v—v EVALUATION OF THE UTILITY OF TWO - DIMENSIONAL FOURIER SHAPE ANALYSIS FOR THE STUDY OF OSTRACODE CARAPACES Thesis for the Degree of M. S. MICHIGAN STATE UNIVERSITY JEAN. KAY YOUNKER ‘ 1971 INK. H! I III; IIZIIIILIII III II I II III III I L ~ LIBRARY Michigan Si no University SE? 3. (a 8; I??4 4 33 ABSTRACT EVALUATION OF THE UTILITY OF TWO-DIMENSIONAL FOURIER SHAPE ANALYSIS FOR THE STUDY OF OSTRACODE CARAPACES BY JEAN KAY YOUNKER Shapes of organisms and of their constituent structures are manifestations of innate genetic limits, modified by environmental conditions. Because fossil organisms are generally represented by a residuum of hard parts, studies of the variation of those structures have been used to construct genetic and environmental models. A group of organisms whose morphologic variation is considered to be an important information carrier are the ostracodes. Reflection of this variation through two-dimensional shape has been qualitatively studied but quantitative determination of the exact nature of shape variation and correct interpretation of the information carried by shape has been difficult. Shape description by means of a Fourier Shape Program deve10ped by Ehrlich and Weinberg (1970), permits Jean Kay Younker evaluation of the relative contribution of shape com- ponents for taxonomic and environmental studies. This shape analysis was performed on lateral outlines of ostracode carapaces of a number of specimens from the families Trachyleberididae, Hemicytheridae, and Bairdiidae. Discriminant analysis using the shape infor- mation indicated that taxonomic information carried by ornamental structures, hingement, and muscle scars is also reflected in two-dimensional shape variation. Be- cause the taxonomic characteristics are considered to be important manifestations of genetic differences, the re- lated shape components must represent similar responses. With this established, it was then possible to describe the inter-specific shape variation caused by sexual dimorphism and the nature of shape changes during growth of the ostracode. These results indicate that shape variation, measured and expressed as a continuous variable, should be a good index of the effects of evolutionary or environmental changes. EVALUATION OF THE UTILITY OF TWO-DIMENSIONAL FOURIER SHAPE ANALYSIS FOR THE STUDY OF OSTRACODE CARAPACES By JEAN KAY YOUNKER A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE College of Natural Science Department of Geology 1971 ACKNOWLEDGMENTS I extend my appreciation and thanks to the following people: Dr. Robert E. Ehrlich, Dr. Thomas A. Vogel, Dr. S. B. Upchurch, Dr. B. H. Weinberg, Dr. Robert Anstey, and Dr. Joseph E. Hazel. ii TABLE OF CONTENTS ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . LIST OF TABLES LIST OF FIGURES . . . . . . . . . . . . . LIST OF APPENDICES . . . . . . . . . . . . . . . . . INTRODUCTION NATURE OF SHAPE VARIATION IN OSTRACODES DESCRIPTION OF SPECIMENS USED IN SHAPE STUDIES DESCRIPTION OF METHOD OF SHAPE ANALYSIS CONSTRUCTION AND USE OF CHI-SQUARE CONTINGENCY TABLES . . . DISCRIMINANT ANALYSIS TECHNIQUES USE OF SHAPE CHARACTERISTICS FOR SEX DISCRIMINATION IN FOSSIL OSTRACODES . . . . . . . . SHAPE ANALYSIS OF MOLT STAGES . . . . . . . . . . INFORMATION CONTENT OF HARMONIC ORDERS USE OF SHAPE VARIABLES IN TAXONOMY. SUMMARY AND CONCLUSIONS REFERENCES CITED . . . . . . . . . . . . . . . . APPENDIX A - Discriminant analysis results for species of Rabilimis . . . . . . . . . . . APPENDIX B - Discriminant analysis results for species of Rabilimis and Echinocythereis APPENDIX C - Discriminant analysis results for genera of Bairdiidae APPENDIX D - Discriminant analysis results for genera of Hemicytheridae iii ii iv vi 15 19 21 24 27 31 35 49 52 54 56 58 61 Table 1. 10. LIST OF TABLES Specimens of Rabilimis and Echinocythereis used in this study . . . . . . . . . . Specimens of Hemicytheridae used in this study . Specimens of Bairdiidae used in this study . Chi-Square results for the Species of Rabilimis. Chi-Square results for the genera of Bairdiidae O O O O O O O O O O O O O O Chi-Square results for the genera of Hemicytheridae . . . . . . . . . . . Discriminant function classification matrix for species of Rabilimis . . . . . . . Discriminant analysis classification matrix for species of Rabilimis +-Echinocythereis . Discriminant analysis classification matrix for genera of Bairdiidae . . . . . . . . . . . . . . Discriminant analysis classification matrix for genera of Hemicytheridae . . . . . . . iv Page 13 33 33 33 35 41 44 46 LIST OF FIGURES Figure l. 2. Comparison of amplitude spectra of Rabilimis mirabilis and Rabilimis septentrionalis . . . . . . 18 Relationship between sex and amplitude of second harmonic . . . . . . . . . . . . . . . . . . 25 Shape change with growth of Qypridopsis vidua; shapes from Kesling (1951) . . . . . . . . . . . . 28 Average outlines of molt stages of over 500 specimens of Qypridopsis vidua (after Kesling, 1951) O O O O O O O O O O O O O I O O O O O O O O O 29 a. Relationship between amplitude spectra of specimens of R, paramirabilis assigned to R, septentrionalis and the mean amplitude spectra of the two Species. b. Relationship between amplitude spectrum of specimen of R, septentrionalis assigned to R, paramirabilis and the mean amplitude spectra of the two species. . . . . . . . . . . 37 Comparison between mean shapes of R:_paramirabilis and g, septentrionalis and the outlines of specimens misclassified by discriminant analysis: a. Specimens of R, paramirabilis classified as R, septentrionalis b. Specimen of R. septentrionalis classified as 3, paramirabilis . . . . . . . . . . . . . . . . . 38 Shape relationships between specimens of g, mirabilis, R, paramirabilis, and g, septentrionalis . . . . . 40 Comparison between mean shapes of R, paramirabilis, R, septentrionalis, and Echinocythereis and the outlines of specimens misclassified by discriminant analysis . . . . . . . . . . . . . . . . . . . . . 42 LIST OF APPENDICES APPENDIX A Discriminant analysis results for species of Rabilimis O O O O O O O O O O O O O O O O 0 APPENDIX B Discriminant analysis results for species of Rabilimis and Echinocythereis . . . . . . . APPENDIX C Discriminant analysis results for genera of Ba irdiidae O O O O O O O I O O O I C O O O 0 APPENDIX D Discriminant analysis results for genera of Hemicytheridae . . . . . . . . . . . . . . . vi INTRODUCTION The shapes of entire organisms and of their con- stituent structures are a manifestation of innate genetic limits modified by environmental conditions. Under uniform growth conditions, shape variations of many varieties of organisms would reflect primarily genetic variability. Variation in environmental history from individual to indi- vidual would impress additional shape variability on the population. Because fossil organisms are generally repre- sented by a residuum of hard parts, studies of the variations of those structures have been used to construct genetic and environmental models. Whereas appearance and disappearance of discrete structures can be readily observed, the nature of a continuous change, such as the shapes of those structures, is less easily monitored. A group of organisms whose shape variation is con- sidered an important carrier of information are the Ostracoda. Morphologic variation in ostracodes serves as a medium for expression of sexual dimorphism and as an important source of information used in classification. Shape ratios which characterize variation in length-height dimensions are a common method for semi- quantitative description of ostracode shapes. Inappreciable change in this ratio with increasing carapace size has been used to demonstrate the nearly allometric nature of ostracode growth. Three dimensional shape ratios have been devised using three shape parameters and triangular graphs (Shaver, 1960). In the absence of a priori information to the contrary, there is no way to assess the relative amount of useful information contained in shape ratios. Evalua- tion of complete shape information necessitates a method which accounts for all aSpects of shape and shape variation. The object of this study is to demonstrate the application of a new method of shape analysis, previously applied successfully in non-organic forms. Developed by Ehrlich and Weinberg (1970), this method yields a mathe- matical description of two-dimensional shape as precise as desired. The potential of this method in taxonomic, growth, and evolutionary studies are evaluated herein. NATURE OF SHAPE VARIATION IN OSTRACODES Both hard and soft-part morphologies provide sources of taxonomic information in living ostracodes but only the carapace is available for study of fossil taxa. As a result, such features as ornamentation on the carapace surface, muscle scar patterns, hingement, and shape of the carapace provide diagnostic criteria in fossil ostracodes. The ontogeny of the individual ostracode is recorded in distinct growth stages or molts. Graphs showing length- height ratios have previously been used to express the rela- tively constant shape-size relationship during growth, and to define groups of specimens representing specific molt stages. Kesling (1951), in a detailed study of the morphology of growth stages in the fresh water cyprid CypridoPSis yidgg_(Muller, 1785), concluded that in his specimens, the early molts appeared more rounded while the seventh and eighth molts were more elongate. Kesling also concluded that adults appeared to be more rounded than the eighth molt, probably furnishing space for the sex organs which reach full develOpment in the adult stage. Results of shape analysis of the outlines of Specimens used in Kesling's study will be presented later in this paper. DESCRIPTION OF SPECIMENS USED IN SHAPE STUDIES Two-dimensional lateral outlines of the following genera and families were used to test the utility of the shape description in paleontological studies. Uniform orientation of Specimens is essential, capecially in the study of specimens of similar shape. A very slight change in orientation may significantly alter the shape description. Comparison of specimens oriented by a single individual would partially eliminate this problem, but the possibility for an additional source of shape variability must be considered. Family TRACHYLEBERIDIDAE Sylvester-Bradley, 1948 Subfamily ECHINOCYTHEREIDINAE Hazel, 1967 Genus RABILIMIS Hazel, 1967 Species of the genus Rabilimis were used in an initial evaluation of the usefulness of the new method of shape description in ostracode study. Species differentiation in Rabilimis is based on subtle differences in shape of lateral outline in addition to variation in position of normal pore canals, surface fossae, and ridges. Rabilimis is important in recognition of frigid or subfrigid climatic conditions in Pleistocene deposits and appears to be a bio- stratigraphically useful taxa in the Plio-Pleistocene of Alaska. One species, 3, paramirabilis (Swain, 1963) is an extinct taxon known from Alaska and Russia (Swain, 1963; Lev, 1964). The other two, 3, mirabilis (Brady, 1874) and R, septentrionalis (Brady, 1866) occur in Pleistocene and Recent deposits in the Attic and northern parts of the North Atlantic. Genus ECHINOCYTHEREIS Puri, 1953 The genus Echinocythereis is closely related to Rabilimis, most likely the progenitor of Rabilimis (Hazel, 1967). Table 1 is a complete list of all specimens of Echinocythereidinae from which tracings of illustrations were taken for use in this part of the study. If the original illustration was a right lateral view of a carapace, the illustration was reversed and treated with left valves and left lateral views of carapaces. This is permissible in this group because left valves are larger; therefore right lateral outlines of carapaces actually represent the left valves. Table 1. Specimens of Rabilimis and Echinocythereis used in this study. Genus Rabilimis 1. Rabilimis mirabilis (Brady, 1868) a. Female carapace. Holocene, East Greenland. Unpublished photograph. b. Female left valve. Pleistocene of Scotland. From Brady, Crosskey, and Robertson (1874, pl. 7, fig. 22) c. Female carapace. Recent, eastern North Atlantic. From Elofson (1943, fig. 2) d. Male carapace. Recent, eastern North Atlantic. From Elofson (1943, fig. 6) e. Female left valve. Recent, Laptev Sea. From Akatova (1946, fig. 6a.) f. Male left valve. Recent, Laptev Sea. From Akatova (1946, fig. 6b) 3. Juvenile? left valve. Recent (very likely Pleistocene), The Minch. From Brady (1868, pl. 29, fig. 7) Rabilimis paramirabilis (Swain, 1963 a. Male left valve. Pliocene, Beringian stage, Sub- marine beach at Nome, Alaska. Unpublished photograph. b. Female carapace. Upper Pliocene, Beringian stage, Submarine beach at Nome, Alaska. Unpublished photograph. c. Female left valve. Upper Pliocene. Beringian stage, Gubik fm., Artic Coastal Plain, Alaska. Simpson Core Test Well # 1, 72-73 ft., Unpublished photograph of holotype. d. Female left valve. Upper Pliocene. Beringian stage, Bering Sea, Bureau of Mines Drill Hole 12, 98-100 ft. Unpublished photograph. e. Female left valve. Pliocene, Alaska. Unpublished drawing by K. G. Mckenzie. f. Female left valve. Pliocene, Alaska. From Swain (1963, text-fig. 12b) g. Male left valve. Pliocene, Alaska. From Swain, (1963, pl. 99, fig. 10c) Table 3. l. (cont'd) Rabilimis segtentrionalis (Brady, 1866) a. Female left valve. Upper Pleistocene, Kotzebuan Stage, Baldwin Peninsula, Alaska. Unpublished photograph. b. Male left valve. Upper Pleistocene, Kotzebuan Stage, Baldwin Peninsula, Alaska. Unpublished photograph. c. Male? left valve. Upper Pleistocene, ? Kotzebuan Stage, Gubik Fm., Artic Coastal Plain, Alaska. Teahekpuk shothole, Line 1-48. Unpublished photo- graph of holotype of Pseudocythereis simpsonensig Swain, 1963. d. Male left valve. Upper Pleistocene, Kotzebuan Stage, Baldwin Peninsula, Alaska. Unpublished photograph. e. Female left valve. Unpublished drawing by K. G. Mckenzie. f. Female? left valve. Recent Hunde Islands, Western Greenland. From Brady and Norman (1889, pl. 16, fig. 13) g. Female left valve. Pleistocene, Alaska. From Swain (1963, text-fig. 12a) h. Female? left valve. Recent, Hunde Islands, Western Greenland. From Brady (1866, pl. 60, fig. 4c) 1. Male left valve. Pleistocene, Alaska. From Swain (1963, pl. 99, fig. 10b) Echinocythereis E, planibasalis (Ulrich and Bassler, 1904. Female left valve. Recent, Gulf of Maine. From Hazel (1967, pl. 6, fig. 5) E, margaritifera (Brady, 1870). Female left valve. Recent, Atlantic shelf east of New Jersey. From Hazel (1967, pl. 6, fig. 6) E, margaritifera (Brady, 1870). Male left valve. From Hazel (1967, pl. 6, fig. 7) E, echinata (Sars, 1865). ? left valve. From Hazel (1967, pl. 6, fig. 11) Family HEMICYTHERIDAE Puri, 1953 Six genera of the family Hemicytheridae from three subfamilies were chosen for shape analysis. Original classi- fication was based on a combination of both hard and soft part features. In general, family level discrimination is based on structure of appendages; muscle scars and shape were subfamily characteristics; hingement, type of duplicature, shape details, muscle scars, and primary ornamentation repre- sent generic criteria (Hazel, 1967). Table 2 lists the specimens used in this generic level study. Table 2. Specimens of Hemicytheridae used in this study. Note: All specimens are taken from Hazel (1967). Subfamily Hemicytherinae 1. Genus Hemicythere Sars, 1865 a. E, villosa (Sars, 1865) Plate 2, # 4 b. E, borealis (Brady, 1868) Plate 2, # 5 c. E, borealis (Brady, 1868) Plate 2, # 11 d. E, borealis (Brady, 1868) Plate 2, # 10 2. Genus Elofsonella Pokorny, 1955 ( = Paracythereis Elofson, 1941) a. E, concinna (Jones, 1857) Plate 4, # 10 b. E,concinna (Jones, 1857) Plate 4, # 11 c. E, concinna (Jones, 1857) Plate 4, # l3 3. Genus Baffinicythere Hazel, 1967 a. E, emarginata (Sars, 1865) Plate 2, # 8 b. E, costata (Brady, 1866) Plate 2, # 13 c. E, costata (Brady, 1866) Plate 2, # l4 4. Genus Finmarchinella Swain, 1963 a. E, finmarchica (Sars, 1865) Plate 1, # 4 b. E, finmarchica (Sars, 1865) Plate 1, # 6 c. E, angulata (Sars, 1865) Plate 1, # 9 d. E, angulata (Sars, 1865) Plate 1, # 10 10 Table 2. (cont'd) Subfamily Coquimbinae 1. Genus Muellerina Bassiouni, 1965 a. E, abyssicola (Sars, 1865) Plate 3, # l E, lienenklausi (Ulrich and Bassler, 1904) Plate 3, # 5 E, abyssicola (Sars, 1865) Plate 3, # 8 E, canadensis (Brady, 1870) Plate 3, # 13 E, canannsis (Brady, 1870) Plate 3, # 19 E, lienenklausi (Ulrich and Bassler) Plate 3, # 4 E, canadensis (Brady, 1870) Plate 3, # 12 E, canadensis (Brady, 1870) Plate 3, # 20 Subfamily Campylocytherinae 1. Genus Bensonocythere Hazel, 1967 a. b. C. .E. americana Hazel, 1967 Plate 5, # l ‘E. whitei (Swain, 1951) Plate 5, # 2 E, whitei (Swain, 1951) Plate 5, # 9 11 Family, BAIRDIIDAE Sars, 1887 A second generic study was carried out using genera of the family Bairdiidae (Sohn, 1960). Bairdia McCoy, 1844 is a smooth genus to which over two-hundred Paleozoic Species have been assigned. Sohn's goal was to demonstrate that several distinct generic categories could be distinguished within the genus Bairdia. This would partially solve the obvious problems encountered when dealing with an unorna- mented genus supposedly containing two-hundred species. Sohn used a punched-card technique calling for sixty-seven features per individual. Experimentation with various com- binations of characters determined which variables most readily divided the Specimens into natural groups. Other problems exist in the classification of Bairdia. Sex differentiation is extremely difficult and as a result, shape variations due to sexual dimorphism are probably hidden in different Specific or generic names. Four genera from Sohn's resultant classification were chosen for shape examination. Bairdia McCoy, 1844; Cryptobairdia Sohn, 1960; Bairdiacypris Bradfield, 1935; and Orthobairdia Sohn, 1960. Choice of the particular 12 genera was solely based on availability of sufficient numbers of photographs from Sohn's plates showing similar views. Exact listing and source of each specimen studied can be found in Table 3. 13 Table 3. Specimens of Bairdiidae used in this study. Note: All Specimens are taken from Sohn (1960). Bairdia 1. E, beedei Ulrich and Bassler, 1906 Plate 1, # 5 E, hisgida? Harlton, 1928 Plate 1, # 6 E, beedei Ulrich and Bassler, 1906 Plate 1, # 8 E, grahamensis Harlton, 1928 Plate 1, # 9 ,E. grahamensis Harlton, 1928 Plate 1, # 16 E, pecosensis Delo, 1930 Plate 1, # 22 E, rhomboidalis Hamilton, 1942 Plate 1, # 27 E, hassi Sohn, 1960 Plate 1, # 29 E, whitesidei Bradfield, 1935 Plate 1, # 30 10. .E. whitesidei Bradfield, 1935 Plate 1, # 31 11. E, girtyi Sohn, 1960 Plate 1, # 33 Orthobairdia 1. Q, oklahomaensis (Harlton, 1927) Plate 3, # 13 2. Q, oklahomaensis (Harlton, 1927) Plate 3, # 15 3. Q, oklahomaensis (Harlton, 1927) Plate 3, # 17 4. Q, oklahomaensis (Harlton, 1927) Plate 3, # 19 5. O. oklahomaensis (Harlton, 1927) Plate 3, # 21 14 Table 3. (cont'd) 6. Q, cestriensis (Ulrich, 1891) Plate 3, # 24 7. Q, cestriensis (Ulrich, 1891) Plate 3, # 27 Bairdiacypris 1. E, bedfordensis (Geis, 1923) Plate 2, # 9 2. E, curvis (COOper, 1941) Plate 2, # ll 3. .E. deloi Bradfield, 1935 Plate 3, # 4 4. E, transversus (Roth, 1929) Plate 6, # 28 5. E, transversus (Roth, 1929) Plate 6, # 20 cryptobairdia l. E, forakerensig (Kellett, 1934) Plate 2, # 2 2. E, recta (Harlton, 1929) Plate 2, # 7 3. E, coryelli (Roth & Skinner, 1931) Plate 2, # l6 4. g;_hoffmanae (Kellett, 1943) Plate 2, # 28 DESCRIPTION OF METHOD OF SHAPE ANALYSIS Two-dimensional maximum projection area is enlarged to approximately two inches long and superimposed on an X-Y grid. Coordinates of points on the periphery are re- corded by an automatic digitizer which punches the values directly on data cards. Fourier shape approximation produces a mathematical description of the shape defined by this set of coordinates. Center of gravity of the Shape is calculated and the rectangular coordinates are converted to polar coordi- nates about this origin. A Fourier series is then used to estimate the shape by an expansion of periphery radius as a function of angle about the center of gravity. Precision of shape description is dependent upon Spacing of initial peripheral points in addition to the number of harmonic orders considered. Harmonic orders in polar coordinates are analogous to harmonic orders in rectangular coordinates ex- cept the waves are closed forms. The zeroth harmonic is a centered circle with an area equal to the total area. The radius of the zeroth harmonic is set equal to unity in this analysis in order to allow shape comparisons independent of size. The first harmonic is an off-set circle, second a 15 16 figure eight, third 3 three-leaf clover, and fourth a four- leaf clover. Qualitatively Speaking, each harmonic represents a figure with "n bumps" where "n" is the number of the har- monic order. The sum of an adequate number of harmonics will completely reproduce the given shape. The mathematical expression obtained gives the ampli- tude of the contribution from each harmonic order and a phase angle orienting the figure in relation to the coordinate system. It is expressed as follows: R(-9-) = R0 +m3‘Rn(cos n9- - Nu) where'RCO) is the radius as a function of polar angle, R0 the average radius, n the harmonic order, Rn the harmonic amplitude, and In the phase angle. The harmonic amplitudes (Rn) for each harmonic order are the important variables. Shape changes are reflected by changes in the relative amplitude contribution of the harmonics. Each shape can be characterized by a unique harmonic spectrum. Comparison with spectra of other shapes may allow location of specific harmonics reSponsible for major simi- larities and differences. 0f taxonomic interest, the shape most representative of a given taxon can be selected by locat- ing the specimen with harmonic amplitude values nearest the median values for the group. Shapes yielding extreme values can also be observed, potentially of value in identification of transitional Specimens. 17 As an example of the application of individual har- monic spectra, Figure 1 shows a specimen of E, mirabilis and its characteristic amplitude spectrum. For comparison, a spe- cimen of E, septentrionalis is also shown. In dissimilar shapes, variation in amplitude spectra corresponds to large scale variations in two-dimensional shape. Amplitude dif- ferences in similar shapes report only minor variations, as demonstrated by the higher second harmonic for the specimen of E, septentrionalis, indicating it is slightly more elon- gate than the specimen of E, mirabilis. A general decrease in amplitude values over the entire spectrum describes a change toward a more circular shape. E, mirabilis 18 E, septentrlgnalis Harmonic Amplitude 1.0 .5 -II- ? E, mirabilis -——-- I\‘ E, septentrionalis -------- " \ I \ ’ \ I l \ 01 ‘h I, I \ I, \ \ I \ .05 -- I k I \ \3-—-- i P. \ \ \ \ ’ \ \ v \ ‘ " \ \ 001 .1. \E L l L l l 1 I r j r I f l 2 3 4 5 6 Harmonic Number Figure 1. Comparison of amplitude spectra of Rabilimis mirabilis and Rabilimis septentrionalis CONSTRUCTION AND USE OF CHI-SQUARE CONTINGENCY TABLES Although the entire harmonic spectrum is required to completely describe a given shape, it is possible the shape information is not uniformly distributed over the harmonic orders. Certain harmonics may carry identical information thus creating redundancy in information content, whereas other harmonics may reflect unique shape character- istics. As a preliminary analysis, harmonic amplitude values, one harmonic at a time were tested for information content by use of a chi-square contingency table. This test is designed to reveal a degree of asso- ciation statistically greater than is likely to occur by chance. Data is arranged in rows and columns, in this case taxonomic categories are placed in row positions and four amplitude intervals make up the column divisions. Contingency tables were set up for both specific and generic taxa. The null hypothesis is that no association exists between the harmonic amplitude values and the taxonomic categories. Expected values baseH on marginal totals are cal- culated and deviations of observed values from expected values are used to calculate a chi-square value for the contingency table. The calculated value is compared with a theoretical 19 20 value which must be exceeded in order for the association between harmonic amplitude intervals and taxa to be declared significant. A level of significance ehosen prior to setting up the table indicates the statistical reliability of the results. If the selected level is .05, a significant chi- square result indicates the interaction observed between harmonic amplitude intervals and taxonomic classes will occur only once in twenty times if they are not in some way asso- ciated. DISCRIMINANT ANALYSIS TECHNIQUES Multivariate discriminant analysis was used to test the potential usefulness of the shape description in taxonomic studies. This is a statistical method for assigning unknown samples to previously defined groups on the basis of a number of variables considered simultaneously. The discriminant analysis used is a maximum likelihood classification which assumes normal distributions for the special case of equal training class covariance matrices. This condition is ob- tained by pooling the individual matrices (Nilsson, 1965). Discriminant analysis can also be used to test the internal consistency of the original categories, thus offering an external means for reinforcing or discrediting a classifi- cation scheme. Amplitude values for the first six harmonic orders were used as the independent variables in this study. A pOpulation defined by the variables may be pictured as a cluster of sample points in six-dimensional space. Dimensions of the cluster are defined by the amount of variance in each parameter. Representative samples from different p0pulations should occupy distinct regions in Six-dimensional space. Some variables may overlap causing the clusters to merge in certain 21 22 directions but in other directions the clusters should be distinct if they actually represent different populations. The "location" of a cluster is described by the six- dimensional coordinates of its multivariate mean. Discrimi- nant analysis is based on computation of the six-dimensional surfaces most efficient in separating the clusters. It operates in such a way as to minimize the distance between the multivariate mean of an unknown and the multivariate mean of the nearest cluster. In this study, specimens representing previously defined taxonomic categories are used to establish clusters, each with a distinct multivariate mean. Once the clusters have been established, the original specimens are then clas- sified by the discriminant analysis. It distributes these specimens throughout the established clusters on the basis of the six shape variables. If the multivariate mean of a specimen belonging to a given taxon falls closer to the multivariate mean of a different taxon, the discriminant analysis assigns this individual to the second group. Correct classification of individuals into the original classes is expected only if the information carried by shape duplicates the information carried by the taxonomic 23 characteristics used to define the categories. If the dis- criminant analysis distributes the specimens randomly through the classes, several interpretations are possible. Two- dimensional shape may not carry the same information as those features used in taxonomic discrimination, or shape alone may not be an important information carrier. On the other hand, if shape information was used in defining the original cate- gories, any specimen reclassified by the discriminant analysis must be carefully examined. USE OF SHAPE CHARACTERISTICS FOR SEX DISCRIMINATION IN FOSSIL OSTRACODES Most species of Ostracoda are sexually dimorphic with respect to shell characteristics. In the major Mesozoic and Cenozoic group, the Podoc0pida, this is usually manifested in relative differences in the elongation of the carapace, with males tending to be more elongate than females. Consequently, sexual dimorphism can be expected to contribute to the range of shape variation within a Species. The range of amplitude values for the second harmonic and correSponding sex of individual specimens of the three species of Rabilimis are shown in Figure 2. Males are char- acterized by higher relative amplitude values within each species. Because of the nature of the second harmonic, a figure eight, higher relative amplitude values represent more elongate shapes. Specimens whose sex was questionable (Hazel, personal communication) are noted by question marks in Figure 2. The intermediate nature of the amplitude values for these indi- viduals verifies their shapes are less elongate than typical male members of the species, yet more elongate than the typical female shape. E, paramirabilis diSplays the most distinct segregation of males and females, with the second harmonic values clearly separated into two groups. E, mirabilis shows 24 Relative Amplitude - 23E Harmonic .23 .22 .21 .20 .19 25 0c? - of 04f 06‘ ‘ ch ’ 0d” of o.r o? O? o? 89 " oJ‘ 9 03 P ()9 of (>3 _ oz _ 09 ()9 (>9 3—. 3.0 LR_. mirabilis paramirabilis septentrionalis Figure 2. Relationship between sex and amplitude of second harmonic 26 a less definite separation, with two females exhibiting harmonic amplitude values lying closer to the male end of the Spectrum. E, septentrionalis exhibits a continuum of shape variation with one extreme representing "maleness" and the other extreme representing "femaleness". No other har- monics were found to carry information useful in sexual discrimination. The continuum observed in the Specimens representing E, septentrionalis suggests that lateral outline is not a good discriminator of sex in this Species. This suggests that other orientations such as dorsal or ventral views may carry information useful in sex discrimination in certain taxa. SHAPE ANALYSIS OF MOLE STAGES As was previously discussed, ostracodes offer an excellent potential for the study of shape alteration with growth. Observed shape changes are generally attributed to addition of appendages in early stages and attainment of sexual maturity in later stages. Figure 3 shows the change in contribution of each harmonic amplitude over the nine molt stages studied by Kesling (1951). Shapes used for the growth study are shown in Figure 4, and represent average outlines of over five- hundred specimens of Cyprid0psis vidua (0. F. Muller, 1785). Continuous increase in the second harmonic through stage eight verifies Kesling's observation that elongation is the major shape change during growth. Increase in the second harmonic relative to the other harmonic orders indicates an increasing "oblateness". The eighth molt reaches maximum elongation and the ninth molt is a slightly more rounded adult shape, shown by the decrease in amplitude of the second har- monic. The strong, monotonous increase in the second harmonic indicates the fundamental shape change is expressed through this harmonic. Other harmonics show less distinct trends, with the fifth harmonic describing a Shape component not 27 Relative Amplitude - 2nd Harmonic 28 Scale break .08 _... n = 3 .07 -_ I .06 —_ .05 ~- 004 Clb n = 4 .03 -r 002 p n = 5 .01 In = 6 -00 1 , I I. J. ‘ l 2 3 4 5 6 7 8 ‘7; Instars Figure 3. Shape change with growth of Cypridopsis vidua; Shapes from Kesling (1951) 29 Figure 4. Average outlines of molt stages of over 500 specimens of CypridOpsis vidua (after Kesling, 1951) 30 changing from instar to instar. Because outlines used in this study represent only a single Species, comparison with shape changes during growth in other closely related and non-related taxa should be made. The nature of variation in shape during growth may represent a unique taxonomic characteristic, or may prove to be similar in other ostracodes. INFROMATION CONTENT OF HARMONIC ORDERS SPECIFIC LEVEL CHI-SQUARE RABILIMIS Computation of chi-square contingency tables for the Species of Rabilimis indicated that only the second and third harmonics were making statistically Significant con- tributions to intra-specific variation. These results were obtained by means of a four by three contingency table with columns designated by amplitude intervals and individual Species occupying the row positions. Special chi-square tables (Craddock and Flood, 1970), designed for small con- tingency tables and small sample sizes were used. Confidence level of .05 was set prior to construction of the tables. For sample size twenty-three, the critical chi- square value to be exceeded is 12.15. Results of the chi- square calculations are given in Table 4, page 33. Chi- square values for harmonic orders two and three are both large enough to be statistically significant at the .05 level. AS shown in the table, no other harmonics yielded significant chi-square values. 31 32 GENERIC LEVEL CHI-SQUARE BAIRDIIDAE A contingency table was prepared for the four genera of Bairdiidae. A four by four table with twenty- Seven Specimens was constructed. Results are shown in Table 5 on page 33. At the .05 confidence level, the criti- cal chi-square value is 16.92, and as indicated in the table, two harmonic orders gave significant results. The fifth harmonic is significant with a calculated chi-square value of 20.75 and the Sixth with chi-square of 21.08. HEMICYTHERIDAE The same procedure was carried out on twenty- Six Specimens from Hemicytheridae. Six generic categories were used and in this case, only the fifth harmonic yielded a significant chi-square result of 31.84, with the critical value for this table at 24.99. Results can be found in Table 6, page 33. 33 Table 4 Chi-Square results for the species of Rabiligig. Harmonic NumbegE. Calculated Chi-Square Results 2 20.15* 3 16.47* 4 7.46 5 4.36 6 9.92 * Values exceed critical chi-square value (.05) of 12.15. Table 5 Chi-Square results for the genera of Bairdiidae. Harmonic Number Calculated Chi-Square Results 2 16.45 3 14.29 4 13.09 5 20.75* 6 21.08* * Values exceed critical chi-square value (.05) of 16.92. Table 6 Chi-Square results for the genera of Hemicytheridae. Harmonic Numbe£__ Calculated Chi-Square Rgsults 2 21.55 3 18.86 4 12.95 5 31.84* 6 18.93 * Value exceeds critical chi-square value (.05) of 24.99. 34 INTERPRETATION OF CHI-SQUARE RESULTS Significant chi-square results indicate association between row categories (taxa) and columns (harmonic amplitude intervals) is greater than would be expected by chance. Because the significant chi-square values occur at different harmonic orders for the taxa considered, it appears shape information is not uniformly distributed over all the harmonics nor is it carried by the same harmonics at different taxonomic levels. Significant chi-square values for the fifth and sixth harmonic orders in Bairdiidae and the fifth in Hemicytheridae indicate that subtle shape differences are Significant in describing generic shape variation in these taxa. The intra- specific variation in lateral outline in Rabilimis is re- flected most strongly by the second and third harmonic orders, suggesting it is the relative "elongate-triangular" character of a specimen that allows Species discrimination within this genus. The nature of the characteristic shape of a taxon and the variation which this basic shape exhibits, determines which harmonic or harmonics contribute the most to taxonomic dis- crimination. Since shape variation is thought to be influenced by both genetic and environmental factors, the Specific harmonics responsible for reporting major Shape variations are probably unique to the given taxon. USE OF THE SHAPE VARIABLES IN TAXONOMY Results of chi-square contingency tables indicated variations in harmonic amplitude Spectra may be useful in the study of shape variation within and between taxonomic categories. To further investigate this possibility, harmonic amplitude values for the first Six harmonic orders were used as independent variables in multivariate discriminant analysis. SPECIFIC LEVEL - RABILIMIS The species of Rabilimis were used to assess the taxonomic significance of the Shape description at the intra- Specific level. Discriminant analysis results are given in Table 7. Table 7. - Discriminant function classification matrix for Species of Rabilimis Number of samples Original classes Number of samples assigned to each class 1 2 3 l. E, mirabilis 7 7 0 O 2. E, paramirabilis 7 0 5 2 3. E, septentrionalis 9 0 1 8 Using only the shape information carried by the first Six harmonic orders, the discriminant analysis assigned all specimens of E, mirabilis to the correct category. Seventy-one 35 36 percent of the specimens representing E, paramirabilis were correctly assigned to this species by discriminant analysis. Two Specimens of E, paramirabilis were classified as E, _g£f tentrionalis, indicating an overlap in the two species clusters and a Similarity in shapes. Specimens representing E, seEten- trionalis were correctly car's-ti; classified, except for one specimen. This individual was assigned to E, paramirabilis, again reflecting an overlap of E, septentrionalis and E, pgramirabilis. Figures 5a and 5b show the mean harmonic amplitude spectra for specimens of E, paramirabilis and E, septentrio- BELLE: In 5a, the two specimens of E, paramirabilis assigned to E; septentrionalis are shown by dashed lines. In 5b, the spectrum of the specimen of E, septentrionalis reassigned to E, paramirabilis is shown by a dashed line. Qualitative comparison of the actual outlines of the reassigned specimens with members of the original class and the new class illustrates the reasons for the discriminant analysis results. In Figure 6, each specimen is compared with a representative of both the original species and the Species to which the specimen was assigned. Hazel (personal com- munication, 1971) indicated the shapes of certain specimens may, in fact, be more Similar to one species whereas type of surface ornamentation and arrangement of pore canals indicate a closer affinity with members of another Species. Figure 5. a. Relationship between amplitude spectra of Specimens of E, paramirabilis assigned to E, septentrionalis and the mean amplitude spectra of the two species b. Relationship between amplitude Spectrum of Specimen of E, septentrionalis assigned to E, paramirabilis and the mean amplitude spectra of the two Species. 37 Figure 5a. "I" l I Mean Amplitude Spectra of: E. paramirabilis __ E, septentrionalis - ------ . Misclassified Specimens ....... I. I» -_ u-n— qL- Mean Amplitude Spectra of: E, paramirabilis —-———I E, septentrionalis -.m-". Misclassified specimen ..... 4“,! 38 Original Reassigned New Classification Specimens Classification . paramirabilis En paramirabilis septentrionalis E, paramirabilis Figure 6a. Original Reassigned New Classification Specimens Classification Ebtentrionalis <———- E, septentrionalis paramirabilis H Figure 6b. 'Figure 6. Comparison between mean shapes of E, paramirabilis and E, Egptentrionalis and the outlines of Specimens misclassified by discriminant analysis: a. Specimens of E, paramirabilis classified as E, septentrionalis b. Specimen or E, septentrionalig classified as E, paramirabilis 39 It was also suggested by Hazel that the extinct E, paramirabilis is the progenitor of both E, septentrionalis and E, mirabilis, and that E, septentrionalis has diverged less morphologically than has E, mirabilis. Discriminant results indicate several individuals of E, paramirabilis and E, septentrionalis may have two-dimensional outlines exhibiting transitional shape prOperties. Because all Speci- mens of E, mirabilis were correctly classified, less infor- mation about the nature of this Species was available. However, chi-square results indicate harmonics two and three are sig- nificant in species discrimination. By inspection of the amplitudes of harmonic two plotted Opposite harmonic three, more information on the character of the individual Species can be obtained. This graph, shown in Figure 7, confirms the overlap of E, paramirabilis and E, Septentrionalis, further supporting the inferred close relationship of the two species. Specimens of Echinocythereis, a genus thought to be a closely related ancestor of Rabilimis, were added as a fourth category for a second discriminant analysis. Table 8 shows the classification matrix obtained. .3... uhu... - - -.u:.< II“! I Ii II..- Relative Amplitude - 3rd Harmonic 4O E. paramirabilis, and E. sgptentrionalis O E. mirabilis O - E, paramirabilis )( E. septentriorLa1_._i_S_ 0 b IO _ C) IO r C) '* - c) $ '. IO x I. t $~‘+ I. OI - + x r I. ‘ o o r CI 1 1 1 1 1 1 1 L 1 Jr l 1 1 J. l 1 I r I I I I I I I I I I I T I j .19 .21 .23 .25 .27 .29 .31 .33 Relative Amplitude - 22E Harmonic Figure 7. Shape relationships between specimens of E,.mirabilis, 41 Table 8. - Discriminant analysis classification matrix for species of RabilimisI+ Echinocythereis Number of samples Original classes Number of samples assigned to each class l. E, mirabilis 2. E, paramirabilis 3. E, septentrionalis 4. Echinocythereis OOOVH oomoflw ONHOOJ bNHOb -I-\\O\l\l All specimens of Echinocythggeis were correctly classified indicating the genus represents a group distinct from Rabilimis. However, one specimen of E, paramirabilis which was previously assigned to E, septentrionaEEg and two specimens of E, septentrionalis were assigned to EgEigocythereis. Figure 8 compares the shapes of reclassified Specimens with representatives of both their original class and the class to which the specimen was assigned by the discriminant analysis. Results of the first discriminant analysis*which excluded Echinocythereis, indicate that E, mirabilis is the specific taxa representing the most distinct cluster on the basis of shape properties. Transfer of specimens between E, paramirabilis and E, septentrionalis suggests that portions of the clusters defining these taxa overlap, further indicating the taxa may be similar in two-dimensional Shapes. The second set of results ** which include Echino- * Complete discriminant analysis data is given in Appendix A. ** Complete discriminant analysis data is given in Appendix B. 42 Original Reassigned New Classification Specimens Classification E, paramirabilis 4%... ;, paramirabilis '. se tentrional S E, paramirabilig Echinocythereis E, septentrionalis ;, septentrgona us Figure 8. Comparison between mean shapes of E, paramirabilis, E, segtentrionalis, and Echinocythereis, and the out- lines of specimens misclassified by discriminant analysis. 43 cythereis provides additional information, indicating both E, paramirabilis and E, septentrionalis are more similar morphologically to the ancestral stock than is E, mirabilis, which remained a completely distinct category in both analyses. 44 GENERIC LEVEL - BAIRDIIDAE Discriminant analysis was carried out on right valves of Bairdia McCoy, 1844, Orthobairdia Sohn, 1960, Baigdiacypris Bradfield, 1935, and Cryptobairdia Sohn, 1960. The classification matrix obtained is given in Table 9. Table 9. - Discriminant analysis classification matrix for genera of Bairdiidae Number of samples Original Classes Number of samples assigned to each £1988 1 2 3 4 l. Bairdia 11 7 2 O 2 2. Orthobairdia 7 0 7 0 O 3. Bairdiacypris 5 O 0 4 1 4. Cryptobairdia 4 l 1 0 2 Seven out of eleven Specimens of the genus Bairdia were correctly classified. Two of the specimens were assigned to Orthobairdia. These were a cotype and a holotype of E, beedei Ulrich and Bassler, 1906, which apparently are more similar in shape to the specimens of Orthobairdia. The other two reassigned specimens were a paratype of E, hisEida (Harlton, 1928) and a specimen of E, whitesidei (Bradfield, 1935). These Specimens were assigned to Cryptobairdia. Specimens belonging to Orthobairdia were correctly classified in all cases. This included five specimens of '9. oklahomaensis, (Harlton, 1927) and two specimens of g, cestriensis (Ulrich, 1891). 45 Specimens of Egirdiacypris were correctly classified in four of five cases, with a paratype of E, transversus (Roth, 1929) reclassified as Cryptobairdia. E, bedfordensis Geis, 1923, E, curvis, 1941, E, deloi Bradfield, 1935, and a holotype of‘E. transversus Roth, 1929 were prOperly assigned to Bairdiacyprgg. The last category, Cryptobaigdia, was not as well defined by shape with only two out of four Specimens correctly classified. 9, forakerensis Kellet, 1934, and Q, 5gg£§_(Harlton, 1929) remained in the Cryptobairdia category, but 9, coryelli (Roth and Skinner, 1931), and g, hoffmanae (Kellet, 1943) were assigned to Orthobairdia and Bairdia respectively. 46 GENERIC LEVEL RESULTS - HEMICYTHERIDAE Genera of the family Hemicytheridae were also used to test the power of discrimination of the Shape information at the generic level. Four genera of subfamily Hemicytherinae, one genus of subfamily Coquimbinae, and a genus from subfamily Campylocytherinae were studied. Harmonic amplitude values were obtained for a total of twenty-six Specimens and twenty- five were correctly classified by discriminant analysis, again using only the shape information carried by the first six harmonic orders. Table 10 gives the classification matrix for this analysis. Table 10. - Discriminant analysis classification matrix for genera of Hemicytheridae Number of samples Original classes Number of samples assigned to each class 1 2 3 4 5 6 l. Hemicythere 4 4 0 0 0 O 0 2. Elofsonella 3 1 2 0 0 0 0 3. Baffincythere 3 0 0 3 0 O 0 4. Finmarchinella 4 0 0 0 4 0 O 5. Muellerina 8 0 0 0 0 8 O 6. Eensonocythere 4 0 O O 0 0 4 All Specimens of Hemicythere were correctly classified. These include E, villosa Sars, 1865, and three specimens of E, borealis Brady, 1868. Elofsonella was represented by three specimens of E, concinna Jones, 1857. Two were prOperly 47 assigned and the third was classified with Hemicythere. Baffincythere was represented by two Specimens of E, emarglpata Sars, 1865, and one of E, costata Brady, 1866. All three were correctly classified as Baffin- cythere by the discriminant analysis. Specimens defining the genus Finmarchinella were two specimens of E, finmar- gElgg (Sars, 1865), and two specimens of E, angulata Sars, 1865. Finmarchinella represented a distinct category with all specimens correctly classified by the discriminant analysis. Muellerina was represented by two specimens of E, abyssicola Sars, 1865, two specimens of E, lienen- klausi Ulrich and Bassler, 1904, and four specimens of E, canadensis Brady, 1870. All of these specimens were correctly classified as Muellerina on the basis of two- dimensional shape. The genus from the subfamily Campylocytherinae was Bensoncythere Hazel, 1967. This genus was represented by one specimen of E, americana Hazel, 1967, and two Specimens of E, whitei Swain, 1951. Again, 100% discrimination indicates the Specimens used to define Bensonocythere represent a dis- tinct shape class separate from the other five generic classes in this analysis. 48 Results of the generic level discriminant analyses* demonstrate that two-dimensional shape carries important taxonomic information at this level as well as the Specific level. If shape information alone allows accurate classi- fication, the original categories must have been established primarily on the basis of shape, or the taxonomic charac- teristics used to construct the original classification are expressed through the medium of two-dimensional shape varia- tion. * Complete discriminant analysis data for Bairdiidae is given in Appendix C. Complete discriminant analysis data for Hemicytheridae is given in Appendix D. 49 SUMMARY AND CONCLUSIONS This investigation indicates two-dimensional shape of the ostracode carapace contains information in addition to that obtained by simple shape ratios. In most cases, the mathematical shape description is Shown to be of equivalent taxonomic value to the qualitative features more commonly used in taxonomic discrimination. Since other taxonomic characteristics such as ornamentation, muscle scars, and hingement are considered to be important manifestations of genetic differences, the related Shape components must repre- sent similar responses. Demonstration of this clear-cut relationship pro- vides confidence that Shape information can be used in at least two additional ways. Because shape variation measured in the manner described is a continuous variable, it can be used as a direct measure of taxonomic similarity or distance. For the same reason, Shape variables can also be used in the study of inter-Specific and intra-specific variation resulting from sex differences, environmental effects, or phylogenetic distance. 50 In addition to the study of fossil taxa, the shape reSponse of recent taxa to environmental gradients can be monitored. Estuaries or bays offer environments where Specimens living under depth or salinity gradients are available. Using the Fourier shape technique, the pos- sibility of a shape continuum in these specimens can be studied, and if one is confirmed, the exact nature of the shape change can be diagnosed. The shape variables produced by this analysis also provide an external means for testing the reasonableness of certain evolutionary models. Many phylogenetic sequences involve morphologic changes which should be reflected through two-dimensional shape. Verification of the existence of a shape continuum over a sequence of morphologic forms repre- senting a prOposed phylogenetic series would corroborate the accuracy of the predicted pattern of evolution. The results discussed above suggest a number of possibilities for further use of this technique. Although the general utility of this method of shape analysis has not been verified, an inspection of ostracode literature indicates that taxonomic descriptions place great emphasis on the shape of lateral and dorsal outlines. This, along with the results 51 already obtained, suggests the Fourier technique should be of general utility over a wide range of ostracode groups. Because similar criteria are used in other groups, notably pelecypods, brachiOpods, and echinoderms, this method may be of general importance in paleontological research. RE FERENCE S C ITED REFERENCE S C ITED Akatova, N. A., K faune Ostracoda Novosibirskogo melko- vod'ya (Ostracodes from the New Siberian Islands shelf waters), in Dreifuyushchei ekspeditsii glavsevmorputi na ledokol'nom parokode "G. Sedov," 1937-1940 3.3., Trudy: Vsesoyusnyy Arkticheskiy Nauchno-Issledovatel'skiy Inst. (Lenigrad), v. 3, p. 224-230, 1946 Brady, G. S., Ostracoda from the Arctic and Scandinavian Seas: Annals and Mag. Nat. History, 45E.ser., v. 2, p. 30-35, 1868 Brady, G. S., Crosskey, H. W., and Robertson, D., A mono- graph of the post-Tertiary Entomostraca of Scotland in- cluding Species from England and Ireland: London, Palaeon- tographical Soc., 274 p., 1874 Craddock, J.M., and C. R. Flood, "The Distribution of the Chi-Square Statistic in Small Contingency Tables", Applied Statistics, Vol. 19, # 2, pgs. 173-181, 1970 Ehrlich, Robert, and B. Weinberg, "An Exact Method for Characterization of Grain Shape", Journal of Sedimentary Petrology, Vol. 40, # 1, pgs. 205-212, March, 1970 Elofson, Olof, Neuere Beobachtungen uber die Verbreitung der Ostracoden an den Skandinavischen Kusten: Arkiv Zoologi, v. 35A, no. 2, p. 1-26, 1943 Hazel, J. E., "Classification and Distribution of the Recent Hemicytheridae and Trachyleberididae (Ostracoda) off North- eastern North America." Geological Survey Professional Paper 564, U. S. Government Printing Office, Washington: 1967 Kesling, R. V., "Morphology of Ostracod Molt Stages", Illinois Biological Monographs, # 21, 1-3, 1951 52 53 Nilsson, Nils J., Learning Machines, McGraw Hill Book Co., U. 8., pg. 56, 1965 Shaver, Robert H., "The Pennsylvanian Ostracode Bairdia Oklahomaensis in Indiana", J. Paleontology, Vol. 34, # 4, July, 1960 Sohn, I. G., "Revision of Some Paleozoic Ostracode Genera", Geological Survey Professional Paper 330, U. S. Government Printing Office, Washington: 1960 APPENDICES APPENDIX A DISCRIMINANT ANALYSIS RESULTS FOR SPECIES OF RABILIMIS DISCRIMINANT ANALYSIS RESULTS FOR SPECIES OF RABILIMIS __ .-..~.——---.l___--... b-— —. ..._—.- v 7 S: S? $3 54 55 so 3? $3 $9 510 c s 3 7 ‘7 9 -o -o -o -n -n -o -o o (sr7.5/) ”‘”TWKTWTNE'CLASS 1 .n1axo .22169 .oe1e1 .ozaos .031os .51966 'Trainins C1888 1: (E, mirabilis "*ifl167D".22‘31"{UTU19 703250 .0336? .01123 “‘JQ3394 .2079n .0662} .02936 .03338 .n1412 :RQEE BI: indixidnal .n15n2 .24775 .0592: .03275 .04152 .n1476 3P°°1m°“3- 00mm“ are the relative har- monic amplitude values .nizpa .2347. .og,6g .0288? .0?599 .0175, for harmonic orders 1-6. .. u—-c— ' 7.01036"319179”}0536F'.03‘37’TD129amTFffi?8 f .n1554 .1980? .0780} .03424 .01794 .n17o7 “muvms CLASS""2'_—’“"”"’_‘" """"'va$ .0085” .25316 00.1531 .03054 003080 .h1713 R. Earamirabilis "”301453“T32753 .UEPHE .03351 .03133 .02310 ,q1gaz_,gfné5 .04335 .03702 .0?905 .h211n .01118 .23997 .05376 .03024 .01806 .02579 “-Tfi-Iman" .UWU7U .UZQU? 902267 0111987 .q1sagmig2717 .osu1o .03977 .0?416 .hoa1a ___-- .——.-._ .n161? .3092? .05951 .0347? .07315 .01791 '“TDIYVING CrAss ‘3 Training CIass’3: ,nn4a1 .24330 .02452 .02559 .02191 .n1611 R. sgptentrionalis '.nnéda‘:31Hav“707167—1U135T”7019fi5 .nznaa .qufip‘.29asn .01695 .0299: .01738 .0295" .n0698..29686 .02293 .03590 .01434 .01847 ‘ -w- -1fi1§1‘ .3052; .can17‘i04033‘7D?6h%“Tfii203 .o14«4..29327 .0917: .03407 .031?8 .60951 .oGBVSf.2632R .0734: .03389 .01361 .n1445 _,91275 .272}: .05367 .03006 .014ns .hoa39 .n1541 .3442! .0403; .05542 .03699 .fl1636 54 55 m7“ _____*, ..——— ——---— __ _ *‘— ~- MEANS '09 EACH IRAIAING CLASS Columns are the three species .n1416 .01790 .00918 '- " 291760“"‘.29190_729179‘°£%m0 no312 .04cxe .03469 ,n3152 .03371 .0365? de "—"——7w2a . ,. . . ”(1399 .019?0 .01573 for harmonic orders 1'6. “‘“COVAPIINCE PATFIX .onno1 .nouos :onoo: .oono? .00001 -.ooOo1 .00006 .0010? 200004 .00014 .00014 -.00000 ”-—*Tnnnn1'——7Wfiv6a .fififiis .UUDUD .Ununz -:UUUHZ .on0n1 .00014 200002 .00004 .ononz -,ooono ,onon1 .0001. :00002 .nOnoa .ononb -.00000 ";;unUnI“‘?TWUFUH -.UUUUZ -.Uunofi -.ononu .0000? INVERSE 0F COVARIANCE -—3°4572“———35a75a7——v7S03IT‘—737IFIT—:TV5UT7T__3P]337—__——_————__—__—_— -1597ss a9?2. 38486. ~6245. 4539. -15447. -925051. 3H486. 227043. 2004. 43144. -33387. “".37fUTT“"652n5. ZUVI. 552S7T——_-K“BST__—:5673. .199n47. 4539. 45144. 6086. 33?44. -91311. 390564. 015447. -FSPB2. '5675. °?1311. 84875. INVEDSE cnccx nETEhrINATE .571695-27 ToTAL sArPLEs 23 1.0nonn .nocor vinoono -.nOnnn -.onono .flOOnO ' ”,0n0nn”“3.00"0U‘ .UUUUU‘"—ZUDNOW‘"’TO"UHD -760000 ' ,onnnn ..nOHOD 1200000 .nonnn .0"000 - .flUOnO ,onnoo ,00000 :00000 1.nonnn noono - 00000 —«—:unnnU—-7fionor-rrtovvuv-—-Tnvvni-—rTUvUvU"7TWUUUU—““‘—“‘_‘—‘—" ,onuno .,00000 .200000 -.0000n - .00000 1 .00000 r|ASS “ACTUIF“”“FYPTFFFT“FUR‘TRIININH“FETTF7‘fiTbeIIIE: 1 1 .ozseono .335E+02 -.?07E¢0? 1 1 .u9se.oo .17SE+02 .?53660? 1 i ,afiafi.tr .191E+G? .VaKEsUV 1 1 .355Eo01 .14BE+02 .7155009 1 1 .724Ee01 .2335o0? .206E907 "‘"_"l-‘_-T_‘TF?UETWT“TT5‘E?D2 .76$F+D7 1 1 ,SBUEoUI ,2335.02 .327Fa07 ”CHSWTUIL—‘TYP'U'FK‘T—W'Ifim. paramffabi I18 2 3 0296.01 7446+01 .611E‘01 2 2 .?57Eo02 .403£+01 .159Fo0? 2“—'"2'*“125F507“"a96ETUI“"759E331 2 2,1035.0? .¢35F+01 .160E¢0? 2 , 2 2 2 .7335.01 .192Eo01 .a67F.01 fl”“376663U7 .Y75FIUI 27inF}UI 2 .906EIH2 .272F901 .121F007 m ,5: — 5cm“ "ryrrM-‘H 'rrn mem 1th” --- 3 a .1946602“"TTDHE709 .471E¢01 'I 3 3 ,4255.u2 .1836‘02 .67IEo01 3 3 .369E.02 .113E*02 .491Fo01 3 3 ,fififiesa? .‘23E+Gr .?03F+UJ 3 3 .1695.02 :955‘01 .16nEo01 -3 2 .129E¢02 ..483E+01 o53§E¢01 - 3>~n-3--.7245.02"-71225.02 .431F301 3 3 .2515.02 .102E¢02 .804E¢01 3 3 .2455.02 .1745s02 .1?7EoO? CIASSIFICATIOK MATRIX CLASS SAMPLFS 1 2 3 . 7 ,. _--_._7--_--.- U' “10“---- __ ___ _mninfld in ‘23:. pg. . 35. . 1 ...... ? 7 ' 0 5 2 3 9 0 1 8 Mean EQUALITY DFGS rr FREE. 1? rs: souane 98. ......._r __- _..._._ APPENDIX B DISCRIMINANT ANALYSIS RESULTS FOR SPECIES OF RABILIMIS AND ECH INOCYTHERE IS DISCRIMINANT ANALYSIS RESULTS FOR SPECIES OF RABILIMIS AND ECHINOCYTHEREIS ”“”V-‘“T”“5I’“S2“_33_—S?”‘$5_—33-—37‘—SU‘—SV‘3IU c 6 4 7 7 9 4 no .0 no .u .0 TRAINING CLASS 1 _.701K35*T?2169 .06151 .02363 003195 00I265 “,91970 .01594 .22431 .07019 .05250 .03362 .0112: .20790 ,06625 ,U2936 .oassa .01412 fawn—.2077; , . , . , _9039?§. .01286 w}0156'4 .112129 .05468 .03437 .01298 .01020 .23470 ,05169 .02802 .02559 .0170: -o 0 (007.51; Training Class 1: R. mirabilis pg. 52, for explanation of output. €TV507 .U7BUST.03020 .01790 .01707 TRAINING CLASS 2 ' ,00860”:25316‘70368I‘703051"703000 .01710 .0I10i_3§gg23 005006 .0346‘ .03145 002640 .0128? .01118' Training Class 2: 03‘035 .04335 .03702 002995 0“211O _R.. aramirabilis 723997"}D‘376 .0302! .UIBVO .0257? .25090 .05070 .02902 .0226! .01997 ,32717 .05010 .0597? .02410 .00618 ’.01632“T3U922"705951"I03172'.023I5 ("1701* TRAINING CLASS 3 ”}00461 -°°9€9 .00356 ":23338 .0205: .0255? .UZIVI (01511 _1SEQQQW102462 .04551 .01905 .02uae Training Class 3: E, septentrionalis ,28859 .01688 .02991 .01/50 .02250 '.onoéa“?29686‘:0§293 .03500 .o1xsw-701301 .01511 001464 --099§Z5 v30526 o°431%-;9€93609025‘°-2941?§_ ,29327';os173”;03407 .03120“}00051* ,26328 .02342 193389 .01361 .01445 001275 001541 .27236 .05367 .03006 .01466 ,00839 ,34423 .04632“:0554?”.03599'{01636 TRAINING CLASS 4 .01121 .01092 001201 001394 Training Class 4: '327942 .04710’;03612 .62774 ,01210 .26851 0043264291925u902151.191943 Echinoczthereis .29340 .04550 .02943 .0207; ,01626 '328796“705281 .03610 .03512 {01009 56 57 MEANS FOR EACH"TRAININU‘CCISSW”"”-' 001416 01790 00918 01202 .21760 :29190 :29179 :25232 for explanation Of output. "W 7.05312'-_M;U4018_Tm9 'UWZU .0615? .03371 .03652 .06022 .02015 .03500 .02185 .02052 .01399“' j03920‘——_:01573.."_701397 COVAPIANCE MATRIX ‘-'3 5“”?8‘E102__—T10E-02_——VI7E+flI—_TEISETDI 3 3 260E602 .13chtnz ”.00001““’{00005‘ .00003' .00001 .00001 ~000001 000005 .00091 .0000} .00013 .00012 ~000000 .00003 ,00003 ,00011 .0000? .00002 -.00002 .00001‘“ .00013 “.00002 ”‘.00004‘“”.00002"3I00000 .00001 -.00012 .00002 .00002 .00006 -.ocuoo 0.00001 -.00000 -.00002 -.00000 c.00000 ,00002 INVtRse 0F COVARIANCL 3658453, .162642, .070750, 71211, .170955, 355595, -162642, “ 9681, 39616;'“"89573." F4000.""315283. ~870750. 39610. 218710. -22562. 36905. -71506. 71911. o9b73 -22562, 46860. ~1469, 9227. .170965 ~*"*4000;"”-369037*—*;1409;*-*33vr0—-:15¢93, 335595 .15283 .71586 9227 .16490, 02120, INVEESE'CHECK'“"DETEFHIN£TE'”{46039E327“”TUTKE'SAHPLES 27’ 1.00000 .00000 -.00000 c.00000 ..00000 .00000 .00000 1.00000 .00000 .00000 .00000 .,03000 v.00000“.500000“”1;00000““’;0c000’ ..00000“”;00000 c.00000- ,00000 .00000 1.00000 .00000 .03000 .00000 ,00000 .00000 .0r000 1.00000 0.00000 ‘ '900000"'d.00000”"3.00000"’3.ULOOEW‘3360000"”1300000 cLAss ACTUAL expowewr FOR TRAINING 5:: ‘R, mirabilis 1'“'”I‘"I955E+00””7150E£02"’:22VETNZ“'TI59ETUZ" 1 1 99OE+00 .190E¢02 .203t+02 .2106002 1 :3565601 ,195E602 ,276t0n2 .1/96602 1 ' 1 ”.4056+01‘”}1625~02 '.24ze+02 ”.1695002"”' 1 1 7452.01 2405.02 .2606602 .2/1to02 1 1 7102.01 ,1625.02 257:002 ,2D9h002 1 "'I"' :757E601‘”'253E.0?" 3736502“ :3332502 cuss ac. UAL r-XPUNENT MR 10.010100 St' 5‘ aramirabili_ _2 4 .105&«oz_ _47815001 _1692t+01 m 597Eoori"“”“""'""‘ “' ' 2 2 2725.02 4166.01 .146E.nz .120Eo02 ' 2 2 3645.02 .5525601 .501Eon1 .7055601 _ _2 P 107E¢02 #1722E001 .1laeonZ .1726002 “"2 '"' .7625301 .2175601 .37IE+01 .633E901 2 3 .3045602 .1095.02 L+n1 .1045002 _‘2 ‘.Wg.n,217e.02 7776.01 .115:+nz .9575601 CLASS ACTUAL EXPONENT FOR TRAINING ShI R. seetentrional J 3 219E602 .1105.02 .060ton1 .6485601 "' ""3“””3""‘457E+02 .1635202‘”T$3?E301 .1632+02 3 3 .41?h¢02 .1255002 .923t601 .1116002 3 3 :347e.02 .1036102 7726601 .LJDEOOZ 3 ““'3“‘;188E602'“;290E601",tvfit+31 ‘;363t601“ 3 4 .147t‘02 .5155*01 .377t901 02‘IE‘U1 3 3 .2bse.02 .1365.02 .478t+01 .154eo02 02385.02 Refer to Appendix A, pg. 53’ is APPENDIX C DISCRIMINANT ANALYSIS RESULTS FOR GENERA OF BAIRDIIDAE DISCRIMINANT ANALYSIS RESULTS FOR GENERA OF BAIRDIIDAE '.02694. ‘004075 N ‘02281.11_-m '.05490 ” v T 51 52 s: 54 55 $60 s7 50 6 4 11 7 5 4 -0 -o -0 no TRAINING c1155 . 1V .V.‘ , 1- .1 i - .0121: .19471 .05150 .0400: .01377 .01072 .23956".03070 V.05500 7.02140' .00022 '.20631 '.0:009”q.06825m”.01319" .0210: ‘‘‘‘‘‘ .01179 .2507: ".00727".06224‘”.02151W .01544 .21934' .06000 .06504 .03057 .03006 .20209‘ .10148 0.00103".04310 .01190 ".2130: .05176_W.06097.‘.0318{- .01305 .10051 ”.06602' .0395: '.02527 .02992 .30504 .00036 .05431 .04164 .02625' .20914‘ .07444 '.o4460".03154" .03200’ .4253: .0501: .12647 .04377' TRAINING cLAss 2 t -' ‘ *- .01440 ..22223_1.05911...0570971.01150 .01469 .21201. .0502201.05699 ..01705 .01270 .22390, .04976_..05791 .0109? _.01451 .26733. .04900 1.06992 .01308 .01722 .23061 105977 .07279 4.01946 .0070? .25070 .02315 .03200 .0137: .01101 .2509: .0357: .06410 .01370 TRAINING CLASS 3 _. .01642 ,35442 .03434 .06555 .02999 .05104- ,04170 .07700 .09167' .06077 .03752 ,40006 .06573. .0020: .05109 .02256 .22034 .09635 .04094 .02907 TRAINING CLASS 4 - .01212 .2050: .0306: .05734 .01300 .00902 .23070 .04020 .04296 .01021 .00919 .14732 -.05302 .03726 .0074: .02000 .33616 .07257 .07070} .04130 00105 raw 5000 70110100 01055 .0103: .0133: .02060 .01500 ’25588 '2‘026 032197 1-025000 .0615: .04702 .06929 .05112 .06422 .05001 .0599: .05406 .02000 .015072' =.03774~4 .01012 .0:032 .02536 .01196 .01617 58 89 510 .10.”- '0 '0 0 "1670.373 '362k52"TfETfiIEE-CIEEEMIi."H‘ hq_"ngzzgigkn..__u_.. .01002 ‘.05523 mm.-.m1 .02751" .02751’-_ .0331: ‘ T‘..1.;1.;.”c‘i.‘.‘. ‘2' '02‘20‘“‘“”“0rthubatrdta“m“‘ 002027 _0. .1. .H.p-.hon-n.-_-_ 002320” ..u "-11-”-“.. 002025 a- i“ "11”-“.h1 003373 002140 002937 Training Class 3: Bairdiaczpria 001103 . .1 ' ' " .01424..--—--ML1_ ._;;_.1;2,__ .01545 001380 .00029 - 0 .0. .. . . . _ . "Trainin3101nsawé;1_uu_ Crzptobairdia .00650 .01007'" ‘T"’"~=- "'"' 001090 I - h" - "N ’ ~" .02034' Refef'fio pg. 53 for -