Cladistics 14, 221 ] 228 (1998)
WWW http:rrwww.apnet.com
Article No. cl980064
The Logical Basis for the use of Continuous Characters
in Phylogenetic Systematics
Todd C. Rae
Department of Anthropology, University of Durham, U.K. and Department of Mammalogy, American Museum of
Natural History, New York, U.S.A.
Received for publication 12 March 1998
It has been argued that continuous characteristics
should be excluded from cladistic analysis for two
reasons: because the data are considered inappropriate;
and because the methods for the conversion of these
data into codes are considered arbitrary. Metric data,
however, fulfill the sole criterion for inclusion in phylogenetic analysis, the presence of homologous character
states, and thus cannot be excluded as a class of data.
The second line of reasoning, that coding methods are
arbitrary, applies to gap and segment coding, but quantitative data can be coded in a nonarbitrary manner by
means of tests of statistical significance. These
procedures, which are both objective and repeatable,
determine the probability that two taxa possess an
homologous character state; that is, if they have inherited a particular central tendency and distribution of
individual variates unchanged from a common ancestor.
Thus, the application of statistical tests to quantitative
data empirically detects the presence of evolutionary change, the raw material of phylogenetic
reconstruction. Q 1998 The Willi Hennig Society
‘‘I advise my philosophy students to develop hypersensitivity
for rhetorical questions in philosophy. They paper over whatever cracks there are in the arguments.’’
ŽDennet, 1995: 178..
Correspondence to: Todd C. Rae, Dept. of Anthropology, Univ.
of Durham, 43 Old Elvet, Durham, DH1 3HN, U.K. E-mail:
[email protected]
0748-3007r98r030221 q 08 $30.00r0
Copyright Q 1998 by The Willi Hennig Society
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INTRODUCTION
The widespread acceptance of computer-assisted
phylogenetic analysis has served to focus debate on
many of the operational aspects of cladistics. Before
the advent of mathematical algorithms for the analysis of phylogenetic data, verbal descriptions of characters and character states were sufficient. Computer
parsimony packages such as Hennig86 ŽFarris, 1988.
and PAUP ŽSwofford, 1990., Žand even some numerical taxonomy procedures; see Kendrick, 1964., on the
other hand, require the translation of raw data andror
verbal descriptions into alphanumeric codes that represent character states. For discrete characteristics,
each recognizable state is represented by a unique
code.
Many characteristics of organisms, however, vary
in a quantitative manner. The appropriate methods
used to derive character state codes from quantitative
data, and the applicability of these data to phylogenetic analysis in general, have been widely disputed
ŽArchie, 1985; Pimentel and Riggins, 1987; Cranston
and Humphries, 1988; Chappill, 1989; Thiele, 1993..
The purpose of the present contribution is to review
the arguments against the use of continuous characters in phylogenetic systematics, and the various
methods for coding these characters, and to outline a
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Rae
theoretical justification for their use in determining
evolutionary relationships between organisms.
In the following account, several phrases Žquantitative or continuous characters, measurement or metric
data, etc. describing quantitative characteristics will
be used synonymously. All of these phrases are used
to describe characters used in phylogenetic analysis
which can be described quantitatively and show some
intraspecific variation. In practice, quantitative or
continuous characters are nearly always synonymous
with metric or measurement data.
OBJECTIONS TO CONTINUOUS DATA
The arguments against the use of continuous characteristics in phylogenetic analysis are diverse. Disotell Ž1994: 51., in describing the characters used in a
previous study, simply states that ‘‘one is not discrete . . . and therefore cannot be used in a cladistic
analysis’’. Similarly, in the estimation of Crowe Ž1994:
78., ‘‘Žc.ontinuously varying characters are not the
stuff of phylogeny’’. Most of the objections, however,
fall into two main categories: those that question
metric data in principle, and those that question the
methods by which codes are derived from quantitative data.
Many of the objections raised against continuous
data as a class of data are phrased as rhetorical
questions. For example, Crisp and Weston Ž1987: 67,
emphasis added. offer the following argument:
‘‘However, most quantitative data are not discontinuous, but
represent series of overlapping values . . . w andx means must
be calculated and statistical tests applied to group those
means into meaningful subsets. But, then what is the cladistic
significance of a mean for a taxon?’’
This particular line of reasoning is repeated by
Pimentel and Riggins Ž1987: 201., who ask, ‘‘what are
the cladistic properties of means, standard deviations, or tests of significance . . . ?’’ Asking questions
merely for effect, however, is no substitute for a
reasoned argument against any class of data.
More explicit grounds for the elimination of metric
data Žin this case, ratios. were offered by Cranston
and Humphries Ž1988: 81., when they ‘‘question even
whether a ratio represents a cladistic character...
Copyright Q 1998 by The Willi Hennig Society
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Žbecause ratios. are clearly phenetic, possessing little
or no cladistic value, since they cannot be scored in
outgroups and cannot be polarized’’. Data themselves, however, cannot be cladistic or phenetic; those
terms apply only to the analysis of data. Further, any
characteristic that can be scored Žread ‘‘coded’’. for
ingroup taxa can be coded for outgroups as well,
since there is no intrinsic difference between ingroup
and outgroup organisms to prevent it. Thus, all characters, once coded, can be polarized in the same
manner, via the outgroup criterion Žsee Maddison
et al., 1984..
None of these objections represent theoretically justifiable reasons for dismissing metric data on their
intrinsic qualities. In fact, there has been no adequate
justification offered for a fundamental differentiation
between quantitative and discrete characters and
there are many suggestions to the effect that ‘‘discrete’’ traits are simply continuous characteristics
hidden in disjunctive terminology ŽBaum, 1988;
Chappill, 1989; Stevens, 1991; Thiele, 1993.. It is still
possible, however, that continuous characteristics
could be inappropriate for cladistic analysis. To
determine if this is the case, the necessary attributes
of characteristics in cladistic analysis must be defined
and then it must be demonstrated that this particular
class of data does not fulfil those criteria.
THE REQUIRED ATTRIBUTES OF
CHARACTER STATES
The definitions of ‘‘character’’ and ‘‘character state’’
used here follow those of Farris et al. Ž1970: 172.:
‘‘A character Ž‘‘transformation series’’ of Hennig. is a collection of mutually exclusive states Žattributes; features; ‘‘characters’’, ‘‘character states’’, or ‘‘stages of expression’’ of
Hennig. which
a. have a fixed order of evolution such that
b. each state is derived directly from just one other state,
and
c. there is a unique state from which every other state is
eventually derived’’
while recognizing that these terms refer to properties
of taxa in cladistic analysis ŽThiele, 1993.. This definition is followed, explicitly or otherwise, by most of
the authors discussed above Že.g. Pimentel and
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Continuous Characters
Riggins, 1987.. Thus, a character state is an observable property of the individual organisms that belong
to a particular taxon Že.g. blue., while a character is a
collection of character states presumed to be homologous Že.g. color.. Operationally, this is analogous to
the traditional genetic definitions of gene Žcharacter.
and allele Žcharacter state.. The implicit argument in
the above definition is that the character states are
alternative representations of the ‘‘same’’ thing; i.e.
that the states are homologues.
This point has been used by Pimentel and Riggins
Ž1987. again to argue against metric data in cladistics.
They cite Patterson Ž1982. to the effect that there are
two types of homology, taxic and transformational,
and conclude Žp. 208. that it ‘‘seems obvious that
most quantitative variables can lead only to . . . transformational homologies, so are useless in cladistic
analysis’’. Patterson Ž1982., who adapted the concept
from Eldredge’s Ž1979. discussion of approaches to
evolutionary theory, distinguishes taxic homology,
used to diagnose hierarchical groups, and transformational homology, which explains organismal similarities by reference to archetypes. Transformational
homology implies no grouping of taxa and is not
useful for the examination of hierarchical structures
of monophyletic groups. But the link between transformational homology and metric data is unclear at
best. For instance, imagine two taxa ŽA and B. coded
as identical Ži.e. homologous. for femur length for
cladistic analysis; the hypothesis is that the length of
the femur is a shared character of these taxa inherited
from a common ancestor. Thus, the hypothesized
homologues will be used to diagnose groups, not to
refer to hypothetical archetypes, and the hypothesis
of homology will be subject to the test of congruence
Žsee below. in the same manner as for ‘‘discrete’’
characteristics.
The only requisite criterion that a character must
fulfill for use in phylogenetic analysis is that character states must be homologous. There are no other
‘‘cladistic properties’’ of characteristics. In fact, there
are no such things as ‘‘cladistic data’’ Ž sensu Pimentel
and Riggins, 1987.; all attributes of an organism can
be used in phylogenetic analysis, provided they can
be described in terms of homologous character states.
If one hypothesizes that taxa A and B share an
homologous character state of a metric character, one
presents the explicit hypothesis that the organisms
Copyright Q 1998 by The Willi Hennig Society
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inherited the character state unchanged from a common ancestor. This hypothesis of homology, based on
similarity, is then tested by parsimony analysis of the
data set; those character states that have fulfilled the
second criterion of homology, congruence, can be
assumed to have arisen from common ancestry.
Therefore, since quantitative data cannot be excluded
from phylogenetic study by reference to the logical
requirements of ‘‘cladistic characters’’, it remains only
to determine if there is a justifiable method of performing the similarity test on metric characters.
METHODS OF CODING CONTINUOUS
DATA
Continuous characteristics possess a more or less
even distribution of individual variates, and the range
of variates of any given taxon may overlap the range
of another taxon. As data for individuals cannot be
entered into phylogenetic analysis, since the method
cannot logically apply to tokogenetic relationships
ŽHennig, 1966., a method must be employed to determine which codes apply to which taxa. In this
sense, it must be determined whether two taxa are
‘‘the same’’ or ‘‘different’’ for a given quantitative
attribute.
First, however, it is useful to make the distinction
between dividing continuous distributions and coding taxa for phylogenetic analysis. Pimentel and Riggins Ž1987: 201. state categorically that ‘‘continuously
varying quantitative data are not suitable for cladistic
analysis because there is no justifiable basis for recognizing discrete states among them’’. Felsenstein Ž1988:
462. also objects to dividing a continuous distribution
into discrete states:
‘‘None of the authors on coding methods has yet faced the
question of how we could test for the presence of underlying
discrete states. Lacking such a test, there is no reason to
discretize w sicx quantitative characters.’’
‘‘Continuous-into-discrete’’, however, is not isomorphic with coding. The continuous distribution of individual variates is not itself divided; the taxa to which
the individual variates belong are assigned character
codes on the basis of whether they are similar to one
another in central tendency and distribution. All deci-
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Rae
sions about taxon membership are made before coding begins. A particular organism from Taxon A may
be indistinguishable from an organism from Taxon B
with respect to character X, but the two taxa may be
significantly different in the distribution of trait X
nonetheless. The entire issue of overlap is a red
herring that results from a fundamental confusion
between attributes of organisms and attributes of
taxa ŽThiele, 1993..
Even given this distinction, opponents of the use of
metric data contend that dividing the continuous
distribution of individual variates into codes is arbitrary and hence not fit for use in scientific inquiry.
For example, Crisp and Weston Ž1987: 67: emphasis
added. advocate:
‘‘the rejection of w continuousx characters for which states can
be circumscribed only arbitrarily . . . Arbitrary decisions cannot be assessed critically and thus cannot be discussed within
a scientific context. Arbitrary character states have nothing to
do with homology.’’
Even some who accept the use of continuous data
have conceded that some aspects of the division of
nondiscrete distributions into codes is essentially
arbitrary Že.g. Archie, 1985.. Since the class of metric
data cannot be dismissed in principle Žsee above., the
case against the inclusion of these data in phylogenetic systematics relies on whether the methods for
assigning character states to taxa are arbitrary.
The two most common coding methods, segment
coding and gap coding, both have arbitrary elements.
Segment coding proceeds by dividing the range of
variates into a number of equally sized segments,
then assigns codes to taxa according to the segment
in which their central tendency Žusually either the
mean or the median. lies. This method is outlined in
Simon Ž1983. and comprises the theoretical basis of
the technique advocated by Chappill Ž1989; see Farris, 1990.1. Gap coding ŽMickevich and Johnson, 1976;
Almeida and Bisby, 1984; Archie, 1985. proceeds by
defining the boundaries between character states at
those points in the univariate distribution where a
‘‘gap’’ equal to or above a certain size occurs between
the central tendencies of any two taxa. In most cases,
1
The method of Colless 1980., although considered segment
coding by both Chappill Ž1989. and Thiele Ž1993., is a scaling or
weighting method that was applied in that work to gap-coded data
Žsee Farris, 1990. and will not be discussed in this context.
Copyright Q 1998 by The Willi Hennig Society
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the critical gap size is some function of the withingroup standard deviation. Both of these methods of
assigning character states to taxa are arbitrary in that
the number of codes, their distribution, and membership depend entirely on either the a priori number of
segments chosen, or the critical gap size. As Crisp
and Weston Ž1987: 67. argue:
‘‘it would be inadmissible to use a length variable that had
been arbitrarily subdivided into two states, one of lengths less
than, the other of lengths greater than the median length,
since it would be just as reasonable to choose any point along
such a continuum at which to delimit states.’’
The arguments put forth against the arbitrary
nature of coding continuous characters are not as
obviously applicable to coding that is based exclusively on statistically significant differences between
taxa. Because statistical methods are based on probability, the number of subsets or size of gaps is not
determined a priori, nor in an arbitrary fashion. The
use of this type of coding, however, has also been
questioned. Again, Pimentel and Riggins Ž1987: 207.
use a rhetorical question to frame their argument:
‘‘How can . . . statistics or tests of significance be applied
to transform quantitative variables into cladistic
variables? . . . We know of no reported theoretical basis for
doing this and can provide none. Rather, such recommendations amount to ‘‘data massaging’’ to the point of inventing
data.’’
The following section is an attempt to answer this
objection by offering justification for using statistical
methods to derive character states from measurement
data.
THE BIOLOGICAL BASIS FOR
RECOGNIZING CHARACTER STATES
Tests of statistical significance determine the probability that the difference between any two samples is
due to random Žor chance. factors. In two-sample
cases, statistical tests determine whether the absolute
difference between the samples, represented by a
measure of central tendency Žusually means or medians., is ‘‘real’’, or whether it can be attributed to
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Continuous Characters
chance variation, such as measurement error. Central
tendencies are used since they are good summaries of
the data and, by analogy with the normal distribution
histogram based on coin-tosses, the central tendency
of a biological attribute for a taxon can be thought of
as the expected value and the variation is attributed
to natural, random variation andror measurement
error.
This type of distribution, approximating the normal
distribution, is an empirically demonstrable property
of many unidimensional metric characteristics in
populations of living organisms. It has also been
shown that the central tendency of metric attributes
in organisms changes over the course of evolution,
often in a short time and in response to known
selection factors Že.g. Bates Smith et al., 1995..
Because organisms possess characteristics of this type
that change historically, the concepts of probability
can be applied to test whether taxa are ‘‘the same’’ or
‘‘different’’ for a given trait. Thus, statistical tests
provide an objective, repeatable, nonarbitrary method
for deciding if two taxa are similar, in terms of some
metric attribute 2 .
In coding characters, every decision of character
state membership is an explicit hypothesis of homology ŽPatterson, 1982.. That is, the act of assigning a
code for a given trait to any two particular taxa is the
equivalent of proposing that the taxa in question
have inherited that condition unchanged from a common ancestor. Codes are thus explicit hypotheses of
relationships between taxa. An examination of the
implicit assumptions of tests of statistical significance
based on the normal distribution illustrates the applicability of these methods to biological questions. The
Student’s t-test, for example, determines the probability that two samples were drawn from populations
with the same central tendency ŽSokal and Rohlf,
1981.. If the null hypothesis of equality is falsified
Žthat is, if the samples are significantly different from
one another., it can be assumed that the absolute
difference in central tendency between the samples is
‘‘real’’ and cannot be attributed to chance. This type
2
Another coding method, finite mixture coding ŽStrait et al.,
1996., operates on differences between the distributions of groups
of species means, and assigns codes to intermediate taxa via a
likelihood algorithm. This potentially has uses for higher level
analyses, particularly after the theoretical connection between statistical ‘‘populations’’ of means and natural populations has been
adequately explored.
Copyright Q 1998 by The Willi Hennig Society
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of biological hypothesis testing is essential to such
endeavors as species designation, as is implied in
Pimentel and Riggins Ž1987: 208..
Objections to the effect that the choice of statistical
test employed is arbitrary, and thus makes the coding exercise arbitrary as well, are negated by recognition that different samples require different methods
of determining significance. For the analysis to be
accepted, however, the particular test chosen must
conform to the hierarchy of methods appropriate for
the particular samples tested developed within the
field of biometry Žsee Sokal and Rohlf, 1981.. Since
sample parameters cannot be known a priori, it is not
possible to prescribe a particular method that will be
relevant in all cases.
If we accept that codes are hypotheses of homology
and that statistical tests of significance are a biologically relevant way to determine whether two samples
are ‘‘the same’’, the evolutionary Žor cladistic . relevance of this method is evident. If two taxa are
statistically indistinguishable for a particular metric
variable, we may hypothesize that they have inherited the central tendency and distribution of that
character from a common ancestor. If two taxa are
not ‘‘the same’’ Ži.e. if there is less than 5% probability that the samples were drawn from populations
with identical statistical sample parameters., then
these taxa have failed the first test of homology
Žsimilarity. and we can infer that an evolutionary
change Žor step. has occurred in that character, necessitating a second character state. In other words,
these taxa could not have inherited the central tendency and distribution of the variates for that character unchanged from a common ancestor.
Two-sample tests Žlike the Student’s t-test., unfortunately, are not applicable for phylogenetic analysis,
because at least four taxa are needed: three members
of the ingroup and one outgroup for polarity determination. In multiple comparisons, however, areas of
overlap in statistical significance can occur. For example, taxa A and B could be indistinguishable from
one another in central tendency, but only taxon A is
significantly different from a third taxon C. This kind
of distribution of significant differences can lead to
situations in which two taxa that are not significantly
different from one another are assigned different
codes; Farris Ž1990: 98. has argued that this procedure can ‘‘create nonsense distinctions’’.
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Rae
These distinctions, however, are based on an evolutionary argument about the distribution of significant
differences in central tendency between taxa. It is
expected that taxa that inherit a particular character
state for a metric attribute from a common ancestor
will inherit the both the central tendency and distribution of the state unchanged. If this is the case, taxa A
and B should be significantly different from exactly
the same taxa. If they are not, it can be hypothesized
that some change has occurred, although it may only
be in the distribution of individual variates. An
example of this type of coding is given by Simon
Ž1983. under the rubric homogeneous subset coding;
only taxa that form a homogeneous subset Ži.e. are
different from exactly the same taxa. are coded as
identical. This coding method provides an answer to
the common criticism of numerical coding voiced by
Trinkaus Ž1990: 7., that the ‘‘amount of within-group
variation obscured and the number of intermediate
forms denied by these w codingx methods are
unknown’’.
DISCUSSION
The preceding argument suggests that continuous
characters cannot be excluded from cladistic analysis,
but are there reasons to believe that these data
improve our understanding of the evolution of life,
outside of the somewhat obvious ‘‘more-datais-better’’ ŽChappill, 1989; Donoghue and Sanderson,
1992. formulation? Although many workers Žincluding the author; Rae, 1995. have provided empirical
comparisons of various methods using actual data, it
is difficult to imagine, in fact, a situation where it
would be possible to demonstrate that either different coding methods or different classes of data have
any specific logical effect on the outcome of phylogenetic analysis. Two comparisons from the literature
highlight this problem.
To date, tests designed to measure the effects of
coding have been flawed in that they only measure
the a posteriori effect, not how well the coding methods reflect the nature of the organisms. Chappill
Ž1989. attempted to determine the differences
between alternative codings of the same data by
comparing the homoplasy andror resolution of the
Copyright Q 1998 by The Willi Hennig Society
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resulting most parsimonious trees. If the parameter
used to evaluate methods is, for example, level of
homoplasy, then the optimum coding method would
be that which reduced between-taxon variation to
zero, since there is no homoplasy if all taxa are the
same. Unfortunately, this would reduce resolution to
zero, as well. Alternatively, Cranston and Humphries
Ž1988. used a partitioning method; they removed
continuous characteristics from their data set and
performed parsimony analyses on the discrete data
only. They found that the tree lengths of the most
parsimonious arrangements decreased and consistency indices ŽCI. increased, relative to the topologies
obtained from the combined data set. This led them
to recommend excluding continuous data from analyses, in the interest of decreasing homoplasy. This
‘‘result’’, however, was merely an effect of removing
characters; all other things being equal, tree length
will decrease as a result of the reduction of the
number of characters in an analysis, while the CI, a
direct inverse function of tree length, will increase 3.
The same result would be obtained if only discrete
characters were removed.
The only way to evaluate the effect of either different classes of data or different coding methods
would be to use ‘‘known’’ phylogenies, such as artificial virus lineages ŽHillis et al., 1992. or computer
simulations ŽHuelsenbeck and Hillis, 1993.. Unfortunately, this kind of test is unavailable for the vast
majority of organisms. A second alternative, at least
for classes of data, is to utilize tests of taxonomic
congruence ŽMickevich, 1978; Mickevich and Farris,
1981; Farris et al., 1994., although this method will
determine only whether the two data sets are congruent, not which of the two is ‘‘better’’ or ‘‘more appropriate’’. In the absence of an appropriate critical test,
the applicability of data or methods must rely on
their theoretical foundations. Since continuous characters are not inappropriate in principle, since they
fulfil the necessary criterion for use in phylogenetic
analysis Žhomologous character states. and, since they
can be coded in a nonarbitrary, biologically appropriate manner, there can be no theoretically justifiable
means for dismissing them from phylogenetic systematics. In fact, techniques such as homogeneous
3
The same phenomenon was demonstrated by Archie Ž1989. for
numbers of taxa in an analysis Žalthough see Farris, 1989, for a
discussion of the history of this finding..
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Continuous Characters
subset coding may even provide a method for the
inclusion of polymorphisms into computer phylogeny analyses, as the percentage presence of alternative expressions of a character in a taxon can be
treated in the same manner as measurement data. As
Donoghue and Sanderson Ž1992: 359. argue, ‘‘Žt.he
temptation to ignore data is evidently great, but
experience suggests that this should be resisted’’.
ACKNOWLEDGEMENTS
I extend thanks to David Strait and Judith Masters for helpful
discussion, and to two anonymous reviewers for careful, critical
reading of this paper. The work was supported by a Research
Fellowship from the Department of Mammalogy, American Museum of Natural History; my thanks go to Guy Musser, Ross
MacPhee, and the staff of the Department for their help and
encouragement.
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