<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 />
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