Lecture 4:
Allometry
and
Scaling
with examples of
correlation and regression
1
Allometry = "different measure" = the study of
how and why properties of organisms change in
regular ways in relation to body size.
Scaling = essentially a synonymous term in
biology, but used more in engineering.
Can be studied at three levels:
1. ontogenetic (intraspecific) = growth
relationships during development,
between two traits or
between one trait and the whole organism:
a. longitudinal = follow individuals
b. cross-sectional = mixed-age sample
2
Allometric Growth in Human Beings:
Juveniles are not Scale Models of Adults
(http://en.wikipedia.org/wiki/Scale_model)
3
Zebrafish:
again, large
changes in
shape are
occurring ...
Randall, D., W. Burggren, and K. French. 2002. Eckert animal physiology: mechanisms and adaptations. 5th ed. W. H. Freeman and Co., New York.
4
Allometric growth is differential rates of
growth of two or more traits.
It is often well described by the equation:
a
Y = bX
where Y is one trait (e.g., metabolic rate),
b is a constant,
a is the "allometric" or "scaling" coefficient,
and X is the other trait (often a measure of
body size, e.g., mass or length).
5
a
Y = bX
This equation describes a logarithmic
relationship.
It can be made linear by taking the logs of the
values measured for each trait, or by plotting on
log-log graph paper.
If we take logs, then the equation becomes:
log Y = log b + a log X
This equation describes a straight line
with a being the slope.
6
"Reptiles" (not a good term, phylogenetically
speaking) are a good model for studies of
ontogenetic allometry because:
1. they have little or no parental care, so newly
hatched or born offspring must fend for
themselves, forage, escape from predators, etc.;
2. huge size range
from juvenile to
adult;
3. no metamorphosis
to complicate the
picture.
7
An Example of Studying Ontogenetic Allometry:
Amphibolurus (Ctenophorus) nuchalis
from central Australia
Australian
National Bird
Recent Hatchling
Adult Male
Adult Female
8
Fowler’s Gap: a Research Station and Working
Sheep Ranch run by the
University of New South Wales
9
Fowler’s Gap: a Research Station and Working
Sheep Ranch run by the
University of New South Wales
10
Area around Fowler’s Gap: Lizards often Bask on
Fence Posts, or use them for Territorial Outposts
11
Fences can be Hazardous to Emu and Kangaroo!
12
The Wet Season can be Hazardous to Vehicles!
13
Lizard Burrows can be in Surprising Places
14
Amphibolurus (Ctenophorus) nuchalis
Convergent Evolution
with North American
Dipsosaurus dorsalis
(desert iguana)
15
Amphibolurus (Ctenophorus) nuchalis
1.6
Liver Mass (grams)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
10
20
30
Body Mass (grams)
40
50
16
Amphibolurus (Ctenophorus) nuchalis
1.6
Liver Mass (grams)
1.4
Y = 0.03355X0.905
1.2
1
0.8
0.6
Exponent (slope of
line) is < 1, so liver
exhibits negative
ontogenetic allometry
0.4
0.2
0
0
10
20
30
Body Mass (grams)
40
50
17
Amphibolurus (Ctenophorus) nuchalis
0.5
log Liver Mass (grams)
log base 10 version
0
Y = - 1.474 + 0.905X
2
R = 0.815
-0.5
-1
95% Confidence Interval
on slope is 0.747 - 1.063
-1.5
-2
0
0.5
1
1.5
log Body Mass (grams)
2
18
Liver Mass (grams)
Logarithmic axes.
1
0.1
Note that the log transform also:
1. shrinks large values, expands small
2. homogenizes variances
0.01
1
10
Body Mass (grams)
100
19
Amphibolurus (Ctenophorus) nuchalis
Thigh Muscle Mass (grams)
1.4
Y = 0.01314X1.158
1.2
1
0.8
0.6
Exponent (slope of line)
is > 1, so thigh muscle
exhibits positive
ontogenetic allometry
0.4
0.2
0
0
10
20
30
40
Body Mass (grams)
50
60
20
Amphibolurus (Ctenophorus) nuchalis
0.5
log Thigh Mass (g)
Y = - 1.881 + 1.158X
2
R = 0.984
0
-0.5
-1
95% Confidence Interval
on slope is 1.104 - 1.212
-1.5
Juveniles are not just
scale models of adults!
-2
0
0.5
1
1.5
log Body Mass (grams)
2
21
Amphibolurus (Ctenophorus) nuchalis
40
Hematocrit (%)
35
30
25
20
15
10
5
0
0
10
20
30
Body Mass (grams)
40
50
22
Amphibolurus (Ctenophorus) nuchalis
1.7
Relationship is non-linear even on log-log scale.
log Hematocrit (%)
1.6
1.5
1.4
1.3
1.2
1.1
1
Again, juveniles are not
just miniature adults!
0.9
0.8
0.7
0
0.5
1
1.5
log Body Mass (grams)
2
23
Non-linear ontogenetic allometries (on log-log
scale) of physiological traits also occur in garter
snakes and water snakes,
and in many amphibians around metamorphosis.
The common garter
snake, Thamnophis
sirtalis
24
Rattlesnake tail-shaker muscle contraction frequency varies across ontogeny
in a highly non-linear fashion.
http://www.sloanmonster.com/images/snake2.jpg
https://c2.staticflickr.com/2/1331/794381822_d015c02a30.jpg
Moon, B. R., and A. Tullis. 2006. The ontogeny of contractile performance and metabolic capacity
in a high-frequency muscle. Physiological and Biochemical Zoology 79:20-30.
25
Muscle contractile frequency is related to muscle
aerobic capacity across ontogenetic development.
Moon, B. R., and A. Tullis. 2006. The ontogeny of contractile performance and metabolic capacity
in a high-frequency muscle. Physiological and Biochemical Zoology 79:20-30.
26
An Altricial Mammal, the house mouse
Randall, D., W. Burggren, and K. French. 2002. Eckert animal physiology: mechanisms and adaptations. 5th ed. W. H. Freeman and Co., New York.
27
Some Differences between Hatchling and Adult
Amphibolurus (Ctenophorus) nuchalis:
Ratio
50/1
1g
50 g
Liver, % body mass
3.4
2.3
0.69
Heart, % body mass
0.27
0.40
1.51
Thigh, % body mass
1.31
2.44
1.86
Hindlimb span/SVL
1.52
1.25
0.82
Hematocrit (%)
6.0
20.7
3.44
How might these differences affect organismal
performance?
Stopped here 13 Jan. 2015
28
Maximal Sprinting Speed (km/h)
Amphibolurus (Ctenophorus) nuchalis
Body Mass (g)
Started here 15 Jan. 2015
Closed circles are juveniles and adult males. Closed triangles are
females. Line is for all of these animals.
Open circles are gravid females. Open squares are long-term captives.
Star is adult male captured missing lower portion of right leg below knee.
29
Amphibolurus (Ctenophorus) nuchalis
Closed circles and solid
regression line are
January animals.
Open circles and dashed
line are March animals.
Open triangles represent
4 gravid females.
Star is adult male
captured missing lower
portion of right leg below
knee.
Garland, T., Jr., and P. L. Else. 1987. Seasonal, sexual, and individual variation in endurance and activity metabolism
in lizards. Am. J. Physiol. 252 (Regulatory Integrative Comp. Physiol. 21):R439-R449.
30
Take-home Messages for the Lizard
Ontogenetic Allometry Example:
1. Juveniles are not "scale models" of adults;
they are not just miniature adults.
2. They differ in both shape and physiological
functions.
3. The scaling of locomotor performance (speed,
stamina) could not have been predicted simply
from the scaling of the lower-level traits
presented (e.g., muscle mass, hematocrit).
4. However, development of mechanistic models
may allow us to get there in the future …
31
Additional levels at which allometry can be
studied:
2. static = size relationships of traits among
individuals of the same age (typically adults)
3. evolutionary (interspecific) = size relationships
among species
This is a perennial favorite of comparative
and ecological physiologists!
It pervades the fields.
Many books have been written.
32
Books on Allometry (mainly interspecific):
Brown, J. H., and G. B. West, eds. 2000. Scaling in
biology. Oxford Univ. Press, New York.
Calder, W. A. 1984. Size, function and life history.
Harvard Univ. Press, Cambridge.
Peters, R. H. 1983. The ecological implications of body
size. Cambridge Univ. Press, Cambridge.
Reiss, M. J. 1989. The allometry of growth and
reproduction. Cambridge Univ. Press, Cambridge.
Schmidt-Nielsen, K. 1984. Scaling: why is animal size
so important? Cambridge Univ. Press, Cambridge.
(You do not need to remember all of these,
but note that you have a reading from the last one.)
33
Allometry also Occurs at Behavioral Levels
log10 Home Range Area (km2)
49 Species of Mammals
3
Carnivora
2
Eat
Herbivores
1
0
ungulates
-1
Slope = 1.26
(from a phylogenetic analysis )
-2
0
1
2
Eat Plants
3
log10 Body Mass (kg)
Garland, T., Jr., A. W. Dickerman, C. M. Janis, and J. A. Jones. 1993. Phylogenetic analysis of covariance by computer simulation. Systematic Biology 42:265-292.
See also: Dial, K. P., E. Greene, and D. J. Irschick. 2008. Allometry of behavior. Trends in Ecology & Evolution 23:394-401.
34
Allometry also Occurs at an Ecological Level
solid line =
slope of -0.75
Damuth, J. 2007. A macroevolutionary explanation for energy equivalence in the scaling of body size and population density.
American Naturalist 169:621-631.
35
Allometry:
Predictions from
First Principles;
A Tool to Understand
Organismal Design
and Adaptation
36
Scaling Relationships Based on First Principles
For geometrically similar objects (scale models):
Relationships with Length
Circumference  Length1
1
(for a circle, circumference  2 * π * radius )
2
Cross-sectional Area  Length
(for a circle, area  π * radius2)
Surface Area  Length2
Volume  Length3
3
(for a sphere, volume  4/3 π * radius )
3
Mass  Length
(assuming that density remains constant)
37
Consider a cube ...
Length
C-S Area
Circumference
Surface
Area
Volume
S.A./Vol.
1
1
4
2
4
8
3
9
12
6
24
54
1
6
8
3
27
2
38
Relationships with Mass
Volume is proportional to Mass1.00
0.33
Length is proportional to Mass
Surface Area is proportional to Mass0.67
Predictions for geometrically similar animals:
1.00
Blood Volume is proportional to Mass
Mass of Organ is proportional to Body Mass1.00
0.33
Limb Length is proportional to Body Mass
Skin, Gill or Lung Surface Area  Mass0.67
39
Allometric Expectations Based on First Principles
for Geometrically Similar Objects (scale models)
serve as Null Models for Comparison with Real
Organisms.
Deviations from "isometry" may indicate how
evolution has modified organisms from geometric
similarity in order to maintain (more or less)
functional abilities across a range of body sizes.
(Only may indicate because this presumes that
evolutionary changes in size because of random
genetic drift, in the absence of selection, would
occur along lines of geometric similarity …)
40
Example: Mammalian Skeletal Mass
As body mass becomes larger, would need bone
strength to keep up.
Strength  cross-sectional area, so would need
bone cross-sectional area  M1.00 to support the
load.
Mass of skeleton would be M1.00 X M0.33 = M1.33
needed c-s area X length
Empirical result: skeletal mass  M1.08+0.04
So, large mammals should not be able to do what
small ones do (unless all are "over-designed").
In fact, large mammals have more upright postures,
less dynamic & less risky locomotor behavior
(elephants don’t jump).
41
Example: Metabolic Rate
Amount of (metabolically active) tissue should
dictate the rate at which an animal produces heat
(or consumes oxygen).
Oxygen consumption at rest or during maximum
exertion should be  M1.00
However, respiratory exchange surface area
should be  M0.67
"Houston, we have a problem …"
42
2
Expected
Surface
Area, e.g.,
lung, gill
 M0.67
Max. O2
consumption
is typically
10-fold higher
than Resting
Expected
Maximal
Metabolic
Rate  M1
1
0
Expected
Resting
Metabolic
Rate  M1
-1
-2
-2
-1
0
1
2
log10 scale,
Body Mass
on X axis
43
What gives?
Metabolic rate typically scales with an exponent
less than unity, often ~ 0.7-0.8 (many exceptions).
44
Randall, D., W. Burggren, and K. French. 2002. Eckert animal physiology: mechanisms and adaptations. 5th ed. W. H. Freeman and Co., New York.
45
What gives?
Metabolic rate typically scales with an exponent
less than unity, often ~ 0.7-0.8 (many exceptions).
Gas exchange surface area often scales with an
exponent greater than 0.67.
But, the slopes do not always match.
Example: mammalian maximal oxygen
consumption (measured on motorized treadmill)
scales as M0.872
46
Scaling of Maximal Metabolic Rate (Exercise) in Mammals
Weibel, E. R., L. D. Bacigalupe, B. Schmitt, and H. Hoppeler. 2004. Allometric scaling of maximal metabolic rate in
mammals: muscle aerobic capacity as determinant factor. Respiratory Physiology & Neurobiology 140:115-132.
47
What gives?
Metabolic rate typically scales with an exponent
less than unity, often ~ 0.7-0.8 (many exceptions).
Gas exchange surface area often scales with an
exponent greater than 0.67.
But, the slopes do not always match.
Example: mammalian maximal oxygen
consumption (measured on motorized treadmill)
scales as M0.872
whereas pulmonary diffusing capacity
(measured morphometrically) scales as M1.084
So, large-bodied species have excess capacity.
48
Why the apparent excess capacity
in large mammals?
Presumably, other things vary allometrically,
such as:
blood volume
blood oxygen carrying capacity
(hemoglobin, hematocrit)
oxygen unloading characteristics (shape of
oxygen dissociation curve, e.g., P50)
capillary densities
capillary transit times
mitochondrial volume densities
49
Resting and Maximal
Metabolic Rates do not
always Scale in Parallel.
Example: Amphibolurus
(Ctenophorus) nuchalis
ontogenetic allometry
VO2max scales as M0.948
95% C.I. on slope = 0.885 - 1.011
S.M.R. scales as M0.830
95% C.I. on slope = 0.783 - 0.877
50
What about fasting endurance?
All animals "fast" for some periods of time, even if it is
just overnight.
Based on what you have learned so far, how would you
expect fasting endurance (how long they can fast) to vary
with body size?
What are the main components of fasting endurance?
rate of energy use (metabolic rate)
energy stores
(stomach or crop contents, fat, glycogen,
but also protein)
In general, large animals should be able to fast for longer
periods of time, and this is supported by empirical
evidence, e.g., Millar, J. S., and G. J. Hickling. 1990. Fasting endurance and the
evolution of mammalian body size. Functional Ecology 4:5-12.
51
Many Suborganismal Traits Vary Allometrically
log Heart = 0.791 + 0.966 log Body Mass
95% C.I. on slope = (0.951, 0.982)
So, the heart is a bit
smaller, relatively speaking,
in larger mammals.
181 Species
of Mammals
C:\Heart\Heart282.ppt
52
Using Allometry as a Tool to Understand
Organismal Design and Adaptation
Deviations from the empirical line of best fit,
for a given kind of animal, may indicate special
adaptations.
Ex.: pronghorn have high maximal rate of
oxygen consumption (VO2max)
Ex.: humans have large brains
As shown on next 2 slides …
53
"Residual" =
vertical
deviation from
least-squares
regression line
Bat
54
Liem et al. 2001,
Focus Box 14-2
55
Using Allometry as a Tool to Understand
Organismal Design and Adaptation
These deviations call for explanations at both
proximate and ultimate levels.
Ultimate:
Why did natural or sexual selection make them
that way, or was it just random genetic drift?
e.g., apparently, increasing intelligence was
favored in human ancestors
e.g., perhaps pronghorn use their high aerobic
capacity when escaping from predators,
or perhaps males "show off" for females
56
Using Allometry as a Tool to Understand
Organismal Design and Adaptation
Proximate:
How do they do it in terms of morphology and
physiology?
e.g., maybe pronghorn have large lungs, high
hemoglobin levels, large hearts, lots of
mitochondria in their muscles ...
57
Metabolic Rate is Often Expressed per unit Mass
of Tissue (e.g., per gram body mass)
If Whole-animal BMR  M0.75
Then Mass-specific BMR  M-0.25
58
Mass-specific metabolic rate scales about as M-0.25
Randall, D., W. Burggren, and K. French. 2002. Eckert animal physiology: mechanisms and adaptations. 5th ed. W. H. Freeman and Co., New York.
59
M0.75
M-0.25
Randall, D., W. Burggren, and K. French. 2002. Eckert animal physiology: mechanisms and adaptations. 5th ed. W. H. Freeman and Co., New York.
60
Log transformation linearizes the relationships.
This is easier to analyze by least-squares linear regression.
Slope gives the "allometric scaling exponent."
Randall, D., W. Burggren, and K. French. 2002. Eckert animal physiology: mechanisms and adaptations. 5th ed. W. H. Freeman and Co., New York.
61
Statistical
Tutorial
62
Correlation (bivariate) - relationship between two
traits or variables:
Can be positive or negative
Ranges from -1 to +1
Usually assumed to be linear for purposes
of statistical testing
Pearson product-moment correlation
assumes bivariate normality;
denoted as r or R
Spearman rank correlation is a
"nonparametric" alternative
By itself, correlation does not indicate causation!
63
3
N = 50
2
1
R = 0.066
0
-1
2-tailed
P = 0.649
-2
-3
-3
-2
-1
0
1
2
3
64
3
N = 50
2
1
R = 0.293
0
-1
2-tailed
P = 0.039
-2
-3
-3
-2
-1
0
1
2
3
65
3
N = 50
2
1
R = 0.517
0
-1
2-tailed
P = 0.0001
-2
-3
-3
-2
-1
0
1
2
3
66
3
N = 50
2
1
R = 0.747
0
-1
2-tailed
P = 5*10-10
-2
-3
-3
-2
-1
0
1
2
3
67
3
N = 50
2
1
R = 0.894
0
-1
2-tailed
P = 3*10-18
-2
-3
-3
-2
-1
0
1
2
3
Stopped here 15 Jan. 2015
68
Regression - one type of line of best fit:
More formally, "least-squares linear regression"
Distinguishes between "dependent" (Y) and
"independent" (X) variable
Typically used when some degree of causality
is presumed
Minimizes sum of the squared vertical deviations
from the line = residuals
Passes through point mean X, mean Y
Coefficient of determination (r2 or R2) indicates
how much of the variance in Y is explained
by variance in X
Started here 20 Jan. 2015
69
Assumes no measurement error in the
independent (X) variable
If have measurement error in X variable,
then underestimates the slope
Not really appropriate in many biological studies
(including allometry) but used anyway
Various alternatives to ordinary least-squares
(OLS) regression lines (sometimes called Model II
regressions):
(For more details, see: http://www.unc.edu/courses/2007spring/biol/145/001/docs/lectures/Nov5.html)
70
Reduced Major Axis:
line minimizes sum of the area of right triangles
whose legs are the horizontal and vertical
deviations
can be computed as geometric mean of OLS slope
of Y on X and X on Y;
can also be computed as least-squares regression
slope divided by r
71
Major Axis (first principal component):
line minimizes sum of the squared perpendicular
deviations from the line
72
3
Least-squares linear regression
N = 50
2
1
0
-1
y = 0.0709x - 0.0487
2
R = 0.0043
-2
-3
-3
-2
-1
0
1
2
3
73
3
Least-squares linear regression
N = 50
2
1
0
-1
y = 0.3187x - 0.0474
2
R = 0.0861
-2
-3
-3
-2
-1
0
1
2
3
74
3
Least-squares linear regression
N = 50
2
Note that a Reduced
Major Axis or Major
Axis line would be
considerably steeper.
1
For RMA,
0.5615/0.5173 =
1.0854
0
-1
y = 0.5615x - 0.0424
2
R = 0.2676
-2
-3
-3
-2
-1
0
1
2
3
75
3
Least-squares linear regression
N = 50
2
1
0
-1
y = 0.7969x - 0.0324
2
R = 0.5579
-2
-3
-3
-2
-1
0
1
2
3
76
3
Least-squares linear regression
N = 50
2
1
0
-1
y = 0.9309x - 0.0213
2
R = 0.7986
-2
-3
-3
-2
-1
0
1
2
3
77
Using Residuals to Factor out the
Effects of Variation in Body Size:
Often a better choice
than using ratios, as shown in the
following lizard example.
78
Habitat Use and Relative Leg Length
in Australian Lizards (skinks)
R2 = 56.3%
P = 0.002
79
Compute Residuals to Remove Relation with Body Size
Vertical
deviations are
"residuals"
80
Compute Residuals to Remove Relation with Body Size
These lizards deviate from geometric similarity
(slope = 1), so computing a ratio of Hindleg
Length/SVL does not factor out body size
Vertical
deviations are
"residuals"
81
Relation is Weaker after Removing
Correlations with Body Size
82
t-test - statistical test based on the t distribution.
Most commonly, a comparison of the mean
(average) value of samples from two populations:
Are the means "significantly" different?
Statistically "significant" traditionally taken
as P < 0.05
1 in 20 chance of getting a difference that big
by chance alone when you sample
randomly from two populations that
actually have the same mean
1 in 20 times you will get a "significant"
difference when there really is not one,
i.e., commit a Type I error
83
ANOVA - statistical test used to compare mean
values of 2 or more populations.
Based on the F distribution.
F = t squared.
84
ANCOVA - like an ANOVA, but simultaneously
"controls" for a continuously valued X variable,
such as body size, body tempeature or age.
Solid lines are ANCOVA results comparing Carnivora with other
mammals (slopes forced to be parallel).
American Naturalist (1983) 121:571-587.
85
Extra Slides
Follow ...
86
Add something on:
learning outcomes
the scientific method
levels of analysis
MPBF
proximate vs. ultimate causation
4 ways to study evolution:
CM
Biology of Natural Populations
Selection Experiments
Theoretical Models
Plasticity? Epigenetics?
https://www.msu.edu/.../Methods&Levels.
ppt from ZOL 313 May 14, 2008
87
4 Ways to Study Physiological Evolution
1. Phylogenetic Comparisons of Species (or populations)
Shows what has happened in past evolution
2. Biology of Natural Populations:
extent of individual variation (repeatability)
heritability and genetic correlations
natural and sexual selection
field manipulations and introductions
Shows present evolution in action
3. Selection Experiments
Shows, experimentally, what might happen during future
evolution
4. Compare Real Organisms with Theoretical Models
Shows how close selection can get to producing optimal
solutions
88
Allometry of Muscular Power (may constrain behavior)
Figure 2. Vertical escape-flight performance measured from high-speed video of species of three orders of birds. Body-massspecific power output is proportional to approximately M-0.3 (R2 = 0.82) across three orders of magnitude of mass.
Galliformes data from Ref. [27]; Columbiformes and passerine data from Refs [38,39]. Highlighted species: ruby-crowned kinglet
(Regulus calendula), American crow (Corvus brachyrhynchus) and American turkey (Meleagris gallopavo).
Dial, K. P., E. Green, and D. J. Irschick. 2008. Allometry of behavior. Trends in Ecology and Evolution 23:394-401.
89
Many Suborganismal Traits Vary Allometrically
Turner, N., K. L. Haga, P. L.
Else, and A. J. Hulbert. 2006.
Scaling of Na+,K+-ATPase
molecular activity and
membrane fatty acid
composition in mammalian and
avian hearts.
Physiological and Biochemical
Zoology 79:522-467.
[DOS PDTREE and independent contrasts with
divergence times]
90
Many Suborganismal Traits Vary Allometrically
Hulbert, A. J., and P. L. Else.
2005. Membranes and the
setting of energy demand.
Journal of Experimental
Biology 208:1593-1599.
91
Borrell, B. J. 2007. Scaling of nectar foraging in orchid bees.
American Naturalist 169:569-580. [independent contrasts in
Mesquite, discussion of measurement error and RMA]
Damuth, J. 2007. A macroevolutionary explanation for energy
equivalence in the scaling of body Size and population density.
American Naturalist 169:621-631.
Millar, J. S., and G. J. Hickling. 1990. Fasting endurance and the
evolution of mammalian body size. Functional Ecology 4:5-12.
Herrel, A., and J. C. O’Reilly. 2006. Ontogenetic scaling of bite
force in lizards and turtles. Physiological and Biochemical
Zoology 79:31-42.
92
Jayne, B. C., and A. F. Bennett. 1990. Scaling of speed and endurance in garter snakes: a comparison
of cross-sectional and longitudinal allometries. J. Zool., Lond. 220:257-277.
Schilling, N. 2005. Ontogenetic development of locomotion in small mammals – a kinematic study. Journal of
Experimental Biology 208:4013-4034.
Schilling, N., and A. Petrovitch. 2006. Postnatal allometry of the skeleton in Tupaia glis (Scandentia: Tupaiidae)
and Galea musteloides (Rodentia: Caviidae) – A test of the three-segment limb hypothesis. Zoology
109:148-163.
Carpenter, F. L., and R. E. MacMillen. 1976. Energetic cost of feeding territories in an Hawaiian
honeycreeper. Oecologia 26:213-223.
MacMillen, R. E. 1981. Nonconformance of standard metabolic rate with body mass in Hawaiian
Honeycreepers. Oecologia 49:340-343.
Hulbert, A. J., and R. E. MacMillen. 1988. The influence of ambient temperature, seed composition and
body size on water balance and seed selection in coexisting heteromyid rodents. Oecologia
75:521-526.
Lawhon, D. K., and M. S. Hafner. 1981. Tactile discriminatory ability and foraging strategies in
kangaroo rats and pocket mice (Rodentia: Heteromyidae). Oecologia (Berlin) 50:303-309.
Price, M. V., and K. M. Heinz. 1984. Effects of body size, seed density, and soil characteristics on rates
of seed harvest by heteromyid rodents. Oecologia 61:420-425.
MacMillen, R. E., and D. S. Hinds. 1983. Water regulatory efficiency in heteromyid rodents: a model
and its application. Ecology 64:152-164.
Morton, S. R., D. S. Hinds and R. E. MacMillen. 1980. Cheek pouch capacity in heteromyid rodents.
Oecologia 46:143-146.
93
94
Amphibolurus (Ctenophorus) nuchalis
Thigh Muscle Mass (grams)
10
logY = -1.881 + 1.158 logX
95% Confidence Interval
on slope is 1.104 - 1.212
1
0.1
Juveniles are not just
scale-models of adults!
0.01
1
10
Body Mass (grams)
95
Amphibolurus (Ctenophorus) nuchalis
log S.M.R. (ml O2/h)
1.188122558
y = 0.8136x - 0.3735
2
R = 0.9419
0.792081706
0.396040853
95% Confidence Interval
on slope is xx
0
Juveniles are not just
scale-models of adults!
-0.396040853
0
0.5
1
1.5
log Body Mass (grams)
2
96
Examples of evolutionary (interspecific) allometry
… we will return to this later ...
97
3
Volume  Length
98