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Degrees of freedom in interspecific allometry An adjustment for the effects of phylogenetic constraint.

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Degrees of Freedom in Interspecific Allometry: An Adjustment for
the Effects of Phylogenetic Constraint
Department of Anthropology, Washington Uniuersity, St. Louis, Missouri
Interspecific allometry, Comparative methods,
Phylogenetic constraint, Degrees of freedom, Nested analysis of
The data used in studies of bivariate interspecific allometry
usually violate the assumption of statistical independence. Although the
traits of each species are commonly treated as independent, the expression of
a trait among species within a genus may covary because of shared common
ancestry. The same effect exists for genera within a family and so on up the
phylogenetic hierarchy. Determining sample size by counting data points
overestimates the effective sample size, which then leads to overestimating
the degrees of freedom that should be used in calculating probabilities and
confidence intervals. This results in an inflated Type 1 error rate.
Although some workers (e.g., Felsenstein [19851 Am. Nat. 125:1-15) have
suggested that this issue may invalidate interspecific allometry as a comparative method, a correction for the problem can be approximated with variance
components from a nested analysis of variance. Variance components partition the total variation in the data set among the levels of the nested hierarchy. If the variance component for each nested level is weighted by the number of groups at that level, the sum of these values is an estimate of an
effective sample size for the data set which reflects the effects of phylogenetic
constraint. Analysis of two data sets, using taxonomy to define levels of the
nested hierarchy, suggests that it has been common for published studies of
interspecific allometry to severely overestimate the number of degrees of
Interspecific allometry remains an important comparative method for evaluating questions concerning individual species that are not similarly addressed by the format of most of the newer comparative methods. With the
correction proposed here for estimating degrees of freedom, the major statistical weakness of the procedure is substantially reduced. o 1994 Wiley-Liss, Inc.
For many years, a key quantitative
method for comparative analyses has used
the mean value of traits for species as the
basic unit of raw data. Typically, data on two
traits for each of several species are transformed to logarithms and then examined by
correlation and bivariate regression. This
methodology is usually identified as “interspecific allometry,” particularly when one of
the traits represents the general body size of
the species. In recent years, workers developing new comparative statistical methods
have sometimes considered interspecific allometry as one example of a group of techniques all characterized by analysis across
species, and have identified either interspecific allometry or this group of methods as
“the nonphylogenetic approach (Felsen-
Received March 4,1993;accepted September 8,1993
Address reprint requests to Dr. Richard J.Smith, Department
of Anthropology, Washington University, One Brookings Drive,
St. Louis, MO 63130.
stein, 1988), “the traditional ‘equilibrium’
analysis” (Martins and Garland, 1991),“species regression” (Pagel and Harvey, 19881,or
“TIPS’ (referring to the fact that the data
are derived from the species across the tips
of the phylogeny when represented as a cladogram) (Martins and Garland, 1991).
Although interspecific allometry has been
widely used since the early 1930s (Huxley,
1932) and insights from studies using it
were the subject of at least six books during
the 1980s (Calder, 1984; Jungers, 1985;
McMahon and Bonner, 1983; Peters, 1983;
Reiss, 1989; Schmidt-Nielsen, 1984), in
recent years interspecific allometry has
fallen into great disfavor with some authors.
Harvey and Pagel (1991) have suggested
that some of the assumptions underlying
the method “should be anathema for anyone
who believes in evolution.” Gittleman and
Luh (1992) declared that methods dependent on a simple correlation of traits across
species are “obsolete.” The fundamental
flaw with interspecific allometry that has
lead to these views is straightforward: since
species may share similarities in traits because of common ancestry, as well as because of convergent and parallel evolution,
the species used in the regression equation
may not be independent examples of the
relationship between traits (Pagel and
Harvey, 1988). Simulation studies (Grafen,
1989; Martins and Garland, 1991) have confirmed the expected problem; the number of
species overestimates the degrees of freedom of the equation, resulting in excessive
Type 1 errors, underestimated standard errors, and underestimated confidence intervals.
Since this problem was first discussed by
Clutton-Brock and Harvey (19771, many
new comparative methods have been developed that better meet the statistical implications of the fact that there are similarities in
traits among species due to shared common
ancestry. These methods are sometimes explicitly presented as replacing interspecific
allometry as a comparative method. For example, in the seminal paper from which several variants of new comparative methods
have been derived, Felsenstein (1985) begins by listing six papers that are examples
of studies using methods that his method is
designed to replace. Four of the six (Armstrong, 1983; Damuth, 1981; Martin, 1981;
Pilbeam and Gould, 1974) are textbook
examples of interspecific allometry. Riska
(1991) suggests that “analyses incorporating phylogenetic information are probably to
be preferred over the traditional allometric
regressions,”and Grafen (1989)and Martins
and Garland (1991) directly contrast the
Type 1 error rates of simple correlations
across species with the newer comparative
The objectives of this work are twofold.
First, I will argue that interspecific allometry is not obsolete, and for the foreseeable
future will not be. The interspecific allometric equation, using phylogenetically related
species as individual data points, remains
an important method for addressing questions that have not been addressed by the
newer comparative methods. Second, the
problem with interspecific allometry-an
inflated number of degrees of freedomdoes not require discarding the method.
What is required is an estimate of degrees of
freedom for each data set that reflects the
extent to which phylogenetic constraints
limit variation in the traits under study. A
simple method for doing so is presented
here. This adjustment is also applicable to
those newer comparative methods that average values of traits for several species in
order to evaluate trends across higher nodes
of the phylogenetic hierarchy. It is emphasized that this study is not a critique of the
new comparative methods, many of which
are important and should be used in preference to interspecific allometry in many circumstances.
Proposed alternatives
Considerable attention has been focused
on a comparison between nonphylogenetic
correlations across species and new comparative methods insofar as they are able to
document independent instances of correlated evolution between two traits. Some of
these comparisons have been made using
simulation studies (Grafen, 1989; Martins
and Garland, 19911, and they demonstrate
unequivocally the inflated Type 1error rate
of the type of data sets used for interspecific
allometric equations. However, the fact of
correlated evolution, which is dependent
upon the calculation of statistical probability, is a different issue than that of describing the pattern or form of correlated evolution. Harvey and Pagel (1991), in
introducing the organization of their recent
book, state (their italics): “The sixth chapter
deals not with whether characters have
evolved together, but with the way in which
they show correlated evolution.” This sixth
chapter concerns allometry, and includes
two of the new comparative methods that
appear to meet the goal of replacing interspecific allometry for describing the pattern
of evolution between traits. The many other
new comparative methods that are preferable to interspecific allometry for documenting the presence of correlated evolution are
not the issue here.
The first new method is a generalization
of the solution proposed by Clutton-Brock
and Harvey (1977). In the “nested” ANOVA/
allometry method” (Harvey and Pagel, 1991;
Pagel and Harvey, 1988; Steams, 19921, a
nested analysis of variance is calculated for
each trait t o determine a taxonomic level a t
which a pragmatic balance can be struck between the independence of data points and
the number of data points that will be left for
use in the regression. Determination of the
taxonomic levels that account for most of the
variation can be used to select a single level
at which it is most reasonable to consider
the data points as “independent.” In their
original 1977 examples, Clutton-Brock and
Harvey proposed that using genera as independent data points by averaging the values
for all species within each genus substantially improved the validity of the degrees of
freedom used to evaluate the allometric relationships under consideration in that
study. Clearly, and as always recognized by
all who have used it, the method improves
the nonindependence problem, but does not
eliminate it. Genera within families may
share phylogenetic constraints, as may families within superfamilies, and so on for any
level of the taxonomy within the next higher
The second new method is described a t
length in Harvey and Pagel (1991) as the
“allometry based on independent comparisons method.” They demonstrate that the
paired contrast values derived by Felsenstein’s (1985) method can be used as data
points in an allometric regression equation,
and that the slope derived from plotting contrast scores of one variable against contrast
scores of a second variable will estimate the
same allometric slope as the individual species data, with the benefit of statistical independence between each set of contrast
scores. Thus, putting aside the problem of
unresolved phylogenies, the analysis proceeds by identifying for every species in the
data set the phylogenetically most closely
related other species. These two species are
paired, and the raw data used in the regression equations are the differences between
each set of paired species.
Limitations of the alternatives
Although Harvey and Pagel (1991) illustrate the application of the “nested ANOVA/
allometry method” and the “allometry based
on independent comparisons method for describing the form of a relationship between
two traits, it does not follow that these methods replace all applications of the traditional
interspecific allometric equation as a comparative method. Comparative biology is a
vast discipline with many objectives (Bock,
1989).Establishing the fact of correlated evolution between two or more traits, or the
pattern of that evolution, does not exhaust
all possibilities. Two simple examples will
be given here of other comparative questions
that have traditionally depended upon interspecific allometry but that are not addressed by these new methods. These are
counterexamples to the proposition that interspecific allometry is obsolete.
With both of the proposed alternatives t o
interspecific allometry, the values of traits
in individual species are lost to evaluation
and interpretation. In the nested ANOVA/
allometry approach, several species are averaged until some higher level is reached. In
the independent comparisons approach, differences between two species become the
raw data. These differences are specific to
the particular pair, i.e., the difference value
changes if one species is held constant but
the other one changes. These types of data
are unsatisfactory for the many problems
within comparative biology for which the
species is the question. Some critics of interspecific allometry appear to view the species
selected for a comparative study exclusively
in terms of the utility of the species for addressing issues about the adaptation or evolution of traits (e.g., Huey, 1987). There are,
however, many comparative biologists with
a long-term interest in understanding a few
particular species, whatever the difficulties
may be. For example, some comparative biologists may not be interested in documenting whether or not there is a relationship
between brain size and lifespan, but simply
in whether or not a specific species has a
large brain relative to its lifespan.
In anticipation of one line of criticism, it is
important to recognize that questions about
the status of traits in individual species do
not necessarily assume that the interspecific allometric equation can reveal why a
species has a relatively large brain, or what
a relatively large brain is an adaptation for.
As Gittleman (1989) has noted, “Comparative analyses are not always meant to infer
causation.” Once features are characterized
(e.g., once relative brain size for a species
has been documented), other methods may
be used to understand why this might be so
or how the trait might be used. The central
role of adaptation in evolutionary biology
(Coddington,1988) does not exclude the possibility of other important questions (Bock,
1989). The traits of animals, whether adaptations or exaptations, primitive or derived,
parallelisms or convergences, homologies or
homoplasies, are of interest for the biological role (Bock and von Wahlert, 1965) or current utility (Harvey and Pagel, 1991) that
they serve in the life of the animal. Ridley
(1991) has commented on the awkward consequences of defining traits as adaptations
only when they first evolve, as is common in
discussions concerning the objectives of the
new comparative methods. He points out
that this definition leads to defining the pos-
session of eyes in humans as a constraint,
not an adaptation. It is obvious that those
interested in the biological role of a structure might find this concept of adaptation of
little relevance. Anthony and Kay (1993)
note that the maintenance of a trait in descendant lineages is useful evidence of biologically interesting stabilizing selection.
A second example of a question for which
the new comparative methods cannot yet
substitute for interspecific allometry concerns the use of the equation in some types
of paleontological inference. It is common to
use fragmentary fossil remains to estimate
body mass of extinct species. The widely accepted method of choice (e.g., Damuth and
MacFadden, 1990) involves generating an
interspecific allometric equation using modern species in which some anatomical feature also available on fossil specimens is the
independent variable and body mass of the
extant species is the dependent variable.
The interspecific allometric equation generated with extant species is then used to predict the dependent variable in the extinct
species. The objective is to predict the value
for an individual species, and the equation is
constructed with individual species. Furthermore, it is usually considered desirable
to take advantage of phylogenetic constraint
in generating the equation and to use species closely related to the fossil species of
interest. For example, the relationship between tooth size and body mass may differ
substantially in carnivores and primates.
An equation based on modern primates
would be irrelevant for prediction of a fossil
carnivore, and vise versa. Therefore, equations are often limited to a group of the most
closely related species possible.
It follows from the preceeding examples
that a comparative method that allows for
the retention of the identity of individual
species and their trait values has a place in
comparative biology. Interspecific allometry
is useful, but statistically flawed. It should
be noted that some of the new comparative
methods can also produce estimates of values for individual species, although I am not
aware of any attempts to use them for that
purpose. Steams’s (1983) phylogenetic subtraction method, the autocorrelation analysis proposed by Cheverud et al. (1985) and
fined as the effective sample size (effective
N) for the data set and trait, as opposed to
the traditionally used observed sample size
(observed N). In Felsenstein’s example
(19851, a set of species highly constrained to
the ancestral values could approach an effective sample size of 2, but could not possibly be lower. A set of species without phylogenetic constraint could approach an
effective sample size of 40,but could not possibly be higher.
The nested analysis of variance has been
used often since initially proposed by Clutton-Brock and Harvey (1977) to partition
A CORRECTION FOR THE DEGREES OF the variation in a trait to levels of a nested
taxonomy. It is now well-established and acALLOMETRY
cepted as an effective method for showing
The problem with using individual species the taxonomic level at which phylogenetic
as data points in an interspecific regression correlation occurs (Gittleman and Luh,
is that the values of individual species are 1992). The method for quantifying taxoonly partially free to vary. The magnitude of nomic effects from a nested ANOVA inthis effect reflects the closeness of the phylo- volves using the observed and expected
genetic relationships between species in the mean squares a t each level to calculate varidata set coupled with the extent to which ance components for each level. These variphylogenetic inertia limits the rate with ance components are then expressed as perwhich the particular trait undergoes adap- centage variance components by summing
tive change. Therefore, any correction for them for all levels and expressing each level
phylogenetic constraint must be specific to as a percentage of the total (e.g., Sokal and
the particular set of species and an individ- Rohlf, 1981; Bell, 1989). The percentage
variance components sum to 100% and parual trait.
In a widely used example of the problem tition the total variation between subjects
presented by Felsenstein (19851, two ances- (i.e., species) to levels of the taxonomic hiertral species each give rise to twenty descen- archy.
In the method proposed here, the percentdant species. Typically, an interspecific allometric equation would treat the 40 species age variance components are used to estias 40 independent observations. Felsenstein mate the effects of phylogenetic constraint
asks whether it might be more realistic to on the freedom of species to vary by weightview the data set as having closer to 2 obser- ing the number of groups at each level of the
vations, with 20 highly constrained similar taxonomic hierarchy according to the pervalues in all the descendants of each of the centage of the total variation explained by
two independent ancestors. Clearly, neither that level of the nested analysis of variance.
extreme estimate is true. There are not 40 For example, in a nested ANOVA of a trait
independent observations, because species with no phylogenetic constraint, virtually
within each set of 20 are constrained by the all variation would be assigned to the speancestral value. Neither are there two ob- cies level, and the percentage variance comservations, because within each set of 20, ponent for that level would approach 100%.
there is variation and descendants differ Multiplying the number of species by 1.0
from the ancestral value. If we consider each would result in an effective sample size
species as some fraction of a free observa- equal to the number of species. On the other
tion, varying between 0 and 1.0, a value hand, in a situation such as Felsenstein’s
could be computed between 2 and 40 that hypothetical radiation, a large component of
woud reflect the balance between constraint variation would be attributed to the level
and independent evolution. This value is de- with two nodes. If all variation was due to
modified by Gittleman and Kot (1990), and
Lynch’s (1991) mixed model or maximum
likelihood approach, all allow for the calculation of values for individual species. In
these methods, the values for species would
represent trait values after the removal of
the portion of the trait attributed t o phylogenetic constraint by each method. Thus, interspecific allometry would continue to
serve a different purpose in that the values
of traits relative to each other represent the
removal of covariation with the independent
variable, usually body size.
the two higher nodes, with no variation between the 20 species in each group, the
higher level would have a variance component of 1.0 and would result in a sample size
of two. Consider an example in which species are organized into superfamilies, families, and genera. With percentage variance
components (PVC) from a nested ANOVA
expressed in decimal form and summing to
1.0, the effective sample size would be:
effective N = (# of superfamilies)
(PVC for superfamilies)
(# of families)(PVC for families)
+ ( # of genera)(PVC for genera)
(# of species)(PVCfor species)
Levels of the nested analysis are defined so
that species within genera (the variance
component for species) is defined by the error (residual)variance.
This approach weights the sample size according to the sources of variation in species
values. The maximum effective sample size
is the number of species, and is reduced according to the extent to which higher phylogenetic relationships (which of necessity
have a smaller number of groups) appear to
explain variation among species.
In order to use interspecific allometry as
a comparative method, the procedures
described here require that two nested
ANOVAs (one for each trait) are calculated
as additional, separate steps after the calculation of the interspecific allometric regression. Workers wishing to use this method
would calculate an allometric equation just
as they always have. However, to interpret
the slope and/or constant of the equation, or
the residuals of individual data points, it is
often desirable to make some statistical inferences about them, such as the significance of the bivariate correlation, or the confidence interval for the slope. Calculation of
these statistics requires a sample size, and
here is where the nested ANOVA is applied.
The same species used in the allometric
equation are now organized as a separate,
second analysis into a nested format, and
the procedure outlined earlier in this section
is carried out for each variable. The output
of these nested ANOVAs is not used for any
comparative insights in and of itself, but
solely to estimate an effective sample size
for the analysis that reflects the effects of
phylogenetic constraint. Thus, although
several workers have proposed comparative
methods that are based on a nested ANOVA
structure (e.g., Bell, 1989; Harvey and Pagel, 1991; Page1 and Harvey, 1988),the purpose and use is entirely different here.
Worked examples
I have evaluated this method on two data
sets. Harvey and Clutton-Brock (1985) published data on life history traits for 135 primate species. They analyzed each trait with
a nested ANOVA, partitioning variation
into family, subfamily, genus, and species
levels. As a result of this analysis, they decided to average species values within genera, and generic values within subfamilies.
This reduced their data set for each trait to
19 subfamily values, and they calculated allometric equations with these 19 data
points. Thus, in spite of the fact that some
variation in the trait existed between species within genera, and between genera
within subfamilies, the information that
might be gained from this variation was excluded before analysis began. I repeated the
nested ANOVA with two traits from the
Harvey and Clutton-Brock study, female
body weight and adult brain size. My estimates for the percentage variance components differ slightly from theirs, which is not
unexpected given possible differences in the
algorithm for calculating this measure. Using the formula for calculating an effective N
and my values for variance components, the
effective sample size when regression equations were generated using all available species as separate data points was 21.5 for female body weight and 17.5 for adult brain
weight (Table 1, part A). In addition to confirming that Harvey and Clutton-Brock
were remarkably percipient in selecting the
subfamily level with an N of 19 for their
analysis, the results document that phylogenetic constraint in this type of data set is
very strong, and that previous studies that
have determined significance or confidence
intervals on the basis of observed sample
size need to be reevaluated. However, in using the 19 subfamilies as the sample size
from which statistical significance was cal-
TABLE 1 . Data from nested analysis of uariance used to calculate effectiue sample size from observed sample size
Part A
Body mass
Brain mass
Part B
Body mass
Brain mass
Part C
Body mass
Brain mass
Part D
Body mass
Molar area
PVC = Percentagevariance component:N = Number of groups at the taxonomic level indicated;* = Level not used in the analysis.
Part A = Data from Harvey and Clutton-Brock(1985). Nested ANOVA calculated using the same levels as in their original study.
Part B = Data from Harvey and Clutton-Brock (1985). Data for species combined into generic means. Generic means then combined into
subfamily means. Nested ANOVA calculated with subfamilyvalues.
Part C = Data from Harvey and Clutton-Brock(1985). Nested ANOVA of species values calculatedwith one additional level (superfamilies)from
the analysis reported in Part A.
Part D = Data from Gingerich et al. (1982).
Nested ANOVA calculatedon 81 data points. Taxonomy for classifying species taken from Harvey and
culated, Harvey and Clutton-Brock did not
take into account the fact that they had lost
the variance in genera and species that was
present in their original nested ANOVA. As
they noted, subfamilies are not independent
of their families. I repeated the calculation
of variance components using a data set in
which trait values for the 19 subfamilies
formed the raw data (Table 1, part B). A
substantial portion of the variation is attributed to the differences between families. Using the formula for calculating an effective
N, the value for adult female body weight
becomes (19" .253) + (12" .747) = 13.8. The
effective N for adult brain weight is also
13.8. Thus, in treating subfamilies as independent when they clearly are not, Harvey
and Clutton-Brock have overestimated the
df for their analyses also. This effect should
be evaluated whenever a higher-nodes approach is used, although the method presented here eliminates a major portion of
the logic that justifies higher-nodes methods.
Using the Harvey and Clutton-Brock
data, I also tested the method for differences
in effective N when different taxonomic lev-
els are used to define the nested levels of a
data set. The data for 135 species were evaluated by a four-level nested ANOVA, in
which each species was classified into a
superfamily, family, subfamily and genus
(Table 1, part C). When effective Ns were
calculated with these percentage variance
components, the value for female body
weight decreased from 21.5 to 21.0, and for
adult brain weight from 17.5 to 17.0. Thus,
the method will produce some differences in
the estimated effects of phylogenetic constraint depending upon the taxonomic levels
used in the nested ANOVA for a particular
data set.
The second data set evaluated was one
published by Gingerich et al. (1982) to estimate body mass of fossil primates from tooth
dimensions. They used tooth size data for 43
extant primate species, and generated interspecific allometric equations in which body
mass was the dependent variable. Unlike
Harvey and Clutton-Brock, whose method
reflected the removal of the lower levels of
variance before analysis, Gingerich et al.
(1982) treated mean values for males and
females within species as separate data
points. Thus, Gingerich et al. (1982) calculated confidence intervals for their body
mass estimates assuming that the 86 data
points from 43 primate species produced 86
degrees of freedom. I calculated a four-level
nested ANOVA with their data for both body
mass and lower first molar area, using the
available 81 data points from 42 species.
Families, subfamilies, genera and species
defined the nested levels, and males and females within species were the residual variance. The calculations indicate that the effective N for this data set is 12.7 for body
mass and 13.1 for lower molar area, rather
than the 81 used in the original study (Table
1, part D). One reason for this reduction is
the inflation of the apparent sample size resulting from using the two sexes as separate
values. Separating males and females
rather than using species values accounts
for only a small proportion of the variance in
the nested ANOVA, and using this percentage variance component within species multiplied by the 81 data points within species
contributes only a small amount to the total
df. It should be noted that neither the
Harvey and Clutton-Brock (1985) nor Gingerich et al. (1982) papers were selected for
evaluation because of any particular problems with sample selection or data analysis.
On the contrary, both are well-done studies
and include a wide diversity of primate species, so that the relationship between effective and observed sample size in these studies should be representative or better than
much of the rest of the primatological literature.
The results pointedly confirm the fact
that interspecific allometry has deficiencies
as a comparative method, at the same time
as they illustrate a solution t o the specific
problem of statistical nonindependence that
has been the basis for the strongest criticism. The method for estimating a n appropriate sample size described here should
largely eliminate the problem of an inflated
Type 1 error rate. However, it requires (unavoidably, I believe) a substantial loss of statistical power in comparison with analyses
using newer comparative methods that generate a sample of independent data points
equal to the number of species in the original data set.
Applications, assumptions, and
Degrees of freedom and effective
sample size
Most discussions of the consequences of
phylogenetic constraint on the statistics of
comparative methods have phrased the
problem as a discrepancy between the numbers of species and the degrees of freedom
(do used in statistical calculations. A slight
modification of emphasis has been used
here. The df for a statistic reflects both the
data set and the statistic. When data points
are independent, each one equals one df, but
then depending on the statistic to be calculated, a certain number of df are subtracted.
Phylogenetic constraint has an effect on
only one component of the calculation of df.
It results in an error in the assumption that
each data point is equal to one df. It does not
alter the number of df that are subtracted
for the particular statistic. Therefore, in this
discussion the effects of phylogenetic constraint have been defined as resulting in an
effective sample size that is less than the
number of data points used to calculate the
regression. After an effective sample size is
calculated that reflects the effects of phylogenetic constraint, the df for a statistic is
determined by subtracting an appropriate
number of df from the effective N, rather
than from the observed N.
Combining estimates of effective sample
size for two traits
The calculation of statistics for a bivariate
regression equation (or a bivariate correlation coefficient) requires a single value for
df, not separate values for the x and y variables. #en a nested ANOVA has been used
to select a higher level for grouping species,
it has been routine to evaluate variance
components only for the dependent (y axis)
trait. For the method proposed here, a conservative approach would be to calculate the
phylogenetically reduced effective sample
size for both the x and y axis traits and select
the smaller of two choices: (1)the effective N
calculated for the y axis trait, or (2) the
mean of the effective Ns for the two axes.
TABLE 2. Variance components from nested analysis of variance of residuals from interspecific allometric equations
Part A
Part B
Eff N/Obs N
Eff N/Obs N
1 3 . m1
* = level not used in the analysis.
Part A = Residuals from least squares regression of In lower molar area (y-axis)against In body mass (x-axis).Nested ANOVA calculated with
residuals in natural log fonn. Data from Gingerich et al. (1982).
Part B = Residuals from least squares regression of In adult brain mass (y-axis)against In body mass (x-axis).Nested ANOVA calculated with
residuals in natural log form. Data from Harvey and Clutton-Brock(1985).
Effectivesample size for analyses
with residuals
Following calculation of the interspecific
allometic regression, it is often desirable to
use the residuals as data in further statistical analyses. The regression line has been
described as a “criterion of subtraction”
(Gould, 1975; Smith, 1984) that allows trait
values t o be partitioned into a component
explained by correlated change with size,
and a component reflecting differences that
are independent of size. The effective N for
analyses with residuals should also reflect
the phylogenetic relationships between species. This can be approximated by proceeding with three steps after obtaining the residuals from the allometric equation. First,
a new nested analysis of variance should be
performed on the residuals. Since part of the
phylogenetic constraint on a trait may reflect a constraint on body size that is manifested in the trait, the pattern of variance
components may differ substantially between the original two traits and the residuals from a regression between them. These
variance components from the nested
ANOVA on the residuals should then be
used t o calculate an effective N for the residuals in an identical manner to the calculation of an effective N for a trait. The final
step is to assign each residual a fractional
value of an observation (between 0.0 and
1.0) depending upon the overall relationship
between the observed N and the effective N
calculated for the set of residuals. For example, if the nested analysis of variance of residuals leads to a calculation of an effective
N of 30 for a data set in which there were 40
observations, further analyses with residuals could be interpreted by considering each
residual to contribute 0.75 (30/40) degrees of
freedom. Obviously, this would not adjust
for differences in the density of the phylogenetic relationship in different portions of the
I examined this procedure by calculating
nested ANOVAs with residuals from two regression equations: (1) brain mass against
female body mass with the data from
Harvey and Clutton-Brock (19851, and (2)
lower first molar area against body mass for
the data from Gingerich et al. (1982). As
shown in Table 2, there is a large change in
the taxonomic location of constraint on residuals compared with the original traits
(Table l), and the effective N may be substantially larger for the residuals than it is
with the original traits.
Random effects in nested ANOVA
One question about the method proposed
here (as well as with any use of variance
components from a nested ANOVA) is the
extent to which a nested analysis of variance
takes advantage of random variation in the
data set, inflating the proportion of variance
assigned to higher levels. This was evaluated by calculating percentage variance
components from nested ANOVAs with randomly assigned values. Using the “shuffle”
option in the random number generator of
J M P 2.0.5 (SAS Institute, Inc), I reassigned
the female body mass from Harvey and Clutton-Brock (1985) and the lower molar area
from Gingerich et al. (1982) to species within
their data set at random. Thus, instead of
using random numbers, the actual species
values were used, but they were assigned
randomly to the incorrect species in the data
set. Ten trials of reshuffled values were calculated for each trait, and a three-level
nested ANOVA was used for female body
mass and a four-level nested ANOVA for
molar tooth area. The results of these analyses indicate that variance components are
heavily weighted to the lowest level (residual) variance in a nested ANOVA with random data, as they should be. It is difficult to
interpret the results as percentage variance
components for each level, since in every
trial some (usually most) of the higher levels
had a negative variance component. In only
six of the twenty trials was the sum of variance components at all higher levels combined (excluding only the residual variance)
greater than zero. Making the most conservative interpretation and treating negative
variance components simply as zero variance explained, the minimum value for the
percentage variance attributed to the residual level [the species level with Harvey and
Clutton-Brock (1985) data and the intraspecific level with the Gingerich et al. (1982)
data] was 77% and 65%, respectively, and
the median values for the ten trials were
89%and 83%.Thus, the nested ANOVA may
find some structure in random data for
which no variance should be attributed to
any higher level. Attributing variance to
lower levels increases the estimate of the
effective N, while variance attributed to
higher levels decreases the effective N. The
fact that some variance was attributed to
higher levels in these trials of randomly assigned values indicates that the method proposed here tends toward a conservative estimate of the effective N for a data set.
Phylogeny, taxonomy, and cladistics
It may appear that this method involves a
theoretical preference for rank-based taxonomic classifications over information concerning the cladistic relationship between
species. This is not the case. The method is
based first and foremost on recognition of
the fact that interspecific allometry as a
comparative method is fatally flawed unless
some way is found to incorporate independent information on phylogeny into the statistical analysis. The method does not require that levels of the nested hierarchy are
defined by taxonomic categories. It is possible, and would be preferable, to organize the
species of an allometric data set cladistically, using higher and lower levels of nodes
from a cladogram to define levels of the
nested ANOVA. For some simplified cladograms this is straightforward, but in general, it is often difficult (or impossible) to
organize “real-life” cladograms into 3 or 4
levels and apply a nested ANOVA to them.
Others who have used nested ANOVA as a
comparative method (Bell, 1989; Page1 and
Harvey, 1988)have also used taxonomic categories to define levels. Taxonomy is wellrecognized to often “only provide crude
representations of phylogenetic distance”
(Gittleman, 1993) and should be used with
that understanding. The key problem with
using taxonomy as a representation of phylogenetics in the setting up of a nested
ANOVA is the assumption that taxonomic
groups are monophyletic. It should also be
noted, as Miles and Dunham (1992) pointed
out, that nested ANOVA models assume
that lineage-specific variation in a trait is
evidence for phylogenetic constraint; in
other words, that trait variation correlated
with phylogeny is due to phylogeny. This
will result in a lower estimate of df than is
Unit of analysis
The routine use of mean values for species
as the unit of data in interspecific allometry
is usually in violation of the assumption of
homoscedasticity (equal variances for all observations) that is required €or regression
analysis. Species means are rarely based on
an identical number of individuals within
each species, and these differences in sample size result in different standard errors
for each mean. The solution to this problem
is the method of “weighted least squares”
regression, which, although well-established in the statistical literature (e.g., Darlington, 1990; Draper and Smith, 1981), has
not received attention in applied allometric
studies. While the use of weighted least
squares (LS) regression is relatively
straightforward, and is readily available in
statistical packages, I am unaware of any
literature that clarifies the management of
weighted values in nested ANOVA. Furthermore, the literature on weighted regression
appears specific to least squares, and it is
unclear whether or not the method is appropriate for other line-fitting criteria, such as
reduced major axis or major axis. Although
the issue has been ignored in literally thousands of interspecific allometric studies to
date, one solution might be to use individuals within species as the data points in interspecific allometry, with the effective sample
size adjusted by the nested ANOVA method.
If this is done, and individuals within species define the lowest (subordinate) level of
the nested design, then the treatment of sex
differences is an important consideration.
There would appear to be two options for
dealing with sex differences: (1) simply list
each individual as a case within a species,
ignoring sex, or (2) within each species, define a nested level with two groups (males
and females) and list values separately by
sex. The latter choice would create a fixed
effect for sex within species, and the overall
nested ANOVA would be a “mixed model”
with the single fixed effect of sex within species and random effects a t all other levels.
This structure is problematical. Sokal and
Rohlf (1981) suggest that it is a “crucial
point” that only the highest level of a nested
ANOVA be fixed, and that all subordinate
levels must be random. Although there may
be computations possible for fixed effects
within random effects (McKone and Lively,
1993), fixed effects in nested ANOVA remain controversial in any case (Zucker,
1990).The only approach that can be recommended is to ignore sex and list all individuals as simple cases within a species.
It should also be noted that the use of a
nested ANOVA to estimate an effective N
allows for yet another permutation in the
selection of data points for interspecific allometry. If multiple studies of a single species allow for several independent estimates
of a species mean, all of these estimates can
be used, with the nested ANOVA con-
structed so that different estimates of a trait
within single species define the lowest level
of the nested hierarchy.
Sample selection
There has been some discussion of the fact
that interspecific allometric equations can
be calculated on a poorly selected set of species. Clutton-Brock and Harvey (1984) use
the example of a study by Millar (1977) in
which the 98 species supposedly representing the entire Class Mammalia included 81
species of rodents. It is routine to find studies of the primates heavily biased toward
species of Old World monkeys. However,
there is nothing unique about interspecific
allometry in being either susceptible or influenced by this problem. Naive selection of
species results in a bias toward oversampled
taxa whether the study involves independent contrasts, directional methods, or interspecific allometry. Consider the case of a
sample of mammals overweighted with rodents. In an analysis using new comparative
methods based on independent or directional contrasts, the number of contrasts between members of Rodentia will bias the
results according to the particular pattern
of pairwise relationship or evolutionary
change experienced in this order, just as the
individual species values will bias a simple
allometric regression line. It occasionally
seems that some of those writing about comparative methods assume naive species selection by those doing allometry, and balanced species selection by those using newer
approaches. Obviously, this is not inherent
in the techniques. The many recent discussions of phylogenetic constraint make it
clear that reasonable balance and diversity
is necessary in the selection of species for
any comparative method.
The issue of balance and diversity in selection of species involves an attempt to appropriately distribute species within each taxonomic level (see also Gittleman, 1989). If a
study is concerned with an entire taxonomic
class, the first step is to evaluate the number of orders that are represented, and then
the number of species within each order.
There should be an attempt to achieve diver-
therefore to find a way to incorporate phylogenetic information into the statistical inferences of an interspecific allometric equation,
and thereby salvage a technique that some
comparative biologists have discarded without concern. The estimation of effective sample size by weighting variance components
from a nested analysis of variance uses a
procedure (nested ANOVA) that has been
applied for other purposes in several of the
newer comparative methods. It results in a
useful estimate of the phylogenetic effects
that interspecific allometry must account
for. The results also indicate, however, that
phylogenetic constraint results in a large decrease in statistical power when typical data
sets are analyzed by interspecific allometry.
There is thus a substantial cost to the continued use of interspecific allometry on
groups of closely related species. Finally, the
For some time now, a body of literature implications of phylogenetic relationships
has been developing in which it is suggested among species on the statistics of comparathat interspecific allometry as routinely pre- tive methods are sufficiently well-recogformed is invalid as a statistical technique nized and accepted that statistical inference
(e.g., Clutton-Brock and Harvey, 1977; from an interspecific allometric equation
Felsenstein, 1985; Harvey and Pagel, 1991). without any consideration of the effects of
Coinciding with this conclusion has been the phylogenetic constraint should be unacceptdevelopment of a large variety of new com- able.
parative statistical methods that take into
account what interspecific allometry does
not; namely, the fact that species resemble
I thank Jim Cheverud, John Gittleman,
each other because of phylogenetic con- and Tab Rasmussen for helpful comments
straint as well as because of convergent and on earlier drafts of the manuscript.
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effect, freedom, constraint, interspecific, degree, adjustment, phylogenetic, allometric
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