Limits of among-group variation in each measurement

    The minimum and maximum values in the among-group variation of each measurement are shown in Table 2.  It was found here that the among-group distributions of craniofacial variables are unimodal when the number of samples was 350 or more (in the case of “Samples with the size of 20 or more” in Table 2), even if the skewness and kurtosis of sample means across various Homo sapienssapiens populations were significantly different from those of normal distributions (Table 2; Figs. 1 to 5). 

    The ultramodern skull of Iyeyoshi Tokugawa (Suzuki, 1967, 1981), one of so-called aristocrats, is placed almost within the range between the minimum and maximum values of Homo s. s. sample means, and definitely within the range between the minimum minus 2SD and the maximum plus 2SD in main craniofacial measurements (Table 3 and Fig. 6).  But the skull of Herto [Homo s. idaltu] (White et al., 2003) has a very large maximum cranial length (larger than the maximum of Homo s. s. plus 4SD), though the other main craniofacial measurements are almost the same as or smaller than the maximum of Homo s. s. (Table 3 and Fig. 6).

Positions of sample means in within-group multivariate distributions

    Two PCAs based on within-group correlations were first carried out as a base for confirming the positions of the vectors of sample means in the within-group multivariate distributions.  The correlation matrices used are of a Japanese and an Australian Aboriginal male sample.  The results of the PCAs are shown in Tables 4 and 5, respectively.  In these analyses, the fourth variable set of the skull was used so that the results from Homo sapienssapiens samples can be compared with the reported data of one of the two special male specimens, Herto [Homo s. idaltu] (White et al., 2003).  The PCs obtained from the Japanese sample (Table 4) may be interpreted as follows: PC 1 is a so-called general size factor; PC 2, a factor associated with cranial length and height and, simultaneously, inversely associated with upper facial and nasal heights; PC 3, associated with nasal breadth and inversely with cranial height; PC 4, associated with cranial breadth and inversely with nasal breadth; and PC 5, associated with cranial length and inversely with orbital height.  On the other hand, the PCs obtained from the Australian Aboriginal sample (Table 5) may be interpreted as follows: PC 1 is a general size factor; PC 2, a factor associated with nasal breadth and bizygomatic breadth and inversely with upper facial height; PC 3, associated with cranial height and length and inversely with cranial breadth and nasal height; PC 4, associated with cranial breadth and inversely with orbital height; and PC 5 is associated with orbital height.  The repeatability of these PCs was examined by comparing the variation patterns of factor loadings on the PCs from the two samples (Table 6).  By this comparison, it is found that PC 2 from the Japanese sample seems to correspond to PC 3 from the Australian Aboriginal sample, and vice versa, though the significance of the Spearman's rank correlation coefficients obtained is not so high.

    Using the coefficients of the simultaneous linear equations for prediction of PC scores obtained from the above two PCAs, PC score vectors for 283 sample mean vectors from various Homo s. s. male populations were estimated on the basis of the most reliable data (Class A in Table 1).  The PC scores are plotted in Figs. 7 and 8 in the form of radar chart.  It must be noted here that the axes of PC 2 and PC 3 are reversed in Fig. 8 so that the results on Japanese (Fig. 7) and Australian Aboriginals (Fig. 8) can easily be compared.  Both Figs. 7 and 8 show that all the samples are located between the maximum and minimum pseudo-individuals in the case of PC 1 (general size factor), but, in the other PCs, scattered not only within but also outside of the range.  It is also found that the PC scores for almost all of the 283 sample mean vectors are located within the ±2SD range of within-group PC scores (Figs. 7 and 8).

    Similarly, two special specimens, i.e., the 160,000-154,000 year-old skull from Herto (White et al., 2003) and the ultramodern skull of Iyeyoshi Tokugawa [1793-1853] (Suzuki, 1967, 1981) were compared with the minimum and maximum pseudo-individuals (Tables 7 and 8, and Figs. 9 and 10).  Comparing Fig. 9 with Fig. 7 (both are based on the Japanese variances/covariances), it is found that Herto is located clearly outside of the Homo s. s. range in PC 2 and PC 5.  In the comparison with the Australian Aboriginal variances/covariances (Figs. 8 and 10), however, Herto is located far outside of the Homo s. s. range only in PC 4.  On the other hand, Iyeyoshi Tokugawa is found to be very close to the lower limit of the Homo s. s. range in the PC 3 scores based on the Japanese variances/covariances (Figs. 7 and 9), while, in the PC scores based on Australian Aboriginal variances/covariances (Figs. 8 and 10), he is close to the lower limit of the Homo s. s. range in PC 2, which corresponds to Japanese PC 3, and, in PC 5, exceeds the upper limit of the Homo s. s. range.

Selection of samples for among-group multivariate analyses

    First of all, it was checked whether or not samples were practically usable in multivariate analyses on the basis of the quality of samples, i.e., Classes A, B, and C in Table 1.  As a result, in the combined set of the first variable sets for the skull and limb bones, the number of male samples available was only 5, 8, and 10 for Class A, B, and C, respectively.  In the case of female samples, this check was not made because the number of the female samples collected was too small, as shown in Table 2.

    In the combined set of the second variable sets for the skull and limb bones, the number of male samples available was zero, two, and three for Class A, B, and C, respectively (the check on female samples was not made for the same reason as the above).  In the combined set of the third variable sets for the skull and limb bones, the number of male samples available was 14, 22, and 27 for Class A, B, and C, respectively.  The number of female samples available was 11 and 20 for Class B and C, respectively (the check on Class A was not made for the same reason as the above).

    As mentioned above (also as shown in Table 2), the samples of limb bones collected was too small to carry out multivariate analyses compared with the number of the variables to be analyzed here.  Therefore, several sets of only craniofacial measurements were also prepared for multivariate analyses to achieve the aim of the present study.

    In the first variable set of the skull, the number of samples available was 117, 140, and 159 in males, and 38, 49, and 69 in females for Class A, B, and C, respectively.  In the second variable set of the skull, the number of male samples available was only 5, 7, and 10 for Class A, B, and C, respectively (the check on female samples was not made for the same reason as the above).  In the third variable set of the skull, the number of samples available was 237, 291, and 325 in males, and 42, 53, and 73 in females for Class A, B, and C, respectively.

Among-group covariations of craniofacial, limb bone, and environmental variables

    The among-group covariations between craniofacial, limb bone, and environmental variables were also examined using PCA.  In either case of the first and second variable sets, however, the number of samples collected was not enough for any multivariate analysis of the three kinds of data under the statistical restriction on sample size given the number of variables.  Only in the third variable sets of craniofacial and limb bone measurements, 27 male samples were available (in this case, the total number of variables is 23, i.e., 9 craniofacial, 8 postcranial, and 6 environmental variables), though the quality of the samples is not so high (Class C in Table 1).  The results of PCA and the rotated solution based on these samples are shown in Tables 9 and 10, respectively.

    It was found that PC I was significantly associated with bizygomatic breadth and, at the same time, with humeral and femoral midshaft thickness measurements (Table 9), and that Fac IV which was significantly associated with average temperature was inversely associated with absolute value of latitude (Table 10).  No association was suggested between craniofacial or limb bone measurements and environmental variables.  Incidentally, 22 of the 27 samples are those from the Japanese archipelago of the Jomon period to modern times.  This means that the above results do not necessarily reflect a global tendency.

Among-group covariations of craniofacial and environmental variables

    Since the number of the limb bone samples collected was too small, the among-group covariations of craniofacial measurements and environmental variables were intensively examined.  The PCA based on the data set with the highest quality for the first variable set (Class A in Table 1) showed a very interesting result (PC I in Table 11 or Fac I in Table 12), suggesting that cranial breadth, upper facial height, bizygomatic breadth, orbital height, and nasal height have a negative correlation with average temperature and a positive correlation with absolute value of latitude.  Strangely, however, any of the factor loadings on PC I or Fac I was not significant at the 5% level.  The significance tests were performed by Efron’s bootstrap method.  Usually, the bootstrap method gives convincing results, even if the form of distribution of the statistic to be tested is deviated from that of a normal distribution, except for an extreme value in an observed sample, as already stated in “Methods.”  In the present study, therefore, PCA was, then, separately applied to the correlation matrices for craniofacial measurements and environmental variables to explore the reason why the bootstrap method did not give an appropriate probability to a high factor loading.  The results from the craniofacial data are shown in Tables 13 and 14, and those from the environmental data, in Tables 15 and 16.  PC I and Fac I from the craniofacial data show that they are highly significantly associated with cranial breadth, upper facial height, bizygomatic breadth, orbital height, and nasal height at the 0.1% level (Tables 13 and 14).  The state of high factor loadings for these original variables is almost perfectly compatible with the results shown in Tables 11 and 12.  On the other hand, the results obtained from the environmental data show that, although some of the factor loadings on PC I and II or Fac I and II have very high values (Tables 15 and 16), the probabilities estimated for them by the bootstrap method are greater than 0.05.

    The same tendency as the above was confirmed also in the analyses based on the third variable set of the skull (Tables 17 to 22).  These findings point to a possible cause hidden in the environmental data.

    In Figs. 11 to 16, the distributions of the six environmental variables are drawn.  Their skewness and kurtosis are indicated in Table 23.  They clearly show their extreme deviation from normal distributions, except for average temperature and absolute value of latitude.  This is considered a cause of the failure in the above bootstrap tests.  In fact, for example, the bootstrap standard deviations for the z-transformed factor loadings of upper facial height and bizygomatic breadth on PC I in Table 11 are 1.16 and 1.40, respectively, while those in Table 13 are 0.12 and 0.11, respectively (not shown in tables).  Namely, the former bootstrap SDs are about ten times as large as the latter.  As a result, it turns out that the former factor loadings are not significantly different from zero, while the latter, significantly different from zero.

    In the present study, therefore, the existence of a factor such as a PC or rotated factor was confirmed mainly by comparing the variation patterns of factor loadings on two PCs or rotated factors, as stated in “Methods.”

    In passing, it is a noteworthy finding in the present study that PC I from among-group correlations between craniofacial measurements is not a so-called general size factor because of the fact that all factor loadings on the PC do not have the same sign (Tables 13 and 19).  In the case of within-group PCA, PC I is usually a general size factor, as seen in Tables 4 and 5.

    The existence of PCs or Facs obtained from the among-group correlations between craniofacial measurements was confirmed using Spearman's rank correlation coefficient (Tables 24 and 25).  At least, the significant Spearman's rank correlation coefficients of 0.90 and 0.98 (Table 24) between PC I’s (Tables 13 and 11) and between Fac I’s (Tables 14 and 12), respectively, suggest that there is a common factor which strongly influence the among-group covariations between cranial breadth, upper facial height, bizygomatic breadth, orbital height, and nasal height.  The significant Spearman's rank correlation coefficients of 0.83 and 1.00 (Table 25) between PC I’s (Tables 19 and 17) and between Fac I (Table 20) and Fac II (Table 18), respectively, also point to the existence of the same factor.  Tables 24 and 25 furthermore suggest the existence of other common factors.

    Similarly, the repeatability of PCs or Facs from environmental variables was also examined (Tables 26 and 27).  The significant rank correlation coefficients of 0.94 and 0.89 (Table 26) between PC I’s (Tables 15 and 11) and between Fac I’s (Tables 16 and 12) suggest that there is a common factor which strongly associated with average temperature and absolute value of latitude in the opposite direction.  The same tendency is also indicated by the significant rank correlation coefficients of 1.00 and, again, 1.00 (Table 27) between PC I’s (Tables 21 and 17) and between Fac I’s (Tables 22 and 18), respectively.

    Hence, PCA and the rotation of the PCs on both craniofacial (the first variable set) and environmental variables were carried out for female Class A samples as well (Tables 28 and 29).

    The PCAs based on Class B samples, both male and female, showed similar results to those based on Class A samples.  Therefore, the description is omitted here.

    The results of PCAs and the rotations based on male and female Class C samples are shown in Tables 30 to 33.

    Table 34 shows Spearman's rank correlation coefficients between the PCs or rotated factors from the Class A male samples and those from the Class C male samples of ten craniofacial and six environmental variables.  These rank correlation coefficients indicate very clear correspondence between the results from the Class A and C male samples.  It is almost true of females (Table 35).  Tables 36 and 37 reveal the degree of correspondence between males and females in Class A and Class C samples, respectively.

    Tables 38 to 47 show the results of analyses for the third variable set of the skull, which were performed following the same procedure as used for the first variable set of the skull (Tables 28 to 37).

    Here, once again, the existence of PCs or rotated factors from the combination of the nine craniofacial and eight postcranial measurements and six environmental variables (Tables 9 and 10) was examined using Spearman's rank correlation coefficients.  The results are shown in Tables 48 and 49, where, however, no clear evidence is found for existence of concrete common factors.

    In Figs. 17 to 22, the factor loadings of major common factors, the existence of which was confirmed by Spearman's rank correlation coefficients, are illustrated.  To know which people have the highest or lowest scores in each major factor, standardized means of craniofacial and environmental variables were checked in the Class A male samples with the highest or lowest scores (Table 50).  As a result, for example, those who have the highest scores of PC I in Table 11 are peoples like Yakuts (Russia; 73 YAKUT in Appendix 3), Buryats (Russia; 71 B-T-B) and Chukchi (Russia; 74 CHUK3), while those who have the lowest scores of PC I in Table 11 are peoples from Lower Nubia (Egypt; 26 L-NB2), Naqada (Egypt; 4 NAQAD) and S. Egyptians (Upper Egypt; 3 S-EGY).  The standardized means of original variables make the characteristics of extracted factors clearer, as shown in Figs. 23 to 28.

Path analysis of craniofacial measurements and environmental variables

    Finally, path analysis was performed to get a piece of information on some unknown factors which may influence the among-group variations of craniofacial measurements.  On a simple model, such as used in Mizoguchi (1978, 1986), craniofacial measurements were treated as endogenous variables, and environmental variables, as exogenous variables.  The results based on the Class A male samples of the first and third variable sets are shown in Tables 51 and 52, respectively.  In both analyses, it was found that residual variables had relatively high values.  This suggests the existence of unknown factors influencing on craniofacial measurements in addition to the factors found in the above PCAs.