
other metabolic
show
size
effect
are
3).
(Table
arthropods
amphibians,
reptiles,
at metabolic
states
ranging
estimates
attempt
heterogeneity.
we used
manual
stepwise
to finer
and finer
relationships
to
3. Effect
vs.
for
this
groupings
and
taxon.
tion
model
shows
partitioning
Table
for
size
effect
also
endotherms
averages
the plot of
the overall
Table
and
and
fish,
account
former
consistent with an exponent of 3/4 (Table 2).
Whether
studies
reptiles,
of
analysis
mammals,
unicells
as
type
thermoregulatory
b values
higher
consistent
of PC
model
adequate
for
birds,
this partitioning
fails to
sizes
sizes. Notably,
effect
rate in
of resting metabolic
in effect
are heterogeneous
mammals,
cases
in several
heterogeneity
(allometric
allometry
2).
Fig.
(Table 2), although slightly higher b values are obtained.
The
remove
(OLS) slope value
regression
of metabolic
(n) for estimates
Discussion
(e.g.,
RMA
from
600
metabolic
for
endotherms
to b values
ic rate. However,
between
b
2). Thus,
(Table
ectotherms
and
differently
and
pertain
interaction
different
ectotherms
results
the
predictors
between
significantly
and
between
endotherms
500
size (A7)
Sample
exponent,
calculated
400
300
2005,
2005,
Suarez
West
with
findings
mechanistic
al.
2001,
and
Darveau
Brown
2005,
2005).
of
and
Agutter
and Konarzewski
and
to
regard
basis
2004,
Weibel
However,
EVIDENCE AGAINST UNIVERSAL ALLOMETRY 319
February 2007
b calculated
1. Mean
allometric
separately
exponents
(slope estimates)
least squares (OLS),
for different
endotherms,
ordinary
regression methods:
correct (PC).
and phylogenetically
(RMA),
Table
OLS
slope
Observed
95% CL
b
MethodEstimate,
Difference
and
axis
for ectotherms
reduced major
t
SE
estimates
means
Ectothermic
Endothermic
0.804
0.704
0.77,0.83
0.68, 0.72
0.100
0.02
6.11***
means
Weighted
Ectothermic
Endothermic
0.809
0.694
0.78,0.83
0.68, 0.71
0.114
0.01
7.53***
0.800f
0.74,
0.66,
0.86
0.77
0.085
0.03
2.50*
Ectothermic
Endothermic
0.843
0.722
0.81,
0.70,
0.87
0.74
0.120
0.02
6.27***
means
Weighted
Ectothermic
Endothermic
0.866
0.711
0.84,0.89
0.70,0.72
0.154
0.01
10.36***
0.860
0.81,0.91
0.67, 0.76
0.144
0.03
4.93***
0.842
0.77,0.91
0.66, 0.71
0.154
0.04
4.32***
0.687f
0.834|
0.683J:
0.72,
0.67,
0.95
0.70
0.151
0.06
2.60*
0.837|
0.670J
0.73,
0.62,
0.94
0.72
0.167
0.06
2.97**
means
Weighted
Ectothermic
Endothermic
with
random
effect
0.714|,i
slope estimates
means
Observed
RMA
means
Weighted
Ectothermic
Endothermic
with
random
effect
0.715|,?
PC
slope estimates
means
Observed
Ectothermic
Endothermic
means
Weighted
Ectothermic
Endothermic
means
Weighted
Ectothermic
Endothermic
with
random
effect
Notes: Means were calculated
from raw b values (observed means),
from b values weighted
by an
estimate
of their variance
and from b values weighted
of their
(weighted means),
by an estimate
variance and including a random effect for study independence
(weighted means with random effects).
*
P < 0.05; ** P < 0.01; *** P < 0.001.
mean
95% confidence
limits of exponent
0.75.
encompass
f Least-square
mean
95% confidence
limits of exponent
0.67.
encompass
%Least-square
while
our
results
that
accommodate
veau
et al. 2002,
are most
scaling
Hochachka
2003), these models
congruent
exponent
with
those
heterogeneity
et al. 2003,
Kozlowski
models
(Dar
et al.
do not provide testable predictions
for
taxa
all
and
the
Additionally,
limited deviations
et
al.
1997),
and
metabolic
fractal
states
geometry
considered
model
here.
predicts
from b = 3/4 at small masses
the
supply-demand
balance
(West
model
Endothermic
n Poikilothermic
Exercise
Fig. 2. Ordinary
states in endotherms
Field
(OLS) slope estimates
least-squares
regression
and ectotherms. There are no scaling exponent
Rest
Thermogenic
Daily
(scaling exponent or b values; means ? SE) for different metabolic
or daily metabolic
rates for ectotherms.
estimates for thermogenic
R. WHITE
CRAIG
ET AL.
Vol.
Ecology,
88, No.
2
or minimum
2. Final
models
for ordinary
least-squares
regression
(reduced
adequate)
correct
reduced major
axis regression
and phylogenetically
(OLS),
(RMA),
regression
(PC)
and interaction.
estimates
of b in terms of metabolic
state, thermor?gulation
mode,
Table
Fixed
least-square
OLS
or metabolic
mode
Thermor?gulation
state
Field
Resting
Thermogenesis
X metabolic
Thermor?gulation
RMA
state
Metabolic
X metabolic
Thermor?gulation
*
state
State
Daily
Rest
Thermogenic
Field
Rest
Rest
Exercise
Rest
Thermogenic
Exercise
Rest
Rest
Rest
4,
102
69.65***
2,
102
9.56***
0.846
0.711
0.79,
0.67,
0.90
0.75
0.648
0.889
0.835
0.777
0.745
0.54,
0.81,
0.77,
0.73,
0.64,
0.75
0.97
0.90
0.82
0.85
12.80***
4, 86
5.08**
interaction
6.44*
0.05;
**
0.837
0.670
P <
accommodates
Future
0.01;
***
values
of
P <
size statistics
sents
of b between
these models
for different
0.73,
0.62,
nested metabolic
a
that
1, 20
accommodate
states,
indicates
birds, mammals,
arthropods
arthropods
amphibians
of
the
refinement
of
the documented
that
these models
thermoregulatory
Taxon
Thermor?gulation
ectothermic
falsification
suggests
and
modes,
Z+
endothermic
endothermic
endothermic
endothermic
endothermic,
ectothermic
endothermic
ectothermic
ectothermic
endothermic
endothermic
ectothermic
ectothermic
ectothermic
0.94
0.72
8.83**
0.001.
may
3/4.
yet
expansion
for the heterogeneity
described
here. Thus,
it is
clear whether
repre
exponent
scaling
heterogeneity
Exercise
Total
P <
et al. 2002)
Effect
12.78***
reduced model
Thermor?gulation
Ectothermic
Endothermic
3.
0.51, 0.69
0.81,0.93
0.72, 0.83
0.71, 0.78
0.58, 0.77
1, 102
state
Resting
Thermogenesis
Table
0.601
0.871
0.776
0.741
0.676
0.83
0.72
interaction
Daily
Exercise
Field
not
0.72,
0.66,
reduced model
Thermor?gulation
Ectothermic
Endothermic
2/3
account
0.775
0.690
state
Daily
Exercise
and
F
df
reduced model
Metabolic
(Banavar
95%CL
Estimate
Thermor?gulation
Ectothermic
Endothermic
PC
Type III tests
of fixed effects
effect
means
reptiles
mammals
4
2.59
4 birds 2.45
12
birds
2.26
2 birds 2.21
11
2.16
10
2.15
reptiles
mammals
32
2.10
4
1.86
10
1.85
2
mammals
1.85
2 birds 1.76
2
mammals
1.76
2
1.70
4
unicells
1.51
110
2.08
"single
exponent"
these models
heterogeneity,
some
sacrifice
models,
is necessary
to
or merely
detail
for the
taxa.
95%CL
2.31, 2.87
2.31, 2.58
2.19,2.33
1.92,2.50
2.07,2.25
2.04, 2.26
2.07, 2.13
1.71, 2.01
1.74, 1.96
1.67,2.03
1.44, 2.08
1.54, 1.97
1.55, 1.85
1.38, 1.65
2.06,
2.11
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
that effect sizes are homogenous
within classes by manually
the classes across
specification
partitioning
each of the nested classification
dimensions
in a stepwise manner
to yield finer and finer groupings.
The order in which classification
were chosen was based on which
dimensions
the most
hypothesis
(when compared with the other remaining hypotheses)
explained
effect-size
within classes for each subsequent
In each case we tested whether
exists
homogeneity
partition.
significant heterogeneity
within classes (x2). Classes
that are italicized are those for which partitioning
resolve their class heterogeneity.
does not significantly
is always significantly
Z+ is the mean effect size, which
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 321
The
last possibility
of generality.
as any
of these models,
shortcoming
ship
species
mental
and
BMR
(e.g.,
variables;
variation
can
some
to
attempting
explain
in our
metabolism,
here
identified
about
are
heterogeneity
used
to quantify
consequence
the
allometric
scaling
by
the
regression
of
scaling
statistical
the precise
regression
as
rate,
and
PC
of
values
the
the
methods.
analytical
assumes
regression
measured
without
Ordinary
that
the
squares
variable
independent
since
this
and
error,
least
assumption
itmay be argued that RMA
violated by body mass,
technique
Since
metabolic
rate
as
long
independent variable
variable.
that OLS
suggests
as
the
is <l/3
variance
in
is
the
of that in the dependent
to our
the error variances
in
knowledge
have never
been assessed
body mass
and
in any
the question
of which
of these
comparative
study,
be
most
in
terms
these
remains
may
techniques
justified
it is a question
to be
that is likely
open. Nevertheless,
moot.
Both OLS and RMA assume that data are
independent, and this assumption is likely to be violated
in any
comparative
are
likely
species
metabolic
rates
two
to be
et
Freckleton
McKechnie
independence
regression
derestimated
context
Assuming
Indeed,
between
not
provide
the norm
related
and
independent
the
between
relationship
in
correlation
phylogenetic
rate
metabolic
(Elgar
and
mass
body
and Harvey
standard
of physiology,
that metabolic
in such
data,
(allometric
errors
(for
see Halsey
rate
is an
correlation
studies
and
of
un
the
[2006]).
character
not
exponent
et al. 1999,
Savage
signifi
et al.
avian BMR, FMR, flight
cold-induced
MMR
all
scale
with
significantly different from 3/4 (Bennett and
exponents
1999, Tieleman
et al.
Rezende
and Wolf 2004, Anderson and Jetz
2002, McKechnie
2005, Bishop 2005, Nagy 2005). Studies of relationships
between
birds
energy
are
of
of
Regardless
better
approach
likely
rate
use
by
of
body
mass
and
1993, Hayes
and
Garcia-Berthou
the
need
in an ANCOVA
can
1988,
body
mass6,
a
introduces
not
1988,
Schonkwiler
2001,
information
by
Boardman
approach
to assume
a
is,
1999).
evident
or making
number
of
an
when
is taken (Atchley and Woodruff
approach
and
by
scaling.
a given
exponent,
effects
body mass
Boardman
are
that
problems
for
effects by dividing metabolic
dividing
residuals
1976, Packard
ANCOVA
for
to include body mass
mass,
ANCOVA
of
applicability
to accounting
body
potential
rate
metabolic
the
Inclusion of body mass
variables
ecological
to be compromised
3/4-power
Packard
(e.g.,
and
expenditure
therefore
Hayes
also has
a value
2001,
1999, Albrecht
et al.
1996, Berges
1997,
Brett
the advantage
for b, while
An
2004).
of obviating
phylogenetic
be
incorporated
using modern
et
statistical
al.
packages
(see Halsey
2006).
In concluding
their meta-analysis,
et al. (2004)
Savage
that a century
was
of science
distorted
suggested
by
readily
to fit observations
to an unsatisfactory
trying
law (b = 2/3). Given
the apparent
widespread
surface
acceptance
and application of b = 3/4, it seems history is in danger
of repeating.
there
is no
metabolic
Our
analysis
of
127 exponents
true
allometric
single
rate to body mass
and no
suggests
exponent
universal
that
relating
metabolic
Acknowledgments
We
thank Andrew
and two anonymous
Clarke
referees for
an earlier version
comments
their constructive
about
of this
Jim Brown
and Charles
Darveau
for helpful
manuscript,
and the many
who have worked
with
discussions,
colleagues
us on scaling,
was
C. R. White
Roger
especially
Seymour.
Environment
Research
Council
supported
by Natural
grant
to G. R. Martin,
P. J. Butler,
and A. J.
NER/A/2003/00542
Woakes.
Literature
allometry
raises
because
exponent,
an
(Nagy
is thus likely to be
model
a 3/4
of
allometry.
of
in
et al.
evolved
of metabolic
a phylogenetic
require
perspective.
for a multiple
Support
exponent
and
exponents)
a discussion
0.75
example,
in mammals
variation
with
the
For
see
2002,
coefficients
use
2005). However,
MMRex,
1987,
et al. 2003,
but
Blomberg
et al. 2006).
This
violates
the assumption
can
to biased
and
lead
estimates
al.
istic, and phylogenetic
widespread
allometry:
closely
masses
similar
body
do
estimating
variables.
relationships
seems
so
and
when
information
these
study
to have
of
2004, Nagy
MR,
where possible,
is the better
error
to
is likely
the problem
of
involved.
of FMR
scales
from
model
to use to quantify
the allometry
of
technique
rate. Reduced
assumes
axis
major
regression
that error variances
in the dependent
and
independent
are equal
to their
variables
true variances.
However,
(1988)
different
taxa
the
FMR
is
the correct
to use
might
mammalian
is
metabolic
McArdle
and
of patterns
make
reasonably
assumptions
approaches
(Table 1), and any given value of b reflects idiosyncrasies
of
exponent,
extent
et al.
1987, Norberg
1996, Nagy
Harvey
et al. 2001,
and Williams
2000, Frappell
method
of metabolic
allometry
for OLS,
observed
RMA,
state
examination
cantly
exponent
the
(Table
1). However,
across
do
differ
exponents
the
of b.
unaffected
are
patterns
while
inevitable
and
the
will depend on the question, which will determine
an
size
other
but
metabolic
significant
heterogeneity
a challenge
to any model
that
effect
or any
3/4,
such studies,
between
et al. 2004),
to explain
these
However,
?
b
2/3,
undermine
environ
the
view,
represents
a single value
predicts
only
Conclusions
similar
that attempt
different.
seems
detail
Rezende
2003,
no
be
necessarily
allometric
relation
variation
important
associated
with
Lovegrove
models
the mechanistic
relationships
of
sacrifice
of
obscures
potentially
a
is not
sake
the
to account
of how properly
for body
important
question
mass
in broad
The
results
effects
analyses.
interspecific
of our analysis
that dogmatic
of b =
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APPENDIX
A
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used
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(Ecological
Archives
E088-019-A1).