COMPARING WING SHAPE OF BATS: THE MERITS OF
PRINCIPAL-COMPONENTS ANALYSIS AND
RELATIVE-WARP ANALYSIS
M. BIRCH
JAMES
Museum of Zoology, University of Michigan, Ann Arbor, MI 48/09
Present address: Department of Biology, Northern Arizona University, Flagstaff, AZ 86011
Size and shape are fundamental features of organisms. Ideally, the methods used to describe
size and shape must be sensitive enough to detect small differences, and at the same time
provide interpretation that is visually satisfying. I compared shape of wings among Eptesicus fuscus, Myotis lucifugus, Pipistrellus hesperus, and Tadarida brasiliensis using three
methods of morphometric analysis; principal-components analysis on traditional measures
of lengths and areas, principal-components analysis of interlandmark distances, and relativewarp analysis. These species were chosen to test the efficacy of each morphometric analysis
at discovering and describing differences in both size and shape. Although principal-components analyses provided similar aggregations of species based on variable loadings, only
relative-warps analysis provided clear pictures as to how species differ. The molossid bat,
T. brasiliensis, had narrower wings than the vespertilionid bats; this shape was not due to
compression of the entire wing, but to a complicated rotation around the fifth digit. The
vespertilionid bats were difficult to discern based on wing shape alone, but the warp analysis showed that P. hesperus and M. lucifugus are more similar in shape of wing than
either is to E. fuscus. Each method of analysis of shape has strengths, but relative-warp
analysis provided both the power to discriminate among minor differences in shape of
species within families, and the graphical capability to display those differences in a visually pleasing way.
Key words:
bats, morphometrics, principal components, wing shape, warp analysis
Size and shape are fundamental features
of an organ,ism that provide clues to help
distinguish species or understand how organisms function. How to describe size and
shape has been studies intensively (Rohlf
and Bookstein, 1990). Ideally, the methods
employed should provide biologically
meaningful information and be sensitive
enough to detect slight variation in morphology of similar organisms. This study
compares how well three methods of analysis of shape reach these goals.
The shape of certain morphologies has
been implicated in determining an animal's
ecology, and this appears especially true for
bats (Findley, 1993; Vaughan, 1970). Aldridge and Rautenbach (1987) have shown
that bats with broad wings are found in
Journal of Mammalogy, 78(4):1187-1198, 1997
1187
dense, cluttered habitats where maneuverability is more important than speed. Other
bats have long, narrow wings designed for
speed. This type of wing is found in fast
fliers that use uncluttered habitats. These
correlations may help answer ecological
questions about why certain species are
found in certain habitats (Wainwright and
Reilly, 1994) or how different species can
coexist in the same habitat (Findley, 1993).
This correlational approach between structure and function is exemplified by Norberg
and Rayner (1987) and Norberg (1994).
Norberg and Rayner (1987) studied 257
species in 16 families in the order Chiroptera to quantify wing shape and correlate it
with style of flight, foraging habitat, and
characteristics of echolocation calls. Others
1188
JOURNAL OF MAMMALOGY
have looked at smaller groups of species
(Aldridge, 1986; Findley, 1993; Norberg,
1981) and have described how differences
in morphology relate to choice of food,
style of flight, and structure of habitat. Imperative to all these studies is a method of
describing wing shape that is informative to
the aerodynamicist, morphologist, and ecologist.
Wings of bats must provide both lift and
thrust. Size of the wing determines lifting
capacity; shape of the wing determines generation of thrust and maneuverability (Norberg, 1994; K. Powell, pers. comm.; WardSmith, 1984). During flapping flight, different areas of the wing generate different
amounts of lift and thrust. In a typical wingbeat sequence, the area from the body to
the wrist (the arm wing) does not rotate as
much as the area distal to the wrist (the
hand wing-Pennycuick, 1972; WardSmith, 1984). Therefore, in flapping flight,
the arm wing is considered a lift-generating
surface, whereas the hand wing, due to its
rotation at the top and bottom of each
stroke, provides both lift and thrust (Norberg, 1990; Pennycuick, 1972, 1975; WardSmith, 1984).
Descriptions of differences in shape
should take these functional areas into account. Measurements of area and lengths of
bones, commonly used in morphometrics of
bats (Burnett, 1983; Farney and Fleharty,
1969; Myers, 1978; Williams and Findley,
1979), do not fully describe differences in
shape and ignore or overlook the covariation of measurements or points on different
areas of the wing.
Ratios of linear distances and principalcomponents analysis (peA) are the most
common tools for describing wing shape independent of size for bats (Farney and Fleharty, 1969; Fenton, 1972; Norberg and
Rayner, 1987). Unfortunately, ratios as variables of shape rarely are independent of
size (Atchley et aI., 1976), making it difficult to recognize changes in shape that result from effects other than changing size.
Principal-components analysis reduces a
Vol. 78. No. 4
dataset with a large number of variables to
a smaller number of uncorrelated variables
(the principal components). These components are arranged hierarchically from those
explaining the most variance in the original
data to those explaining the least. Most
variation in a set of morphological variables
is assumed to come from size differences
among groups and the first principal component is taken to represent size; the remaining principal components represent
shape differences (Jolicoeur, 1963; Jolicoeur and Mosimann, 1960). This interpretation of the principal components, and
even what the terms size and shape describe
has been debated extensively (Bookstein,
1989b; Somers, 1989; Sundberg, 1989).
Despite both the semantic difficulties and
the sometimes enigmatic loading patterns of
each principal component, principal-components analysis is widely used to describe
shape of wings in bats (Findley, 1993; Norberg and Rayner, 1987).
This paper examines the efficacy of a
principal-components analysis on both traditional morphological data and distances
among landmarks on the wing. These principal-components analyses are then compared to another method, the decomposition
of the thin-plate spline by relative warps, to
determine how each method locates and describes differences in shape among four
species of bats.
MATERIALS AND METHODS
I compared wing shapes of Eptesicus fuscus,
Myotis lucifugus, Pipistrellus hesperus, and Tadarida brasiliensis. Although E. fuscus. M. lucifugus, and P. hesperus have similarly shaped
wings, differences in size allow easy discrimination in the field. These three species were chosen because they differ in size, but shapes of
their wings are superficially similar. T. brasiliensis lies within the size range of the above
three, but has distinctive narrow, pointed wings.
It was included to test the descriptive power of
each analysis when presented with obvious differences (Fig. la).
I measured 10 E. fuscus and M. lucifugus. 9
T. brasiliensis. and 5 P. hesperus. Only adult (as
BIRCH-WING SHAPE OF BATS
November 1997
a
E.fuscus
b
FIG. I.-a, Left-wing outlines of Eptesieus
fuseus, Myotis lueifugus, Pipistrellus hesperus,
and Tadarida brasiliensis; all are to same scale;
bar = 30 mm. b, The 13 landmarks selected for
the interlandmark-distance principal-components analysis and the warp analysis.
judged by ossification of the epiphyseal plates
in the fifth digit) males were used. Members of
each species were captured at the same locality
(E. fuse us near Corvallis, Oregon; M. lueifugus
near Cheboygan, Michigan; T. brasiliensis near
Phoenix, Arizona; P. hesperus near Yakima,
Washington) by mist net or harp trap. Bats were
either killed and frozen until an outstretched
wing was photographed in the lab, or anesthetized, photographed, and released in the field.
The right wing of each bat was photographed
with the wing extended until the angle between
the humerus and radius approximated the angle
1189
during mid-downstroke and the leading edge
was ca. 90° relative to the midline of the body
(similar to photographs in Norberg, 1976). To
reduce error associated with possible positioning
bias of the outstretched wing, each wing was
extended and photographed four times.
Body mass was measured to the nearest 0.1 g
upon capture. Wingspan was calculated as two
times the distance from the midline of the body
to the wing tip; wing area was two times the
entire membrane area, including the body and
tail membrane. Measurements of length and area
were acquired from two photographs of each
specimen using Bioscan Optimas software and
image acquisition system (v. 3.l4---0ptimas
Corporation, Bothell, WA). Mean wingspan and
wing area for each specimen was used. From
these measurements, I also calculated two measures of wing shape, wing loading (mass X acceleration due to gravity/wing area) and aspect
ratio (wingspan2/area of wing).
Three analyses were performed. First, I recreated a PCA of Norberg and Rayner (1987:
369) on the four species by using mass, wingspan, and wing area as variables. I used the correlation matrix of log-transformed variables in
this PCA. Although this three-variable analysis
is simplistic compared to more variable-rich
analyses, it provides an example of how previously used methods compare to the analysis of
shape using landmarks.
In the second and third analyses, I digitized
13 landmarks, points of maximum and minimum
curvature or junctions of tissue that sample homologous portions of the wing. These landmarks
provided a realistic representation of the wing
and could be identified on each specimen (Fig.
lb).
For the second analysis, I calculated each interlandmark distance for the 13 landmarks, generating a total of 78 distances. Mean inter landmark distances were calculated for each specimen from the four replicate photographs. I performed a PCA using the covariance matrix of
the log-transformed means. Because all interlandmark distances were in the same units (mm),
I chose the covariance over the correlation matrix to directly interpret the character loadings,
rather than view these loadings through the filter
of standardized variables (Bookstein et aI.,
1985). All statistics were calculated using SYSTAT (Wilkinson, 1990).
Although uneven sizes of samples are rou-
1190
JOURNAL OF MAMMALOGY
tinely used in PCA, sampling artifact may be
introduced if different sizes of samples affect the
variance of each group. Equal samples counter
any bias that may be introduced from over (or
under) sampling one group. In both PCAs, using
group means yields the same scatter of points,
but ignores intragroup variation. Although using
groups with differing numbers of individuals
may give undue weight to better-sampled species, intragroup variation in this sample was inconsequential compared to differences among
species and, thus, did not influence the subsequent rotation and loadings of axes.
The third analysis was a decomposition of the
thin-plate spline by its relative warps (Bookstein, 1991). To begin, I calculated a mean wing
shape from the four replicate photographs of
each animal by decomposing locations of landmark into triangles, all scaled to the same baseline. Differences in shape were then retained in
the position of the third vertex of comparative
triangles. Size was calculated before this scaling
as the root of the sum-of-squared distance between each landmark and the centroid (center of
mass) of the form. This multivariate measure of
size is called centroid size. I calculated a mean
centroid size and a mean landmark configuration
(based on the shape coordinates) from the four
photographs of each specimen. Bats were tested
for differences in centroid size by analysis of
variance and the Tukey-Kramer multiple comparison test. I used Lilliefors and Bartlett tests
to check the assumptions of normality and homogeneity of group variances, respectively. All
assumptions were met.
The relative-warp analysis is in essence a
principal-coordinates analysis on a Procrustes
similarity matrix. A Procrustes analysis optimizes the goodness of fit between landmark configurations. There are many ways of optimizing the
goodness of fit between two forms (Chapman,
1990). Generally, a similarity coefficient is calculated by superimposing one form over another
and jiggling landmarks of each form to minimize the distance between as many landmarks
as possible. This coefficient usually is calculated
as the sum-of-the-squared differences in the coordinates of the two specimens (Rohlf, 1990).
The coefficients generated from comparisons of
many specimens to another, or comparisons to a
consensus specimen, are then assembled in a
matrix suitable for principal-coordinates analysis.
Vol. 78. No. 4
A principal-coordinates analysis simplifies
this similarity matrix by calculating orthogonal
components that explain the variation (Le., interspecimen distance) in the similarity matrix.
Principal-coordinates analysis is similar to principal-components analysis, except that instead of
maximizing distances among variables (component analysis), principal-coordinates analysis
maximizes a set of distances among objects
(Marcus, 1990; Reyment and JOreskog, 1993).
Components from the principal-coordinates
analysis are called the relative warps (Bookstein,
1991). The first relative warp explains the greatest variability in the scatterplot of specimens
when specimens are plotted by their principalcoordinate scores. The second relative warp is
orthogonal to the first, and explains the next
greatest amount of variation, and so on.
The differences discovered through principalcoordinates analysis are then depicted by the
thin-plate spline. In woodworking, a spline is a
narrow, flat piece of wood that fits into a groove
or slot to join two larger pieces. In an analogous
fashion, two forms under comparison can be
joined by a mathematical spline. This spline is
similar to Thompson's (1942) transformational
grid, and can be modeled as an infinitely thin,
infinitely large, steel plate. Certain properties of
this spline allow description of shape change at
and between landmarks (Bookstein, 1991).
The deformation of this spline can be subdivided into two types of change; a uniform component and a non-uniform component. The uniform component represents how a grid of rectangles superimposed on one shape would
change to another shape if the grid were altered
such that the same alteration occurred on all
rectangles. Parallelograms on the starting grid
remain parallelograms on the ending grid. The
uniform component of shape change represents
the most global difference between shapes, or
change that is identical over the entire form.
The non-uniform component represents differences that are not the same everywhere on the
form. If we superimpose two points using a
steel-plate interpolant, more energy is necessary
to bend the plate when the points are close together than when the points are far apart. This
energy partials the non-uniform component into
subdivisions describing changes from large scale
(distant points) down to small scale (near
points). The amount and direction of change at
each landmark (determined by the comparison
1191
BIRCH-WING SHAPE OF BATS
November 1997
of two forms), along with information on the
energy necessary to make those changes, generate the relative warps.
Uniform and nonuniform changes in shape are
incommensurate and describe two separate kinds
of differences in shape. Uniform changes discovered by this method are not necessarily allometric (differences due to size), as size is removed during the calculation of shape coordinates. Instead, uniform changes describe largescale and identical-integrated change. Nonuniform differences are not the same everywhere
on the form and are represented by small-scale,
localizable change.
Bookstein (1989a, 1991) gives a technical explanation of the statistical methods employed
when using the thin-plate spline. A less technical
use of the spline can be found in Swiderski
(1994) and Zelditch et al. (1992).
I estimated the uniform component using the
factor-approximation method in Bookstein
(1991 ;279). The mean landmark configurations
were entered into the Thin-Plate Spline-Relative Warps program (TPSRW) developed by F.
J. Rohlf. This program computed a mean shape
from the 34 specimens, performed Procrustes
and principal-coordinates analyses, then depicted the results via the thin-plate spline.
RESULTS
Traditional measurements.-General
morphological data from measurements of
wings (Table 1) concurred with earlier published data (Farney and Fleharty, 1969;
Norberg and Rayner, 1987). The principalcomponents analysis of mass, wingspan,
and wing area identifies differences in size
as the major source of variation for this
sample of 34 bats (Table 2). The loadings
of all variables are positive and about equal
on the first component, which explains almost 95% of the variation in the three variables. Plotting the scores for this component (Fig. 2a) shows tight clusters of species. The order of these species along PC 1I
follows their distribution in size from
smallest to largest; P. hesperus, M. lueifugus, T. brasiliensis, and E. fuse us.
The second principal component loads
heavily on wing area and explains 4% of
the variation among these variables. This
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1192
Vol. 78, No.4
JOURNAL OF MAMMALOGY
nent provides little information on shape
(Fig. 2b).
Principal-components analysis of traditional measurements distinguishes species
on size, but loadings within the second and
third components make it difficult to identify exact locations of difference in shape,
due most likely to the use of only three
variables to describe shape.
Interlandmark distances.-When all interlandmark distances (n = 78) were subjected to a peA, the first principal component explains 86.4% of the variance with all
loadings positive and of similar magnitude.
The scores for the first component again
show distinct clustering of species (Fig. 2c),
with T. brasiliensis placed at the positive
TABLE 2.-The three component loadings for
the principal-components analysis of traditional
measures of wings of bats.
PC1
PC2
PC3
Wing area
Mass
Wingspan
0.954
0.982
0.981
-0.299
0.143
0.148
0.001
-0.125
0.124
Eigenvalues
Variance
explained (%)
2.8
0.13
0.03
94.6
4.4
1.0
Variable
component separates T.
the other three species.
nent loads heavily on
body mass, but accounts
variation in the sample.
a
brasiliensis from
The third compowingspan and on
for only I % of the
The third compo-
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FIG. 2.-Scores for the principal-components analysis of traditional measurements and interlandmark-distance measurements. Traditional measurements: a, PC] versus PC2:b, PC2 versus PC3. Principal component ] explains 94.6% of the variance, component 2 explains 4.4%, and component 3
explains 1.0%. Interlandmark distances: c, PCI versus PC2: d, PC2 versus PC3. Principal component
1 explains 86.4% of the variance, component 2 explains 8.5%, and component 3 explains 1.8%.
BIRCH-WING SHAPE OF BATS
November 1997
a
b
11
12
_e
13
FIG. 3.-Loadings for interlandmark distances
>2 SD from the mean of each component: a,
variables from PC2. Note how all distances relate to the chord or breadth of the wing: b, variables from PC3.
end of the second component (Fig. 2d). For
the second and third principal components
(explaining 8.5 and 1.8% of the variance,
respectively), I diagrammed the interlandmark distances that had loadings >2 SD
from the mean for that component. Variables (interlandmark distances) with large
loadings along principal-component 2 represent the breadth or chord of the wing (Fig.
3a), whereas variables of principal-component 3 load heavily on the distance between
landmarks 2 and 12 and landmarks 5 and 6
(Fig. 3b).
Principal-components analysis of interlandmark distances again shows that inter-
1193
species variation in interlandmark distances
is much greater than the variation within
species. In addition, the peA identifies locations of high variability (corresponding to
breadth or chord of wing), mainly due to
the more informative nature and larger
number of variables.
Warp analysis.-In the warp analysis,
size is calculated as the distance from each
landmark to the centroid. Multiple contrasts
show all four species are significantly different in centroid size (F = 235.08, d.! =
3,31, P < 0.001; Tukey-Kramer test, P <
0.05). These results confirm the absolute
sizes described by the wing area measurements; E. fuscus has the largest wings, followed by T. brasiliensis, M. lucifugus, and
P. hesperus.
Analysis of shape shows the location of
most of the variation on the wing, and how
this variation is realized. If all landmarks
on the wing move such that parallel lines
in an overlain grid remain parallel, the
changes can be displayed as a shear or as a
compression of the grid (with a resulting
shear or compression of the wing.) These
changes in shape constitute the uniform
component (Fig. 4). The x-axis in Fig. 4
represents shearing of one comer of the
overlaid parallelogram. The y-axis represents change resulting in compression or
expansion of the wing breadth. Most species (except P. hesperus) cluster near the
y-axis origin, which represents no uniform
compression or expansion of the wing.
Most of the uniform change in these species
is in the amount and direction of shear by
landmarks on the wing.
Performing a principal-coordinate analysis (i.e., the warp analysis) on the results of
a Procrustes analysis yields scores for each
animal. These scores are based on orthogonal axes that explain successive degrees of
variation in shape among individuals: A
plot of these scores (Fig. 5) shows that the
greatest variation in shape is along the molossid-vespertilionid axis.
To visualize the changes in shape across
relative-warp 1, I have indicated the direc-
Vol. 78. No.4
JOURNAL OF MAMMALOGY
1194
0.2
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FIG. 4.-The uniform component of difference in shape among the four taxa. The x-axis represents
the movement of one corner of a parallelogram in the (+) or (-) direction. This results in a shear
along the long axis of the wing. The y-axis represents the vertical movement of that corner, resulting
in compression or expansion of wing breadth.
(landmarks 1,2 and 7,8) along with an opposite translation in the wrist and fifth digit
(landmarks 4,12). The next greatest amount
of variation (relative-warp 2) is in the arm
wing, namely an expansion of the main lifting surface between the fifth digit (landmarks 12,13) and the body (landmarks 1,2;
tion and magnitude that each landmark
must move to change a consensus of all 34
individuals into a specimen with the largest
scores for relative-warp 1 (Fig. 6a). The
first relative warp shows that most of the
variation in shape of these four species is
due to translation of the wing tip and base
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Relative warp I
FIG. 5.-Relative warp scores of the 34 individuals. Relative warp I explains the greatest amount
of variation of shape in the data, and Tadarida brasiliensis appears distinct from the vespertilionids
although Pipistrellus hesperus is intermediate. Relative warp 1 explains 58% of the variation of shape
in this sample, whereas relative warp 2 explains 18%.
BIRCH-WING SHAPE OF BATS
November 1997
1195
a
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as in Fig. I b: a, vectors in relative-warp 1 identify the magnitude and direction of movement at each
landmark that separates Tadarida brasiliensis from the vespertilionids. These two families differ not
from mere compression of the wing chord in Tadarida brasiliensis, but by a complicated rotation
around the fifth metacarpal. This warp describes movements of the base and tip of the wing in the
same direction, but opposite to the movement of the fifth digit: b, warp 2 describes an expansion of
the arm wing (the area between landmarks 1 and 13), a reduction in the hand wing (the area between
landmarks 13 and 8), and the shortening of the fourth digit due to differences in landmarks 10 and
11.
Fig. 6b). Relative-warp 1 explains 58% of
the variation in shape of the sample, whereas warp 2 explains 18%.
DISCUSSION
Not surprisingly, different methods of
analysis provide varying amounts of information on shape. All three methods separate these four species on size, but in the
principal-components analyses, the information that remains often is enigmatic in
pattern and difficult to apply to the original
data. This point is most important. A method that clearly shows the wheres and hows
of change in shape has distinct advantages
over inferences from loadings and scores.
Because of its ability to describe and represent graphically the areas of change in
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JOURNAL OF MAMMALOGY
shape, relative warps are clearly an improvement over principal-components analysis of traditional measurements. There is
no need to infer change from differences in
surface area or wing span. These two variables may provide a picture of size and
breadth of wing, but how components of
the skeleton of the hand, for example,
change the shape of the hand wing remains
unknown. By grounding the biological explanation in landmark locations, plots using
the thin-plate spline as an interpolant provide this positional precision.
In their peA of traditional measures,
Norberg and Rayner (1987:349) assumed
that the first principal component accounted
for differences in size among species and
that the second and third components acted
as size-free measures of shape. The loading
pattern on the second and third components
of their three standardized variables followed the algebraic form of wingloading
and aspect ratio, respectively. Attributing
these morphological parameters to the second and third components provided segregation of 16 families of bats into groups
with generalized wing shape and flight
characteristics.
In my analysis of three morphologically
similar vespertilionids and a distinct molossid, a peA of traditional measures provides
only superficial information on shape. Size
(component 1) accounts for 94% of the
variation, whereas <6% of the variation in
mass, wingspan, and area of wing was explained by the second and third principal
components. Principal-components analysis
of traditional measures may be adequate for
studies concerned with large-scale differences among diverse families of bats, but
this technique may be less useful for describing slighter differences. How wing
shape is different in the four species of this
study is difficult to discern from these principal-component loading patterns, but it
would be unrealistic to assume clear separation between the similar vespertilionids
when only three IPorphological variables
are analyzed.
Vol. 78, No.4
In the second analysis, although there
were 26 times the number of variables as
in the first peA, plots of the component
scores provide segregation of species similar to the first peA (Fig. 2). All variables
load positively with about the same magnitude on the first component, therefore,
this component is interpreted to represent
size. The second component is dominated
by four variables, all related to the chord or
breadth of the wing. In this sample, the
lengths of the tibia (landmarks 1 and 2) and
the fifth digit (landmarks 4 and 12) are
highly variable, as are two distances from
the leading edge of the wing to points on
the fourth digit (Fig. 3). This analysis, compared to the analysis of traditional measures, grounds variables in landmark locations that subsequently provide general locations of variation in shape. The interlandmark distances that load heavily on the
second component help distinguish T. brasiliensis from the other species, but how the
vespertilionids differ remains obscure. Although the third component loads heavily
on the distance between landmarks 2 and
12, and landmarks 5 and 6, the effect of this
variability on overall difference in shape
also is difficult to discern.
The third analysis shows both where and
how these species differ. Sizes, as measured
by centroid size, are significantly different
and mirror traditional measures of wing
size (Table 1). Differences in shape that are
identical across the wing (the uniform component) result mainly from shearing (Fi~.
4). This is especially interesting when explaining differences between the narrowwinged T. brasiliensis and the broaderwinged vespertilionids. Wings of T. brasiliensis are not uniformly compressed in an
anterior-posterior direction relative to the
vespertilionids, as both E. fuse us and M. lueifugus also occupy similar positions along
the y-axis in Fig. 4. Rather, different
changes in different parts of the wing result
in the high aspect ratio (i.e., narrowness) of
wings of molossids.
The nonuniform part of differences in
November 1997
BIRCH-WING SHAPE OF BATS
shape shows that areas of greatest variability in these four species are near the body,
along the fifth digit, and at the tip of the
wing (Fig. 6). Narrowness of wings of T.
brasiliensis results from opposite movements of the middle with respect to the base
and tip of the wing. The vespertilionids are
different in the relative areas of arm-wing
membrane (the plagiopatagium) and handwing membrane (specifically, the dactylopatagium latus). It is unknown how these
morphological differences might influence
flight or behavior of bats.
After photographing many E. fuscus and
M. lucifugus, I developed a qualitative impression of differences in shape between
these two species, and described E. fuscus
as having a more paddle-like hand wing,
due (I believed)' to lengthening of the fifth
digit. The only analysis to confirm this subjective impression was the warp analysis.
But instead of coming from the fifth digit,
the changes came from the addition of
membrane between the fourth and fifth
digits, along with the increased length of
digit four (Fig. 6). Whereas the fifth digit
moved medially, it did not lengthen. This
information, if present, was indecipherable
in both principal-components analyses.
If description of shape is the goal, visualization is paramount. When only mass,
area, and length are measured, it is difficult
to translate principal-component loadings
into graphical representations of difference
in shape. Analysis of interlandmark distances gives more information than analysis
of traditional measures regarding the location of differences, but this type of analysis
does not provide directions of movement at
these specific locations. In addition, adding
landmarks to a complicated form can quickly produce a huge number of interlandmark
distances, making it even more difficult to
interpret which variables with high loadings
result in which patterns of difference in
shape. The use of the relative warps, diagrammed by use, of the thin-plate spline, is
the only method that provides quantitative
output on differences in shape, along with
1197
the means to graphically display those differences.
The clue to interpreting differences in
shape lies in determining how structures
vary. All three analyses maximize variation
according to the variables used as input.
Principal-components analysis maximizes
variation of measured variables, and the
loadings generated from this maximization
are used to interpret how the morphology
(based on the variables measured) changes.
The relative-warps analysis maximizes variation between shapes of individuals, not
simply the variables used to construct the
shapes. Maximizing variation between individuals, and using the thin-plate spline as
the graphical interface for display of this
variation, provides a clear picture of how
shapes differ. For straight aerodynamic interpretation, aspect ratio and wingloading
are required for calculating and determining
the cost of transport, power requirements,
and efficiencies of flight. But if the interest
is in a more detailed description of how
wings differ, for systematists to discover areas containing potential characters, morphologists to gain insights into constraints
placed on kinematics, or developmental biologists to understand the sequence and
timing of growth of wings, warps provide
the necessary level of description.
ACKNOWLEDGMENTS
I thank F. Bookstein, M. Foote, B. Lundrigan,
S. Meagher, P. Myers, K. Nishikawa, S. Swartz,
D. Swiderski, M. Zelditch, and an anonymous
reviewer for providing fruitful comments and
ideas during the formulation and writing of this
project. This project was supported in part by a
Hinsdale-Walker award from the University of
Michigan Museum of Zoology.
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Submitted 18 January 1996. Accepted 10 December
1996.
Associate Editor was James G. Hallett.