Phylogenetic approaches in comparative physiology
1 Department of Biology, University of California, Riverside, CA 92521,
USA
2 Department of Ecology and Evolutionary Biology, University of California,
Irvine, CA 92697, USA
* Author for correspondence (e-mail: abennett{at}uci.edu)
Accepted 13 June 2005
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Summary |
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Key words: allometry, comparative method, evolutionary physiology, model of evolution, phylogeny, statistical analysis
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Introduction |
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Comparative methods have been radically restructured over the past two
decades, and now routinely incorporate both phylogenetic information and
explicit models of character evolution. Indeed, Sanford et al.
(2002) suggest that this new
emphasis be termed the `comparative phylogenetic method'. As outlined
in Blomberg and Garland
(2002
), this revolution in
comparative phylogenetic methodology followed from several conceptual
advances: (1) adaptation should not be casually inferred from comparative
data; (2) the incorporation of phylogenetic information increases both the
quality and even the type of inference from comparative data alone; (3)
because all organisms are differentially related to each other, taxa cannot be
assumed to be independent of each other for statistical purposes; (4)
statistical analyses of comparative data must assume some model of character
evolution for effective inference; (5) taxa used in comparative analyses
should be chosen in regard to their phylogenetic affinities as well as the
area of functional investigation; and (6) even phylogenetically based
comparisons are purely correlational and inferences of causation drawn from
them can be enhanced by other approaches, including experimental
manipulations.
To expand on some of these points, `quality' in point 2 includes the simple
fact that adding an independent estimate of phylogenetic relationships to a
comparative analysis increases often greatly the amount of
basic data that is brought to bear on a given question, whereas `type' refers
to analyses that are simply impossible without a phylogenetic perspective,
such as reconstructing ancestral values or comparing rates of evolution among
lineages. Although phylogenetic information and a suitable analytical method
may allow any comparative data set to be `rescued' from phylogenetic
nonindependence (e.g. avoid inflated Type I error rates; point 3),
phylogenetically informed choice of species (point 5) can accomplish more,
such as actually increasing statistical power to detect relationships among
traits (Garland et al., 1993;
Garland, 2001
). Finally, we
note that point 6 was recognized long ago, but has been re-emphasized as
phylogenetically explicit methods of statistical inference have been developed
(e.g. see Lauder, 1990
;
Garland and Adolph, 1994
;
Leroi et al., 1994
;
Autumn et al., 2002
).
The intent of this commentary is to provide a review of some advances that
have occurred in the comparative method, with an emphasis on their place in
comparative physiology. We examine the underlying reasons for the
incorporation of phylogenetic information into comparative studies. In an
Appendix, we give a brief overview of the three most commonly used and best
understood phylogenetically based statistical methods: independent contrasts
(IC; worked example in Fig. 5), generalized least-squares (GLS) models, and
Monte Carlo computer simulations. These methods apply mainly to analysis of
continuously varying (or at least quantitative) traits, which is the nature of
most physiological traits (e.g. blood pressure, metabolic rate, enzyme
activity). However, they can also easily incorporate independent variables
that are treated as discrete categories, such as diet (e.g. insectivore,
frugivore, sanguivore) or habitat (e.g. fresh or salt water). Discussions of
methods for categorical traits and computer programs to implement them are
available from Mark Pagel (e.g. see
Pagel, 1999), in MacClade
(Maddison and Maddison,
2000
), and in Mesquite
(http://mesquiteproject.org/mesquite/mesquite.html;
see also Paradis and Claude,
2002
). For a general listing of phylogeny-related programs, see
the website maintained by Joe Felsenstein
(http://evolution.genetics.washington.edu/phylip/software.html).
We discuss when phylogenetically based statistical methods should be used
and give some practical examples of where a phylogenetic perspective has
improved our understanding of comparative data and evolutionary processes. We
also discuss some of the practical and theoretical limitations of such
methods. Throughout, we try to emphasize that the incorporation of phylogeny
can greatly enhance comparative studies, deliver new insights, and open new
areas for research. This is of necessity only a brief summary and readers are
directed to more extensive discussions of the topics and issues raised here
(e.g. Ridley, 1983; Lauder,
1981
,
1982
,
1990
;
Harvey and Pagel, 1991
;
Garland et al., 1992
,
1999
;
Garland and Adolph, 1994
;
Harvey, 1996
;
Ricklefs and Nealen, 1998
;
Ackerly, 1999
,
2000
,
2004
;
Pagel, 1999
;
Purvis and Webster, 1999
;
Diniz-Filho, 2000
;
Feder et al., 2000
;
Garland and Ives, 2000
;
Maddison and Maddison, 2000
;
Garland, 2001
;
Rohlf, 2001
;
Autumn et al., 2002
;
Blomberg and Garland, 2002
;
Brooks and McLennan, 2002
;
Blomberg et al., 2002, 2003
;
Rezende and Garland, 2003
;
Housworth et al., 2004
). We
have intentionally not cited some `forum' and `perspective' type papers
because we felt that their rhetoric was misleading, and in some cases they
contain outright errors.
The empirical examples cited here are idiosyncratic, reflecting mainly our
own research interests. Thus, we emphasize examples that involve physiological
phenotypes, but include others when they are lacking. Our enthusiasm for
phylogenetic approaches in comparative physiology should not be taken to
imply, however, that we think they are more important than other approaches,
such as measurement of selection acting in natural populations, experimental
evolution (e.g. see Garland and Carter,
1994; Bennett and Lenski,
1999
; Ackerly et al.,
2000
; Feder et al.,
2000
; Garland,
2001
,
2003
;
Bennett, 2003
;
Swallow and Garland, 2005
),
or more purely mechanistic investigations (e.g.
Mangum and Hochachka, 1998
;
Hochachka and Somero,
2002
).
We are concerned that some of our discussion of assumptions and intricacies
of phylogenetically based statistical methods may be off-putting to those who
simply want to analyze their data (see also
Felsenstein, 1985). However,
it must be acknowledged that statistical analyses in general are not always
simple and have underlying assumptions that cannot be ignored. Most of the
tools that we use in everyday research (e.g. correlation, regression, analysis
of variance, analysis of covariance) have been around for 50 years or even a
century. Nonetheless, the field of statistics (both theoretical and applied)
continues to refine these methods. Such questions as what type of line is best
for describing functional relationships (e.g.
Rayner, 1985
; chapter 6 in
Harvey and Pagel, 1991
;
Riska, 1991
;
McGuire, 2003
;
Garland et al., 2004
), how to
deal with non-linear relationships
(Quader et al., 2004
) or
random effects in ANOVA models, when to include or exclude interaction terms,
how best to transform data, or when to employ nonparametric methods, still do
not have simple, general answers. Moreover, new statistical methods continue
to be developed, including computer-intensive approaches that were not
possible 50 years ago (e.g. see Lapointe
and Garland, 2001
; Roff, in
press
). For many statistical parameters, including comparative
methodologies, several different approaches (and attendant algorithms) may be
used for estimation, none of which performs `best' in all situations. We
believe that it is important that a comparative biologist understand the
assumptions and approaches underlying these methodologies, and does not just
resort to their rote application, and that is the basis for our more detailed
presentation.
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Phylogeny and modern (statistical) comparative methods |
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Concern about the possible influence of phylogeny in comparative and
ecological physiology antedated Felsenstein's
(1985) publication. For
example, explicit comparisons of marsupial with placental mammals
(MacMillen and Nelson, 1969
;
Dawson and Hulbert, 1970
) and
of passerine with non-passerine birds
(Lasiewski and Dawson, 1967
)
were motivated by cognizance of phylogeny, and some workers tried to partition
the effects of phylogeny on physiological relationships (e.g.
Andrews and Pough, 1985
).
Moreover, some workers voiced concerns about specific adaptive interpretations
of characters shared more widely in their clades (e.g.
Dawson and Schmidt-Nielsen,
1964
; Dawson et al.,
1977
). What those earlier studies lacked was not necessarily a
general perspective on the importance of phylogeny, but rather a formal
logical and statistical methodology for incorporating detailed phylogenetic
information. Analytical techniques have been greatly expanded and modified
since 1985 (see below and Appendix), but Felsenstein's IC method is still the
most widely used and his insights were pivotal to modernization of the
comparative method. Moreover, the realization that IC is a special case of
generalized least-squares (GLS) methods (see Appendix) means that the former
can always serve as a useful entry point for the latter, and one that retains
the major heuristic of `tree thinking' (sensu
Maddison and Maddison,
2000
).
Traditional interspecific comparative analyses applied conventional
statistical methods to test for associations between traits (e.g. metabolic
rate and body size), or between a trait and an environmental variable (e.g.
blood oxygen carrying capacity and altitude). This approach treats all data
points (e.g. mean values for a series of species) as statistically independent
of each other. Unfortunately, mean phenotypes of biological taxa usually will
not be statistically independent because they are all related through their
hierarchical phylogenetic history. Empirically, more closely related species
do indeed tend to resemble one another; put simply, hummingbirds look like
hummingbirds, and turtles look like turtles, and the same is true for
physiological traits (Blomberg et al.,
2003; see below). This general tendency exists for several good
biological reasons (Harvey and Pagel,
1991
), including time lags for change to occur after speciation,
occupation of similar niches by close relatives, and conservative
phenotype-dependent responses to selection. Thus, the extent of these
phylogenetic relationships and hence the expected degree of
resemblance must also be figured into comparative analyses. Analytical
techniques that do not incorporate phylogenetic information make the tacit
statistical assumption that all the species studied are equally distantly
related to each other, that is, that they descended along a `star phylogeny'
(Fig. 3A), when in fact their
ancestral associations are hierarchical
(Fig. 3C).
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Second, it is important to consider what is meant by the `branch lengths'
of a phylogenetic tree that is used for analysis. In general, proponents of
phylogenetically based comparative methods assume that analyses of
physiological and other traits will involve use of a phylogenetic tree that
was inferred from other data, such as variation in DNA sequences, which is
presumed to be independent of the data being analyzed. Otherwise, it seems
intuitively obvious that analyses may involve some circularity. However, this
is actually a complicated subject and beyond the scope of the present paper
(Felsenstein, 1985;
de Queiroz, 2000
). Leaving
aside the general issue of having available a phylogeny that is independent of
the characters under study, the branch lengths of the working phylogenetic
tree are confounded with the model and rates of character evolution that will
be assumed for statistical analyses of most real data sets (see Figs
1,
2). In other words, we usually
do not have independent information on, for instance, divergence times
and selective regimes that may have prevailed along various branches
of the tree. In any case, all of the main phylogenetically based statistical
methods require branch lengths in units proportional to expected variance of
evolution for the characters(s) under study (see Felsenstein,
1985
,
1988
; Garland et al.,
1992
,
1993
,
1999
;
Garland and Ives, 2000
;
Rohlf, 2001
;
Blomberg et al., 2003
;
Housworth et al., 2004
).
Branch lengths essentially indicate our a priori expectations for how
likely a given trait was to change (increase or decrease in value) from one
node to another along a phylogenetic tree, and thus become an integral
component of our statistical null model. Under a simple Brownian motion model,
those branch lengths would necessarily be proportional to divergence times.
Under any other model, such as the OrnsteinUhlenbeck (OU) process,
which is like Brownian motion while tethered to an elastic band and is used to
model stabilizing selection or constraints on trait space
(Felsenstein, 1988
;
Garland et al., 1993
;
Diaz-Uriarte and Garland,
1996
; Martins and Hansen,
1997
; Blomberg et al.,
2003
; Freckleton et al.,
2003
; Butler and King,
2004
; Housworth et al.,
2004
), they would be more-or-less different from divergence
times.
A simple hypothetical example can illustrate this distinction. Many traits
evolve within limits set by physical or biological properties. Some of these
are trivial. For example, body mass cannot evolve to be as small as 0 g.
Others are more interesting. Apparently, for example, activity body
temperatures (Tb) of squamate reptiles (lizards and
snakes) cannot evolve to be more than about 42°C. We do not know the
ancestral activity Tb of squamates, but it was probably
substantially lower that 42°C. Thus, during their initial radiation and
diversification, Tb would have been free to evolve,
perhaps in a fairly Brownian motion-like fashion, with an increase or decrease
about equally likely to occur along any branch of the phylogeny. However,
lineages that `explored' the climate space towards higher
Tb would eventually be constrained by the reduction in
Darwinian fitness that can be caused by exceedingly high temperatures (e.g.
via failure of spermatogenesis or outright death). Thus, if we were
to depict a phylogenetic tree of squamates with branch lengths proportional to
expected variance of Tb evolution, then we would need to
know the Tb at the start of each branch segment and also
have the branches be, in effect, different if the lineage was near a thermal
limit, either upper or lower. That is, a lineage near an upper limit would
have a low probability of evolving a higher Tb, but a
`typical' probability of evolving a lower Tb, and vice
versa. It should be obvious that our ability to specify such detailed
branch-length information for any trait in any group of wild organisms is
severely limited. Thus, for simplicity and/or analytical tractability,
phylogenetically based statistical methods usually begin with an assumption of
Brownian motion evolution along whatever branch lengths are specified in a
working phylogeny (e.g. Fig.
1). And in many cases (e.g. see reviews of published studies in
Blomberg et al., 2003;
Ashton, 2004a
), these will be
arbitrary values, such as setting all segments equal to unity in length or by
some other simple rule (e.g. Fig.
4B). In such cases, it is often prudent to perform computations
with more than one set of branches as a sensitivity analysis for the
conclusions (e.g. see Ashton,
2004b
; Hutcheon and Garland,
2004
; Laurin,
2004
). Similarly, some studies use multiple phylogenies
(topologies) (e.g. Bauwens et al.,
1995
; Symonds and Elgar,
2002
; Hodges,
2004
).
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These points have suggested to some that phylogenetically based analyses
are so fraught with pitfalls that we should stick with non-phylogenetic ones.
But a conventional statistical analysis actually has as many assumptions as a
phylogenetic one. For example, it assumes that the species under analysis have
not been interacting, e.g. as by character displacement
(Hansen et al., 2000). It
assumes that each species should be equally weighted, which is equivalent to
saying that the heights of each branch from the root of the tree (assumed to
be a star) are equal. And so forth.
In any case, it has become increasingly clear that, because we never know
the true branch lengths and/or model of character evolution, we should pay
careful attention to the branch lengths used, employing methods that can
consider options ranging between a star and our working hierarchical
phylogeny, and possibly something even more hierarchical. Thus, recent methods
emphasize estimation of optimal branch length transformations as an essential
part of phylogenetic analyses of comparative data (e.g. see
Grafen, 1989; Diaz-Uriarte and
Garland, 1996
,
1998
;
Pagel, 1999
;
Harvey and Rambaut, 2000
;
Freckleton et al., 2002
;
Martins et al., 2002
;
Blomberg et al., 2003
;
Housworth et al., 2004
).
Although some researchers may be uneasy with such transformations of branch
lengths, they are analogous to use of a BoxCox procedure to find the
optimal transformation of data (e.g. best approximation of normality) in
conventional statistical procedures (for instance, use of a BoxCox
procedure to transform branch lengths;
Reynolds and Lee, 1996
).
Moreover, aside from its benefits with computer-simulated data, such careful
attention to branch lengths can sometimes improve statistical power to an
important extent with real data (see below).
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An example of how phylogeny can affect statistical analyses |
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The answer depends on what is assumed about the phylogenetic relationships
of the 12 species. If we assume that species are unrelated, then we can refer
to conventional tables of critical values for correlation coefficients. For a
one-tailed test with 12 data points (and hence 10 degrees of freedom for
testing a correlation), the critical value is +0.497, so a value of +0.585
would be considered significant at P<0.025. If, instead, we want
to assume that the species are related in a hierarchical fashion, then we
cannot use the conventional tables. Fortunately, however, we can incorporate
phylogenetic information as follows
(Martins and Garland, 1991;
Garland et al., 1993
). We can
construct different working phylogenies, model the uncorrelated evolution of
these traits by a Monte Carlo computer simulation that assumes random,
Brownian-motion like trait change, and calculate a correlation coefficient for
each simulated data set. We can then determine the critical 5% level for the
correlation coefficient for each distribution. If we assume that all species
are completely independent or related by a star phylogeny
(Fig. 4A), then the one-tailed
probability for obtaining a correlation as large as +0.585 is 0.023 (based on
this particular set of 1000 simulated data sets), so the relationship is
statistically significant at P<0.05. In fact, if we do a very
large number of simulations, then we will obtain exactly the same
results as when referring to conventional tables.
If, however, we simulate data along our best estimate of the phylogeny of these lizards (Fig. 4C), then a correlation of +0.585 would be observed much more frequently than 5% of the time and would not be considered very unusual, hence not statistically significant (P>0.15). If a hypothetical phylogeny with different branch lengths, involving fewer deep roots, were assumed (Fig. 4B), then a value of +0.585 would have a lower probability of being observed, but in this case would still be non-significant. Thus, the assumed pattern of the relationships among the species crucially affects the statistical significance of the observations: the more the phylogeny departs from a star, the lower is the number of effectively independent observations and the more likely we are to observe an extremely large (or small) correlation just by chance. If the working topology, branch lengths, and simulation model are somewhat realistic, then we will claim significance too often if we ignore phylogeny.
Another important point is that if the simulated data of Fig. 4B or C are analyzed with IC, using the corresponding phylogenies, then the resulting distribution of correlation coefficients will be the same as in Fig. 4A (results not shown). Thus, as discussed in the Appendix, the IC method uses the specified phylogenetic information to transform the data to make them independent and identically distributed. This then allows one to refer to conventional tables of critical values for hypothesis testing.
All of the simulations shown in Fig.
4 were done under a simple Brownian motion model
(Fig. 2), but the model used
can have a large effect on the resulting distributions of statistics (e.g. see
Garland et al., 1993;
Diaz-Uriarte and Garland,
1996
; Price,
1997
; Harvey and Rambaut,
2000
; Martins et al.,
2002
; Freckleton et al.,
2003
). Brownian motion is a very simple model of character
evolution, and its analytical tractability was exploited by Felsenstein
(1985
) to develop IC. It is a
good model for traits that evolve solely by random genetic drift, and may also
be adequate for some types of `fluctuating' selection (i.e. when the direction
of selection changes from generation to generation). As a basis for
statistical methods to estimate and test character correlations, it may also
be an adequate model for traits that are subject to certain types of selection
(Felsenstein, 1985
,
1988
;
Grafen, 1989
). But most traits
probably evolve in ways that are too complicated and idiosyncratic to be
modeled realistically by Brownian motion
(Felsenstein, 1988
;
Hansen et al., 2000
).
Fortunately, simulations can use arbitrarily complex models of character
evolution, limited only by one's ability to write computer programs and
imagination (e.g. Garland et al.,
1993
; Diaz-Uriarte and Garland,
1996
,
1998
;
Price, 1997
;
Harvey and Rambaut, 2000
;
Freckleton et al., 2003
). Of
course, whether more complicated models lead to more accurate analyses depends
on whether they are actually a better descriptor of past evolution by the
characters under study, and that is something very difficult to know.
Moreover, finding a model that fits a set of data reasonably well does not
necessarily mean that it is the correct model, and other models can probably
be found that would provide equally good fit (see also
Blomberg et al., 2003
). (As
always, it is risky to attempt to infer process from pattern.) Given the near
impossibility of knowing how traits actually evolved in the distant past,
simulation studies are also used to gauge how robust analytical methods such
as IC are to violation of their assumptions (e.g. Brownian motion, accurate
knowledge of branch lengths), how diagnostic tests can alert one to such
violations, and how well remedial measures (e.g. transformations of tip data
or branch lengths) can rescue statistical performance when assumptions are
violated (e.g. Martins and Garland,
1991
; Purvis et al.,
1994
; Diaz-Uriarte and Garland,
1996
,
1998
;
Harvey and Rambaut, 2000
;
Diniz-Filho and Torres, 2002
;
Freckleton et al., 2003
).
Still, physiologists may sometimes be able to improve the accuracy of assumed
models by their knowledge of how organisms work (or could have worked), as in
the case of limits to body temperature evolution discussed above. Similarly,
paleontological information can be used to improve the realism of simulations
(Garland et al., 1993
).
We close this section by emphasizing that the use of computer simulations
to obtain `phylogenetically correct' (PC) null distributions for testing
hypotheses about comparative data is a very general tool that can be used for
virtually any analysis (Martins and
Garland, 1991; Garland et al.,
1993
), including bivariate (e.g.
Ricklefs and Nealen, 1998
) or
multivariate analyses of evolutionary diversification. For example, the
PHYLOGR (available at
http://cran.r-project.org/)
program allows one to test hypotheses about canonical correlation or principal
components analysis (PCA) in relation to computer-simulated data (R.
Diaz-Uriarte and T. Garland, manuscript in preparation).
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When and why to use phylogenetic information in comparative studies |
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What kinds of characters demand a phylogenetic analysis? Although we may
generally expect that most characters will tend to `follow phylogeny', this is
an empirical question. The simplest general test for whether related organisms
actually do tend to resemble each other more than they resemble those that
might be chosen randomly with respect to phylogenetic position uses
randomization procedures (see also Abouheif, 1994;
Ackerly, 2004;
Laurin, 2004
;
Rheindt et al., 2004
).
Specifically, once phylogenetically IC have been computed, it is possible to
calculate the variance of those contrasts. The lower the variance of the
contrasts, the better the fit of the phylogeny (topology and branch lengths)
to the character in question. To determine whether a given variance indicates
the presence of statistically significant `phylogenetic signal' (i.e. more
closely related species tend to resemble each other more than they resemble
randomly chosen species), one can compare it with the distribution of
variances for a large number of data sets that have been randomized (shuffled)
across the tips of the phylogeny (Blomberg
and Garland, 2002
; Blomberg et
al., 2003
). For studies with 20 or more species (for which
statistical power should be reasonably high), more than 90% of the traits
examined to date (including behavioral, physiological, morphological, life
history and ecological/environmental traits) do exhibit a significant
phylogenetic signal (P<0.05:
Blomberg et al., 2003
;
Al-kahtani et al., 2004
;
Ashton,
2004a
,b
;
Rezende et al., 2004
;
Ross et al., 2004
;
Muñoz-Garcia and Williams, in
press
; see also Freckleton et
al., 2002
).
The empirical finding of pervasive phylogenetic signal implies that
hierarchical phylogenies as presented and used in numerous
publications provide a better fit to the data under analysis than does
a star phylogeny (Figs 3A,
4A). This sends a strong
message that we should routinely consider phylogenetic information in
statistical analyses of comparative data. However, this does not necessarily
mean that, for any given set of data, we should simply obtain a phylogenetic
tree, perform an IC, GLS or Monte Carlo simulation analysis, and automatically
presume that the results will be more reliable than the comparable
conventional statistical analysis. As we and others have emphasized for more
than a decade, analyses using a given topology and branch lengths can perform
relatively poorly if their assumptions are severely violated (e.g. see
Grafen, 1989;
Martins and Garland, 1991
;
Diaz-Uriarte and Garland,
1996
,
1998
;
Price, 1997
;
Garland and Diaz-Uriarte,
1999
; Harvey and Rambaut,
2000
; Diniz-Filho and Torres,
2002
; Martins et al.,
2002
; Freckleton et al.,
2003
; Housworth et al.,
2004
). Thus, we urge practitioners to apply robust tests for
phylogenetic signal, diagnostic checks, and branch length transformations as
warranted (for recent discussions and methods, see
Freckleton et al., 2002
;
Blomberg et al., 2003
), and
sensitivity analyses by varying branch lengths and/or model of evolution (e.g.
see Garland et al., 1993
;
Ashton, 2004b
;
Hutcheon and Garland, 2004
;
Laurin, 2004
;
Muñoz-Garcia and Williams, in
press
).
What sorts of branch lengths should be used? Given the uncertainties
regarding branch lengths (see above), many workers have reported results with
multiple branch lengths to explore consistency or the lack thereof (e.g.
Ross et al., 2004). Although
it is often the case that conclusions are relatively robust (insensitive) to
the branch lengths used, this is not always true, and the importance of
attempting to use `optimal' branch lengths transformations can be illustrated
with two empirical examples. Garland et al.
(1993
) analyzed home range
areas in relation to body size for 49 species of carnivores and ungulates.
Conventional analysis of covariance (ANCOVA) indicated a highly significant
(P<0.001) different in size-adjusted home range areas of the two
groups. Analysis via IC (or Monte Carlo simulations), however,
revealed no statistically significant difference (two-tailed P=0.126
for IC). The branch lengths used for the phylogenetic analyses were estimates
of divergence times, derived from various sources. They passed the diagnostic
`lack of fit' tests as described in Garland et al.
(1992
). However, power to
detect a difference is apparently improved by applying the transformations of
branch lengths as proposed by Blomberg et al.
(2003
) to mimic particular
models of character evolution. Using the branches transformed under the OU
model for log body mass, the P value is reduced to 0.099, and using
their AcceleratingDecelerating (ACDC) model the P value is
reduced to 0.044, thus crossing the typical threshold of <0.05 to be
considered statistically significant (degrees of freedom were reduced by one
in both cases to reflect the additional parameter estimated in these models;
see also Diaz-Uriarte and Garland,
1996
,
1998
;
Garland and Diaz-Uriarte,
1999
). Similarly, in a recent comparison of the generic average
body sizes of `megabats' and `microbats,' Hutcheon and Garland
(2004
) found statistical
significance in the IC analysis only when using branch lengths transformed
under the OU or ACDC models. We suspect that such increases in power may be
more likely to occur in comparisons between groups that are fairly highly
phylogenetically confounded (i.e. the independent variable of interest, such
as diet, is highly clumped with respect to phylogeny, as in comparisons of
clades; e.g. see Garland et al.,
1993
; Vanhooydonck and Van
Damme, 1999
; Perry and
Garland, 2002
; Rezende et
al., 2004
) than in tests of correlations between two traits.
How do we choose species for study? Traditionally, animals were chosen for
comparative studies for any number of reasons, including convenience (e.g.
local availability or an existing literature data base), possession of an
interesting biological trait (e.g. the long neck of the giraffe), occupation
of an extreme environment (e.g. a hot dry desert, the Arctic), or
characteristics that make it well suited to study a particular physiological
process (Garland and Adolph,
1994; Garland and Carter,
1994
; Bennett,
2003
). Frequently, a particular species or group living in a
particular environment has been the key that originally sparked interest in
the project. It is now clear that phylogenetic information should also be
considered when choosing species for study (for a simulation study on the
effects of taxon sampling when testing for correlated evolution, see also
Purvis and Webster, 1999
;
Ackerly, 2000
).
To increase analytical power, it is a good idea to include other species
that experience a very broad range of the environmental (`independent')
variable. You might then randomly sample species from a broad taxon (e.g.
mammals) or focus exclusively on a particular lineage, such as bats or
rodents. From a design perspective, the latter strategy is preferable because
the broader comparison will involve distant relatives that vary in many
traits, potentially complicating the analysis of particular traits of
interest. From a phylogenetic point of view, comparisons of distant relatives
are like an experiment with multiple uncontrolled variables
(Garland and Adolph, 1994;
Garland, 2001
). To quote
Felsenstein (1985
, p. 465),
`Comparative biologists tend to suspect comparisons of distantly related
species; they hope to base their comparisons on recent evolutionary events
that have not been overlaid by much subsequent change'. In principle, it
might be possible to control for confounding traits that differ in distant
relatives by including additional independent variables in the analysis, but
it is often difficult to know a priori what those traits might be,
let alone actually obtain quantitative data for them. In any case, an example
in which casting too broad a net seems to reduce statistical power is provided
by a study of body mass evolution in birds
(Garland and Ives, 2000
, p.
354). A comparison of passerines with their sister clade indicates that the
former have significantly smaller log body masses, on average, whereas a
comparison of passerines with all birds (including their sister clade) does
not. (It should be noted that the identity of the sister clade of passerines
is controversial, and the foregoing example may well change as improved
phylogenetic information becomes available.) A related topic is whether one
might a priori exclude certain subclades from a comparative analysis
because they are `unusual' as compared with the larger clade in general. For
example, many studies of lizards (e.g.
Perry and Garland, 2002
)
exclude snakes. A recent study by Bininda-Emonds and Gittleman
(2000
) suggests that this sort
of a priori data exclusion may be less warranted than is often
presumed.
A particularly powerful comparative design is one that has several
different pairs of closely related species that differ in the variable of
interest (e.g. high and low temperature) and has these species pairs
relatively distantly related to each other (i.e. in different branches of the
phylogeny). As noted by Garland
(2001), a particularly
favorable distribution of this sort has the power to detect significant
associations even when conventional statistical methods fail to do so.
However, some workers choose species in this way, but then analyze only the
pairs of tip species rather than performing a full analysis of the entire
phylogeny (e.g. Lavergne et al.,
2004
). If that is done, Type I error rates should be correct, and
the analysis should be robust with respect to errors in branch lengths and/or
model of evolution, but statistical power will likely be lost (see
Ackerly, 2000
). A more extreme
analytical variant is to perform a sign test on the tip pairs
(Felsenstein, 1985
), thus not
using any information on branch lengths, but this comes at the extreme loss of
statistical power (Ackerly,
2000
).
The worst, that is, the least powerful, comparative design is one in which
all species on one side of the root of the tree share, say, high values for an
independent variable of interest (e.g. high temperature) and those on the
other side of the root share low values (e.g. low temperature) (e.g.
Garland et al., 1993;
Garland, 2001
). Although some
methods can enhance inferential power in such situations (e.g.
Schondube et al., 2001
), it
is not an attractive comparative scenario.
How many species or other taxa need to be included in a comparative study?
In general, the statistical power of phylogenetically based analyses, when
applied with an accurate phylogeny and model of character evolution, is the
same as for conventional statistical methods, so standard power calculations
can be employed (e.g. see fig. 5 in
Garland and Adolph, 1994).
However, it is also true that phylogenetic analyses sometimes uncover
relationships that were not apparent in conventional analyses (see below).
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Examples of the utility of incorporating a phylogenetic perspective |
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We will now review just a few examples from the literature in which phylogenetic information has added to our understanding and interpretation of comparative data. The first two of these deal with the evolution of lower metabolic rate in endotherms as part of their adaptation to desert environments.
It has long been recognized that low metabolic rates, low body
temperatures, and an ability to become torpid would be beneficial to
endotherms in hot, arid environments, in order to minimize heat load and
energy (and water) demands in environments of low productivity (e.g.
Dawson and Bartholomew, 1968;
Dawson and Hudson, 1970
;
Williams, 1996
;
Tieleman et al., 2003
;
Rezende et al., 2004
). When
these traits were first discovered in desert caprimulgid birds (e.g. poorwills
and nighthawks), they were initially interpreted as adaptations to desert
conditions (e.g. Bartholomew et al.,
1962
). That is, they were seen as part of the evolutionary changes
that permitted the occupation of hot, arid environments. Subsequently,
however, it was recognized that other species of caprimulgids from more mesic
and even tropical areas also had low metabolic rates and could also become
torpid (e.g. Dawson and Schmidt-Nielsen,
1964
; Lasiewski et al., 1970). A recent phylogenetic analysis
(Lane et al., 2004
) confirms
that some of these traits exist in even the most basal members of the clade
(but not the sister group, owls). Therefore, this constellation of
thermoregulatory traits might be more general for this group and not an
evolutionary adaptation to desert environments per se. Thus, while
the possession of low metabolic rates and torpor ability may have facilitated
the occupation of arid environments by caprimulgids, and thus constituted a
`preadaptation', these traits do not appear to have evolved as adaptations to
them.
The hypothesis of low resting metabolic rates during adaptation to desert
environments was also tested in the group Procyonidae (raccoons and their
relatives). Chevalier (1991)
measured metabolic rates of individuals from desert and mesic populations of
ringtails, and from single populations of four other procyonids. These data
can be analyzed via a conventional least-squares linear regression of
log metabolic rate on log body mass. The procedure is to exclude the desert
ringtail population, fit the regression line, and then compute the one-tailed
95% prediction interval for a new observation (see fig. 4B in
Garland and Ives, 2000
). The
desert ringtail population falls below the regression line, consistent with
the hypothesis of adaptation, but not outside the prediction interval, and
thus not `significantly' so. If the same procedure is followed with
phylogenetically independent contrasts, the ringtail datum falls far
outside the prediction interval (fig. 4C in
Garland and Ives, 2000
; see
also Garland and Adolph, 1994
)
and thus the low metabolic rate of the desert population can be associated
with desert occupation by this group.
Why the large difference in results? With a conventional analysis, each of the five data points is weighted equally for both computing the regression line and the prediction interval, and the place of the datum to be predicted (the desert ringtail's metabolic rate) is not considered in the sense that a star phylogeny (e.g. Figs 3A, 4A) is assumed, mathematically speaking. In the phylogenetic approach, two differences occur.
First, the data points are weighted differentially when the regression line
is computed, so it differs somewhat from the conventional line. Second, for
computing the prediction interval, the algebra specifically recognizes that
the desert ringtail population has a very close relative, the mesic ringtail
population, which has a fairly high metabolic rate, and thus the prediction is
`pulled' to a higher value. This makes intuitive sense because we would
generally expect a close relative to be a better predictor of an unmeasured
organism's phenotype as compared with the prediction derived from one (or
several) less closely related species. The two effects together, but
particularly the second (see fig. 4 in
Garland and Ives, 2000),
weight the comparison to be primarily between the desert and mesic ringtail
populations (i.e. between the tip of interest and its closest relative in the
data set), whereas the conventional analysis just compares the focal tip to
all other values in a general, unprincipled way, thus losing statistical
power.
We realize that sometimes it seems that phylogenetic methods only reduce
analytical power and may obscure real relationships. However, the procyonid
example is an instance where incorporating phylogenetic information actually
supports an adaptive hypothesis that would not be found with a conventional,
non-phylogenetic analysis. For another example, see Al-kahtani et al.
(2004) on the correlation
between size-corrected kidney mass and habitat aridity in rodents.
Turning from an analysis of specific adaptive patterns to more general
issues in comparative physiology, phylogenetically based methods can be
equally useful there too. Perhaps one of the most familiar relationships in
comparative data is that between basal metabolic rate (BMR) and body size, the
famous `mouse-to-elephant' curve. The allometric slope of this relationship
and its interpretation have been debated endlessly in the literature. However,
most calculations of that relationship are based on the same incorrect
assumption of the independence of observations that historically characterized
other comparative data. This can be particularly problematic for such data
sets as that on BMR, where certain groups (e.g. rodents) tend to be
over-represented in the observations and others (e.g. cetaceans) tend to be
greatly under-represented. As discussed above, failure to account for the
relationships among the taxa will overestimate effective sample size and
underestimate error limits. This situation is not unique to the allometry of
metabolism, and equally applies to all compilations of scaling relationships
(e.g. Calder, 1984;
Peters, 1984
). Recent
phylogenetically based recalculations of these relationships (e.g.
Garland and Ives, 2000
;
Cruz-Neto et al., 2001
;
Hosken et al., 2001
;
Symonds and Elgar, 2002
)
often differ significantly from those of conventional analyses, including the
value for slope of the allometric equation relating BMR to body size. For
instance, conventional statistical methods produce a (loglog) slope of
0.670 for 254 species of birds, but four different calculations involving
phylogenetically independent contrasts have slopes ranging from 0.709 to
0.759, and none of the 95% confidence intervals for these latter slopes
include the value of 0.670 (but see reanalysis of a refined data set by
McKechnie and Wolf, 2004
).
Clearly, debates about the interpretation of allometric slopes rest on infirm
ground if the values in question have been incorrectly calculated (see also
Nunn and Barton, 2000
).
In further regard to the allometry of avian metabolism, there has been a
longstanding debate (e.g. Lasiewski and
Dawson, 1967) as to whether separate equations should be used for
the allometric relationship of BMR in passerine and non-passerine birds (note
that the later taxon is paraphyletic and liable to cause apoplexy among
cladists). (This is just one of many examples in which possible `grade shifts'
are of interest, i.e. differences among clades; see also
Garland et al., 1993
;
Ackerly, 1999
;
Purvis and Webster, 1999
;
Ackerly et al., 2000
;
Nunn and Barton, 2000
; fig. 1
in Garland, 2001
.) The
incorporation of phylogeny into data analysis has resolved that debate: these
two groups do not show a statistically significant difference in
mass-corrected BMR (Reynolds and Lee,
1996
; see also Rezende et
al., 2002
). In addition, further analysis of those data
(Garland and Ives, 2000
)
revealed a very interesting evolutionary pattern in the passerines: the rates
of evolution of both body size and size-corrected metabolic rate within this
group are significantly less than those in other birds (see also
McKechnie and Wolf, 2004
).
This finding may indicate that passerines are under more size and energetic
constraints than other avian taxa in general. This result is an example of the
utility of phylogenetic methods to our understanding of the evolution of
physiological characters; uncovering this result would have been impossible
without them.
Here it is worth noting as an aside that ANCOVAs and related techniques
(reviewed in Harvey and Pagel,
1991) were traditionally applied to examine metabolic scaling and
whether `grade shifts' may be present. These analyses are `phylogenetic' in
the sense that taxonomic groupings, such as families or orders, are used as
factors (main effects). If one presumes that these taxa are separate
evolutionary lineages (clades), then phylogeny is being partly considered.
However, orders, families, and even genera themselves contain hierarchical
relationships of their constituent species, and so a taxonomy-derived ANCOVA
cannot capture the entire richness of phylogenetic information that may be
available. This is why we consider Felsenstein's IC
(Felsenstein, 1985
) to be the
first fully phylogenetic comparative statistical method.
A final example involves the dietary, latitudinal and climatic correlates
of BMR and maximal metabolic rate (under cold exposure) in rodents
(Rezende et al., 2004).
Although conventional multiple regression analyses indicated that diet,
latitude and temperature could explain significant amounts of the
interspecific variation in mass-corrected BMR, a phylogenetic analysis
indicated that only latitude was a significant predictor. As most traits
showed substantial phylogenetic signal, the latter analyses should be more
reliable. Those authors point out that whereas several interspecific
comparisons of mammalian BMR have reported a significant association with diet
using conventional statistics, this association has not yet been supported
with phylogenetically based methods. Diet, at least when scored in crude
categories, tends to be strongly associated with phylogeny in mammals,
including rodents, so it is conceptually and statistically difficult to
analyze dietary effects separate from phylogeny. As noted by Garland et al.
(1993
), quantitative
information on diet should increase statistical power to detect associations
with other traits. Indeed, a significant relationship between BMR and diet,
scored quantitatively, was found in a recent phylogenetic analysis of the
Carnivora (Muñoz-Garcia and
Williams, in press
).
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Some notes of caution |
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First, one needs a phylogeny. If one does not already exist, then you need
to derive it for your organisms of interest. This is obviously no trivial
matter, especially if your principal interest is physiology and not
systematics. One possible first step is to infer phylogeny from taxonomy, but
this is especially risky for groups where the existing taxonomy was not
derived from actual phylogenetic information (i.e. information about the
branching order of past branching events). Even if existing taxonomic
information is not positively misleading with respect to phylogeny, it will
generally lead to working phylogenies that contain numerous soft polytomies
unresolved nodes depicting several taxa differentiating simultaneously
rather than as a series of discrete bifurcations
(Fig. 3B). Because they reflect
uncertainty in our phylogenetic information, soft polytomies cause analytical
problems for phylogenetic approaches. Analytical methods that adjust degrees
of freedom (Purvis and Garland,
1993; Garland and
Diaz-Uriarte, 1999
; empirical example in
Tieleman et al., 2003
) or
employ more sophisticated computer simulations (Housworth and Martins, 2001)
are available, but result in lowered statistical power as compared with an
analysis that used a fully resolved tree. Of course, it is also possible,
perhaps in collaboration with a bona fide systematist, to construct a
phylogeny using appropriate data and well-defined inferential procedures (e.g.
Felsenstein, 2004
). Indeed,
the ready availability of DNA sequencing technology and computer programs
(e.g. see
http://evolution.genetics.washington.edu/phylip/software.html)
has democratized the process, and many physiologists are now making their own
trees (e.g. Block et al.,
1993
; Johnston et al.,
2003
; Tieleman et al.,
2003
).
In addition to the statistical and analytical concerns with regard to
branch lengths and models of character evolution mentioned above, the basic
accuracy of topological information will affect results. Phylogenies are only
estimates of (hypotheses about) true but unknown (and probably unknowable)
branching relationships. Any conclusions drawn from a study are ever
susceptible to future modification or falsification by a revision of the
phylogenetic hypothesis. This is not simply a theoretical concern. For
example, the study on lizard running speed and body temperature discussed
above (Garland et al., 1991)
revised the conclusions of an earlier (and partially phylogenetic) study
(Huey and Bennett, 1987
),
partly because of a subsequent phylogenetic revision. It is possible that the
conclusions will be revised again if the phylogeny is further revised. Along
these lines, recent methods are making it possible to incorporate phylogenetic
uncertainty directly into comparative analyses that include simultaneous
estimation of the phylogeny from DNA sequence data
(Huelsenbeck and Rannala,
2003
; see also p. 693 in
Butler and King, 2004
), as was
originally suggested by Felsenstein
(1985
).
Next is the issue of the number of taxa appropriate for a comparative
analysis. One of us co-authored a paper entitled `Why not to do two-species
comparative studies...' (Garland and
Adolph, 1994), which pointed out that the absolute minimum number
of taxa required is three, in order to provide at least one degree of freedom
for hypothesis testing (and to allow deduction of the direction of character
evolution). In practice, many more taxa will generally be required to achieve
adequate statistical power and the desired level of coverage of both
phylogenetic and putatively adaptive states (for discussions, see
Garland and Adolph, 1994
;
Garland et al., 1997; Garland,
2001
). Some of these taxa may be difficult or impossible to
obtain. For example, one of Bennett's students
(Eppley, 1996
) wanted to
include crab plovers for a study of the ontogeny of endothermy in charadriform
birds. Unfortunately (for multiple reasons), crab plovers nest by the Persian
Gulf in Iraq and Iran, which were then at war. These and such other issues,
such as obtaining collecting permits for many species in different
geographical locations, can make comparative studies difficult in practical
terms. In addition, a trade-off must exist between the number of species that
can be studied and the depth of investigation for each species, although the
latter can be overcome with time if investigators publish their methods and
raw data in sufficient detail to allow subsequent cumulative comparative
studies that combine new data with data mined from the literature
(Mangum and Hochachka, 1998
).
When a comparative study incorporates literature data for multiple traits, it
is often the case that missing data severely limit analyses (e.g. see
Bininda-Emonds and Gittleman,
2000
). Some recent comparative analyses have employed fairly
sophisticated methods for dealing with missing data (e.g.
Fisher et al., 2003
), but
phylogenetically based methods for maximizing effective sample size with
missing data need to be developed (S. P. Blomberg, personal communication; see
also related methods in Garland et al.,
1999
; Garland and Ives,
2000
).
Although confidence intervals can be computed for estimates of ancestral
states under simple evolutionary models, when the number of species is small
these can be so wide that they include or even exceed the range of observed
states at the tips of the phylogeny (see fig. 8 in
Schluter et al., 1997; fig. 2
in Garland et al., 1999
).
Narrower limits can be calculated for larger phylogenies (e.g. see
Laurin, 2004
; K. E. Bonine, T.
T. Gleeson and T. Garland, Jr, manuscript submitted for publication), but it
must be kept in mind that most analytical procedures assume a simple
evolutionary model, such as Brownian motion. If this assumption is invalid, as
when evolutionary trends have occurred, then estimates may be quite misleading
(for some empirical examples, see Garland
et al., 1999
; Oakley and
Cunningham, 2000
; Webster and
Purvis, 2002
). Comparative historical analysis can thus never
really know ancestral or intermediate states, but only conjecture about them.
Only experimental evolutionary analyses, which establish ancestral state and
observe intermediate states, can have that certainty (e.g.
Bennett and Lenski, 1999
;
Oakley and Cunningham, 2000
;
Garland, 2001
). However, it is
also possible to include fossil taxa directly in a phylogenetic analysis (e.g.
see Polly, 2001
;
Laurin, 2004
;
Ross et al., 2004
;
Hone et al., 2005
), although
rarely if ever for physiological traits. Although reaching decisions about
inclusion/exclusion of taxa can be problematic (Garland et al., 1997), as in
any comparative study, the inclusion of fossil taxa has great potential to
increase both the accuracy and precision of estimates of ancestral states
(e.g. see Laurin, 2004
). For
example, `fossil' taxa can be added anywhere on a phylogeny, with branches of
any length, including length of zero. Thus, one can, if desired, set the value
of a trait at any node on a phylogenetic tree by adding to it what amounts to
a `ghost node' with a specified tip value. All of this can be done in our
PDTREE program, and we encourage further theoretical and empirical work in
this area (see also Laurin,
2004
).
Conventional statistical analyses of comparative data typically treat mean
values for species (or populations) as if they were estimated without error.
This is often unavoidable if the data set includes values from the literature,
as often only mean values have been reported. Nonetheless, it can cause
problems. For example, as noted above, allometric slopes are often of interest
in comparative physiology (e.g. see
Calder, 1984;
Peters, 1984
;
Reynolds and Lee, 1996
;
Clobert et al., 1998
;
Ricklefs and Nealen, 1998
;
Garland and Ives, 2000
;
Hosken et al., 2001
;
Symonds and Elgar, 2002
;
McGuire, 2003
;
McKechnie and Wolf, 2004
;
Muñoz-Garcia and Williams, in
press
), and it is well known that least-squares linear regressions
will tend to underestimate the slope when the independent variable (e.g. log
body mass) includes error variance (e.g. see
Rayner, 1985
;
Riska, 1991
;
Nunn and Barton, 2000
). If
information on the within-species variation is available (e.g. estimates of
standard errors associated with each tip value), then `measurement error
models' can be employed (e.g. Fuller,
1987
). An important area of current research is developing such
methods for phylogenetic analyses (chapter 6 in
Harvey and Pagel, 1991
;
Christman et al., 1997
;
Martins and Hansen, 1997
;
Martins and Lamont, 1998
;
Felsenstein, 2004
;
Garland et al., 2004
;
Housworth et al., 2004
). In
the context of phylogenetically independent contrasts, it is possible, in
effect, to use estimates of tip standard errors to first lengthen terminal
branches of the working phylogeny, then perform analyses
(Garland et al., 2004
). As has
been noted by several workers (e.g. Purvis and Rambaut, 1985;
Ricklefs and Starck, 1996
;
Purvis and Webster, 1999
;
Nunn and Barton, 2000
),
contrasts between two tips (as opposed to those involving deeper nodes, whose
branches are lengthened as part of the contrasts algorithm) that are connected
by short branches fairly commonly appear as `outliers' in analyses. Rather
than reflecting a truly high rate of evolution since the tips in question
diverged, such a pattern may instead reflect disproportionate effects of
measurement error in the tip values (or errors in estimates of the branch
lengths). Incorporation of information on the error associated with estimates
of tip values thus has the potential to reduce this problem and hence allow
greater confidence in terminal contrasts. This is important because many
comparative studies intentionally include one or more species (or populations)
of particular interest plus their closest available relatives, which are
necessarily connected by relatively short branches. Interpretation may hinge
critically on whether a given contrast between two tips is large in magnitude
[e.g. see the ringtail example of Chevalier
(1991
) as discussed in Garland
and Adolph (1994
) and Garland
and Ives (2000
)]. Another
point is that data quality is particularly important when close relatives are
compared (Purvis and Webster,
1999
).
Finally, we must remember that all comparative studies (phylogenetic or
not) are inherently correlational and, taken alone, cannot demonstrate
causality of relationships (but see Autumn
et al., 2002). Only experiments can demonstrate causality. This
also raises the issue of inference when the `independent' variable of interest
is highly confounded with phylogeny (clumped in particular parts of the tree).
This situation often arises in studies of diet, which is usually categorized
fairly crudely, e.g. as carnivore, omnivore, herbivore (e.g. see
Garland et al., 1993
;
Perry and Garland, 2002
;
Rezende et al., 2004
; but see
also Muñoz-Garcia and Williams, in
press
). This sort of diet categorization often shows a high degree
of phylogenetic clumping. Diet is also often significantly associated with
some `dependent' variable, such as body size-corrected metabolic rate, in a
conventional statistical analysis, while a phylogenetically based statistical
analysis shows much less support for the relationship (i.e. a higher
P value). Two things must be noted. First, strong phylogenetic
clumping of an independent variable (e.g. when diet tends be uniform within
clades but differ among clades) leads to low statistical power to detect its
effect on a dependent variable (Garland et
al., 1993
; Vanhooydonck and
Van Damme, 1999
). Second, even if an effect is detected (e.g.
P<0.05), it cannot be logically attributed to diet without good
reason to dismiss possible effects of other shared derived features
(synapomorphies) of one or more of the clades. In the limit, a comparison of
two clades suffers from many of the same inferential problems as does a
comparison of two single species (Garland
and Adolph, 1994
).
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Summary and perspectives |
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We further believe that it is important to concentrate on the positive aspects of including phylogeny in the comparative method. It is the best way to remind ourselves continually that all functional characters are the products of evolution. This is the essence of evolutionary physiology: characters are not stable through time but are continually susceptible to modification. Phylogenetic comparative methods are a principal tool of evolutionary physiology to examine patterns and to infer processes of evolutionary change. They are uniquely positioned to permit us to speculate about ancestral conditions, as well as rates and patterns of evolution in historical time. Incorporation of phylogeny into the comparative method can and has served to expand the kinds of questions that biologists are capable of studying.
A final point is that phylogenetically based statistical analyses often
suggest that evolutionary adaptation is just not as common as we once thought
in comparative physiology (or at least it is hard to find strong empirical
support for it). If these results are confirmed once adequate meta-analyses
are performed (e.g. P. Carvalho, J. A. F. Diniz-Filho and L. M. Bini, personal
communication), including due consideration of effects of errors in
phylogenies, it could lead to an important reorientation of perspectives,
given that earlier generations of comparative physiologists routinely
assumed the presence of evolutionary adaptation in most if not all
traits they studied, rather than seeing adaptation as one possible explanation
for possession of a character (Feder et al.,
1987,
2000
;
Garland and Adolph, 1994
;
Garland and Carter, 1994
;
Bennett, 1997
; Autumn et al.,
2000).
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Appendix |
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The IC method is an algorithm that transforms data to account for the
differential relatedness of the taxa within a study
(Felsenstein, 1985). It also
turns out that analyses with contrasts yield numbers that are mathematically
identical to those obtained through equivalent GLS analyses
(Garland and Ives, 2000
;
Rohlf, 2001
; see below).
Thus, IC can be viewed simply as a clever algorithm to avoid the need to
invert large matrices (Freckleton et al.,
2003
), and the algorithm was originally developed by Felsenstein
(1973
) in the context of
attempting to estimate phylogenetic trees from continuous-valued
characters.
Independent contrasts converts the original N measurements (which
were non-independent of each other if they represent mean values for
hierarchically related taxa) into N1 contrasts of the
measurements between pairs of related taxa or (estimated) ancestral nodes in
the phylogeny. Computations are done for one trait at a time, as shown in Fig.
5. If multiple traits are involved in the analysis, then the contrasts
calculated separately for each trait are used to compute a correlation,
regression, multiple regression, etc. Worked examples for bivariate
correlations can be found in Garland
(1994, fig. 11.2) and Garland
and Adolph (1994
, fig. 2; also
reproduced in box 9.2, pp. 348-349, of
Freeman and Herron, 2004
).
Readers are cautioned that some published examples of the calculations
(including in text books) are incorrect because they are oversimplified and do
not properly use branch lengths as described in Felsenstein
(1985
). In addition, some
publications have failed to calculate correlations and regressions through the
origin (see below). Thus, readers are encouraged to validate any new computer
program by running through one of the published examples listed above.
The goal of IC is thus to transform the original data into independent and
equally distributed contrasts (assuming its assumptions are met) that are then
amenable to standard statistical comparisons and analyses. The only constraint
on statistical analyses of contrasts is that all models are forced through the
origin (Garland et al., 1992).
But this is actually just a requirement of the IC algorithm
(Rohlf, 2001
), and so it is
also possible to calculate, for example, phylogenetically correct
y-intercepts for regression equations
(Garland et al., 1993
;
Garland and Ives, 2000
).
Although first presented in the context of correlation and regression, and
most commonly used for such analyses, the algebra of IC also allows such
univariate analyses as computing phylogenetically correct mean values (and
standard errors) for clades (also interpretable as hypothetical ancestors:
Garland et al., 1999
),
comparing average values of clades or ecologically defined groups
(Garland et al., 1993
;
Rezende et al., 2004
), and
comparing average rates of evolution between clades
(Garland, 1992
;
Garland and Ives, 2000
;
Hutcheon and Garland, 2004
;
McKechnie and Wolf, 2004
). IC
analyses can also be used for multivariate purposes, such as principal
components analysis (PCA; e.g. Clobert et
al., 1998
; see also Ricklefs
and Nealen, 1998
).
Although not intuitively obvious, if one collapses a phylogeny to be a star
by shortening all internal branches to zero length, while lengthening all
terminal branches so that all tips remain or become contemporaneous (as in
Figs 3A and
4A), then all of the results of
IC calculations will be identical to those of conventional `non-phylogenetic'
analyses (Purvis and Garland,
1993). In fact, this is a good exercise to perform to verify that
a particular computer program for computing IC is actually working correctly.
In any case, a conventional analysis, which mathematically assumes a star
phylogeny with contemporaneous tips, can be viewed as a special case of a
phylogenetic analysis (Garland et al.,
1999
).
|
It is important to reiterate that given the same tip data, phylogenetic
information (topology and branch lengths), and assumed model of evolution, IC
and GLS methods yield identical results. IC is thus a special case of GLS
models. Worked examples of a univariate GLS analysis can be found in
Cunningham et al. (1998, box
3) and Freckleton et al.
(2002
). Because phylogenetic
GLS analyses are not as familiar as IC, we will briefly explain how they work.
First, the phylogenetic tree is converted to a symmetrical matrix that is
intended to represent the expected variances and covariances of the tip data
or, if in a regression model, those of the residuals. The diagonals of this
matrix indicate the expected variances, and are taken simply as the total
branch length distance from the root to each tip. If the tree has
contemporaneous tips (e.g. as in Fig.
4), then all these values will be the same. These values thus
represent the putative total opportunity for evolutionary change that each
species has experienced (since the basal split of the tree). The off-diagonals
are taken as the amount of branch length that is shared by any two
tips, i.e. from the root of the tree to last common ancestor.
Once the phylogenetic tree has been converted into a variancecovariance matrix, its incorporation into standard statistical analyses is actually rather intuitive. For example, in a standard linear regression model it is assumed that the expected variancecovariance matrix of the residuals is the identity matrix, which has values of unity for all diagonals and values of zero for all off-diagonal elements. Standard `weighted regression' is simply a variant of this in which the diagonal elements need not be the same value, which has the effect of giving the data points different `pull' in computing the regression equation. Many common statistical packages allow one to perform weighted regression. The phylogenetic GLS regression is essentially the same, except that the specified matrix can now have off-diagonal elements that are not all zeros. Note that if one specifies the phylogeny to be a star, then the matrix is the identity matrix, so GLS methods as with IC can yield standard statistical results.
Both IC and GLS analyses can be modified to incorporate some other models
of character evolution. In essence, this is done by transforming the branch
lengths of the working phylogenetic tree. As with data transformations in
conventional statistics, transformations of branch lengths can be done from a
purely statistical perspective (e.g.
Grafen, 1989;
Garland et al., 1992
;
Diaz-Uriarte and Garland,
1996
,
1998
;
Freckleton et al., 2002
) or in
a fashion intended to mimic some model of character evolution, such as the
OrnsteinUhlenbeck (OU) process, a more complex model that has been used
to mimic characters under stabilizing selection (see
Felsenstein, 1988
;
Garland et al., 1993
;
Diaz-Uriarte and Garland,
1996
; Martins and Hansen,
1997
; Blomberg et al.,
2003
; Housworth et al., 1994). No studies have yet examined
whether one approach or the other yields better statistical performance, but
transforms designed to mimic explicit models facilitate inferences with
respect to parameters of those models.
Although they are functionally equivalent for most purposes, IC and GLS
approaches differ in how intuitive they are for certain analyses, and also in
terms of what software is available for their implementation. For example,
with IC, `tree thinking' is retained, which facilitates graphical analyses,
identification of places (bifurcations) in a phylogeny where rapid
evolutionary events occurred, and also suggests more intuitively such
procedures as rerooting along branches to reconstruct hypothetical ancestors
as direct descendants or to predict values of unmeasured species (see
Garland et al., 1999;
Garland and Ives, 2000
;
Reynolds, 2002
;
Ross et al., 2004
). With IC,
it is easier to employ different sets of branch lengths for different traits
(e.g. Garland et al., 1992
;
Lovegrove, 2003
;
Rezende et al., 2004
), which
may be particularly useful when one trait does not actually show phylogenetic
signal (e.g. Tieleman et al.,
2003
; Rheindt et al.,
2004
) and/or for traits that are only nuisance variables, such as
details of measurement or calculation methods that differ among studies (e.g.
Wolf et al., 1998
;
Perry and Garland, 2002
;
Rezende et al., 2004
). With
GLS, on the other hand, once the phylogenetic variancecovariance matrix
has been constructed, a variety of commercial and free statistical and matrix
algebra packages can be used (e.g. SAS, as in
Butler et al., 2000
; the
Matlab PHYSIG package of Blomberg et al.,
2003
; PHYLOGR and APE in the R language, available at
http://cran.r-project.org/).
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Acknowledgments |
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