BRIEF COMMUNICATION Associations among Cranial Traits ROBERT A. BENFER University of Missouri, Columbia, Missouri ABSTRACT Multivariate analysis of associations among seven cranial characteristics originally presented by Hertzog ('68) suggests that the appearance of the circumparietal accessory ossicles, parietal foramina, and form of the fronto-temporal suture are largely independent of each other both within and across geographical races. Hertzog ('68) has presented associations of seven cranial characteristics judged by presence or absence. Five sites where accessory ossicles occur, the presence of parietal foramina, and the form of the fronto-temporal suture were recorded for 366 crania and presented for the total sample as well as by six geographic racial subsamples. It was suggested that the number of significant associations varies racially. Hertzog also observed adjacent sites to be more highly associated than those further apart. In order to examine possible racial differences in the pattern of appearance of these seven characteristics, I converted the contingency coefficients (presented in table 2, Hertzog, '68) to standard Pearson product moment correlation coefficients, which in the case of two by two contingency tables are often called phi coefficients. I intended to use image analysis, a variation of principle components analysis, to compare nonredundant dimensions of variation among the subsamples. Image analysis proceeds from a matrix of correlation coefficients to an image-covariance matrix in which the squared multiple correlation coefficients of each variable with all the others are placed in the principle diagonal and the off-diagonal entries are adjusted to reflect only shared variation (Guttman, '53). Principle axes analysis, to compare nonredundant dimensions result in components reflecting no unique variance. The program used was Factor, a listing and writeup of which may be found in Veldman ('67). Output includes the amount of common, or shared, variance in the original correlation matrix. Table 1 presents the percent common variance and the trace of the imagecovariance matrix, which is the sum of AM. J. PHYS. ANTHROP.,32: 463464. the squared multiple correlation coefficients for each variable (the Polynesian sample was not analyzed since lambda ossicles in these crania are totally lacking). As can be seen from the table, the percent common variation ranges from 4.74 to 11.29. The sums of the squared multiple correlation coefficients are uniformly very low. Apparently, while individual assessments of probability of the contingency coefficients do reach traditional levels of sigmficance (although the comparisons are not independent), the actual amount of shared variation is very small; too small, in fact, for meaningful comparisons of differences across races. Therefore I do not present the results of the principle axes analyses as they are based on too little common variation to be meaningful. Slight differences in pattern which may exist within the 5 to 12% shared variation could be explained by sexual dimorphism or age changes related to the traits of interest and differing in the subsamples. Results of this analysis suggest that the seven discontinuous cranial traits analyzed by Hertzog are almost completely independent of each other, and racial variation TABLE 1 Interrelationships among seven cranial characteristics, data from Hertzog ('68) Per cent shared vanance Total sample Asian subsample European subsample Amerindian subsample African subsample Indian subsample 6.14 4.74 6.90 11.29 9.30 6.63 ~~~~~f c o ~ a ~ n ~ e 0.430 0.332 0.483 0.790 0.651 0.464 463 464 ROBERT A. BENFER in their pattern of Occumence has not been demonstrated. LITERATURE CITED Guttman, L. 1953 Image theory for the stmcture of quantitative variates. Psychometrika, 18: 277-296. Hertzog, K. P. 1968 Associations between discontinuous cranial traits. Am. J. Phys. Anthrop., 29: 3 9 7 4 0 3 . Veldman, D. J. 1967 Fortran Programming for the Behavorial Sciences. Holt, Rinehart and Winston, New York.