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<br />ative!y low within-group to total variance ratio <br />(i.e., the potentially more discriminating char- <br />acters). The variance-covariance matrix of stan- <br />dardized variables was analyzed using stepwise <br />multiple discriminant analysis (SMDA program <br />7~1 0'[ BMDP; Dixon, 1990), to identify those <br />morphological variables (if any) that best distin- <br />guished the two sexes. Males and females were <br />allocated to group using a jackknifed classifi- <br />cation function (where individuals were succes- <br />sively left out of the derivation of the classifi- <br />cation function and used later to test the <br />function's efficacy). Jackknifed group totals by <br />sex were then tested against chance-corrected <br />expectations using Cohen's kappa statistics (Ti- <br />tus et aI., 1984). A histogram of discriminant <br />scores was plotted to illustrate the magnitude <br />of sexual dimorphism (if any). <br />Standardized data were evaluated using mul- <br />tiple group principal component analysis <br />(MGPCA) of NT-SYS. Scores for individuals <br />were plotted in component space to illustrate <br />this multivariate summary, and loadings for each <br />character (i.e., correlations of characters with <br />components) were tabulated. <br />PCA [as conceived by Hotelling (1933)] is a <br />computational methodology applicable to but a <br />single group. If a given studv involves multiple <br />groups (as in Jolicoeur and Mosimann, 1960: <br />this, and probably most studies), the simplest <br />approach is to generate individual PCAs for each <br />group (as in Rising and Somers, 1989). How- <br />ever, in most circumstances, researchers tend <br />to ignore group structure and instead apply or- <br />dinary peA to data pooled over all groups (re- <br />viewed by Airoldi and Flury, 1988). But pooling <br />of data in this manner is dearly inappropriate. <br />Directionality of a given component (i.e., its ei- <br />genvector) is determined by both between- and <br />within-group variability, and pooling over all <br />groups inextricably mingles both, thus con- <br />founding the component. Thorpe (l 983) in- <br />stead advocated derivation of principal com- <br />ponents after pooling the variance-covariance <br />matrices of all groups (a procedure called mul- <br />tiple group PC A; MGPCA). This approach has <br />its advantages, in that variation between groups <br />is not confounded by that found within groups. <br />However, before variance-covariance matrices <br />can be pooled, eigenvectors must be identical <br />within groups. If this is not so, the group with <br />the greatest variability will often determine the <br />directionality of extracted components (i.e., <br />dominate the eigenvectors). Thus, when using <br />MGPCA, it becomes imperative to test equality <br />of within-group covariance matrices (i.e., their <br />eigenvalues and eigenvectors), as was done in <br />this study. This has led some researchers (i.e., <br /> <br />~ <br /> <br />DOUGLAS-SEXUAL DIMORPHISM IN GILA CYPf[A <br /> <br />339 <br /> <br />Airoldi and Flurry, 1988) to argue that :YIGPCA <br />is less applicable in most circumstances, because <br />of its more stringent requirements (but see <br />Thorpe, 1988). <br />Sheared PCA (Bookstein et aI., 1985; with <br />corrections outlined in Rohlf and Bookstein, <br />1987) was also employed to evaluate size-free <br />body-shape relationships amongst individuals <br />I"rom each sex. The sheared PCA \vas calculated <br />from standardized data using PROC :VIA TRIX <br />(SAS, 1985: modified from an algorithm writ- <br />ten by L. Marcus). <br /> <br />RESULTS <br /> <br />UIl il 'a ria II' alla{ysl's.-All transformed variables <br />were normally distributed, and the variance-co- <br />variance matrices for both sexes did not differ <br />significantly (P > 0.10: Chi-square test of ho- <br />mogeneity of within-group covariance matrices, <br />Proc Discrim; SAS, 1985). Correlations among <br />the 53 pooled characters were positive, ranging <br />from 0.031-0.995. No statistical differences <br />were found between the slopes of the regres- <br />sions for each sex (ANCOV A of Proc GLM: <br />SAS, 1985). Only two characters [(48): tip of <br />snout to descent of pupil, and (53): peduncle <br />length] exhibited significantly different adjust- <br />ed means (or intercepts) between males and fe- <br />males [(48): F = 4.40, P < 0.04; (53): F = 4.40, <br />P < 0.04]. Table I lists all 53 variables, with <br />their untransformed means, standard devia- <br />tions, and F-values for the ANCOV A. <br /> <br />Discriminant allalysis.- The discriminant func- <br />tion separating males and females incorporated <br />three characters [i.e., (22): distance from ver- <br />tical pupil to midpoint between vertical pupil <br />and nape; (33): distance from anterior insertion <br />of dorsal to anterior insertion of anal; and (48): <br />distance from snout to vertical descent from <br />pupil]. This function (not presented) was based <br />on equal probability of a particular specimen <br />being male or female. A jackknifed classifica- <br />tion procedure demonstrated that only 58.6% <br />of the males (17/29) and 61.8% of the females <br />(21/33) could be correctly classified to sex. <br />These group totals are not significantly differ- <br />ent from chance alone (Kappa = 0.203, Z = <br />1.61, P > 0.11). The average number of correct <br />classifications for both sexes was only 60.3%, <br />slightly greater than half. A histogram of ca- <br />nonical scores for individuals of both sexes is <br />provided in Figure. 4. <br /> <br />PrinciPal component analysis.- The PCA re- <br />vealed general trends in morphological varia- <br />tion over the 53 characters. The first four com- <br />