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 SERIAL PUBLICATIONS OF THE SMITHSONIAN INSTITUTION The emphasis upon publications as a means of diffusing knowledge was expressed by the first Secretary of the Smithsonian Institution. In his formal plan for the Insti- tution, Joseph Henry articulated a program that included the following statement: "It is proposed to publish a series of reports, giving an account of the new discoveries in science, and of the changes made from year to year in all branches of knowledge." This keynote of basic research has been adhered to over the years in the issuance of thousands of titles in serial publications under the Smithsonian imprint, com- mencing with Smithsonian Contributions to Knowledge in 1848 and continuing with the following active series: Smithsonian Annals of Flight Smithsonian Contributions to Anthropology Smithsonian Contributions to Astrophysics Smithsonian Contributions to Botany Smithsonian Contributions to the Earth Sciences Smithsonian Contributions to Paleobiology Smithsonian Contributions to Zoology Smithsonian Studies in History and Technology In these series, the Institution publishes original articles and monographs dealing with the research and collections of its several museums and offices and of professional colleagues at other institutions of learning. These papers report newly acquired facts, synoptic interpretations of data, or original theory in specialized fields. These pub- lications are distributed by mailing lists to libraries, laboratories, and other interested institutions and specialists tbroughout the world. Individual copies may be obtained from the Smithsonian Institution Press as long as stocks are available. S. DILLON RIPLEY Secretary Smithsonian Institution 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 2.0 1— 1.5 -I— DISTANCE 1.0 0.5 —t— 20 I— 15 DISTANCE 10 05 —I— £ d d IA 2 3 9B 4 8B II 9A 13 14 16 8A 7 I2B0 10 5 l2Aa l2Bb 6 l2Ab 15 18 22 l2Ca l2Cb 2IAf 26 2IB 21 Al 23 20 29A -r17 I 30 30A 19 24 28 27 29E c 29B 29C 29F 25 29D IB IC 30B 31 ■c ■c cE € L-d 10 8A 9A SB II 9B l2Ca 13 14 16 l2Cb l2Bb l2Ab 15 6 18 22 17 30A 20 29A 19 2IAI 23 2IAf 2IB 26 24 28 29B 29C 27 29D 29E 25 29F 30B 31 IB IC 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