A Vector Approach to Size and
Shape Comparisons among Zooids in
Cheilostome Bryozoans
ALAN H. CHEETHAM
and
DOUGLAS M. LORENZ
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
NUMBER 29
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SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY • NUMBER 29
A Vector Approach to Size and
Shape Comparisons among Zooids in
Cheilostome Bryozoans
Alan H. Cheetham
and Douglas M. Lorenz
ISSUED
JUL 8 1375
SMITHSONIAN INSTITUTION PRESS
City of Washington
1976
ABSTRACT
Cheetham, Alan H., and Douglas M. Lorenz. A Vector Approach to Size and
Shape Comparisons among Zooids in Cheilostome Bryozoans. Smithsonian Con-
tributions to Paleobiology, number 29, 55 pages, 37 figures, 19 tables, 1976.—
Although zooid size and shape have long been used in comparative studies of
cheilostome bryozoans, procedures for measuring these properties have been
little investigated. Predominantly intussusceptive growth of buds suggests a
method of comparing zooid outlines based on (1) correspondence of principal
growth direction (proximal-distal axis) and (2) size and shape properties express-
ing differential growth about this axis.
Vector properties of a wide variety of autozooidal outlines (in frontal view)
were studied by principal components. Size (area within the outline) accounts
for more than one-third of the variation and tends to vary less within colonies
than shape, even in severely disturbed budding patterns. The portion of shape
independent of size is divisible into three components. Each of the first two
components accounts for about one-fourth of the total variation, the third for
less than five percent. One shape component is associated with asymmetry of
outline, as measured both by departure of the mean vector direction from the
proximal-distal axis and by inequality of vector lengths on either side of the
axis. The amount of asymmetry is small, can be either antisymmetry or fluctuat-
ing asymmetry, and varies greatly within colonies apparently with microenviron-
mental effects on budding patterns. The second shape component is associated
with elongation (concentration of vector lengths near the mean growth direction)
and distal inflation (proportion of area distal to the midpoint of the proximal-
distal axis). These two variables seem less affected by microenvironment than is
asymmetry. The third component accounts for only the small part of variation
in elongation and distal inflation that is not positively correlated. Variation in
this component suggests that distal inflation is slightly more sensitive to micro-
environment than is elongation. Estimates of intrapopulation variation in one
fossil species suggest that size and that part of elongation varying in opposition
to distal inflation are sufficiently consistent within single populations, under the
same conditions of ontogeny, astogeny, and polymorphism, to form a basis for
taxonomic discrimination. Within the range of colony means for each of these
two properties among the variety of outlines examined, at least three and possibly
four potentially taxonomically distinct intervals can be recognized. The number
of measurements per colony needed to detect differences between these intervals
is surprisingly small.
OFFICIAL PUBLICATION DATE is handstamped in a limited number of initial copies and is recorded
in the Institution's annual report, Smithsonian Year. SI PRESS NUMBER 6134. SERIES COVER DESIGN:
The trilobite Phacops rana Green.
Library of Congress Cataloging in Publication Data
Cheetham, Alan H.
A vector approach to size and shape comparisons among zooids in cheilostome bryozoans.
(Smithsonian contributions to paleobiology ; no. 29)
Bibliography: p.
Supt. of Docs, no.: SI 1.30:29
1. Cheilostomata, Fossil. 2. Invertebrate populations. 3. Vector analysis. I. Lorenz, Douglas
M., joint author. II. Title. III. Series: Smithsonian Institution. Smithsonian contributions to
paleobiology ; no. 29.
QE70I.S56 no. 29 [QE799.C5] 560'8s 75-31747 [564'.7]
Contents
Page
Introduction 1
Acknowledgments 3
Growth of Zooid Outline 3
A Vector Diagram of Zooid Outline 5
Quantitative Measures of Outline Geometry 7
Study Methods 10
Comparison of Outlines 11
Size 14
Asymmetry 15
Elongation and Distal Inflation 19
Variation among and within Colonies . 19
Nature of Variation in Outlines ... 24
Ontogenetic Variation 25
Astogenetic Variation ... 27
Polymorphism ... ... 29
Microenvironmental Variation .... 32
Intrapopulation Variation 34
Taxonomic Implications of Properties of Outline 37
Summary of Characters . . 37
Preliminary Evaluation of Characters within Taxa 39
Summary and Conclusions 45
Appendix A: Derivation of Vector Statistics 47
Appendix B: Summary of Data Used 51
Literature Cited 53
A Vector Approach to Size and
Shape Comparisons among Zooids in
Cheilostome Bryozoans
Alan H. Cheetham
and Douglas M. Lorenz
Introduction
Many groups of colonial invertebrates, including
the cheilostome bryozoans, commonly display a
greater complexity of morphologic variability than
do most groups of solitary animals. Such complex
variation is useful in evolutionary biology, for it
provides a wide array of phenotypes in which
many microevolutionary processes may be ex-
pressed which are not detectable in populations of
solitary animals. On the other hand, such com-
plexity has been a major problem in cheilostome
taxonomy (Stach, 1935), for there have proved to
be few characters sufficiently invariant that their
taxonomic utility can be easily determined.
In order to cope with the problem of complex
variation, cheilostome taxonomists are in the proc-
ess of reevaluating the entire taxonomic structure
of their group. One important result of this re-
examination has been the realization that single
characters sufficiently diagnostic to provide a con-
sistent basis for classification are unlikely to be
found. Indeed, many characters vary even within
colonies among zooids of the same ontogenetic,
astogenetic, and polymorphic state. Consequently,
some taxonomists now consider a polythetic ap-
Alan H. Cheetham, Department of Paleobiology, National
Museum of Natural History, Smithsonian Institution, Wash-
ington, D.C. 20560. Douglas M. Lorenz, Department of
Geology, University of California, Los Angeles, California
90024.
proach to cheilostome classification not only de-
sirable, but indispensable in order to make prog-
ress toward taxonomic stability.
In contrast to monothetic classification systems,
for which single taxonomic characters are rela-
tively invariant within taxa and distinctly different
between taxa, polythetic systems require considera-
tion of the patterns of variation and covariation
among all available, potentially useful characters.
Redundancy, or statistical covariation among poly-
thetic characters, is particularly troublesome be-
cause it tends to mask independent genetically
controlled variation. From a theoretical stand-
point, reducing observed phenotypic covariation
by removing undesirable environmentally corre-
lated and genetically redundant effects should em-
phasize independent genetically controlled com-
ponents of the phenotype and permit improved
estimates of true genetic similarities and differ-
ences. Therefore resolution of independent vari-
ance components should improve the phylogenetic
fidelity of the resulting classification.
Variation and covariation in observed quantita-
tive characters can be estimated directly from meas-
urements. Once the observed covariation has been
eliminated and the total variation resolved into a
number of statistically independent components,
the proportions of each component attributable to
sources of variation within colonies, between col-
onies, and between populations can be estimated.
Under certain not-too-restrictive conditions, hy-
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
potheses of statistical difference between colonies
and/or populations can be tested for each inde-
pendent component.
For each component, such an analysis provides
an estimate of the minimum independent differ-
ences by which taxa can be distinguished. Each of
these differences can be used to recognize two or
more intervals, or variable states, of the corre-
sponding component. The set of components for
which such variable states can be recognized then
serves as a basis for classification. But in contrast to
a monothetic system, not all taxa can be expected
to have different states of each component, nor can
all taxa having the same state of a given compo-
nent be expected to have like states for each of the
other components. It is the entire set of independ-
ent taxonomic components which serves to dis-
tinguish among taxa in a polythetic classification.
Statistical analysis also provides an estimate of
the number of zooids per colony and the number
of colonies per population that must be measured
to detect the minimum taxonomic difference in
states of a character with a given degree of confi-
dence. These numbers are obviously important in
the allocation of resources in a taxonomic study.
Precise estimation of taxonomically significant
states of variable characters requires precise meas-
urement and analysis. Once such states have been
estimated with a high degree of confidence, they
commonly can be sufficiently distinguished in the
study of additional taxa by more approximate, per-
haps even qualitative methods. The recognition in
the present study of four or five size classes of
cheilostome autozooids, for example, does not de-
mand precise measurement and analysis, but pre-
cision was necessary to establish that differences of
this magnitude and no smaller are highly likely to
reflect taxonomic differences. For less quantifiable
characters, more approximate methods must be
used throughout, but under guidelines provided by
the analysis of quantitative characters. The meth-
ods used here thus should be applicable, at least in
principle, to a wide variety of morphologic char-
acters in cheilostomes, as well as answering the
question of taxonomic usefulness of quantitative
characters themselves.
Size and shape of autozooids in cheilostomes are
examples of variable characters, the taxonomic sig-
nificance of which has been controversial. In past
studies, measurements of "standard" dimensions
(e.g., length, width) and qualitative comparisons
of zooid outlines with idealized shapes (e.g., rec-
tangle, hexagon, pear-shape) commonly have been
employed. The question of how size and shape can
be expressed in mutually independent series of
linearly arranged numerical states by which taxa
can be compared and contrasted has been little
investigated.
For a few cheilostome species, principal com-
ponents analysis has been used on sets of standard
dimensions to extract independent components
("characters") which, although mathematical ab-
stractions, usually can be interpreted as "size" and
"shape" factors. Studies employing this method
(Cheetham, 1968a; 1973; Malmgren, 1970) suggest
that there is considerable redundancy among the
autozooid dimensions commonly measured. The
particular dimensions which are redundant or
which are associated with "size" or "shape" com-
ponents, however, differed among the species
studied. Principal components analysis of standard
dimensions can be expected to indicate whether all
of the dimensions measured need have been in-
cluded, but it is possible that other dimensions, not
measured, should have been included for adequate
characterization. For example, if zooids differ only
in size, any one of a number of linear dimensions
may be sufficient to distinguish between them. For
zooids of different shapes, length and width, or
length-width ratios, etc., may be insufficient for
distinguishing them.
In addition to possible redundancy and insuffi-
ciency, standard dimensions suffer from another
drawback. Each dimension is at least implicitly,
sometimes explicitly, assumed to measure the dis-
tance between corresponding morphologic points
(h-points of Sneath, 1967) on different zooids and
therefore to express the same morphologic prop-
erty, no matter how different in shape the zooids
may be. In the same polymorph in closely related
species, there are probably many such points recog-
nizable on the zooid outline. For zooids of very
different shapes in widely divergent species, there
may be few such points. Ideally, morphologic cor-
respondence should be based on growth properties
in addition to simple geometric similarity.
Recent advances in understanding the mode of
growth of cheilostome autozooids and in applying
multivariate analytical techniques to the study of
size and shape now make it possible to try new
NUMBER 29
approaches to the evaluation of these characters
with a minimum of assumptions of morphologic
correspondence and maximum expectation for
adequate characterization. As in past studies of
autozooid size and shape in cheilostomes, we have
attempted to analyze the frontal outline (Figures
1-3), but the approach explored here should apply
to other orientations as well.
Fourier series analysis provides a characteriza-
tion of shape that avoids problems of sufficiency
and redundancy and involves no assumption of
morphologic correspondence except for an initial
orientation (Ehrlich and Weinberg, 1970). This
method was employed by Anstey and Delmet
(1972, 1973) to characterize cross-sectional shapes
of zooecia in some trepostome bryozoans by com-
puting the contributions of successively more intri-
cate geometric figures to the shape analyzed.
Geometric and biologic interpretations were then
attached to these contributions by comparing them
with those for ellipses, equilateral pentagons, equi-
lateral hexagons, etc. (Anstey and Delmet, 1973:
1955).
In this paper we explore a slightly different ap-
proach to characterizing autozooid outlines, follow-
ing the suggestion of D'Arcy Thompson (1942:
1044) to " . . . look on the outline ... as a vector-
diagram of its own growth." In particular, we at-
tempt to analyze the frontal outline of cheilostome
autozooids with respect to relative growth about a
principal (proximal-distal) axis. We employ this
approach to examine differences in autozooid size
and shape both within and among colonies in a
variety of genera with a wide range of autozooid
morphotypes. Some polymorphic and ontogeneti-
cally and astogenetically differing autozooids have
been included in the analysis for comparison, but
for the most part we have concentrated on the
ordinary, fully formed autozooids in primary zones
of astogenetic repetition, the size and shape of
which have figured most prominently in past dis-
cussions of taxonomic significance. Our results in-
dicate that the highly significant independent tax-
onomic aspects of size and shape can be separated
from those aspects that are too highly modified by
the environment to have much taxonomic poten-
tial, and that this can be done by measurement of
surprisingly few antozooids in surprisingly few col-
onies with a high degree of confidence. This sug-
gests that these important taxonomic characters
should be amenable to more approximate char-
acterization without an unacceptable risk of losing
or obscuring important taxonomic information.
Before approximate methods are applied to all
variable morphologic characters in cheilostomes,
however, more of them should be subjected to simi-
lar statistical analysis.
ACKNOWLEDGMENTS.—We are indebted to JoAnn
Sanner for measuring specimens, preparing data
for the computer, plotting results, and drafting
most of the figures; to L. B. Isham for preparing
the interpretive diagrams in Figures 1 and 2; to
Charles Roberts for providing some of the calcula-
tions; to H. V. Andersen, Louisiana State Univer-
sity, Baton Rouge, E. Buge, Museum National
d'Histoire Naturelle, Paris, H. Kollmann, Natur-
historisches Museum, Wien, H. Richards, Academy
of Natural Sciences, Philadelphia, and J. D. Soule,
Allan Hancock Foundation, Los Angeles, for loans
of specimens; and to W. C. Banta, R. S. Boardman,
M. A. Buzas, P. L. Cook, and J. E. Hazel for tech-
nical criticism of the manuscript.
Research for this paper was supported by grants
(to AHC) from the Smithsonian Research Foun-
dation (funds 427206 and 430005) and the Na-
tional Museum of Natural History ADP Program.
We also wish to thank the Smithsonian Institution
for a postdoctoral fellowship (to DML) and the
Information Systems Division for support (to
DML), both of which permitted completion of this
research.
A portion of this paper was presented in the
T. G. Perry Memorial Symposium on Bryozoa at
the 1974 Annual Meeting of the Society of Eco-
nomic Paleontologists and Mineralogists in San
Antonio, Texas. We are grateful for the opportu-
nity to participate in this memorial to Tom Perry,
who with his students so enthusiastically applied
quantitative methods to the study of Bryozoa.
Growth of Zooid Outline
Viewed frontally, zooids in cheilostome bryo-
zoans are generally outlined at some or all onto-
genetic stages by cuticular boundaries marking the
junctions of their frontal wall with their vertical
walls. Zooids thus outlined originate as hollow,
bladderlike buds, which grow by swelling (intus-
susception) of their membranous exterior walls
(Figures 1, 2; Silen, 1944a; Lutaud, 1961; Banta,
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
1969). This mode of growth prevails even in spe-
cies in which one bud is partitioned to form mul-
tiple zooids (Silen, 1944b; Lutaud, 1961; Hakans-
son, 1973) or multiple buds fuse to form one zooid
(Silen, 1944b; Banta, 1969; Gordon, 1971). (Silen,
1944b, and Banta, 1969, considered lateral com-
munication organs, present in most species investi-
gated, to be tiny buds which have fused with the
main bud; the contribution of communication
organs to the zooid outline, however, is assumed to
be of minor consequence.) In addition to exterior
walls, a fully formed zooid has interior walls grown
from exterior walls to cut off the zooidal cavity
from those of neighboring zooids in the same
budding series (Silen, 1944a, 1944b; Banta, 1969).
Interior walls may comprise no more than pore
plates forming parts of communication organs
(Silen, 1944b) and thus not contribute significantly
to zooid outlines, or these walls may comprise ex-
tensive partitions between zooidal cavities with a
corresponding reduction in the extent of exterior
walls. Even in species having all their vertical
zooid walls grown as interior walls, the vertical
walls reach and are attached to outer cuticles
which define the zooid outline in frontal view, and
upgrowth of vertical walls is preceded in these spe-
cies by intussusceptive growth of a bladderlike bud
having approximately the shape of the fully de-
veloped zooidal unit (Hakansson, 1973). Intus-
susceptive growth thus appears to be the primary
factor conditioning cheilostome zooid outlines.
According to Thompson (1942:346-363), intus-
susceptive growth of membranes, whether on or-
ganisms or colloids, leads to curved boundaries,
which tend to assume a spherical shape. Flattening
or embayment of boundaries results from inter-
ference with the expanding membrane, localized
changes in internal pressure, or the solidification
to a rigid state of parts of the membrane as other
parts of it continue to expand (Thompson, 1942:
346). Walls at the frontal surface of a cheilostome
zooid are largely exposed, partly or wholly uncalci-
fied, and more or less swollen and bubblelike. Basal
walls are pressed against the substrate in encrusting
growth or against those of other zooids in most
forms of erect growth; they are commonly com-
pletely calcified and generally flattened. Vertical
walls are calcified and vary in curvature. Calcifica-
tion generally proceeds close behind the growth of
the bud, except for transverse walls in species hav-
FIGURE I.—Idealized diagram to illustrate predominantly in-
tussusceptive growth in a hypothetical anascan cheilostome.
An autozooid budded at the distal end of one lineal series
from a multiserial colony is shown from just after budding
(A) to just before completion of its walls (D). (Growth direc-
tions are indicated by arrows on basal projection of com-
pleted zooid outline.)
ing multizooidal buds (giant buds of Lutaud,
1961). The shape of the vertical walls and there-
fore to a large extent the shape of the zooid in
frontal view appear to depend upon the direction
of expansion of the bud, the influence of the
NUMBER 29
FIGURE 2.—Idealized diagram of three adjacent lineal series
of hypothetical cheilostome shown in Figure 1. Buds in ad-
jacent lineal series form a coordinated growing edge, along
which more than one ontogenetic stage is represented because
of typical quincuncial arrangement of zooids.
growth of adjoining zooids, and external
interference.
GROWING
EDGE B
The type of budding most commonly described
in cheilostomes results from intussusception at the
distal end of a lineal series of zooids (Figure 1; pri-
mogenial budding of Banta, 1972). A bud grows
principally on a proximal-distal axis, but a variable
FIGURE 3.—Frontal view of outlines of autozooids near grow-
ing edge (A) of hypothetical cheilostome shown in Figure 2.
Note changing relationships between growth directions of
shaded zooid and those lateral to it as growing edge advanced
from position A to position B. (DLZ=distolateral zooid;
PLZ=proximolateral zooid.)
proportion of growth usually also takes more lat-
eral directions, in accordance with the influence of
adjoining lineal series (Figures 2, 3), external in-
terference, and internal control. The directions in
which, and distances to which, a bud expanded to
produce the zooid outline determine a set of vec-
tors characterizing the size and shape of the zooid.
It must be emphasized that the different lengths of
these vectors do not in most cases represent differ-
ent growth rates but rather may be more or less
proportional to growth duration. The distal por-
tion of a zooid normally continues to expand after
walls bounding more proximal portions of the out-
line have calcified and stopped growth. The
greater lengths of the vectors intersecting the distal
margin reflect the continued expansion.
Zooids budded in other directions are also com-
mon in some cheilostomes, generally in addition to
those formed by distal budding. Their characteri-
zation by the vector representation used here
would involve the same arbitrary initial orienta-
tion as application of the terms "distal" and "prox-
imal" to the description of their morphology
(Banta, 1972).
A Vector Diagram of Zooid Outline
The proximal margin of a cheilostome zooid
normally is colinear with the distal margin of the
preceding zooid. Since budding is initiated some-
where along this margin, all growth in the distal
zooid must originate there and if the zooid outline
is to be geometrically represented as a " . . . vector
diagram of its own growth" (Thompson, 1942), so
also must a system of vectors. In some simple
cheilostomes having pyriform zooids with very nar-
row proximal margins consisting of little more
than points (Figure 28, specimen 28; Silen, 1944a,
figs. 5, 6; Thomas and Larwood, 1960, figs. 1, 4;
Pohowsky, 1973, pl. 1: figs. 1-3, 5, 6), the place-
ment of this vector origin is unambiguous. In
many other cheilostomes zooids have wider prox-
imal margins, but although the detailed directions
of early growth in such forms clearly are complex,
overall zooid growth can be represented to a close
approximation by placing the vector origin at a
point on the proximal margin midway between the
proximolateral walls of the fully formed bud (Fig-
ure 3). Hence the proximal part of the outline is
represented by vectors which, although not parallel
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
to actual early growth directions, nevertheless ac-
curately reflect zooid size and shape.
The vector whose tip falls on the vector origin
of the next successive zooid, or on the midpoint of
the combined proximal margins of twinned zooids
where lineal series bifurcate, is called the "Prox-
imal-Distal Axis." Its azimuth is termed the "Prin-
cipal Growth Direction" and is defined to be the
direction from which the remaining vector argu-
ments are measured. By this definition, it is ap-
parent that the vector origin ("point of budding")
is the only point of morphological correspondence
that need be assumed for this measurement system.
To avoid directional bias in size-shape measure-
ments, vector azimuths should be symmetrically
distributed about the principal growth direction
(Figure 4). Maintaining a constant angle between
all adjacent vectors immediately suggests itself as
the simplest scheme of azimuth spacing and would
be appropriate if zooid frontal outlines tended to
be semicircular. Most shapes we have encountered,
however, are longer than they are wide, and spac-
ing vectors at equal angles would emphasize the
proximolateral margins at the expense of the more
distal portions of the outline. In addition, the
proximolateral margins are highly sensitive to un-
systematic irregularities in budding pattern and
early growth direction (Figures 5, 6), at least partly
because newly budded zooids are often crowded by
more fully developed, laterally adjacent zooids
(Figure 3). To deemphasize these irregularities,
vectors measuring outline geometry on or near the
distal margin should be more closely spaced than
those in the proximal part of the outline. We have
more or less arbitrarily chosen a roughly geometric
rate of increase in azimuth spacing with the vectors
distributed symmetrically about the principal
growth direction (Figure 4).
A complete vector representation of a cheilo-
stome zooidal outline can be constructed as follows
(Figures 4, 33):
1. Determine vector origins on proximal margins
of both the zooid to be measured and the next
succeeding zooid in lineal series.
2. Connect these points by a vector to form the
proximal-distal axis.
3. Construct remaining vectors at geometrically
increasing azimuths distributed symmetrically about
the principal growth direction (azimuth of the
proximal-distal axis). Vectors terminate at inter-
FIGURE 4.—Frontal outline of autozooid of hypothetical
cheilostome in Figures 1-3 showing spacing of reference di-
rections for vector representation (0°, ±2°, ±5°, ±10°, ±16°,
±26°. ±42°, ±64°, ±90°).
section with zooid margin (intersection farthest
from the vector origin in cases where vector inter-
sects margin more than once).
4. Connect vector termini with straight-line seg-
ments.
The resulting polygon is the approximation of
true frontal outline that is used in further
computations.
The precision with which a particular vector
system represents the zooid outline depends on the
number of vectors employed, the complexity of the
outline to be measured, and the curvature of the
NUMBER 29
FIGURE 5.—Frontal outlines of autozooid of hypothetical
cheilostome in Figures 1-3 under different conditions of
growth, i.e., lineal series parallel, or converging or diverging
at small angles. Area within outlines has been held constant.
(Proportional changes in vector lengths in different parts
of the outline are indicated.)
proximal margin. Spacing at angular intervals that
are too great to detect significant irregularities in
the outline will introduce serious errors in the esti-
mates of various size-shape factors such as area,
asymmetry, and distal inflation (see page 10). Re-
entrants along the lateral margins can result in
overestimated areas (Figure 7). And because por-
tions of the proximolateral margin proximal to the
90° vector (Figure 4) are ignored, areas of zooidal
outlines with highly concave proximal margins
will be slightly underestimated.
To estimate the error introduced in our esti-
mates of zooidal area (as an example) by using the
vector system of measurement described above, the
area enclosed within the outline was measured by
a different method and compared with that calcu-
lated from the vector measurements (see Appen-
dix A). We obtained this independent measure-
ment by counting (at X 100 magnification) the
number of 0.01 mm squares contained within the
entire outline. (The average-sized zooidal outline
encloses about 1850 such squares.) For the 129
autozooecia studied in the initial analysis, the area
calculated from vectors is correlated at 0.999 with
that obtained by counts (Figure 7). Differences be-
tween the two estimates were greater than the em-
pirically determined limits of precision of the
counting method in only 30 of the 129 outlines. We
conclude that for the range of shapes studied, the
error introduced by using the vector system of
measurement is of minor consequence. For zooids
having long, narrow, irregular caudae or proximal
extensions (e.g., some species of Hippothoa; see
Harmer, 1957, pl. 73: figs. 25, 27) the error can be
much greater, but such shapes are not common
and can be treated individually as they occur.
Quantitative Measures of Outline Geometry
In the vector representation of zooidal growth,
only a single point, the vector origin, and direc-
tion, the principal growth direction, are assumed
to correspond morphologically among different
zooids. Vectors from different zooids that might be
considered geometrically analogous because they
fall in the same positions relative to their respec-
tive principal growth directions are expressly not
considered to correspond and hence are not com-
pared directly. It is only the entire set of vectors
which is assumed to express properties of growth
that are comparable from zooid to zooid. Geomet-
ric properties of these vector sets which reflect
various intuitive aspects of outline size and shape
are defined in this section and treated as measured
characters in the subsequent taxonomic analyses.
Measurement precision is discussed in the follow-
ing section (Study Methods)
A measure of size is required both as a geometric
property of autozooidal outline and as a reference
with which to standardize size-independent shape
variation. The area contained within the zooidal
outline is certainly the most direct measure of the
total amount of growth and can be computed trig-
onometrically from the vector measurements (see
Appendix A for computational details). In order
to preserve dimensional consistency with the vari-
ous measures of shape, we have converted this area
estimate into an equivalent linear metric, the ra-
dius of a semicircle with the same area, which we
have denoted ra. In the 129 autozooecia studied
initially, ra ranges from 0.18 to 0.57 mm, with a
mean of approximately 0.35 mm. An alternative
measure of size that might be used is the arithme-
tic mean of the vector lengths, but in a random
sample of 11 from the 129 autozooecia, its correla-
tion with the estimated area obtained by counting
squares (see above) was only 0.918, suggesting
that this measure would yield less precise estimates
of area than ra.
In addition to the single estimate of size, ra, we
have defined six vector statistics reflecting various
aspects of what can be considered outline shape.
Four of these shape measures have proved useful
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
B
FIGURE 6.—Variation within each of three cheilostome colonies in the halves of autozooecial
outlines proximal and distal to the midpoint of the axis: a, Electrina lamellosa d'Orbigny
(Appendix B, Id. no. 17); b, Houzeauina parallela (Reuss) (Appendix B, Id. no. 24); c, Tretosina
arcifera Canu and Bassler (Appendix B, Id. no. 27).
in the taxonomic analyses; the other two are re-
dundant. The three conceptual aspects of shape
which were most useful are elongation, asymmetry,
and distal inflation.
Elongation can be visualized as the tendency of
zooidal growth to be concentrated in a "preferred"
direction. In the cheilostomes we studied, this di-
rection lies near, but does not coincide with the
principal growth direction. Elongation can be
measured by the standard coefficient of vector con-
centration (Appendix A), herein denoted p. The-
oretically p ranges from 0 to 1, corresponding to
single straight lines representing growth in op-
posing or a single direction, respectively. But for
the zooid shapes encountered, its range is 0.86 to
0.96, which roughly corresponds to shapes ranging
from semicircles (0.77) to straight lines (1-0). The
arithmetic mean of p for all 129 outlines studied is
0.91.
Asymmetry is also related to the "preferred" di-
rection of growth. It reflects the departure of this
direction from the principal growth direction, and
is measured herein using two vector statistics. The
tangent of the vector mean azimuth, tan Q, has
intuitive appeal in that it is theoretically closely
related to p (Appendix A). In addition, its value is
NUMBER 29
//•
/ *'
0.50--
0.40--
0.30--
0.20--
-■ /*'
REGRESSION (WITH 95% CONFIDENCE INTERVALS)'
ra by vectors = (0.976 to 0.995)x (ra by count) +
(0.002 to 0.004 mm)
ra by count = (1.002 to 1.022) x (ra by vectors)*
(-0.003 to-0.001 mm)
0.20
0.30
0.40
ra by count (mm)
0.50
FIGURE 7.—Comparison of area enclosed by autozooecial outline (expressed as radius of equiva-
lent semicircle, ra) determined by vector representation and by direct measurement (counts).
The coefficient of correlation between the two methods is 0.999. Dots are 129 autozooecia meas-
ured in 32 zoarial fragments (Appendix B, specimens used in principal components analysis).
(Dashed lines are limits of precision (empirically determined) of the counting method. Actual
outlines (dotted lines) and vector representations (solid lines) are shown for the two zooecia that
yielded the extremes of difference between the two methods.)
signed and hence distinguishes between "left" and
"right" asymmetrical outlines. Unfortunately, tan
6 is much more sensitive to slight variations in out-
line width along the proximolateral margins than
to variations of similar magnitude along the distal
margin. Consequently, a second measure of asym-
metry was devised to reflect variation in vector
magnitudes rather than just their width compo-
10
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
nents. This measure, denoted a, is a weighted root-
mean-square of proportional differences between
the lengths of pairs of vectors at equal angular
intervals on either side of the proximal-distal axis
(Appendix A). Although it is more sensitive than
tan 6 to departures from symmetry in the distal
portion of the outline, a has the disadvantage that
it is always positive and hence cannot convey di-
rectional information. Because these two measures
of asymmetry convey complementary information,
both were included in the analyses. Neither a nor
tan 6 reflect large departures from symmetry in the
outlines studied (maximum 0 = 4.06°).
Finally, the term "distal inflation" refers to the
relative concentrations of lateral growth compo-
nents in the distal and proximal parts of the
zooidal outline. Its measure, di, is defined as the
proportional area that is distal to the midpoint of
the proximal-distal axis (Appendix A). Zooid
shapes examined range from distinctly proximally
inflated (di = 0.38) to distinctly distally inflated
(di — 0.64); the average of 129 autozooecia lies at
about di = 0.5.
In addition to these five measures of size and
shape, we initially considered two others, which
later proved to be redundant. The length of the
mean vector (r) is obviously related to the "pre-
ferred" direction of growth, but is also highly cor-
related with size. The maximum vector length
(rmax), which has been used as a measure of zooid
length in some previous studies, also has high cor-
relations with both size and shape variables. For
this reason, and because its use would require an
assumption of morphologic correspondence in ad-
dition to the vector origin, it was dropped from
consideration very early in the analyses and is not
treated further.
Means and standard deviations of all size and
shape measures except rmax are given in Appen-
dix B for each of the 32 cheilostome zoaria studied.
Study Methods
Thirty-two colonies representing 30 species (Ap-
pendix B) were selected for the initial analyses to
give a wide range of autozooidal shapes and sizes.
Most specimens studied are type-specimens of type-
species and thus represent, for the most part, dif-
ferent genera. On each specimen, an attempt was
made to measure outlines of five autozooids, a
number which subsequent analysis indicated to be
adequate for recognizing minimum taxonomic dif-
ferences (Table 18). A total of 129 autozooecia
were included in the initial analyses, the results of
which are described in the next section (Compari-
son of Outlines).
Vector lengths were measured from projections
of frontal outlines from X 50 or X 75 photographs
onto the semistarburst pattern (Figure 4) at a
standard magnification of X 100. A precision of
measurement of 0.01 mm (to which all vector
lengths used in the analyses were recorded) was
estimated from separate trials of projecting, ori-
enting, and measuring the same and separate
photographic outlines of the same zooecium
(Table 1). Replicate measurements from the sep-
TABLE 1.—Precision of measurement of 17 vectors of same autozooecial outline with three meas-
urement techniques (standard deviation calculated as square root of weighted average variance)
Method of measurement
Standard deviation (mm)
Projection from photograph to semistarburst pattern
Separate measurements, same projection
Separate projections, same photograph, same orientation
Separate orientations, same photograph
Separate photographs, same zooecium
Projection of camera lucida image to semistarburst
Direct measurement with rotating stage, ocular micrometer...
0.00097
0.00146*
0.00306
0.00449
0.01113
0.00792**
* two replications only; all others three **excludes deviations in measurement of angles
NUMBER 29
11
arate photographic projections were more consist-
ent than either replicate measurements made with
an ocular micrometer directly from the specimen
on a rotating mechanical stage or replicate meas-
urements made from camera lucida outlines pro-
jected onto the semistarburst pattern (Table 1).
The size and shape coefficients ra, p, tan 6, a, di,
and f were computed from the measured vector
lengths and the a priori determined vector direc-
tions using a program (VECSTAT) written by
one of us (DML). The program includes principal
components analysis of the correlations among the
vector coefficients and computes normalized coor-
dinates for each zooecium in eigenvector space.
Procedures in which variables and components are
standardized were used throughout because of the
differences in units of the vector coefficients.
Principal component axes were subjected to two
separate orthogonal rotations to measure the re-
lationships among certain vector coefficients. Both
rotations were made in 4-component space, in
which well more than 90 percent of the total vari-
ation and of that in each vector coefficient is ac-
counted for (Tables 3-5). The first axis, lying in
the direction of maximum variation, was rotated to
coincide approximately with the direction of the
size coefficient, ra, in order to examine variation
in size and to remove size effects from the analysis
of variation in shape. The rotated second axis was
further rotated to a position near the directions of
two of the shape coefficients, p and di, in order to
study the relationships among independent aspects
of shape.
For each of the four rotated components, vari-
ation within and among the 32 colonies used in
the initial analysis was compared by analysis of
variance, after first determining that nonnormality
of data, or of transformed data, and heterogeneity
of variances do not preclude use of this method.
Certain parts of the sample of 32 colonies used
in the initial analyses were supplemented by addi-
tional material in order to examine aspects of
intracolony and intrapopulation variation. Onto-
genetically and astogenetically differing autozooids
were measured in one colony each; dimorphic
autozooids were measured in each of three con-
specific colonies; and autozooids were measured in
each of six colonies inferred to be from the same
fossil population. Variation within each of these
subsamples was studied by, first, transforming vari-
ation in the six vector coefficients to the same ro-
tated 4-component space developed in the initial
analyses and, then, performing analyses of vari-
ance (or, where data were significantly nonnormal
or variances were significantly heterogeneous, sub-
stituting appropriate nonparametric methods) on
each rotated component. The analyses including
the additional material (a total of 51 zooecia) are
described in a following section (Nature of Varia-
tion in Outlines).
Comparison of Outlines
The interrelations among the six size and shape
variables calculated for the frontal outlines of 129
cheilostome autozooecia (see Appendix B), from
measured vector lengths and chosen vector direc-
tions (see Appendix A), are summarized in Tables
2-6 and Figures 8-11.
Correlations between pairs of variables (Table
2) include three high values, 0.957 for ra and f,
0.813 for p and di, and 0.724 for a and tan 6. Two
of these highly correlated pairs comprise concep-
tually related variables, ra and f primarily express-
ing size and a and tan § primarily expressing asym-
metry. The high correlation between p and di is
TABLE 2.—Correlation coefficients of six vector variables for 129 cheilostome autozooidal out-
lines (underlined values significantly different from 0 at P = 0.01; other values not significantly
different from 0 at P = 0.05)
Variable
r
P
di
a
tan 6
.957
-.433
-.171
-.321
-.130
.813
-.241
-.132
.516
■ 479
-.001
..
.018
.080
di
.019
.724
12
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
not conceptually required, and some cheilostome
autozooids can differ in these two variables inde-
pendently. All the shape variables except tan $ are
significantly, but not highly, correlated with size
(which we have defined to be measured by ra). Al-
though the significant size-shape correlations are
all negative, the directions of the shape variables
relative to size are arbitrary. For example, elonga-
tion could be redefined as squatness or distal infla-
tion could be redefined as proximal inflation, and
their resulting correlations with size would be
numerically unchanged but positive rather than
negative. Three of the shape variables—p, di, and
a—are significantly, but not consistently highly
intercorrelated. The remaining shape variable, tan
9, is significantly correlated only with a, not with
any other variable.
The rather complex pattern of intercorrelations
among the six variables is resolved by principal
components analysis into as few as three orthog-
onal components (Table 3), each of which ex-
presses an independent linear combination of the
variables. Although these three components ac-
count for almost 95 percent of the total variation
in the outlines examined, they account for appreci-
ably less of the variation in p and di (see com-
munalities, Table 3). Moreover, the correlations
between ra and other variables are reproduced con-
siderably less closely in three-component space
than in four-component space (see Tables 4 and 5
TABLE 3.—Normalized eigenvectors from principal components analysis of six vector variables
based on correlation matrix in Table 2
.590
.706
-.382
.076
-.008
-.040
.847
.993
.998
Component
F(l)
F(2)
F(3)
F(4)
F(5)
F(6)
Communality
F(l)-F(2).
F(l)-F(3).
F(l)-F(4).
ra
.754
.615
-.222
-.058
-.015
.044
.946
.995
.998
Eigenvector coefficients (loadings)
a
.750
.172
.560
.280
.124
.003
di
-.811
.148
-.472
.312
.014
.015
.679
.902
.999
-.724
.570
.270
-.077
-.269
-.001
.593 .849
.906 .922
.985 .928
tan 9
-.335
.651
.643
.042
.222
.000
.536
.949
.951
Eigenvalue
2.777
1.675
1.218
.192
.137
.004
Prop, of
variance
.462
.279
.203
.032
.023
.001
Cum.
prop.
.462
.742
.944
.976
.999
1.000
TABLE 4.—Factors resulting from orthogonal rotation of first three normalized eigenvectors in
Table 3 so that F(l)' approximately coincides with ra (correlations of ra with other variables
in parentheses)
Factor coefficients (loadings)
Factor
F(l)'
F(2)'
F(3) '
Communality
F(l) '-F(3) '
.000
.000
di
a
tan 5
.966 -.416 -.336 -.256 .005
(.957) (-.433) (-.321) (-.247) (-.001)
.174 .627 .608 .899 .716
-.171 -.580 -.651 .218 .661
Sum of
squared coeff.
2.114
1.273
Prop, of
variance
.380
.352
.212
Cum.
prop.
.380
.732
.944
NUMBER 29
13
TABLE 5.—Factors resulting from orthogonal rotation of first four normalized eigenvectors in
Table 3 so that F(l)' approximately coincides with ra and F(2)" lies near p and di (correlations
of ra with other variables in parentheses)
Factor
Factor coefficients (loadings)
a
di
tan
Sum of
squared coeff.
Prop, of
variance
Cum.
prop.
F(l)"
P(2)"
F(3)"
F(4)'
Communality
F(l)'-F(4)'.
.999 .960 -.434 -.319 -.251 .003
(.957) (-.433) (-.321) (-.247) (-.001)
-.000 .244 .853 .890 .482 .039
-.000 .002 .034 -.030 .790 .973
.001 .133 .286 -.299 -.092 .043
.379
1.813
.302
.681
1.574
.262
.943
.199
.033
.976
SHAPE
«-H—I—I—i—U—i—i—i—i—" F(l)
and discussion of rotation below). Therefore, the
first four components were employed in all further
calculations, even though the fourth component
accounts for a small percentage of the total
variation.
The principal components axes do not lie very
near the directions of any of the size or shape
variables and thus have low morphologic interpre-
tability (Table 3; Figures 8-11). The direction of
greatest variation, F (1), is subequally distant from
ra, p, di, and a, disregarding whether their direc-
tions are positive or negative. To examine inde-
pendent aspects of size and shape differences
among the autozooid outlines measured, it is im-
portant to separate the effect of size from those of
shape while keeping each aspect of shape differ-
ences independent of the others. To improve mor-
phologic interpretability within these guidelines,
we rotated the first four component axes orthog-
onally so that the first axis coincides with the
direction of the size variable, ra. As this rotation
was made in the reduced space of four components,
the coincidence of the rotated axis F (1)' with ra is
approximate but close (Table 5; Figures 8-10).
Although F(l)' is no longer in the direction of
maximum variation defining F(l), its variance is
not much less than that of F(l) (Table 5).
Rotation of F (1) to coincide approximately
with ra placed F (2) among the shape variables as
a generalized measure of shape (Figure 8, F (2)').
In order to separate aspects of shape, axes F(2)'
and F (3)' were rotated again, this time in the
plane which contains them both, to new positions
F(2)", lying near the directions of p and di, and
FIGURE 8.—Relation of six vector variables to first two prin-
cipal components axes showing separation of size and shape.
(Orthogonal rotation (^ = 39.2°) of F (1) to approximate
coincidence with ra results in placement of F(2)' amid the
shape variables.)
F (3)", lying near tan § (Figure 11). This rotation
preserved the orthogonal relationships among all
axes, and left the positions of F(l)' and F (4)' un-
changed. The new directions F (2)" and F (3)"
have variances little different from those of com-
ponents F(2) and F (3) (Table 5). Thus the two
rotations produced a set of orthogonal axes with
variances little different from those of the unro-
tated components, but lying in directions near
those of size and shape variables. The transforma-
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
F(3)'
14
F(3)
ASYMMETRY
ASYMMETRY
ELONGATION +\di
DISTAL INFLATION
DISTAL INFLATION
FIGURE 9.—Relation of six vector variables to principal com-
ponents axes F(l) and F(3) showing separation of asymmetry
from elongation and distal inflation. (Orthogonal rotation
(#2= —12.85°) of F(l) to approximate coincidence with ra
results in little change in position of F(3).)
FIGURE 10.—Relation of six vector variables to principal com-
ponents axes F(l) and F(4) showing partial separation of
elongation and distal inflation. (Orthogonal rotation (4>3 =
— 3.38°) of F(l) to approximate coincidence with ra results
in little change in position of F(4).)
tion matrix for the two rotations is shown in
Table 6.
The especially high correlation between ra and
r (Table 2) suggests that one or the other of these
ELONGATION +
DISTAL INFLATION
FIGURE 11.—Additional orthogonal rotation (f,= — 45°) of
F(2)' (Figure 8) to new position F(2)" near p and di. (Rota-
tion results in displacement of F(3)' to F(3)", which is nearer
tan e and a.)
variables could have been dropped from the prin-
cipal components analysis with little or no loss of
information. The principal components analysis
itself takes redundancy into account, but it is in-
teresting to compare the results of the four-
component representation based on the correla-
tion matrix with f dropped out to that based on all
six variables and used throughout the analyses.
Although the first four eigenvectors have different
coefficients and different proportional eigenvalues,
the cumulative proportion of their eigenvalues
and their communalities for each variable are prac-
tically identical (maximum difference = 0.003).
This means that a similarly orthogonally rotated
four-component representation based on the five
variables excluding r would have almost exactly
the same correlations with variables and propor-
tions of variance as the four-component represen-
tation based on all six variables.
SIZE.—Among the 129 autozooecia included in
the initial analyses, almost 40 percent of the total
variation in outlines is in the direction of the ro-
tated first principal component, F(l)', which was
made to coincide approximately with ra, the size
variable. The size variation in outlines is greater
than that in any single aspect of shape but less
than that in all aspects of shape combined.
The plot of F(l)' against F (2)' (Figure 12)
NUMBER 29
15
TABLE 6.—Transformation matrix of direction cosines used to rotate first
four components in Table 3 to factors in Table 5
Unrotated
components
Rotated factors
F(l)
F(2)
F(3)
F(4)
F(l) '
.7542
-.5688
-.3250
.0446
.6152
.4486
.6474
.0364
-.2220
-.6894
.6894
-.0131
-.0590
F(2) "
0
F(3)"
0
F(4) '
.9983
shows within-colony variation polygons to be ap-
preciably more elongated parallel to the shape axis
(F (2)') than parallel to the size axis (F (1)'). This
suggests that differences between colonies (most of
which belong to different species and genera in this
sample) are better expressed by size than by shape.
However, the shape axis, F (2)', is a composite of
the several shape variables, and the within-colony
variation in any one shape variable can be con-
siderably less than that in F (2)'. The separate
aspects of shape variation are examined in the
following sections.
The plot of F(l)' against F(2)' also provides an
opportunity to show graphically the relative "dis-
tortion" of the rotated four-component representa-
tion, i.e., the differences between the values of a
variable and those of the component which has
been rotated to coincide approximately with it. As
a very high percentage of the total variation and
of the variation in each variable (97.6 percent of
total; 92.8 to 99.9 percent of that in each variable;
see Table 5) is accounted for, the distortion can be
expected to be small. This is also suggested by the
almost perfect correlation (0.999; see Table 5) be-
tween variable ra and its representation in four-
component space, F(l)'. Even with a lower per-
centage of the total variation accounted for (see
Rohlf, 1972), one could expect the zooecia to be
arranged in the direction of F (1)' in their actual
rank order with respect to ra. In Figure 13, the
actual values of ra for the 129 zooecia used in the
initial analyses have been plotted at their positions
ordinated in the F(l)', F (2)' plane, and lines of
equal values of ra have been interpolated between
points. With no distortion, the lines of equal ra
would be parallel and evenly spaced. The distor-
tion of ra by the four-component representation is
obvious, especially for zooecia of approximately
mean size and mean shape (center of Figure 13),
for which it can amount to 0.6 standardized unit
(= 0.05 mm). For colony means, however, the dis-
tortion should be less (central limit theorem), and
is indicated graphically (Figure 14) to be less than
0.2 standardized unit (< 0.017 mm).
ASYMMETRY.—The two orthogonal rotations,
F(3) to F(3)' (Figure 9) and F (3)' to F (3)"
(Figure 11), placed the third axis near the direc-
tions of tan 9 and a as a measure of asymmetry
which is independent of size and other aspects of
shape. Thus expressed, asymmetry accounts for
about one-fourth of the total variation in outlines
among the 129 autozooecia included in the initial
analyses. As the absolute value of tan 9 was used in
the calculation of F (3)", this axis measures the
amount but not the direction (handedness) of
asymmetry.
The amount of asymmetry, as indicated by a and
by the absolute value of tan 9, is small for all
zooecia examined (Table 7). The maximum ob-
served value of tan 9 corresponds to an angle of
about 4°. The plot of F (3)" against F(2)" (Figure
15) further suggests that asymmetry accounts for
most of the high within-colony variation in shape.
The nature of the asymmetry exhibited by the
129 autozooecia examined is suggested by the
signed values of tan 9 (Table 7). Of the three
kinds of asymmetry distinguished by Van Valen
(1962), that deriving from the systematic preva-
lence of one side over the other (directional asym-
metry) appears not to be important except possibly
in localized regions of colonies, such as in a branch
with a diverging budding direction (e.g., Figure
16
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
FIGURE 12.—Variation in 129 autozooecial outlines in 32 zoarial fragments for first two rotated
principal components, F(l)' (size) and F(2)' (overall shape). (Scatter polygons are numbered as
in Appendix B. Directions of vector variables are indicated by arrows arranged as in Figure 8.
Colony 1 (dashed polygon) exhibits astogenetic variation in shape. F(l)' and F(2)' scales are in
standardized units normalized to their respective factor variances (Table 4). One unit on F(l)'
is approximately equivalent to 0.06 mm of ra, with smaller zooecia lying to the left and the
center at the grand mean of the 129 zooecia. The variables related to F(2)' are dimensionless.)
NUMBER 29
17
FIGURE 13.—Lines of equal values of ra for the 129 autozooecial outlines of Figure 12 at their
positions ordinated by the rotated four principal component representation. (Units of ra are
standardized, but not normalized, one unit equivalent to approximately 0.09 mm.)
18
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
F(2)'
FIGURE 14.—Lines of equal values of ra for 31 of the within-colony means of Figure 12 (omitting
colony 1). (Representation is as in Figure 13.)
TABLE 7.—Comparison of asymmetry statistics (tan e), using absolute and signed
values, for 105 zooecial outlines in 22 zoarial fragments
Statistic
Signed value
Absolute value
-0.040 to 0.071
0.000 to 0.071
0.52
-0.010 to 0.032
0.004 to 0 041
0.64
0.003
0.014
0.00002 to 0.0018
0.00001 to 0.0008
0.0004
0.0002
NUMBER 29
19
24, D). Although the data tend to have more posi-
tive than negative values of tan 9 (i.e., to be
slightly "right-handed"), the pooled mean is not
significantly different from zero (P > 0.05). In ad-
dition, none of the individual zoarial fragments
studied have means significantly different from
zero (P > 0.05). (The slight apparent directional
asymmetry was removed from further calculations
by standardization as described in "Study Methods.")
Antisymmetry, that is, the tendency to be asym-
metrical but "right " or "left" with about equal
frequency, is suggested in zoarial fragments having
near-zero means and high variances. The most ob-
vious example is furnished by Thalamoporella
biperforata (Figure 23, A; mean, 0.0008; variance,
0.0018). All such examples appear related to ar-
rangement of asymmetrical zooids in more or less
bilaterally symmetrical budding patterns. Most of
the asymmetry in outlines appears to be fluctuating
asymmetry, that is, the generally small, subequally
"right" and "left" departures from a general tend-
ency toward bilateral symmetry. This is suggested
by the near-zero means and low variances for most
of the zoarial fragments studied. Fluctuating asym-
metry has been taken as a measure of "develop-
mental noise" (Van Valen, 1962), and in the zo-
oecial outlines examined it seems ascribable chiefly
to microenvironment.
It must be emphasized that the asymmetry ex-
amined here is that of the outline only. Asymmetry
of other zooidal structures, such as placement of
the orifice or orientation of adventitious avicularia
(see Cheetham, 1973), can be exactly reversed in
handedness and different in amount from that dis-
played by the outline.
ELONGATION AND DISTAL INFLATION.—The rotated
second axis F (2)" (Figure 11; Table 5), accounting
for almost one-third of the total variation in out-
lines, represents the size-independent, positively
correlated portion of the variation in p and di.
The plot of F(2)" against F (3)" (Figures 15,
16) suggests that elongation and distal inflation
are generally less variable within colonies than is
asymmetry. Although there is only a weak correla-
tion between asymmetry and the other two shape
variables, Figure 15 further suggests that depar-
tures from symmetry can be greater in more elon-
gate and distally inflated outlines. This is to be
expected because the greater inequality of vector
lengths in different parts of elongate or distally
inflated outlines can emphasize inequalities on
either side of the proximal-distal axis. Both squat,
proximally inflated and elongate, distally inflated
outlines, however, vary from nearly symmetrical
to distinctly asymmetrical.
Most of the remaining size-independent varia-
tion in p and di is associated with the rotated
fourth axis F (4)' (Figure 10; Table 5), which ac-
counts for only a small part of the total variation
in outlines. The variation of p and di on F (4)',
unlike that on F (2)", is not correlated.
Almost all of the size-independent variation in
p and di is thus expressed in the plot of F (2)"
against F (4)' (Figure 17). That part of the varia-
tion in p that is independent of di, as well as of
size, is expressed at right angles to the direction of
di in that plot (Figure 17a). Conversely, size- and
p-independent variation in di is expressed at right
angles to the direction of p (Figure 176). Ob-
served ranges of outlines within colonies measured
in those directions suggest that part of p is less vari-
able within colonies than the corresponding part
of di. This suggests that, even though p and di are
strongly correlated among the outlines examined,
these two variates in part measure different aspects
of shape and that p may be slightly more impor-
tant in characterizing the shapes of zooids within
a colony.
The principal variations in elongation and dis-
tal inflation are summarized on Figure 18. Ten
zoarial fragments from which autozooecial outlines
were measured for the foregoing analysis are ar-
ranged approximately as in Figure 17 with respect
to the F (2)" and F (4)' axes. With differences in
size and asymmetry disregarded, these ten groups of
autozooecia illustrate shape variation as a whole.
It can be noted that the mean shape of the 129
outlines (Figure 18, specimen 18) is similar to that
of the hypothetical cheilostome shown in Figures
1-4.
VARIATION AMONG AND WITHIN COLONIES.—Rela-
tions between variation among colonies and that
within colonies for the four orthogonally rotated
axes (factors of Table 5) are suggested by the
ranges of variation plotted in Figures 12, 15, 16,
and 17. These relations were further examined by
single-classification analysis of variance (Tables 8,
9). One zoarial fragment, Tetraplaria simata
(1A-C), was excluded because of obvious astoge-
netic heterogeneity in autozooecial outlines. Con-
20
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
F(2)"
30B
-•—
27
290
-•—
28
24
26
^•20
9B
2IA
—•—
"22
-•-
_F»r
"9A
8B
8A
4
FIGURE 15.—Means of 31 of the 32 zoarial fragments of Figure 12 for rotated principal com-
ponents F(2)" and F(3)". Individual values are shown for specimen 1. (For colonies in which
three to five zooecia were measured, 95 percent confidence intervals for F(3)" are indicated by
horizontal lines. Directions of vector variables are shown by arrows arranged as in Figure 11.
Dotted line marks empirically determined position of perfectly symmetrical shapes, the value of
asymmetry increasing to the right. F(2)" and F(3)" scales are in standardized units normalized to
their respective factor variances (Table 5). The variables associated with these two components
are dimensionless. Portions of confidence intervals for specimens 7 and 28 extending to left of
symmetry line are hypothetical.)
sistent deviations from normality or significant
heterogeneity (Fmax test, Sokal and Rohlf, 1969)
of within-colony variances, which would have made
questionable the application of analysis of variance
to this problem, were not found for factors (F(l)',
F (2)", or F (4)'.
For F (3)", the plot of colony means and their con-
fidence intervals (Figure 15) suggests a positive cor-
relation between means and variances. This is not
surprising because the measures a and absolute
value of tan 9, on which F (3)" is primarily based,
permit high variances only with large means. Low
variances are possible with either large or small
means, but the predominance of fluctuating asym-
NUMBER 29
21
TFO)"
SYMMETRICAL SHAPES
I2A
-•—
VI
9AI4
9B '
••
20
23
27
2IA
F(2)"
Jj-
22
24
26
-•-
26
8A
2*5
29D
-•—
tan9
30B
FIGURE 16.—Same plot as Figure 15 but turned so that 95 percent confidence intervals for means
of F(2)" are indicated by horizontal lines. (The values of elongation and distal inflation increase
to the right, with the grand mean for the 129 outlines at the intersection of the F(2)" and
F(3)" axes.)
metry in the data, as discussed above, results in
association of low variances almost entirely with
small means. To reduce this apparent departure
from normality, the appropriate logarithmic trans-
formation (Sokal and Rohlf, 1969) was used on
F (3)". Within-colony variances are not significantly
heterogeneous, so analysis of variance was per-
formed on log (F (3)"). For comparison, an analy-
sis of variance was made on the untransformed
data, with practically identical results.
For all four factors, the among-colonies com-
ponent of variance is highly significant, indicating
that all factors are potentially important in dis-
tinguishing autozooecial outlines in different col-
onies. Even though each factor expresses highly
significant differences among colonies (Table 8),
the relative importance of the four factors appears
to differ appreciably, not only with regard to their
portion of the total variance as revealed by the
foregoing principal components analysis, but also
with regard to the portion of each attributable to
among-colonies differences over and above within-
colony variation (Table 9).
The importance of size is further emphasized by
the extremely high proportion of the variance in
F(l)' that is attributed to the among-colonies vari-
ance component. The distinction between a high
among-colonies and a low within-colony variance is
demonstrated by the wide gap between their 95
percent confidence intervals.
Because of possible distortion of within-colony
variances introduced by the rotated principal com-
ponents representation (see page 15), another
analysis of variance was made directly on values of
22
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
Sr
■\ftf
INFLATED
DISTALLY
INFLATED
PROXIMALLY
SQUAT • ELONGATE
26 27/
-*— 2j *'
2IA 23 /
\ 22 9B
ttL
20 \ 18V'''
•.17/ ■**:
J_ r
~F(4y
••\ 13 12*
.:I6^ • «1
6A\ »7 6
FIGURE 17.—Means of 31 of the 32 zoarial fragments of Figure 12 for rotated principal com-
ponents F(2)" and F(4)': a, plot turned to allow observed range in each zoarial fragment in
which three to five zooecia were measured to be represented by horizontal line perpendicular to
the direction of di (dotted line separates shapes having values of p greater than about 0.90
(elongate) from those with values less than about 0.90 (squat)); b, plot turned to allow observed
variation ranges in zoarial fragments to be represented by horizontal lines perpendicular to the
direction of p (dotted line separates shapes with values of di greater than about 0.50 (distally
inflated) from those with values less than about 0.50 (proximally inflated)). (Individual values
are shown for specimen 1. Directions of vector variables are shown by arrows. Scales are in
standardized units normalized to factor variances (Table 5). The variables associated with the
components except f, are dimensionless.)
ra. The proportions of variance within and among
colonies are virtually the same as those obtained
for F (1)' (Tables 8, 9).
Elongation and distal inflation, as measured by
F (2)" and F (4)', also show high among-colonies
variance components, although not so proportion-
ally high as that associated with size. The gap be-
tween 95 percent confidence intervals for among-
colonies and within-colony variances in F (2)" is
small, and the intervals for variances in F (4)'
partly overlap. The importance of these two as-
pects of shape in distinguishing outlines in differ-
ent colonies thus seems to be slightly less than that
of size.
Asymmetry, as measured by F (3)", is the only
one of the four factors for which the within-colony
variance component is larger than that among
colonies (Table 9). The 95 percent confidence
interval for within-colony variance, however, is
completely overlapped by that for among-colonies
variance, making interpretation of the relation-
ship more uncertain than that in other factors.
NUMBER 29
23
FIGURE 18.—General variation in shapes of autozooecia, independent of size and asymmetry, in
10 of the 32 zoarial fragments on which principal components analysis was based. Directions of
vector variables are indicated by arrows. Frontal views (X 25) are arranged approximately as
in Figure 17 with respect to F(2)" and F(4)' axes, and are numbered as in Appendix B: 30B,
Wilbertopora mutabilis Cheetham; 28, Pyripora texana Thomas and Larwood; 27, Tretosina
arcifera Canu and Bassler; 24, Houzeauina parallela (Reuss); 18, Antropora? oculifera (Canu
and Bassler); 15, Ogivalia elegans (d'Orbigny); 10, Schismoporella schizogaster (Reuss); 7, Metra-
rabdotos unguiculatum pacificum (Osbum); 4, Floridina sp. 1: Cheetham and Hakansson; 3,
Entomaria spinifera (Canu).
24
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
TABLE 8.—Single-classification analysis of variance in factors (Table 5) and in ra (in brackets)
for 126 autozooecial outlines measured in 31 colonies
Factor
F(l)\
[ra]..
F(2)".
F(3)"*
F(4) '.
Degrees
Source of
of
Variation
freedom
Among colonies
30
Within colonies
95
Among colonies
[30]
Within colonies
[95]
Among colonies
30
Within colonies
95
Among colonies
30
Within colonies
95
Among colonies
30
Within colonies
95
Sum of
squares
274.110
11.624
[119.782]
[4.636]
175.710
29.472
9.018
8.228
19.791
4.505
Mean square
9.137
0.122
[3.993]
[0.049]
5.857
0.311
0.301
0.086
0.660
0.046
F ratio
74.893**
[81.818**]
18.833**
3.471**
14.348**
* log transformation
** significant, P < 0.001
TABLE 9.—Components of variance in factors (Table 5) estimated by single-classification analysis
of variance (Table 8) of 126 autozooecial outlines in 31 colonies (confidence intervals of variance
calculated from subsample of 17 colonies in each of which five outlines were measured)
Total
variance
Among
colonies
Within
colonies
Proport
ions
Factor
Variance
95% conf. int.
Variance
95% conf. int.
Among
Within
F(l)'
2.347
2.225
0.971-4.207
0.122
0.092-0.180
0.948*
0.052*
F(2)"
1.680
1.369
0.684-3.119
0.311
0.237-0.462
0.815
0.185
F(3)"**
1.643
0.729
0.411-2.344
0.914
0.636-1.239
0.379
0.621
F(4)'
0.199
0.153
0.050-0.242
0.046
0.036-0.071
0.761
0.239
* proportions from anova of variable ra 0.952 and 0.048, respectively ** partitioned from log transformatic
Nature of Variation in Outlines
From the foregoing analysis of autozooecia hav-
ing a wide variety of outlines, it can be inferred
that size and aspects of shape expressed by elon-
gation and distal inflation are significant in dis-
tinguishing colonies of cheilostome bryozoans in
which measurements are made on zooids of approx-
imately the same ontogenetic, astogenetic, and
polymorphic condition. How are these distinctions
likely to be affected by measuring zooids belonging
to different asexual generations in a zone of asto-
genetic change or to different autozooidal poly-
morphs, or by measuring zooids at different onto-
genetic stages, unlikely though that may be in
many cheilostomes? And are these distinctions as
evident if the variation among colonies belonging
to the same population is taken into considera-
tion? To explore these questions, we enlarged parts
of the sample used in the foregoing analysis to esti-
mate intracolony and intrapopulation variation in
factors F (l)'-F (4)' in some of the species examined
in this study.
To evaluate factors F (l)'-F (4)', it is of course
necessary to distinguish zooids ontogenetically, as-
togenetically, or polymorphically by means of
morphologic criteria separate from those being
evaluated. These criteria enable one to recognize
the ancestrula and the growing edge of the colony
and the sequences of morphologically differing zo-
oids from each, and to distinguish among poly-
morphs. Differences in the factors expressing size
NUMBER 29
25
and shape of the frontal outline can then be tested
against the series or groups of autozooids so dis-
tinguished. Although the examples studied here
were selected for the distinctiveness with which
they exhibit criteria on which to identify their
ontogenetic, astogenetic, and polymorphic states,
variation in factors F(l)'-F(4)' cannot be pre-
cisely assigned to each possible source of intracol-
ony variation. A difference in size or shape be-
tween two zooids which are obviously in different
ontogenetic stages or which obviously belong to
different asexual generations in a zone of astoge-
netic change can, and probably generally does, also
reflect, less obviously, difference in microenviron-
mental conditions. The evaluation of ontogenetic,
astogenetic, and polymorphic differences in F(l)y-
F (4)' could well yield new and possibly more pre-
cise criteria for recognizing these kinds of intra-
colony variation, but its purpose here was rather
to compare these kinds of variation with potential
taxonomic differences. How much will the unlikely
eventuality of overlooking ontogenetic, astoge-
netic, and polymorphic heterogeneity in the ma-
terial at hand obscure possible taxonomic
distinctions?
The vector properties obtained for these en-
larged subsamples were transformed to the same
set of rotated axes as in the foregoing analysis to
hold constant the previously inferred relation-
ships among variables. For each subsample, plots
were made of F (1)' against F (2)' (size vs. overall
shape), F (2)" against F (3)" (elongation + distal
inflation vs. asymmetry), and F (2)" against F (4)'
(elongation vs. distal inflation). Among-colonies
and within-colony components of variation were
examined through analysis of variance, or, with
contraindication of normality or homogeneity of
within-colony variances, with nonparametric
analogues.
ONTOGENETIC VARIATION.—Although the size and
shape of an autozooidal bud change obviously dur-
ing growth to the complete zooid, the pattern of
change in size and the various aspects of shape can
be expected to vary, and the stages at which there
is no further change can be expected to differ for
different aspects of the outline. To illustrate onto-
genetic changes in factors F (l)'-F (4)', we selected
a specimen of Metrarabdotos unguiculatum Canu
and Bassler (Figure 19) in which the shapes of
buds at the growing edge appear similar to those
most commonly observed among fossil and modern
cheilostomes, and in which the ontogenetic stages
are obvious from the degree of development of the
transverse wall and the frontal shield. The distal-
most, uncalcified margins of the buds are missing,
but, by analogy with observed living forms, prob-
ably lay a very short distance beyond and parallel
to the preserved margins of the calcified walls. The
shapes of the calcified parts of the buds thus can
be expected to represent actual shapes of whole
buds at slightly earlier ontogenetic stages.
The five zooecia measured (Figure 19, A-E) rep-
resent a sequence of ontogenetic stages at increas-
ing distances proximally from the growing edge.
These zooecia are members of two contiguous lin-
eal series (A-C-E and B-D) and alternate in po-
sition. Differences in size and shape within each
series and in the two series combined (Figures 18,
20; Table 10) can be expected to follow compar-
able sequences. Differences between these se-
quences are small and generally can be attributed
to microenvironment.
From zooecium A to zooecium D, F (1)' increases
progressively, but not uniformly. Zooecium D (and
possibly also zooecium B) is slightly larger for its
ontogenetic stage than are C and E. As zooecia D
and E both represent fully formed zooids, the dif-
ference in their size appears to be part of the gen-
eral, microenvironmentally controlled fluctuation
to be expected throughout the zone of astogenetic
repetition. The similar difference between zooecia
B and C, although these two represent slightly dif-
ferent ontogenetic stages, suggests that microenvi-
ronmental effects may have been similar within
lineal series, but different between series. It also
seems possible that the development of the trans-
verse wall, which began to grow in zooid C at a
stage just later than that represented by zooecium
B, affected the size difference. Initially, growth of
the transverse wall would have decreased the size
of the zooid by partitioning off the distal part of
the bud. With continued growth of this wall ob-
liquely upward (stages represented by zooecia C to
D), size would again have increased until, once
the wall was fully developed (stages represented by
zooecia D and E), there was no further increase in
size.
For the most part, changes in F (2)" parallel
those in F (1)', i.e., as buds became larger, they also
became generally more elongate and distally in-
26
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
F(2)'
F(l)'
: ♦*
F(2>"
K3>"
MM PROX. WALL ZOOID - DIST. MARGIN COLONY
F(l)'
F(2)"
F(4)'
FIGURE 19.—Variation in size and shape of autozooecial outlines near growing edge of colony of
Metrarabdotos unguiculatum Canu and Bassler (Appendix B, Id. no. 6). (Frontal view X 40.)
TABLE 10.—Changes in factors (Table 5) with increasing distance (in mm) from proximal wall
of zooid to growing edge of colony for five autozooecial outlines in Metrarabodotos unguicu-
latum Canu and Bassler, Recent, Brazil, expressed as proportions of greatest difference measured
Factor
0
37 - 0.59
0
59 1.05
1
OS 1.45
1.45 1.90
F(l)'
+ 0.637
+ 0.054
+ 0.309
0.137
F(2)"
+ 0.526
0.160
+ 0.432
+ 0.202
F(3)"....
0.173
+ 0.221
0.018
+ 0.798
F(4)'
f 0.401
+ 0.463
+ 0.136
0.000
flated. However, the rate of increase is reversed
(Figure 20) near the middle of the sequence, re-
sulting in a decrease in F (2)" between zooecia B
and C, presumably as a result of the introduction
of the transverse wall and of microenvironmental
factors. One increase in F (2)" apparently not re-
flected by an increase in size is that between the
proximal zooecia (D and E). This appears related
NUMBER 29
27
2.0
3JOT
2.0--
1.0--
..x'F(3)"
1.0
MEAN DISTANCE FROM GROWING EDGE
FIGURE 20.—Rates of change, in standardized units, in F(l)'
to F(4)' between ontogenetic stages within lineal series repre-
sented by zooecia A to E of Figure 19. (Rate of change is
the difference between contiguous zooids within series divided
by the mean of their distance from growing edge (Table 10).)
to development of the peristome and avicularian
rostrum which project slightly beyond the trans-
verse wall.
Changes in F (3)" from zooecium A to zooecium
D appear to be minor fluctuations related to micro-
environment. That between zooecium D and zo-
oecium E, however, is larger and seems to have
resulted from development of the peristome and
avicularium, accentuating the slight initial asym-
metry. The amount of asymmetry in fully devel-
oped zooecia is variable but generally low.
The uniformly progressive increase in F (4)'
from zooecium A to zooecium D seems to reflect
that part of the gradient of increasing elongation
least affected by microenvironmental "noise," al-
though the rate of change within series is close to
that for F(l)' (Figure 20). F(4)' thus appears to
be potentially the best indicator of ontogenetic
change among the characters examined. In the
plot of F (2)" against F (4)', most of the change is
near the direction of p and approximately perpen-
dicular to that of di.
In summary, the largest ontogenetic changes in-
ferred for M. unguiculatum (Figure 20) are size
increases, generally obvious and rapidly decelera-
ting but slightly complicated by microenviron-
mental "noise," and variations in shape, expressed
by that part of elongation contrasted with distal
inflation and which are also large and rapidly de-
celerating; these form the smoothest gradient and
reach the most characteristic completed state.
Changes in asymmetry are small, the most impor-
tant increase occurring late in ontogeny during
development of the peristome and avicularium.
These ontogenetic changes were inferred from
zooids within two and one-half to three lengths of
the growing margin. In other colonies of this spe-
cies, this interval can be observed to differ in
length, probably under environmental as well as
genetic control, but the pattern of changes is other-
wise similar.
ASTOGENETIC VARIATION.—Astogenetic differences
in size and shape of autozooids are generally less
obvious than ontogenetic changes in these char-
acters. In primary zones of astogenetic change a
common pattern in cheilostomes is a general in-
crease in average size for a variable number of
asexual generations from the ancestrula. The asto-
genetic increase in size is generally accompanied by
an increase in morphologic complexity, commonly
including the introduction of polymorphism
(Abbott, 1973). In Wilbertopora mutabilis Cheet-
ham, the first five asexually produced generations
of zooids have previously been recognized as be-
longing to the primary zone of astogenetic change,
chiefly from the dimensions of the zooecia, al-
though the first appearance of ovicelled zooecia in
a colony coincides with the beginning of repetition
of nonovicelled zooecial morphology. To illustrate
astogenetic changes in factors F(l)'-F(4)', we se-
lected a specimen in which the ancestrula and the
asexual generations succeeding it are readily iden-
tifiable and measured nonovicelled zooecia in the
first seven generations (Figure 21, I-VII). The
ancestrula was excluded from measurement be-
cause its growth directions cannot be expected to
28
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
IT
■T
UI o
I i
-•
i '.
1+4
«r
NUMBER 29
29
have conformed to those of the vector representa-
tion.
The plots of factors F (1)' to F (4)' for the seven
generations examined (Figure 21) suggest an asto-
genetic gradient of increasing size and slightly
increasing elongation from generation I to genera-
tion V, with generation VI forming the first repeti-
tion of size and shape, although with a wide range
of variation. The great variability of size and par-
ticularly of shape within generations further sug-
gests a high level of microenvironmental "noise"
complicating the astogenetic pattern.
The high within-generation variability of this
colony of W. mutabilis is indicated by the fact that
fully one-half the within-generation variances ex-
ceed the 95 percent confidence intervals for within-
colony variances for the general sample of cheilo-
stomes (Table 9). Another one-third fall within
the confidence intervals. Because this heterogeneity
of within-generation variances is significant for all
factors (Fmax test; Sokal and Rohlf, 1969), we
have examined differences between generations en-
tirely with nonparametric tests (Table 11). With
the exception of F (1)', we found no significant
differences, even though the pattern of changes of
within-generation means in F (4)' is similar to that
in F(l)' (Figure 21).
For F (1)', the uniformly progressive increase of
within-generation means from generation I to gen-
eration V (Figure 21) is reflected in a significant
overall difference between generations (Table 11).
In no case, however, is the increase from one gen-
eration to that immediately succeeding it signifi-
cant. Instead, significant increases skip one or two
generations. After generation IV, no differences
are significant.
In summary, the obvious astogenetic gradient in
the colony of W. mutabilis is an increase in mean
size distributed over the first five asexual genera-
tions, with no significant changes in shape. Vari-
ation within generations precludes significant size
differences between contiguous generations.
POLYMORPHISM.—In addition to the obvious qual-
itative differences between polymorphic autozooids
(e.g., presence or absence of ovicells), size and
shape of autozooidal outlines can be expected to
show discontinuous, but less obvious differences in
correlation with qualitative characters. A well-
TABLE 11.—Variation in factors (Table 5) for 23 autozooecial outlines in first seven asexual
generations of Wilbertopora mutabilis Cheetham, holotype, Albian, Ft. Worth Limestone, Krum,
Texas (differences among all generations tested by Kruskal-Wallis method, those between suc-
cessive generations by Mann-Whitney U-test)
Variance
Differences between generations
Within
generation
Factor
Total
Generation II III IV V VI
VII
F(l) '
0.341
0.169
j ns * ** ** **
**
0.035
II ns ns * *
*
0.064
III ns * *
*
0.244
IV ns ns
ns
0.038
V ns
ns
0.298
VI
ns
0.439
VII
F(2) "
0.555
0.121
(all differences ns)
1.710
F(3)"
1.715
0.264
10.404
F(4)'
0.095
0.001
0.168
* significant, P < 0.05
** significant, P < 0.01
ns not significant
30
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
known example is Steganoporella magnilabris
(Busk), in which autozooids are dimorphic, one
set (a-zooids) having opercula with the main
sclerite inverted U- or V-shaped and the zooecium
with a thin oral arch and a narrow postoral shelf,
and the other set (b-zooids) having opercula with
an inverted Y-shaped main sclerite, augmented op-
ercular musculature, and the zooecium with a
thickened oral arch and a broad postoral shelf
(Harmer, 1900; Cook, 1964). The b-zooids in this
species are reportedly usually larger than the
a-zooids, although specimens have been described
in which some a-zooids are longer than b-zooids
(Cook, 1964).
To examine size and shape of outlines in di-
morphic autozooids, we chose three modern col-
onies of 5. magnilabris from Puerto Rico (Figure
22, A-C). In each colony, a- and b-zooids are dis-
tinguished by their opercula, oral arches, and pos-
toral shelves. In two colonies (Figure 22, A, B),
the budding pattern is regular; in the third (Fig-
ure 22, C), it had apparently been disrupted by
breakage and subsequent regenerative budding.
Morphologic differences between zooids in colony
C on the one hand and colonies A and B on the
other can thus be expected to relate in part to the
environmental causes of budding pattern disrup-
tion, although such morphologic differences can
include the genetic differences between colonies as
well.
The plots of F(l)' to F (4)' for the three col-
onies (Figure 22) suggest that b-zooids are gen-
erally larger, more elongate, and more inflated
distally than a-zooids, and that there is no system-
atic difference in the highly variable asymmetry of
both dimorphs. For the two colonies having regu-
lar budding patterns, no overlap was observed be-
tween a- and b-zooids for F(l)' or F(4)', but over-
lap is considerable in F (2)". Colony C, however,
complicates these relationships so that the di-
morphs, considered in all three colonies, overlap
in all four factors. This seems to suggest that the
function of dimorphism in this species was not re-
lated to space filling.
The variation in size and shape of each dimorph
is quite large, and a considerable portion appears
to be a morphologic effect of the disruption of the
regular budding pattern. In the two colonies show-
ing regular patterns, less than one-fourth of the
within-dimorph, within-colony variances exceed
the 95 percent confidence intervals for within-
colony variances for the general sample of cheilo-
stomes (Table 9). In colony C, on the other hand,
fully three-fourths of the variances exceed these
intervals. The heterogeneity of variances in S.
magnilabris is significant (Fmax test; Sokal and
Rohlf, 1969) for F (3)" and F (4)', but not for
F (1)' and F (2)". In keeping with these results, we
examined the differences between dimorphs and
among colonies with two different sets of tests
(Tables 12, 13).
For F (1)', the difference by which b-zooecia on
the average exceed a-zooecia is significant. Differ-
ences between dimorphs are in the same direction
and of about the same magnitude in all colonies, as
indicated by the insignificant interaction between
colonies and dimorphs (Table 12). Differences
among colonies, however, are also significant, and
for the zooecia measured account for most of the
within-dimorph variance. Tests on colony means
(Student-Newman-Keuls test, Sokal and Rohlf,
1969) reveal significant difference between colony
C and the other colonies for b-zooecia but not for
a-zooecia. Differences between colonies A and B
are not significant. This suggests that variation in
size associated with difference in budding pattern
is accommodated principally by the b-zooids.
The only shape factor in which the difference
between dimorphs is unequivocally significant is
F (4)'. Differences among colonies are not signifi-
cant, suggesting that this factor is less sensitive to
disruption of the budding pattern than is size.
The difference between dimorphs in F (2)", al-
though barely significant, is not strong enough to
yield a significant overall among-groups difference
(Table 12). The difference between dimorphs in
F (3)" is not significant. For F (2)", among-colonies
variance is small and insignificant. That of F (3)",
however, is highly significant and again appears re-
lated to the difference in budding pattern and to
have been felt more strongly by b-zooids.
In summary, a- and b-zooecia in the three col-
onies of S. magnilabris differ significantly but over-
lap in size and shape. On the average, b-zooecia
exceed a-zooecia in size and in that part of elonga-
tion contrasted with distal inflation. Size and shape,
especially asymmetry, of both dimorphs are highly
variable in the colony in which the budding pat-
tern is disrupted, and the b-zooids seem to have
NUMBER 29
31
I il
to oT
flj ^-.
C ..
o M
u
u X
V
M en
■fi *
o
•3
°= §3
" IV.
CO = *
§ *;■
O H
« fl 3
rt a, a
fi wj rt
•~ S TJ
Jo
■S J; ii
J? -2 T3
§ £ g
32
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
TABLE 12.—Two-way (dimorphs vs. colonies) analysis of variance in factors F (1)' and F (2)"
(Table 5) for 18 autozooecial outlines in Steganoporella magnilabris (Busk), three a-zooecia and
and three b-zooecia measured in each of three colonies, Recent, Puerto Rico
Degrees
of
Sum of
Factor
Source of variation
freedom
squares
Mean square
F ratio
F(l)"
Between dimorphs
1
6.682
6.682
15.610**
Among colonies
2
8.574
4.287
10.015**
Interaction
2
1.218
0.609
1.422 ns
Within dimorphs,within
colonies
12
5.137
0.428
F(2)"
Between dimorphs
1
2.632
2.632
4.833*
Among colonies
2
3.038
1.519
2.789 ns
Interaction
2
0.361
0.180
0.330 ns
Within dimorphs, within
colonies
12
6.535
0.545
* significant, P < 0.05, but overall anova not significant
** significant, P < 0.01
ns not significant
TABLE 13.—Summary of variation in all factors (Table 5) for 18 autozooecial outlines in
Steganoporella magnilabris, three a-zooecia and three b-zooecia in each of three colonies (variance
components for F(l)' and F(2)" based on two way analysis of variance (Table 12); variation in
F(3)" and F(4)' tested by Kruskal-Wallis method)
Difference
between
dimorphs
Variance within
dimorphs
Factor
Variance
Among-colonies
proportion
Within-co lony
proportion
Interaction
proportion
F(l)'
F(2)"
**
+
1.124
0.707
0.572
0.230
0.381
0.770
0.047
0.000
Factor
Difference
between dimorphs
Mean variance
within dimorphs
Difference
between colonies
F(3)"
F(4)'
ns
7.159
0.047
**
ns
* significant, P < 0.05
k* significant, P « 0.01 ns not significant
+ see Table 12
been more sensitive to this disturbance than the
a-zooids.
MlCROENVIRONMENTAL VARIATION. In all the
foregoing examples, outlines of autozooecia meas-
ured within the same zoarial fragment and inferred
to be in the same condition of ontogeny, astogeny,
and polymorphism were found to vary in size and
in all aspects of shape. This variation is ascribed
to microenvironment (Boardman et al., 1970) on
the assumption of genetic uniformity throughout
a colony.
Microenvironmental variation, as measured by
within-colony variances, is itself variable among
the specimens and characters examined. Two col-
onies can show different amounts of microenvi-
ronmental variation because of differences in the
sets of microenvironmental conditions each expe-
rienced and/or differences in the latitudes of phe-
notypic variation permitted by each genotype.
These sources (compare with the "'nongenetic
(microenvironmental)" effect and "nongenetic-
genetic'' interaction term of Farmer and Rowell,
1973) apply to differences between colonies,
whereas variations within a colony all must have
NUMBER 29
33
"5T
o ■-•
-.5
-1.0
J3*"Y H55T
3.0j
2.5 -
zxy-
1.5-■
1.0 -
.5 -
0 ■■
-.5-
-1.0--
-I.5--
.5 -
0--
-.51
"I—
F(l)1
F(2)"
F(3)"
F(4)'
FIGURE 23.—Variation in size and shape of autozooecial outlines in two zoarial fragments of
Thalamoporella biperforata Canu and Bassler (Appendix B, Id. no. 8A-B). (Frontal views X 25;
symbols as in Figure 21.)
34
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
resulted from the different microenvironmental
conditions experienced by that colony, if its zooids
are genetically uniform. It is thus possible to state
that the microenvironmental variation in one col-
ony is greater than that in another, whether the
sets of microenvironmental conditions under which
the two colony variances developed were different
or the same. In some cases at least, it is further
possible to infer that the sets of microenviron-
mental conditions encountered by different col-
onies differed in one or more factors such as
crowding, substrate irregularity, injury, etc. (Ab-
bott, 1973). Then, the difference between colony
variances, and that between colony means, is very
likely to reflect further environmental modifica-
tion as well as the genetic difference in latitudes of
phenotypic variation permitted by different
genotypes.
Comparison of two zoarial fragments of Thala-
moporella biperforata Canu and Bassler from the
same Miocene locality suggests the relative effects
of environmental modification of size and the
three aspects of shape expressed by factors F (1)'
to F (4)'. In one specimen (Figure 23, A), an ir-
regular budding pattern apparently resulted from
crowding of several divergent lineal series into
fewer, convergent ones. In the other (Figure
23, B), the regular budding pattern indicates ab-
sence of crowding.
Differences in the amount of variation in size
and all aspects of shape are immediately suggested
in the plots of F (1)' to F (4)', and their significance
is substantiated by Fmax test (Table 14; Sokal and
Rohlf, 1969). The differences between the values
of the factors in the two specimens, however, are
not significant, except for F(3)". Thus, the in-
ferred difference in the sets of microenvironmental
conditions which acted on the colonies represented
by these specimens is significantly reflected in size,
elongation, and distal inflation of autozooecial out-
lines by modification of the amount of variation,
but not of the within-colony mean. For F (3)",
modification of the mean as well as the variance
was expected from the interdependence of these
two paremeters as noted above.
INTRAPOPULATION VARIATION.—To estimate the
variability of size and shape of autozooecial out-
lines in a fossil population, we selected six zoarial
fragments of Coscinopleura angusta Berthelsen
(Figure 24, A-F) from a single mound in the
Danian of southern Sweden. (Evidence suggesting
that sediments in this mound incorporate bryo-
zoans and other biotic remains approximately
where the organisms grew has been summarized by
Cheetham, 1971.) Measurements on these speci-
mens were restricted to nonovicelled autozooecia,
and generational differences were not apparent in
any of the specimens studied. Although ontoge-
netic differences are evident between zoarial frag-
ments in the thickness of the cryptocyst (Figure
24, A, D, and E vs. B, C, and F), we did not discern
an ontogenetic gradient for the autozooecial out-
lines measured, either within or between speci-
mens. (Zoarial fragment A, Figure 24, includes at
its distal end a preserved growing edge, at which
occur several zooecia with partly formed crypto-
cysts. These zooecia were not included in this ex-
amination of autozooecial shape.)
The plots of factors F(l)' to F (4)' for the six
zoarial fragments (Figure 24) suggest a greater
TABLE 14.—Comparison of variation in factors (Table 5) for five autozooecial outlines measured
in each of two colonies of Thalamoporella biperforata Canu and Bassler, Miocene, Cercado de
Mao, Dominican Republic (colony A = budding pattern disturbed; colony B = budding pattern
regular)
Factor
Within-co
Colony A
lony
variances
Colony B
Fmax test
between
variances
Mann-Whitney
U-test between
colonies
F(l)'...
0.164
0.008
*
ns
F(2)"...
1.130
0.020
**
ns
F(3)"...
1.689
0.064
*
*
F(4)'...
0.210
0.014
*
ns
* significant, P < 0.05
** significant, P < 0.01 ns not significant
NUMBER 29
35
±-^
■+ i —I H
T Rl CM -
\ 1 1 1 I 1 1-
I •
■+—'—I L) 1 (-
1*'°
$m mm
if
M .-
3 x
.s """'
.2 CM
u
__tions
variance
Variance
95% conf. int.
Variance
95% conf. int.
Within
0.280
0.173
0.047-1.145
0.107
0.064-0.156
0.617
0.383
0.519
0.145
0 -1.024
0.374
0.223-0.702
0.279
0.721
0.949
0.354
0.038-2.720
0.595
0.354-1.116
0.204
0.796
0.059
0.026
0.005-0.189
0.033
0.020-0.062
0.446
0.554
* partitioned from log transformation
NUMBER 29
37
range in mean size for all cheilostomes examined
(Figure 28). The one other significant difference
between within-colony means does not change the
generally overlapping relationship among the
other five specimens, i.e., most of the zoarial frag-
ments do not differ appreciably in mean size of
autozooecial outlines.
Although among-colonies variation in both
F (2)" and F (4)' is significant, none of the six zo-
arial fragments examined differs in these factors
from all others. In F (2)", there are no significant
differences among the six zoarial fragments.
In F(3)", zoarial fragment D is significantly
more asymmetrical than any of the others, which
do not differ significantly among themselves. The
overall difference among all six zoarial fragments is
not significant, as noted previously. The high asym-
metry of zoarial fragment D appears to be related
to the general obliquity of budding direction (Fig-
ure 24, D).
In summary, the six specimens of C. angusta
studied suggest that intrapopulation variation
among colonies in size and shape of autozooecial
outlines is small, although significant for all prop-
erties but asymmetry. The low within-colony varia-
tion in size and the part of elongation contrasted
with distal inflation makes these two properties ap-
pear particularly significant for characterizing pop-
ulation morphology as a basis for taxonomic
interpretation.
are not separable from those of microenvironment,
although in most cases these kinds of variation
should be easily recognizable. Some assessment of
these factors in taxonomic discrimination of cheilo-
stomes can be inferred from the foregoing analyses.
SUMMARY OF CHARACTERS.—From estimates of in-
trapopulation variation in Coscinopleura angusta,
whose within-colony component is close to that for
most of the wide range of cheilostomes examined
(see above), the average minimum recognizable
taxonomic difference can be calculated for size and
each aspect of shape (Table 18). We have based
this calculation on the interpretation of the 75 per-
cent rule of Mayr et al. (1953), assuming normal
distributions within populations. (Contraindica-
tion of normality for F (3)" discussed above pre-
cludes use of the value calculated for this factor
for any purpose but comparison with the other
factors.)
From the average minimum recognizable taxo-
nomic difference so calculated, it is further possible
to calculate the minimum number of measure-
ments needed to detect a difference this small
(Sokal and Rohlf, 1969). Calculations were made
for the number of zooecia per colony needed to
distinguish between two colonies and the number
of colonies per population needed to distinguish
between two populations (Table 18). Although
graphic comparison shows the minimum taxo-
nomic difference for every factor to be exceeded by
Taxonomic Implications of Properties of Outline
The variation of autozooecial outlines in the
wide range of cheilostomes examined greatly ex-
ceeds that in the sample of Coscinopleura angusta
for size and all aspects of shape (Figure 25).
Among-colonies variances for all factors except
F(3)" are significantly less in C. angusta than in
the multispecies sample (Table 17). All factors,
with the possible exception of asymmetry, then,
seem potentially useful as bases for taxonomic in-
terpretation of cheilostome morphology. Differ-
ences in the proportions of within-colony variance
(primarily microenvironmental "noise") among
zooids inferred to be in the same condition of
ontogeny, astogeny, and polymorphism affect the
efficiency of the four factors to different degrees.
Ontogeny, astogeny, and polymorphism can fur-
ther affect taxonomic interpretations if their effects
-B—
AMONG COLONIES
WITHIN COLONIES
F(l)'
F(4)'
F(2f
F(3)"
FIGURE 25.—Comparison of proportional differences in vari-
ances of F(l)' to F(4)' within and among colonies in 31
zoarial fragments of a wide range of cheilostome species
(unshaded) and in six zoarial fragments of Coscinopleura
angusta Berthelsen (shaded).
38
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
TABLE 17.—Comparison of variance (F-test) of 31 colonies in multispecies sample
(Table 9) with those of six colonies of Coscinopleura angusta (Table 16)
Amonq-
colonies variance
Within-
■colony variance
Factor
Multispp.
c.
angusta
Diff.
Multispp.
C. angusta
Diff.
F(l) ' ..
2.225
0.173
**
0.122
0.107
ns
F(2)"..
1.369
0.145
**
0.311
0.374
ns
F(3)"..
0.729
0.354
ns
0.914
0.595
ns
F(4) ' ..
0.153
0.026
*
0.046
0.033
ns
* significant, P < 0.05
** significant, P < 0.01
ns not significant
the expected range of variation within a colony
(Figure 26), it is noteworthy that the numbers of
measurements needed to detect these differences
are quite small and on the average about the same
as used in this study.
Because of their small within-colony variances,
F (1)' (size) and F (4)' (part of elongation con-
trasted with distal inflation) appear to express po-
tentially taxonomically important characters that
can be distinguished with the greatest efficiency,
i.e., in the greatest number of intervals with the
fewest measurements per colony.
The relationships between the intervals of mini-
mum taxonomic difference and the effects of ontog-
eny, astogeny, and polymorphism also appear to
differ for size and the three aspects of shape (Fig-
ure 27). The maximum difference in each factor
except F (3)" with distance from the growing edge
of the colony in Metrarabdotos unguiculatum is,
in general, greater than the minimum taxonomic
difference. The maximum difference between gen-
eration means in the zone of astogenetic change in
Wilbertopora mutabilis barely exceeds the mini-
mum taxonomic difference for F (1)' and is not
significant for other factors. Finally, the average
difference between dimorphic autozooids in Stega-
noporella magnilabris is less than the minimum
taxonomic difference for F(l)' and F (4)', and not
significant for other factors; moreover, with dis-
ruption of the budding pattern, size and shape of
the more specialized set of autozooids are more ob-
viously affected, leaving the ordinary autozooids
relatively unmodified. These relationships suggest
that ontogenetic differences can be expected to
TABLE 18.—Intervals of minimum taxonomic difference recognizable in observed ranges of colony
means and minimum numbers of zooecia per colony and of colonies per population to detect
that difference with 95 percent certainty (factors as in Table 5; observed ranges and taxonomic
differences in standardized units; millimeter equivalents for F(l)' in parentheses; calculations
based on within-colony estimates from Table 9 and within-population estimates from Table 16)
Factor
Observed range of
within-colony means
Minimum
taxonomic
difference *
No. intervals
distinguished
within observed
range
Minimum nc
replications needed to
No. zooecia/colony No.
• of
detect
colom
min. diff.
es/population
F(l)'
6.057
(0.36)
1.354
(0.08)
4.5
3
4
F(2)"
4.941
1.846
2.7
4
3
F(3)"
4.064
2.493**
1.6
6
3
F(4)'
2.021
0.622
3.3
5
3
* 2.56 times estimated average standard deviation of population ** condition of normality not met
NUMBER 29
39
F(l)
F(2)
. 1
■' v::::::j r- i
mmimUMmmhi l
r—
— 3-
.. i
i a
1
i"1—
..i—
3
F(3)
,c
X3
F(4)
,c
FIGURE 26.—Comparison of distribution of variation in F(l)'
to F(4)' in all specimens studied (Appendix B). (Horizontal
bars are observed ranges of within-colony means divided into
intervals representing minimum recognizable taxonomic dif-
ference (see Table 18). Horizontal lines are average 95 per-
cent ranges of variation within colonies, with minimum
numbers of measurements per colony needed to detect aver-
age difference between intervals (Table 18).)
F(l)
xz
F(2)" '
» ' r
F(3)" '
I ,, ■ ~3
F(4)
■ I fcw^i MM
\ " i:
F(l)'
F(IV
~i
F(4)'
FIGURE 27.—Comparison of distribution of variation in all
specimens studied (shown by horizontal bars as in Figure
26) to that in selected specimens illustrating within-colony
sources of variation: a, maximum observed difference between
zooecia at growing edge and more proximal ones in colony
of Metrarabdotos unguiculatum (Figure 20); b, average dif-
ference between means of successive asexual generations in
primary zone of astogenetic change in colony of Wilbertopora
mutabilis (Figure 21); differences in factors other than F(l)'
not significant; c, average difference between means of a-
and b-zooecia in three colonies of Steganoporella magnilabris
(Figure 22); differences in F(2)" and F(3)" not significant.
have the most pronounced effects on F (1)' and
F(4)', the two factors with the greatest inferred
taxonomic potential. It is thus very important to
make every effort to recognize ontogenetic differ-
ences, and this is generally easy to do from the de-
velopment of frontal structures, etc. Because of
their magnitude, ontogenetic differences in F (1)'
and F (4)' can be expected to be obvious, and, be-
cause of the manner of growth of the outline, these
differences can be expected to lose their signifi-
cance for autozooidal outlines not in the vicinity
of the growing edge of the colony. Astogeny and
autozooidal polymorphism in the examples studied
seem to be of generally smaller magnitude, about
on the same order as the microenvironmental
"noise" expected within colonies (95 percent
ranges of variation, Figure 26) and thus not very
important as possible sources of confusion with
taxonomic differences.
For F(l)', it is possible to compare the average
minimum recognizable taxonomic difference with
the precision with which the within-colony mean
can be estimated. The principal source of "error"
(Table 19) is the confidence interval for the mean,
which is about three times as large as the next most
important source, the distortion of the principal
components representation. These two sources of
"error" of course are not unique to the methods of
measurement and characterization of the outline
employed here. The distortion due to the vector
representation and the error of measurement
together are less than the principal components
distortion and much less than the confidence inter-
val for the mean. The cumulative effect of all of
these sources of "error" at a maximum is about the
same in magnitude as the minimum taxonomic
difference. On the average, the "error" would not
be expected to mask important taxonomic distinc-
tions in size. There is no reason to believe that
"error" in the important shape variables would
differ in kind from that in size.
PRELIMINARY EVALUATION OF CHARACTERS WITHIN
TAXA.—The taxonomic significance of size and
shape of autozooid outlines in cheilostomes ulti-
mately depends on how consistent these characters
are among colonies within taxa. Such taxa should,
of course, be based on all available independent
morphologic characters that are likely to reflect
genetic differences correlated with patterns of dis-
tribution in time and space. The data examined
40
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
TABLE 19.—Precision of representation of mean autozooecial size within colonies through vector
measurements, calculation of area, representation by factor F(l)', and calculation of colony mean
Source of variation or error
Amount (mm)
Error of measurement from photograph
(twice average standard deviation
0.009
Maximum distortion of vector representation
(mean for colony with most distorted
0.006
Maximum distortion of rotated 4-component
0.017
0.032
95 percent confidence interval for average
0.052
here were not intended to suggest a classification
based on autozooidal outlines but to test whether
aspects of size and shape can be expected to be
generally consistent within taxa. These data sug-
gest that size and aspects of shape expressed by
elongation and distal inflation are generally con-
sistent within species and even higher taxa, al-
though there are exceptions in some taxa.
Division of the F(l)', F (2)", and F(4)' axes into
intervals of minimum taxonomic significance (Fig-
ures 28, 29) suggests that minimally overlapping
groups of colonies having autozooidal outlines of
similar size and shape can be recognized. To ex-
plore this possibility among the range of outlines
studied, we arranged all of the specimens listed in
Appendix B in a series of dendrograms (Figures
30, 31) based on different combinations of the four
factors, weighted according to their total variances.
It must be emphasized that these dendrograms,
based as they are on characters of the autozooidal
outline only, cannot be expected to reproduce an
arrangement based on characters from the whole
morphology. The placement in these dendrograms
of specimens inferred to be conspecific or conge-
neric on the basis of characters from the whole
morphology can suggest how important taxonomi-
cally the characters of the outline might be.
The scattered distribution of conspecific speci-
mens in Figure 30a (note especially Coscinopleura
angusta, 29A-F), based on all four factors, is prin-
cipally the result of the high variability in asym-
metry. With F (3)" removed (Figure 306), con-
specific specimens are less scattered. With only
F (1)' and F(4)' included (Figure 316) taxonomi-
cally related specimens generally cluster close to-
gether (see especially Poricellariidae, 21, 26, 31).
However, in some species (e.g., Thalamoporella
biperforata, 8A, B) colonies seem to cluster more
closely with just F (2)" and F (4)' considered (Fig-
ure 31a) than they do with F (1)' included. This
suggests that, although the combination of size and
the aspect of shape contrasting elongation with
distal inflation yields generally consistent taxo-
nomic groupings, some taxa are distinctly hetero-
geneous with respect to this combination of char-
acters. This does not diminish the importance of
these characters, however, in contrasting the com-
binations of states in different taxa.
Average among-colonies differences in outlines
within taxa, for which this study provided pre-
liminary data, are summarized in Figure 32.
Within the species examined, most combinations
of factors, except that including asymmetry (Fig-
ure 32c), tend to give small differences, below the
level of minimum taxonomic difference. Either
shape (Wilbertopora mutabilis, Figure 32a, a) or
size (Metrarabdotos unguiculatum, Figure 32e, e),
however, may exceed the level of minimum taxo-
nomic difference within a species. (The size differ-
ence in M. unguiculatum has previously been in-
terpreted as a taxonomic difference by Osburn,
1952, and Cheetham, 1968a.) The combination of
F(l)' and F (4)' seems to provide the most con-
sistent characterization of all species considered.
Within the few genera for which among-colonies
differences in outline have been estimated, size
NUMBER 29
41
rF(2)'
FIGURE 28.—Intervals of minimum taxonomic difference between within-colony means for F(l)'
(Figure 26) superposed on plot of F(l)' vs. F(2)' for 32 zoarial fragments (Figure 12).
42
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
>(2)M
\C 1
o
\fe I
IHlflHI IIIIIUMMpHIIIIIHI|Hlllfliilliaimillllll||||||||IUIIpl
VA
I90J
s>*
IV
s<*
totf
<*m
^i
I I + w>* I
HHAiiiiiiiiiiiiiiiiiiiiiiiniujiiiiiiiniiiiiiiiiiiH(iiiiiiiiiiiiiiiiiaiiiHi
iiiiiiiiiiiiiiiiimiiiiiiiiiiiiii|iiiiiiiiiiiii|iiiiiiiin^iiiii
1
F(4)'
FIGURE 29.—Intervals of minimum taxonomic difference between within-colony means for F(2)"
vs. F(4)' (Figure 26) superposed on plot of F(2)" vs. F(4)' for 32 zoarial fragments (Figure 17A).
NUMBER 29
43
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FIGURE 30.—Dendrograms of 48 groups of zooecia (Appendix B) based on factors expressing
properties of autozooidal outlines: a, dendrogram based on,all four factors, F(l)'-F(4)'; b, dendro-
gram based on factors F(l)', F(2)", and F(4)'. (Average taxonomic distances between colony means
clustered by unweighted pair-group method using arithmetic averages, Sneath and Sokal, 1973.)
44
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
2.0
1.6
H—
DISTANCE
1.0
0.5
—I—
2.0
I—
1.6
H—
DISTANCE
1.0
0.5
—I—
IA
2
3
4
5
10
7
8A
8B
l2Aa
l2Ba
13
9A
14
II
l2Ab
15
6
l2Ca
18
16
17
30A
9B
19
l2Bb
2IAI
2IAf
2IB
20
29A
29F
22
l2Cb
24
25
26
23
29B
29C
27
28
29D
29E
30B
IB
IC
31
j—a
r—
H:
- HZ
M
E
A
"1—rz
r-HZ
^Z
L HI
r
L
FIGURE 31.
-Dendrograms of 48 groups of zooecia as in Figure 30: a, dendrogram based on fac-
tors F(2)" and F(4)'; b, dendrogram based on factors F(l)' and F(4)'.
NUMBER 29
45
(Figure 32e) apparently is less consistent than
within species, but shape (Figure 32a), including
F (2)" and F (4)', seems consistent enough to char-
acterize taxa at this level. Within one genus, Diplo-
didymia (Figure 32, 1), and the family to which it
belongs, Poricellariidae (Figure 32, i), both size
and shape appear consistent enough to warrant
retention as taxonomic characters at these higher
levels in a polythetic classification.
directions (the principal growth directions) be-
tween zooids. Other commonly used measurements
normally assume at least two pairs of correspond-
ing points for each measured character. Such as-
sumptions increase the chances for both operator
bias and biologically meaningless variation intro-
duced by the measurement system itself; hence the
growth-vector representation of zooidal outline al-
Summary and Conclusions
The sizes and shapes of frontal outlines of chei-
lostome autozooids are so highly variable both
within and among colonies of the same taxa that
their use to characterize and distinguish taxa has
been a matter of some controversy. However, vari-
ous aspects of size and shape can be quantified, and
the derived coefficients then statistically evaluated
for taxonomically significant patterns of variation.
Our approach has been to try to reduce opera-
tional bias by treating the frontal outline as "a vec-
tor diagram of its own growth" (Thompson, 1942).
Such a representational system of outline geometry
has at least two important advantages over other
measurement systems. First, it is designed to repre-
sent as closely as possible the directional compo-
nents of cheilostome zooidal growth. Consequently,
the vectors relate morphology directly to a func-
tionally important biological process (growth), a
desirable property of any biometric measurement.
A second advantage is that the growth-vector sys-
tem requires but a single pair of morphologically
corresponding points (the points of budding) and
FIGURE 32.—Mean differences among colonies within taxa for
various combinations of factors of autozooidal outlines: a,
F(2)" and F(4)'; b, F(l)', F(2)", and F(4)'; c, F(l)'-F(4)'; d,
F(l)' and F(4)'; e, F(l)' alone. (Means of the average taxo-
nomic differences, Sneath and Sokal, 1973, between colonies
and the calculated minimum taxonomic difference, M.T.D.,
Table 18, have been normalized to the greatest observed dif-
ference among colony means for each combination of factors;
bars entirely below M.T.D. level, especially where that level
is low, suggest consistency of a combination of factors within
a taxon. a = Wilbertopora mutabilis (30A, B); b = Metra-
rabdotos helveticum (9A, B); c = Steganoporella magnilabris,
b-zooecia (12A-C); d = Coscinopleura angusta (29A-F); e =
Metrarabdotos unguiculatum (6, 7); f = Steganoporella mag-
nilabris, a-zooecia (12A-C); g = Diplodidymia ratoniensis
(21A, B); h = Thalamoporella biperforata (8A, B); i =
Poricellariidae (21, 26, 31); j = Metrarabdotos (6, 7, 9, 25);
k = Coscinopleura (23, 29); 1 = Diplodidymia (21, 26).)
46
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
lows less chance for measurement error.
Size of autozooids, as measured by the area en-
closed within the frontal outline, is consistent
within colonies to a higher degree than previous
studies based on "standard" linear dimensions
(e.g., length, width) have indicated. Variation in
size among comparable zooids, although not negli-
gible, is relatively low, even in colonies with se-
verely disturbed budding patterns. Significant dif-
ferences in size can result from comparing zooids
at different ontogenetic or astogenetic stages, dif-
ferent polymorphs, or zooids which reflect the
effects of different microenvironments. Contribu-
tions from each of these sources of variation, how-
ever, can be more or less identified and accounted
for prior to making taxonomic comparisons. Al-
though differences in zooidal size among colonies
within single populations are statistically signifi-
cant, the range of autozooid sizes among a wide
variety of cheilostome genera is so much greater
that we have recognized at least four size intervals
("variable states"), each of which is larger than
the observed within-population variation. These
variable states have been successfully used as a
basis for taxonomic distinctions among some chei-
lostomes. Populations differing by at least one
such interval can be distinguished with a high de-
gree of confidence by measuring as few as three
zooids per colony and as few as four colonies per
population, despite the appreciable within-colony
and within-population variation in size.
The taxonomically significant aspects of shape,
independent of size, are expressed by two derived
characters. Each of these characters consists of a
different linear combination of two direct meas-
ures of frontal outline shape: elongation, which is
the concentration of relative growth in a "pre-
ferred" direction near the proximal-distal axis, and
distal inflation, which is the proportion of relative
growth concentrated in the distal half of the out-
line. The two derived shape characters are mu-
tually independent and reflect two distinct patterns
of covariation between elongation and distal infla-
tion: (1) that part of their joint variation which is
positively correlated, and (2) that part which is
not correlated. Although the positively correlated
variation in elongation and distal inflation is much
greater in overall magnitude, the uncorrelated
variation is much smaller within colonies. Among
the cheilostomes examined, three intervals of mini-
mum taxonomic difference can be recognized in
each character, for a total of nine states when the
two independent characters are considered simul-
taneously. Populations differing by at least one
such interval are distinguishable by measuring as
few as four or five zooids per colony and as few as
three colonies per population. Astogenetic and
polymorphic differences in these characters are
relatively smaller than those in size, but ontoge-
netic differences are about the same magnitudes.
Asymmetry of the frontal outline (unequal rela-
tive growth on either side of the proximal-distal
axis) is also independent of size and of elongation
and distal inflation, but proved to have little or no
taxonomic significance among the cheilostomes ex-
amined. Asymmetry is so highly variable within
colonies that not even two full intervals of mini-
mum taxonomic difference could be recognized
among the wide variety of shapes studied.
The results of this study demonstrate that sensi-
tive, taxonomically decomposable information is
contained in the autozooidal outline. Statistically
significant patterns of variation, both between col-
onies and between populations, can be recognized
in zooidal size and at least two components of
shape, elongation and distal inflation. Several nu-
merically computed dendrograms using different
combinations of the three independent size-shape
characters group specimens in previously estab-
lished taxa, although no single combination is best
for all the groups studied. In general, however,
dendrograms based on size and the negative com-
ponent of covariation between elongation and
distal inflation resemble "recognized" higher taxo-
nomic groupings, while colonies within popula-
tions in at least some genera are more compactly
clustered through the shape characters alone.
Although the frontal outline geometry of chei-
lostome autozooids is taxonomically important, its
potential utility is masked by redundancy and non-
trivial covariation among the measured characters.
By empirically eliminating redundancy and ex-
tracting variance components from ontogenetic,
astogenetic, and microenvironmental sources, a
small set of taxonomically independent polythetic
characters has been obtained. We suggest that simi-
lar procedures can profitably be applied to the
problem of cheilostome taxonomy using other
(hopefully larger) sets of complexly interdepend-
ent morphologic features as well.
Appendix A
Derivation of Vector Statistics
In the vectoral treatment of size and shape used
herein, all statistics depend on two sets of vari-
ables: the set of vector azimuths, 6^, which are
measured as departures from the principal growth
direction (pgd) and determined a priori, and the
set of corresponding vector magnitudes, rt (i's in-
crease from left to right; Figure 33). Consequently,
minor irregularities in zooid outline occurring be-
tween adjacent vector intercepts are not reflected
in the statistics. However, we have found that the
17-vector system employed in the present study is
sufficient to measure faithfully all but the most
minor irregularities (see p. 7; Figure 7).
SIZE.—Zooid size is defined as the frontal area
bounded by vertical walls. To obtain an approxi-
mation of area from the vector representation, the
tips of the vectors are connected by straight lines,
forming a 17-sided polygon (Figure 33). The area
of a triangular wedge of this polygon (Figure 34)
is easily seen to be
At = lf2(rth)= l/?(r,rl+lrin^
where ^t=|^+ll-^.|.
The area of the entire polygon of (n-1) segments
(we have used n = 17) is thus
»-l n-1
(1) 4 = 2^ = 1/2 2r0.
2 r^^x sin $t
<=i
n-1
2 sin ,
FIGURE 33.—Vector representation of an ideal zooid outline
(see Figure 4). (Distribution consists of 17 vectors, 8 distrib-
uted symmetrically on each side of the proximal-distal axis
(0°). Azimuth (#,) and magnitude (r,) indices (i) increase
from left to right (-90° to +90°). The statistic A is the
area contained within the 17-sided polygon produced by
connecting the tips of the vectors with straight-line segments.)
47
48
SMITHSONIAN CONTRIBUTIONS TO PALEOBIOLOGY
*i= •! + !-•(
FIGURE 34.—Geometric relations of a single triangular wedge
between two adjacent vectors in Figure 33. (The sum of areas
of all such wedges is the area of the entire zooidal out-
line, A.)
ELONGATION.—Simple trigonometric measures of
central tendency and dispersion for vector variables
are well known (e.g., Batschelet, 1965:7-20) and
correspond to the azimuth and magnitude, respec-
tively, of the resultant vector obtained by graphical
addition (Figure 35). For unequally weighted vec-
tor variates (r/s not equal), the standard statistic
for dispersion can be written
the distal margin. The statistic P is a useful meas-
ure of elongation in that it reflects relative con-
centration of vector lengths; larger values of p in-
dicate longer vectors in the principal growth
direction (pgd) than elsewhere (greater elonga-
tion), while smaller p's mean more or less equi-
dimensional zooids (less elongation). As noted
above (p. 8), cheilostome zooids normally range
in shape between roughly semicircular (P = 0.767)
and elongate and narrow (p — 1.0).
ASYMMETRY.—Asymmetry of outline with respect
to the principal growth direction implies that vec-
tor magnitudes tend to be distributed unequally
on either side of the pgd. An intuitively appealing
measure of this property is the tangent of the vec-
tor mean, defined as (Figure 35)
2r( sin 0(
i=\
tan 0-
(4)
(-oot- (rq/2)2 sin 2*,,
[(r/n+iJ^-^cosg,,]2
+ (r /2)r,sin0,+ - ?—« tan R
where q is the smallest index, ^>\^-) satisfying the ii
equality, <•< cos e> \^f
'm'z {
ft
' /
/
z7
r /
/
/
/
/
/
/
b
/ A
1 jfl
faff \
*y i
f/ *\
.y>
1^ wj
+90°
FIGURE 36.—Geometry required to compute area of the proxi-
mal half of a zooidal outline to the left of the proximal-
distal axis. (Formulas for computing the areas of regions W,
X, Y, Z are given in the text.)
FIGURE 37.—Geometry required to compute area of the proxi-
mal half of a zooidal outline to the right of the proximal-
distal axis. (A formula for computing the area of the region
W' + X'+Y' + Z' is given in the text.)
The sign changes in all but the first term of ex-
pression (8) from those in (7) result from the
change in the sign of sin 9.
Finally, our expression for distal inflation is
given by
(10) r=-
(f>0).
„—=(!/«) \ / f 2 rfun ««!*+/2r(a» 0(j *.
(9) rf/= 1-(area, +area,.)/^, (O^dKl)
This measure is obviously correlated with size, and
is related to elongation (p) through the arithmetic
mean vector length,
where, from expression (I) above,
• = (l/n2r,V
A=l/22r,rM1 sin__