AMER. ZOOL., 20:679-693 (1980)
Multivariate Analysis and the Study of Form, with Special
Reference to Canonical Variate Analysis1
GENE H. ALBRECHT
Department of Anatomy, University of Southern California,
Los Angeles, California 90033
SYNOPSIS. This paper presents a simple, geometric, largely intuitive introduction to the
use of multivariate analysis for the study of biological form. Rather than presenting rigorous mathematical formulations of the theory and practice of statistics, the emphasis is
on multivariate analysis as a descriptive, data-analytic approach which represents the
quantitative version of classical comparative anatomy. One method—canonical variate
analysis—is chosen for special attention because of its wide applicability to problems in
which multidimensional data are used to compare differences between two, several, or many
populations with proper regard for individual variation within the populations. The theory, practice, and utility of canonical variate analysis are presented by way of simple,
bivariate examples. Particular attention is given to how the results of canonical variate
analysis are affected by alterations in the within-group dispersion when the relationships
among groups are held constant. Although the discussion focuses on canonical variate
analysis, many statements transfer directly or with appropriate modification to the use of
other multivariate procedures in comparative morphology.
INTRODUCTION
Biological form, its diversity, and its
meaning can be studied from a variety of
approaches and viewpoints. This symposium provides a brief prospectus of some
modern mathematical, physical, and theoretical possibilities. My intent is to discuss
multivariate statistical analysis as one approach to the study of morphology. The
general scope of multivariate analysis, and
its application to biological problems, is extensively reviewed by numerous authors
(see below). Rather than duplicate these
previous commentaries, I have chosen one
method—canonical variate analysis—for
special attention. The emphasis is on simple, descriptive examples which demonstrate how canonical variate analysis works,
rather than detailed statistical formulations.
Numerous reviews present the principles, practices, and problems of multivariate analysis from the viewpoint of the morphologist (Blackith, 1965; Sokal, 1965,
1974; Howells, 1969; Oxnard, 1969, 1972,
1973ft, 1978, 1979; Reyment, 1969, 1973;
Blackith and Reyment, 1971; Rohlf, 1971;
Goodman, 1972; Gould and Johnston,
1
From the Symposium on Analysis of Form presented at the Annual Meeting of the American Society of Zoologists, 27-30 December 1979, at Tampa,
Florida.
1972; Kowalski, 1972; Sneath and Sokal,
1973; Atchley and Bryant, 1975; Bryant
and Atchley, 1975; Corruccini, 1975;
Campbell, 1978). As complements to these
biological perspectives, Anderson (1958),
Seal (1964), Morrison (1967), Dempster
(1969), Rao (1970), Cooley and Lohnes
(1971), Tatsuoka (1971), and Harris
(1975) are recommended for their statistical formulations of multivariate analysis.
These readings provide the interested
worker an introduction to the range of
multivariate methods and their specific
adaptability to the spectrum of morphological problems. They also provide the
theoretical and practical background for
the following discussion of canonical variate analysis.
Canonical variate analysis is applicable
to problems in which multidimensional
data are used to characterize morphological relationships among two, several, or
many populations with proper concern for
the effects of individual variation within
the various populations. When just two
populations are involved, the purpose is to
combine the multiple descriptor variables
into a single function which maximizes the
difference, or discrimination, between the
two populations. The two-group case is
commonly known as discriminant or discriminant function analysis. When more
than two populations are involved, the
679
680
GENE H. ALBRECHT
purpose is to combine the multiple de- and 547 research papers published before
scriptor variables into a reduced number 1972. As a rough but conservative estiof functions which maximize, or most ef- mate, this bibliography would double in
ficiently summarize, the overall differences size if updated to include all of the past
among the populations. The multiple- decade. Research applications to just biogroup case is commonly known as dis- logical problems must also number in the
criminant function, multiple discriminant, hundreds (Sneath and Sokal, 1973, listed
multiple-group discriminant, step-wise dis- about 600 applied multivariate studies in
criminant, canonical, or canonical variate the area of numerical taxonomy published
analysis (the latter two not to be confused between 1956 and 1971).
with canonical correlation analysis).
Historical perspectives
Classification, discrimination,
and identification
Canonical variate analysis dates from the
development of discriminant functions by
Fisher (1936) as a solution to the problem
of finding the maximum difference between two populations when characterized
by multiple measurements. Subsequent
generalization to multiple populations is
usually credited to Rao (1948) although
Hotelling (1936), Bartlett (1938), Fisher
(1938), Brown (1947), and others contributed to the multiple-group solution.
The first two research applications of
canonical variate analysis predate but were
carried out at the suggestion of Fisher
(1936). Barnard (1935) studied temporal
changes in skull measurements and Martin
(1936) studied sexual differences in mandibular measurements of four populations
of Egyptians. Fisher (1936) himself used
flower measurements to study diploid, tetraploid, and hexaploid species of Irus. The
use of canonical variate analysis was limited for the next two decades until modern
computers were developed to handle the
large number of calculations involved
when even a few variables are considered.
Accordingly, the practical application of
canonical variate analysis began with the
exploratory, illustrative, and now classic
studies of 25 years ago (see Blackith and
Reyment, 1971, for examples). Today, canonical variate analysis is common in many
areas of biology (e.g., systematics) yet essentially unknown in others (e.g., cytology)The literature on canonical variate analysis is overwhelmingly voluminous. In an
exhaustive bibliography of just the statistical theory of canonical variate analysis,
Cacoullos and Styan (1973) listed 26 books
Using the term canonical variate analysis
partially alleviates confusion between biological and statistical terminology. In biology, classification means ordering individuals into classes, discrimination means
finding differences among classes, and
identification means assigning an unknown individual to its correct class (see
Sneath and Sokal, 1973). In statistics, classification also means ordering individuals
into classes, but discrimination has taken
the meaning of assigning an unknown individual to its correct class (see Kendall,
1973). Thus, discrimination in statistics has
taken the meaning of identification in biology. The matter is further confused because canonical variate analysis can be
used to both maximize the separation
among many populations (discrimination
in the biological sense) and maximize the
probability of correctly assigning an unknown to one of many populations (identification in the biological sense but discrimination in the statistical sense). While
biological applications (and this paper) emphasize the former usage, statistical theory
emphasizes the latter.
Failure to understand these differences
leads to conceptual inaccuracies with regard to the applications of canonical variate analysis. Corruccini (1975, p. 3) stated:
"Fisher's (1936) linear discriminant function was only designed to minimize probability of misidentification of unidentified
specimens into previously defined samples," and "is not designed to indicate relative affinities of unrelated individuals or
the parent populations." Rightmire (1979,
p. 24) stated: "this use of the functions in
assignment [identification] is the role for
CANONICAL VARIATE ANALYSIS
which discriminant statistics were originally designed by Fisher in 1936." Errors of
fact as well as interpretation are involved
since Fisher (1936) designed the discriminant function specifically for differentiating populations (p. 17, italics added: "special interest attaches to certain linear
functions of the measurements by which
the populations are best discriminated").
Fisher (1936) devoted only two paragraphs
to the problem of assigning individuals to
their parent populations (p. 182, italics
added: the discriminant function "is of interest also in relation to the probability of
TOwdassification," not identification as so
often stated).
The descriptive use of canonical
variate analysis
Canonical variate analysis can emphasize
the rigorous statistical theory and practice
of inferential hypothesis-testing about differences among populations; this involves
matrix algebra, probability distributions,
test statistics, and so forth. Alternatively,
canonical variate analysis can emphasize:
(1) the geometric representation of the algebraic formulations used to express the
statistical functions; (2) the heuristic value
of the statistical techniques in terms of
their underlying rationale and what they
are designed to accomplish; and (3) the
descriptive aspects of statistics in terms of
ordering and summarizing data to facilitate the display, interpretation, and communication of numerical results.
Some workers question the applicability
of multivariate analysis, including canonical variate analysis, to morphological problems (e.g., Kowalski, 1972; Corruccini,
1975). Multivariate analysis is presented as
an inferential methodology with strict requirements of statistical theory and hypothesis-testing. There is a perceived need
for empirical data to be highly conforming
to the structural models and underlying
assumptions of multivariate statistical theory. However, these critiques ignore the
equally valid use of multivariate analysis as
a data-analytic procedure designed to ordinate and efficiently describe multivariate
observations. Thus, the descriptive use of
multivariate analysis emphasizes generat-
681
ing rather than testing hypotheses about
relationships among individuals, populations, and variables (see the comments by
Blackith and Reyment, 1971; Harris,
1975; Campbell, 1978).
MULTIVARIATE ANALYSIS AND
COMPARATIVE ANATOMY
Multivariate analysis as applied to morphological problems represents the quantification of the classical methodology of
comparative anatomy. The size and shape
of an animal, or a particular element of its
anatomy, can be quantified by designing
measurements to characterize the morphological features of interest. These measurements, or variables, can be thought of
as a set of rectangular coordinate axes
which describe a multidimensional space.
The actual values of the measurements for
each individual specimen represent Cartesian coordinates which express the morphology of each individual as a single point
in the multidimensional space. It may be
that individuals are unique in their morphology and, therefore, occupy distant
parts of the data space, or it may be that
numbers of specimens are similar in their
morphology and, therefore, occupy adjacent regions of the data space. Biological
hypotheses are then constructed by analyzing the relative positions of, and distances among, the individuals.
It is critically important that nearness (or
distance) in the multidimensional space be
translated directly as similarity (or difference) in morphology, and not as communality (or differentiation) in terms of
behavior, function, systematic position,
phylogenetic history, or other putative
statements which depend on the interpretation of morphological relationships. The
geometric representation of the individual
specimen in the multidimensional space is
an objectively derived, empirical statement
about the morphology of that particular
specimen; it is invariant within the limits
of inter- and intra-observer measurement
error. However, there very well may be
significant differences in how the morphological position and relationships of an individual are interpreted when viewed by
different investigators. Of course, differ-
682
GENE H. ALBRECHT
molar
length
0 ffl
©
species A
23
mm
S3
species C
83
0 S®
I
m
8#»
B»
•
IB»
EB
species B
molar width (mm)
FIG. 1. Molar tooth morphology in three hypothetical species of mammals. Each specimen is depicted
by a drawing of the occlusal surface of the tooth located at the coordinates which represent the actual
measurements for molar width and length.
ent or additional measurements may
change the metric estimates of morphological relationships, and, accordingly, biological inferences derived from these relationships may be affected.
Consider the hypothetical example of
Figure 1 in which measurements of molar
length and width characterize the tooth
morphology of 60 individuals belonging to
three different species of mammals. The
measurements represent orthogonal axes
of a two-dimensional data space, and the
morphology of each individual is depicted
at the coordinate point which corresponds
to its measured values for the two variables. Clearly, individuals which are similar in morphology are likewise similar in
their position in the metrical space. Examination of the relative positions of the
individuals reveals that members of the
same species occur in elliptical-shaped
clusters along a gradient of size from small
individuals in the lower left to large individuals in the upper right. Moreover,
species B and C are characterized by similar, overlapping morphological plans such
that their teeth are square in shape with
molar length approximating molar width.
In contrast, members of species A have
FIG. 2. Univariate and bivariate representation of
the hypothetical data on molar tooth morphology (see
Fig. 1). Individuals plotted as lowercase letters and
group centroids as uppercase letters; approximate
90% confidence ellipses are given for the withingroup dispersions. Univariate statistics are summarized along each axis; group means are marked by
capital letters, ranges by heavy bars, and theoretical
frequency distributions by bell-shaped curves.
teeth which are rectangular in shape with
molar length exceeding molar width; there
appears to be no overlap in the morphology of species A with that of the other two
species.
Figure 1 is an objective representation
of the morphological relationships among
three mammalian species with respect to
overall surface dimensions of a molar
tooth. Determining the biological significance of these morphological relationships
is an act essentially independent of the
morphology itself. Different readers of
this paper would probably assign different
interpretations depending on their particular expertise, beliefs, and background,
and on the nature and extent of the available outside information about the biology
of these three species. For example, different biological interpretations would
probably result if these were three extant
species which vary in geographic origin as
opposed to three fossil species which vary
in temporal origin. Again, consider that
additional morphological information may
CANONICAL VARIATE ANALYSIS
change the metric relationships and the
biological interpretations which follow
therefrom (e.g., species B and C may be
discriminated on the basis of tooth height).
CANONICAL VARIATE ANALYSIS
Covariation and within-group dispersions
in multivariate analysis
Multivariate methods offer the advantage of simultaneously considering the
joint distribution of variables (Fig. 2). If
the bivariate plot of tooth dimensions is
momentarily disregarded, the univariate
distribution of specimens along the horizontal axis suggests that species A and B
are essentially identical in molar width,
while species C is differentiated from the
other two in its mean value but only partly
so in its distribution. The univariate distribution of specimens along the vertical
axis suggests that species A and C are essentially identical in molar length, while
species B is now differentiated from the
other two in its mean value but only partly
so in its distribution. An overall estimate
of the morphological relationships could
be obtained by simply adding the differences for molar width to those for molar
length. Summation of univariate data suggests that the greatest metric distance, and,
therefore, the greatest morphological differentiation, is between species B and C;
species A is less distinct and approximately
equidistant from these two species.
Comparison of the variables to one
another, however, suggests that the dental
dimensions covary in such a way that wide
molars are correspondingly long. The bivariate plot of Figure 2 demonstrates the
joint distribution of the dental variables
and, thereby, reveals the nature of the covariation. Now, in contradiction to the
univariate results, the group centroids
(multivariate group means) of the three
species are found to be approximately
equidistant from one another (in fact, the
distance between B and C is about 14%
greater than either is from A). Consideration of the within-group population dispersions in the bivariate data space further
changes the estimates of morphological relationships among the species. As indicat-
683
ed in the preceding section, species B and
C are similar in morphology as judged by
the amount of distributional overlap; indeed, some individuals of the two species
are indistinguishable in their molar morphology. Species A is now the most morphologically distinct of the populations
since its members show no morphological
similarity {i.e., no distributional overlap)
with either of the other species.
Figure 2 emphasizes the misleading interpretations which may result when multiple variables are considered univariately
without regard for their joint distribution.
Furthermore, knowledge of the withingroup dispersions is important for evaluating the morphometric relationships
among the populations. Unless the withingroup dispersions are uniformly symmetrical about their respective group centroids, distances in the data space defined
by the descriptor variables are not equivalent in all directions (non-isometric)—that
is, distances among individuals and among
group centroids are not necessarily congruent with normal judgements of morphometric relationships when evaluated by
standards of the within-group dispersions.
For example, in terms of evaluating morphological affinities among populations,
distances parallel to the line joining the
centroids of species B and C are less meaningful than perpendicularly directed distances which separate species A from B
and C. The bivariate case of Figure 2 is a
trivial example of the importance of considering joint distributions. In the case of
multiple variables which describe multiple
populations in an n-dimensional hyperspace, the complexities of covariation within and between groups most likely preclude systematic interpretations of the
actual morphological relationships under
study.
Geometry of canonical variate analysis
Canonical variate analysis is applicable
when multiple, jointly-distributed variables are used to characterize the relationships among many different populations.
The power of canonical variate analysis is
in probing the multivariate data space for
a relatively few dimensions which, never-
684
GENE H. ALBRECHT
theless, efficiently summarize the majority
of the information originally distributed
molar
widely among the descriptor variables. In
length
addition to maximizing the variation in a
reduced number of axes, the transformed
dimensions—i.e., the canonical variates—
are constructed so as to be uncorrelated
with respect to both (1) the distribution of
individuals about their respective group
centroids, and (2) the overall distribution
of the group centroids. Thus, canonical
variate analysis achieves a multivariate
ROTATE TO MAXIMIZE
data space which is isometric with respect
WITHIN-GROUP VARIATION
to the within-group distributions; accordingly, distances among the group centroids
are equivalent in all directions of the multivariate data space. Reducing both the dimensionality and correlations of the original multivariate data space—and obtaining
a situation in which distances have a simple, uniform meaning—enhances the
probability of an investigator being able to
make biological interpretations of morphometric relationships.
The transformation from the complexRESCALE TO STANDARDIZE
ity of the original data space to the relative
WITHIN-GROUP VARIATION
simplicity of the canonical variate space is
sometimes thought to be accomplished by
a mathematical sleight-of-hand which
changes the basic character of the original
measurements, and, thereby, destroys
their biological meaningfulness. Typical of
this misconception is the statement by
Hershkovitz (1977, p. 64; in response to
multivariate studies of primate anatomy):
"evidently, the more extensive the mathematical interventions, the more complicated
become the statistical analyses and the
ROTATE TO MAXIMIZE
more sterile, futile, and unrealistic their
BETWEEN-GROUP VARIATION
results." Rather, canonical variate analysis
is easily understood as a series of rotational
cvn
and rescaling transformations of the original variables which preserve the integrity
of the data while making them more amenable to interpretation. Figure 3 demonstrates the geometry of this sequence
(Rempe and Weber, 1972, presented a
similar graphic demonstration which includes the matrix algebra corresponding
CVI
to each of the transformations).
CANONICAL VARIATES
Canonical variate analysis is not usually
FIG. 3. The geometry of canonical variate analysis. resolved into a series of distinct transforThe sequence of intermediate procedures and steps
is shown for the transformation of raw data to ca- mations as shown in Figure 3. The transformation from raw data to canonical varinonical variates. Data the same as Figure 2.
RAW DATA
CANONICAL VARIATE ANALYSIS
ates can be directly accomplished by
forming linear combinations of the original variables; geometrically, this is equivalent to rotating the original coordinate
axes without the restriction of maintaining
their orthogonality. Campbell (1979&, b),
however, demonstrated that certain practical advantages result when canonical
variate analysis is decomposed geometrically and algebraically into the equivalent
of a two-stage principal component analysis as shown in Figure 3.
The uppermost plot of Figure 3 presents the molar tooth data in a simplified
format. The total variability of the multivariate data space can be partitioned into
(1) within-group variability related to the
distribution of individuals around their
own group centroid, and (2) betweengroup variability related to the overall distribution of the group centroids. The first
step in canonical variate analysis is to rigidly rotate the original coordinate axes so
as to be parallel with the major and minor
axes of the within-group dispersion ellipses. This will normally be a variance-maximizing rotation such that the first (horizontal) axis of the transformed data space
coincides with the major axis of the withingroup dispersion ellipses, and the second
(vertical) axis of the transformed data
space coincides with the minor axis of the
within-group dispersion ellipses. The axes
of the within-group dispersion ellipses are
uncorrelated and the transformed variates
will likewise be uncorrelated. Since this
transformation involves a rigid rotation of
the original coordinate axes, all morphological relationships and morphometric
distances are preserved. The only effect is
that the data space is viewed from a new
vantage point that emphasizes the major
axis of within-group dispersions with subsequent axes arranged in perspective by
criteria of independent and decreasing
within-group variation. This first step is
equivalent to a principal component analysis based on the within-group dispersion
matrix.
The second step in canonical variate
analysis is to rescale the transformed variates so as to standardize the within-group
dispersions along all axes of the transformed data space (this is equivalent to the
685
common statistical practice of forming
standardized z-scores). If the second plot
of Figure 3 is imagined to be drawn on a
rubber sheet, the rescaling can be visualized as simultaneous shrinkage along the
horizontal axis and stretching along the
vertical axis until the within-group dispersion ellipses become circular. The deformation of the transformed data space affects equally the within-group and the
between-group differences. Thus, the distance between species B and C along the
horizontal axis is reduced while the distance between species A and the other two
species along the vertical axis is augmented. The morphometric distances among
the populations now accord with earlier interpretations of morphological relationships based on the relative distributions
and amount of overlap of the withingroup dispersions. If the within-group dispersions in the original data space are uncorrelated {i.e., the dispersion ellipses are
circular), then neither of the first two steps
is necessary.
The third step in canonical variate analysis is to rigidly rotate the rescaled, standardized axes so as to be parallel with the
major and minor axes of the betweengroup dispersion. This is a variance-maximizing rotation such that the first canonical variate coincides with the major axis of
the between-group dispersion, and the second canonical variate coincides with the
minor axis of the between-group dispersion. If more than two-dimensional, the
canonical variates are constructed to be
mutually orthogonal and statistically independent with each successive axis accounting for a smaller proportion of the
total between-group variation. Since this
involves a rigid rotation, all morphological
relationships and morphometric distances
of the rescaled, standardized data space
are preserved in the transformation to the
canonical variate data space. The only effect is that the data space is viewed from
a new vantage point which emphasizes the
major axis of between-group dispersion
with subsequent axes arranged in perspective by criteria of independent and decreasing between-group variation. This final step is equivalent to a principal
component analysis based on the between-
686
GENE H. ALBRECHT
CANONICAL VARIATES
RAW DATA
Y
*33.8
38
H
12
D
1.25 1.00
B
1.00 1.25
[2238.42]
35
-16
X
l
66.2
16
338
H
5.0
4.0
4.0
5.0
[559.60]
35
Y
35
GH
20.0 16.ol
16.0 20.0 I
^66.2
35
Fic. 4. The effect on canonical variate analysis of changing the size of the within-group dispersion ellipses
of the raw data without changing their shape or orientation. The within-group variances are increased from
s2 = 1.25 (top) to s2 = 5.0 (middle) to s2 = 20.0 (bottom); covariances are increased proportionately to give a
constant correlation coefficient of r = 0.80 (see within-group dispersion matrix in lower right of raw data
CANONICAL VARIATE ANALYSIS
group dispersion matrix calculated from
the rescaled, standardized variates.
In summary, no extraordinary "mathematical interventions" are involved in canonical variate analysis. The multivariate
data space of the original descriptor variables is altered only by a rescaling transformation which yields an isometric canonical variate space—that is, between-group
distances in all directions are measured
with respect to a common, unit scale equal
to one standard deviation of the withingroup variation. This rescaling is intuitively attractive because it approximates what
the mind does naturally in evaluating morphometric relationships in the bivariate
plot of the original data. Otherwise, the
only difference between the original and
the canonical variate spaces is the perspective from which they are viewed. It is as if
the coordinate system defined by the original descriptor variables is suspended in air
such that the investigator can walk around
it until the most favorable vantage point is
located for viewing the differences among
the populations. Canonical variate analysis
simply defines the most favorable vantage
point as being related to the greatest statistical separation among the populations.
By analogy, consider that no question
would be raised about picking up two objects, orienting them in common, and moving them about in tandem until a favorable
perspective was obtained for viewing their
principal differences in form (see Oxnard,
1973a, for an interesting pictorial example
of this "by eye" process).
Other criteria, besides that of maximizing statistical variation among populations,
are possible for determining how to view
the multidimensional data space. The importance of looking for oblique or rotated
solutions was emphasized by Albrecht
687
(1979) who demonstrated a simple way to
reveal vectors of biological variation which
may not be concordant with the major axes
of statistical variation. The principles of
canonical correlation analysis could be
used to extend Albrecht's (1979) approach
to the construction of variates which are
orthogonal to and statistically independent
of known axes of biological import. Burnaby (1966) described a similar method
of constructing canonical variates orthogonal to variation which is extraneous to the
desired comparisons.
Within-group versus
between-group dispersions
The above discussion emphasizes the
within-group dispersion as an appropriate
standard against which to judge differences among populations. Figures 4-6
demonstrate how different relationships
of the within-group and between-group
dispersions affect the results of canonical
variate analyses. The between-group dispersion of the raw data is held constant
while the size (Fig. 4), shape (Fig. 5), or
orientation (Fig. 6) of the within-group
dispersion ellipses are modified independently.
Figure 4 demonstrates the result of differences in the within-group variation of
the descriptor variables when all other parameters of the raw data are held constant.
The relative positions of and distances
among the populations in the canonical
variate space are unaffected, and, therefore, the canonical variates are identically
constructed. The canonical variate results
differ only in the absolute distances among
the populations, and, accordingly, in the
amount of distributional overlap among
populations. The magnitude of the absolute differences is inversely proportional
plots). The raw data shown on the left consist of two measurements for eight hypothetical populations of
twenty individuals each. Group centroids and an approximate 90% confidence ellipse for the within-group
dispersion are plotted (identical ellipses could be drawn for the other seven populations). Between-group
relationships are identical for all plots. Results of the canonical variate analyses are shown on the right (note
change of scale for the three plots). Group centroids and 90% confidence circles for the within-group dispersions are plotted. Total between-group variation is given in lower left of each plot, and the percentage of
this variation accounted for by each canonical variate is subscripted to the variable name. Shading enhances
recognition of changes in the metric relationships among populations.
688
GENE H. ALBRECHT
CANONICAL VARIATES
RAW DATA
Y
H
35
5.0
4.5
4.5
5.0
-6
35
[760.98]
-8
FIG. 5. The effect on canonical variate analysis of changing the shape of the within-group dispersion ellipses
of the raw data without changing their orientation. The within-group correlations are decreased from r = 0.90
(top) to r = 0.80 (middle) to r = 0.70 (bottom). See Figure 4 for conventions.
CANONICAL VARIATE ANALYSIS
RAW DATA
689
CANONICAL VARIATES
Y
35'
H
5.0
4.0
4.0
5.0
-6-
[559.60]
l
66.2
35
-12
20
FIG. 6. The effect on canonical variate analysis of changing the orientation of the within-group dispersion
ellipses of the raw data without changing their size or shape. The within-group dispersion ellipses are rotated
by 15° in the middle plot and 90° in the bottom plot. See Figure 4 for conventions.
690
GENE H. ALBRECHT
to changes in the within-group variation of
the raw data. Thus, the successive fourfold increases in the within-group variation relative to constant between-group
differences result in successive fourfold
decreases in the total between-group variation; linear distances in the canonical
variate space decrease by successive factors
of two. The proportionalities of the results
are explained by the decision to scale between-group differences relative to the
within-group dispersions. In the original
data space, the increasing size of the within-group dispersion ellipses is accomplished without changing the within-group
correlation (otherwise, both the size and
shape of the ellipse would change). Since
the major and minor axes of the withingroup dispersion ellipses are affected
proportionately, morphometric distances
parallel to the within-group axes remain
proportionately meaningful for interpreting between-group differences.
Figure 5 demonstrates the result of differences in the within-group correlation
between the descriptor variables when all
other parameters of the raw data are held
constant. The canonical variate results differ in both the absolute and relative positions of and distances among the populations, and, therefore, in the construction
of the canonical variates. For example,
populations A and D are increasingly allied with the cluster formed by populations
B, C, E, and F (the clusters themselves
demonstrate small shifts in their internal
arrangements); as a result, the first canonical axis increasingly reflects the separation
of the outlying populations G and H from
all others. The disparities in the results are
explained by the decision to scale betweengroup differences relative to the withingroup dispersions. In the original data
space, the decreasing within-group correlation between the descriptor variables results in an increasingly circular withingroup dispersion ellipse. The minor axis
of the within-group dispersion ellipse is
affected more than the major axis, and,
accordingly, morphometric distances parallel to the minor within-group axis become less meaningful for interpreting between-group variation.
Figure 6 demonstrates the result of differences in the orientation of the withingroup dispersion ellipses when all other
parameters of the raw data are held constant (maintaining the size and shape of
the within-group dispersion ellipses necessitates changing the within-group variance
and covariance of the original descriptor
variables). The canonical variate results
differ in both the absolute and relative positions of and distances among the populations, and, therefore, in the construction
of the canonical variates. A minor difference of 15° results in a substantial alteration of the canonical variate results; for
example, populations A and D become
more distant from populations G and H
than is the cluster formed by populations
B, C, E, and F. A rotation of 90° dramatically alters the between-group relationships; note that the clusters formed by
populations A and D and by populations
B, C, E, and F are disassociated. The radical changes in the results are explained
by the decision to scale between-group
differences relative to the within-group
dispersions. Rotating the within-group dispersion ellipses correspondingly skews
morphometric distances in the original
data space in terms of their meaningfulness for interpreting between-group differences. Since, in the extreme case of a
90° rotation, the minor axis of the rotated
within-group dispersion ellipses is oriented in the same direction as the unrotated
major axis, then, accordingly, morphometric distances parallel to the rotated minor
axis become significantly more meaningful
for interpreting between-group differences (and vice versa). Lubischew (1962)
noted that taxonomic discrimination is
maximized when within-group and between-group correlations are of the opposite sign.
Figures 4-6 reiterate the importance of
considering within-group dispersions when
investigating differences among populations. A common practice in morphometric studies is to base multivariate analyses
on either single, representative specimens
or group means without regard for withingroup dispersions. The exemplar method
of using a single specimen to represent a
CANONICAL VARIATE ANALYSIS
morphologically variable population suffers from the potential problem of the chosen specimen not being characteristic of its
taxa. Even if the exemplar is representative (the group centroid being ideally so),
morphometric distances among exemplars
do not necessarily accord with distances
among populations when the size, shape,
and/or orientation of the within-group dispersions are taken into account (Figs. 4-6).
Biological interpretations based on exemplars alone may be spurious since the addition of within-group information substantially changes the morphometric
estimates of the between-group relationships. Only in the special case of uncorrelated within-group dispersions (i.e., circular within-group dispersion ellipses in the
original data space) are the relative, but
not necessarily absolute, distances among
exemplars the same as distances among
group centroids in the canonical variate
space.
Interpretation of canonical variate results
Canonical variate analysis is a powerful
tool whose descriptive usage is most profitably viewed as a data-summarizing technique which maximizes the between-group
variation of the original multidimensional
data in just a few transformed variates. It
must be remembered that while canonical
variate analysis is an efficient method for
revealing the basic structure of complex
data sets, the biological interpretation of
the morphometric results is separate from
the methodology itself. Whether or not a
canonical variate analysis leads to meaningful biological inferences depends on:
(1) the particulars of the problem (e.g., are
the data from local populations or genera,
bacteria or elephants, neurons or foot
bones); (2) the goals of the study (e.g., is
the purpose related to phylogenetic relationships, intergradation among taxonomic units, the broad spectrum of functional
morphology, or differences among experimental groups); and (3) the knowledge,
background, and expertise of the investigator (e.g., did the training involve anatomy or ecology, monophyly or polyphyly,
biological or morphological species concepts). The arbiters of any particular
691
multivariate study are those biologists who
have a critical interest in that problem.
Albrecht (1979) emphasized the difference between biological and statistical variation when interpreting the results of
multivariate analyses. The canonical variates should be thought of as a convenient
reference system used to most efficiently
describe the greatest differences among
the populations under study. While the
major axes of statistical variation may have
direct biological meaning, there is no a
priori reason to believe that these statistical
constructs must be treated separately, or
that any one of them has a precise or
unique biological significance. The biological determinants of morphological variation should be sought with respect to the
relative position of and distances among
populations in the full canonical variate
space (or an appropriate subset of the full
space if the relevant information can be
displayed in fewer dimensions). To date,
many morphometric practitioners do not
fully appreciate that biological and statistical variation are not necessarily concordant, and that information of biological
import may be obliquely related to the
major axes of statistical variation.
The results of canonical variate analyses
are usually communicated in the form of
bivariate plots of one canonical variate
against another, or in the form of perspective three-dimensional drawings. The
utility of canonical variate analysis is negated, however, when plots of canonical
variates are presented either without scale
or with unequal scaling for the different
axes. Using different scales for the different axes transforms the within-group dispersions from circular as achieved in the
canonical variate space back into an elliptical form. The result is that the isometry
of the canonical variate space is lost and
distances between populations no longer
have equal meaning in all directions. If the
scale of the drawing is not given, then
there is no way to determine whether the
morphological relationships correspond to
those, for example, of the upper or lower
plots of Figure 3. While the relative positions of the groups are the same in both
plots, the absolute distances among group
692
GENE H. ALBRECHT
centroids and the amount of distributional
overlap among populations may affect the
biological interpretations involved. Perhaps, the upper plot reflects generic differences while the lower plot reflects subspecific differences.
CONCLUSIONS
Limitations of space dictate that not all
aspects of canonical variate analysis receive
attention here. Research examples are
omitted because they are so plentiful in the
literature (see Blackith and Reyment,
1971), and canonical variate analysis is not
integrated with other multivariate methods (see Blackith and Reyment, 1971;
Atchley and Bryant, 1975; Bryant and
Atchley, 1975) nor with other physical and
experimental methods for the analysis of
form (see Oxnard, 19736). The biological
significance of applying within-group variation as a standard by which to judge between-group relationships is not fully examined.
Sampling, sample sizes,
distributional characteristics of the withingroup data (e.g., normality and homogeneity), and linear versus quadratic functions are some of the practical topics which
are important for both the statistical and
descriptive use of canonical variate analysis.
This paper emphasizes the descriptive
fundamentals of multivariate analysis, particularly canonical variate analysis. The
underlying theme is to demonstrate how
morphology can be represented by numbers on paper and points on graphs. Manipulations of the data must then correspond to reasonable comparative processes
which might be performed on the animals
themselves albeit with improbable difficulty. The numerical abstractions provide the
opportunity to alter, control, model, summarize, and otherwise analyze morphology
in search of answers to biological questions. Thus, multivariate analysis is a modern comparative anatomy that maintains
the classical purpose of analyzing variation
and covariation within and among populations in order to explain biological form,
its diversity, and its meaning.
ACKNOWLEDGMENTS
I thank Charles E. Oxnard (University
of Southern California) for his comments.
Norm A. Campbell (CSIRO, Australia)
kindly made his dissertation available to
me; the ideas contained therein, in his
published work, and in other personal
communications provided many statistical
complements to my largely intuitive formulations of canonical variate analysis.
The work was supported by a National Institutes of Health Biomedical Research
Support Grant (#5-S07-RR05356) to the
School of Medicine of the University of
Southern California.
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