On the Validity of Principal Component Analysis as a Method of Analysis of the Variations in Human Physique

Transcription

1 On the Validity of Principal Component Analysis as a Method of Analysis of the Variations in Human Physique I. Selection of the Measurement Items Based on the First Three Principal Components* Makiko KOUCHI Department of Anthropology, Faculty of Science, The University of Tokyo Abstract Principal component analysis (PCA) and cluster analysis were applied to a correlation matrix based on 44 somatic measurements taken on 112 males aged from 18 to 27 years, and the results were compared with those of previous studies. The purposes of this study are: 1) to clarify the reason why the 3rd principal component is varied in meaning from case to case, and 2) to confirm that PCA offers effective criteria for the grouping of the measurement items. Initial selection of the measurement items is considered to be responsible for the varied manifestation of the 3rd principal component. All the studies reviewed show similar grouping of the measurement items based on the first three principal components, except for the location of sitting height. Cluster analysis supported the grouping based on PCA on the whole. Selection of measurement items was attempted on the basis of above-mentioned grouping, and tentatively 20 items were selected. INTRODUCTION The human physique and described has been classified by means of anthropometry or anthroposcopy. Anthropometry has the advantage of objectivity, but the informations carried by measurements are not always fully utilized. For example, indices, which have been used to give a concise numerical expression of the body * Part of this paper was presented at the 32th joint meeting of Anthropological Society of Nippon and Japanese Society of Ethnology, shape, can describe only limited aspects of the physique. A large number of measurements of varied somatological meanings must be taken into consideration simultaneously to express and analyse the variations in human physique. It is, however, difficult to get a simple outline of physique from a large number of measurements unless an appropriate analytical method is applied. For this purpose, principal component analysis (PCA) is recommended, since it is a useful method to summarize the inf ormations carried by a number of mutually correlated variables.

2 424 M. KoucHI However, there are some problems to be solved before PCA is utilized. In the first place, the reproducibility the results of PCA should be investigated. The 1st and the 2nd principal components obtained in many previous studies are comparable to each other, but the 3rd principal component is different in interpretation from case to case. The reason why such difference brought about must be clarified and the reproducibility of the 3rd and the following principal components must be confirmed. Secondly, it is necessary to establish the consistency in interpretation of the results of PCA on the basis of biological ground. PCA is also a useful method of grouping measurement items (OKUNO et al., 1971), and it is expected that the consistency will be found in the manner according to which the measurement items are classified, of so far as such measurement items that belong to the same group have similar somatological meanings. For example, all the measurement items that are mainly determined by limb bone length actually belong to one group (KOUCHI, 1977). Therefore, if the consistency in such grouping is confirmed, the usef ullness of PCA as a method of classification of human physique would be appreciated, because variations in human physique may be expressed through the relationship of these groups. Thirdly, since it is indicated that the results of PCA are influenced by the initial selection of the measurement items(kou- CHI, 1977), the measurement items must be carefully selected not to be too biased in their somatological contents. If one succeeds in establishing the consistent grouping of the measurement items on the basis of PCA, the groups obtained would serve as a basis for the reasonable selection of the measurement items. The selection of the measurement items is also required from the practical viewpoint. MATERIALS AND METHODS The present data consist of 44 somatic measurements taken on 112 subjects, all trainee policemen, aged from 18 to 27 years. The measurements were taken in September 1977 by the author, most according to MARTIN (1957), with some by methods proposed by WEINER-LOURIE (1969). A list of measurement items is given in Table 1. The measuring methods of items not commonly used were as follows: 5) thigh length: iliospinal height minus tibial height, 7) upper limb length (45) : direct measurement, 13)chest breadth 2 (36a) : thoracic breadth measured at the xyphoid level, 18) chest depth 2: thoracic depth measured at the same level as No.13, 22) elbow breadth (52(3)) : bicondylar breadth of the humerus, measured with the cubital joint flexed, 23) wrist breadth (52(4)) : bistyloid breadth, 26) ankle breadth : bimalleolus breadth, i. e. distance between the most medial point of the malleolus medialis and the most lateral point of the malleolus lateralis, 30) chest girth 2: thoracic girth measured at the same level as No. 13, 44) cheek : the skinfold thickness in the centre of the cheek was measured. The thickness of the ski of old was m eas-

3 On the Validity of Principal Component Analysis 425

4 426 M. KoucHI ured with a Harpenden skinfold caliper and the values obtained were transformed according to the following equation in order to normalize the distribution (TAN- NER and WHITEHOUSE, 1955), z= 100 * logio(reading in 1/lOmm-18) Since correlations between cephalof acial and somatic measurements are very low (HOSHI and KOUCHI, 1978; KoucHi, 1977), only somatic measurements were dealt with in the present study. The correlation matrix of the 44 measurements was analysed with PCA. Grouping of the measurement items was carried out according to their factor loadings for the first three principal components. Since the 4th and the following principal components contributed only to a few measurement items, they are considered to be relatively unimportant in the grouping of the measurement items, although they are sometimes useful to explain the character of the groups produced. In order to examine the validity of the grouping based on PCA, the correlation matrix was also analysed by cluster analysis (average linkage method). In order to examine the meaning of the 3rd principal component, and to examine whether the grouping is altered or not according to the initial selection of the measurement items, the present results were compared with those of the PCA studies of several authors. These studies were by: 1)INOUE and YANAGISAWA (1978), 2) KOUCHI (1977), 3) VANDENBERG (1968), 4) FUKUSHIMA (1967), 5) MASUDA (1965), 6) HAMMOND (1953), and 7) HEATH (1952). Since VANDENBERG, HAMMOND and HEATH used various methods of factor analysis, their original correlation matrices were re-analysed with PCA to make direct comparison possible. All the calculations were processed by the HITAC OS7 electronic computer system of The University of Tokyo Computer Centre, Subprogram EIGENS in the HSAP* program package was used to calculate eigenvalues and eigenvectors, and SPSS** was used for cluster analysis. RESULTS AND DISCUSSIONS Factor loadings of each measurement item for the first three principal components are shown in Figs. 1-a and 1-b as two-dimensional illustrations. Fig. 2 shows the contribution rates of the first nine principal components. The first 39 eigenvalues were shown to be significant by the Bartlett-Lawley test (LAWLEY, 1956). Seven principal components had eigenvalues greater than 1. 0, and their cumulative proportion was 79.5%. (1) The first three principal components The 1st principal component (CP1), socalled "general size factor ", has extremely high correlation with body weight, and is considered to represent the variation in body mass. Mass seems to be somewhat different sort of size from height or length, since the contribution rates of CP1 to girth measurements are high, but to such measurements as those which represent length or breadth of the * Hitachi Statistical Analysis Program ** Statistical Package for the Social Sciences

5 On the Validity of Principal Component Analysis 427 skeletal framework amounted to only 20 to 30%. The 2nd principal component (CP2) is Fig. 1-a. Principal component analysis of 49 somatic measurements. Factor loadings for the first and the second principal components. usually considered to contrast a tall slender man with a short obese man. However, it is interesting to note that the measurements which depend on the skeleton have negative factor loadings, while those which depend on the soft tissue have positive factor loadings, and therefore, that CP2 should be considered to contrast the size of the skeletal framework with the development of the soft tissue. Though the cumulative proportion of CPl and CP2 amounted to 58.3% in the present study, this proportion is not enough to obtain sufficient inf ormations about the human physique. For the 3rd and the following principal components, however, a great variety of interpretations have been put forward. In Table 2, the measurement items with high factor loadings for the 3rd principal components (CP3) ever reported are listed. If similar combination of measurement items appears Fig. 1-b. Factor loadings for the second and the third principal components.

6 428 M. KoucHI Fig. 2. Contribution rates of the first nine principal components.

7 Table 2. The 3rd principal component. On the Validity of Principal Component Analysis 429 M : male, F : female, N : number of subjects, NM: number of measurements * garment measurements were analysed original correlation matrix was + re-analysed by the present author using PCA in several studies, and if a reasonable somatological interpretation of such combination is possible, then CP3 could be considered to represent one of the meaningful variations in the human physique. Of the three studies which have dealt with anthropometric measurements, both the present study and that of HAMMOND analysed a variety of measurement items, and their results for CP3 approximately accorded with each other: in both, limbjoint breadths and limb girths are contrasted with limb lengths, girths of the trunk, and skinfold measurements. This principal component seems to relate to the stoutness of the limbs. However, it should be noted that this principal component has two aspects: one is to contrast limb-joint breadth with limb bone length, and the other is to contrast limb girths with trunk girths including skinfold thickness. The former aspect was shown by

8 430 M. KoucHI the CP3 results of VANDENBERG, and the latter in the CP3 results of HEATH. On the other hand, 15 measurement items analysed by KOUCHI(1977) did not include limb-joint breadth, abdominal or skinf old measurements, but did include trunk length and sitting height. Her 3rd principal component was quite different from those of other studies and expressed the shape of the trunk, i, e, it contrasted the length of the trunk with bicristal breadth. Five studies have dealt with garment measurements. Those of INOUE and YANAGISAWA and of FUKUSHIMA analysed a large number of measurement items. The former authors dealt with many surface distances along the trunk above the waist line level, calculated frontback differences, and angles of body profile, but the total length of the trunk was not included. CP3 of MASUDA distinguished a man with rounded shoulders from a man with a thrown-out chest. In FUKUSHIMA's study, the contrast was between a woman with wide, sloping and rounded shoulders and a woman with narrow, square shoulders and a thrownout chest. In the case of INOUE and YANAGISAWA, the contrast was complicated because the lengths of the limbs were included. Since the correlation between limb-length and the measurements concerning shoulder-shape is very low, and since the relationship between them cannot be explained reasonably, CP3 of INOUE and YANAGISAWA may have a two-fold character like CP3 of the present study. In these three studies, however, CP3 was concerned with shape variation of the upper trunk which seems to be influenced not only by the fleshiness, but also by habitual posture of the individual. It is clear that when garment measurements are analysed, the meaning of CP3 differs from that obtained from the analysis of the anthropometric measurements. But so far as anthropometric measurements are analysed, and if the selection of measurement items is not too different, similar results are obtained as shown by the present author and by HAMMOND. Because of the difference in purpose, some garment measurements are never taken in anthropometry, and vice versa, while others, e. g. girths and breadths of the trunk, are always taken in both. In this point, the measurement items used by HEATH and by VANDENBERG were rather similar to those used in ordinary anthropometry, and they included measurement items of the distal segments of the limbs. In the studies by INOUE and YANAGISAWA, by FUKUSHIMA, and by MASUDA, the measurement items that were largely contributed to by CP3 were those taken only in dressmaking. Similarly, in the case of the analyses of anthropometric measurements, the limb measurements usually taken only in anthropometry were highly contributed to by CP3. In other words, the reason for difference in meaning of CP3 from case to case is considered to essentially lie in the initial selection of the measurement items, and appropriate selection of the measurement items is indispensable for the reproducibility of CP3. The relationship between such variations that are

9 On the Validity of Principal Component Analysis 431 Fig. 3. Cluster analysis of 44 somatic measurements. Average linkage dendrogram. expressed only by garment measurements and such variations that are expressed only by anthropometric measurements remains unknown until both sorts of measurements are analysed simultaneously. It seems probable that some important variation in human physique cannot be expressed by ordinary anthropometric measurements. (2) Grouping of the measurement items The results of PCA and cluster analysis are shown in Figs. 1-a, 1-b, 2 and 3. Group A (Nos. 2, 4, 5, 6, 7, 8 and 9) : This

10 432 M. KoucHI group is clearly a cluster of measurement items concerning the limb bone length, including stature (No.2). This sort of grouping was observed in all the reports reviewed. Thigh length (No. 5) is a little apart from other measurements. This could be because it is a projectional distance and does not accord with the real length of the femur. It should be noted that trunk length (No.3) does not belong to this group. This group can be called the long bone length group. Group B (Nos. 22, 23, 24, 26 and 27) : This group consists of measurements of limbjoint breadth, which is known to grow in a different way from the length of the diaphysis. Knee breadth (No. 25) is not included in this group, probably because it is largely influenced by the soft tissue. The correlation coefficients between measurements belonging to this group are not so high as in other groups. Group C (Nos. 10 and 21) : This group is located between groups A and B, possibly because metacarpals, metatarsals and phalanges have large epiphyses relative to the length. Groups similar to B and C, and relationships among groups A, B and C were observed in other studies (VANDENBERG, 1968; HAMMOND,1953). Group D (Nos. 11,15 and 25) : These three breadth measurements are put into one group because of their extremely small factor loadings for CP2 and CP3, thus the somatological similarity among them is not evident. No close relationship between biacromial and bicristal breadths has ever been reported. Knee breadth (No.25) has a relatively large factor loading for CP4, biacromial breadth (No. 11) for CPS, and bicristal breadth (No. 15) for CP6, though the cumulative contribution rate of the first seven principal components to these items were not much greater than 50% (Fig, 3). Cluster analysis showed knee breadth to be closer to group B than to biacromial or bicristal breadths. Group E (Nos. 1, 16 and 33) : Most of the variances of these measurements are explained by CP1, i. e. 60% of the variance of hip breadth (No.16), 80% of hip girth (No.33) and 90% of weight (No.1). In all the studies that analysed both weight and hip girth, these are closely located. According to cluster analysis, weight and hip girth showed an close relationship, but hip breadth was separated from them to some extent. and hip girth are considered Weight to be representative measurements of general mass of the body, and this group can be called the general Group includes size group. F (Nos. 36 and 39) : This group the girths of the upper and lower limbs that are considered to be least influenced by the soft tissue. Group G (Nos. 34,35 and 38) : These girths of the upper and lower limbs are thought to be dependent on the development of the musculature rather than on subcutaneous fat. Thigh girth (No. 37) is not included (see group H). This group can be called the musculature group, especially in adult males. Cluster analysis does not support this grouping, (F and G), but divides the upper limb group from the lower limb group. HAM-

11 On the Validity of Principal Component Analysis 433 MOND showed the same kind of grouping as F and G, but in the case of HEATH, upper arm girth was included in the same group as trunk girths, while thigh girth was included in the same group as hip girth. Difference in sex of the subjects studied may be responsible for these differences. Group H (Nos. 12, 13, 29, 30 and 37) : This group consists of breadths and girths of the thorax, and can be called the thoracic size group. No reasonable explanation could be found for the inclusion of thigh girth, which would seem to belong to group G and is in fact quite close to this group. Cluster analysis indicates a close relationship between thigh girth and group E. Group J (Nos. 17 and 18) : These are chest depths measured at two different levels. It is interesting to note that depth is separated from breadth and girth of the thorax. This separation is affirmed by cluster Group analysis. K (Nos. 19, 20, 31 and 32) : This group consists of depths and girths measured at two different levels: waist line and navel. Waist breadth (No. 14), which is completely dependent on soft tissue, is located between groups H and K, and bicristal breadth(no.15), which is mainly determined by the skeleton, is included in group D. In the case of the thorax, breadth and girth were put together, while for the abdomen, depth and girth were grouped together. The existence of the rib cage in the thorax may be responsible for this difference. INOUE and YANAGISAWA, and FUKUSHIMA gave the following grouping : at the bust and waist levels, breadth, depth and girth were all put into one group. At the hip level, however, they were separated from each other. It must be noted that the measurement, "depth at the bust level ", was influenced by the development of mamma and subcutaneous fat (examinees were adult females). As well as that, the anthropometric definition of depth is different from the definition in dressmaking. For these reasons, INOUE and YANAGISAWA's and FUKUSHIMA's results cannot be compared directly with the present author's. For the waist, however, the results of the cluster analysis support the grouping by these authors. Group L (Nos. 40, 43 and 44) : These are skinfold measurements of limbs and cheek. They have also a rather high correlation with CP4. Group M (Nos. 41 and 42) : This group comprises skinf old measurements of the trunk. It is interesting to note that the skinf old measurements are separated into two groups, trunk and limbs, notwithstanding high correlation coefficients among them. It is also interesting that the cheek measurement belongs to the limb group. Correlation coefficients between cephalof acial and somatic measurements are generally very low (HOSHI and, 1978), but the skinfold thickness KOUCHI of the cheek has a close relationship to the somatic skinf old measurements. The results of a cluster analysis, however, indicate that this relationship should not be overestimated. There are three measurement items

12 434 M. KoucHI which belong to none of above-mentioned groups, i. e, trunk length (No. 3), waist breadth (No. 14) and neck girth (No.28). Since no principal component other than the 1st contributed significantly to waist breadth and neck girth, the nature of these items remains unknown. The uniqueness of trunk length is obvious from the fact that only the 7th principal component had markedly high correlation with, and only with, trunk length. The 3rd principal component of KOUCHI (1977) also indicated the uniqueness of trunk length. On the other hand, sitting height (sitting cervical height in the case of FUKUSHIMA), which includes the length of the trunk portion, was placed in different groups from case to case. It had an independent position in the studies of KOUCHI and HEATH; close to the limb bone length group in the studies of MASUDA and FUKUSHIMA ; and in the case of HAMMOND, it was included in the group of hand and foot lengths. Trunk length is considered to represent the vertical size of the trunk itself better than sitting height, since it is less influenced by other factors soft tissue of the buttocks of the examinee. such as head size, and the posture In conclusion, all the studies reviewed show a similar tendency in the grouping of measurement items regardless of sex, age or race except for the location of sitting height. It would seem reasonable to consider that the present grouping comes close to being a universal one. (3) Selection of measurement items Since the measurement items dealt with in the present study are basically ordinary anthropometric measurements, the present attempt is limited to select a set of measurement items which sufficiently represents the inf ormations carried by the ordinary anthropometric measurements. The selection was carried out on the basis of the grouping discussed in the previous section. Measurement error, easiness of measurement, and popularity were also Table 3. Selection of measurement items.

13 On the Validity of Principal Component Analysis 435 taken into consideration. As summarized in Table 3, four categories of measurements were considered for the skeleton, and six for the soft parts. Iliospinal height and upper limb length were selected from the long bone length group (A). Hand breadth and foot breadth were the most commonly used measurements in the group B. Since items of group C (hand length and foot length) seem to have mixed character of groups A and B, lengths may substitute for breadths of the hand and the foot when breadths are not taken. Trunk length had to be included because of its uniqueness of information. This is equally true in the case of biacromial and bicristal breadths (group D). Wrist and ankle girths (group F), having negative factor loadings for CP2, could be omitted because they are located close to skeletal measurements, and are considered less important in proportional expression of the body shape. Thoracic measurements taken at either of the two levels are equally useful for indicating thoracic size (groups H and J) at least in case of the present subjects, so that the more commonly used items were selected (chest girth 1, chest breadth 1, and chest depth 1). For abdominal size (group K), measurements taken at the level of the navel (abdominal girth and abdominal depth) were considered to be better than those taken at the waist line level, because it is not always easy to define the waist line in male subjects, and because the former have higher correlation with skinfold measurements than the latter. Of the five skinfold measurements, triceps and calf of the limb group (L) and subscapula of the trunk group (M) are usually taken. If only one item is to be selected from the limb group, calf skinf old was chosen, since it is taken at the distal segment of the lower limb. The three limb girths of the musculature group (G) were all adopted (upper arm girth, forearm girth and calf girth), and thigh girth was also adopted on the basis of the results of cluster analysis. Weight, a very popular measurement item, is considered to be the best indicator of the general mass (E). If weight is not taken, hip girth is the best substitute. Hip girth is considered to be a better representative of hip size than hip breadth. Twenty measurement items were thus selected and are listed in Table 3. In the next report, the extent to which the new set of measurements represents the inf o- rmations contained in the original set of measurements will be discussed. As the present results were based only on male subjects aged between 18 to 27 years, further analysis on female subjects and on younger or older subjects would be necessary to examine sex or age differences. Similar analysis on female subjects is being contemplated. ACKNOWLEDGEMENTS The author wishes to express her sincere gratitude to Dr. H. HosHI, Department of Anatomy, the University of Tokyo, for his valuable suggestions and criticism.

14 436 M. KOUCHI REFERENCES FUKUSHIMA, T., 1967: A Note on the classification of human physique by principal component analysis. J. Anthrop. Soc. Nippon, 75: HAMMOND, W. H., 1953: The determination of physical type in children. Hum. Biol., 25: HEATH, H., 1952: A factor analysis of women's measurements taken for garment and pattern construction. Psychometrika, l7: HOSHI, H. and M. KOUCHI, 1978: Anthropometry of adult male Japanese with remarks on correlation coefficients (in Japanese). Acta Anat. Nipponica, 53: INOUE, K. and S. YANAGISAWA, 1978: Principal component analysis of the somatic measurements for a contribution to lady's garment designing (in Japanese). J. Home Economics of Japan, 29: KOUCHI, M., 1977: Principal component analysis of somatometric data with special reference to the effect of the selection of measurement items. J. Anthrop. Soc. Nippon, 85: LAWLEY, D. N., 1956: Tests of significance for the latent roots of covariance and correlation matrices. Biometrika 43: MARTIN, R. and K. SALLER, 1957: Lehrbuch der Anthropologie. Bd. I. G. Fischer, Stuttgart MASUDA, T., 1965: Application of principal component analysis and factor analysis to the measurements of the human body- a trial for designing ready-made garments. Rep. Stat. Appl. Res., 12: OKUNO, T., T. RAGA, H. KUME and T. YOSHI- ZAWA, 1971: Multivariate Statistical Methods. Nikkagiren, Tokyo (Japanese text). TANNER, J. M. and R.H. WHITEHOUSE, 1955: The Harpenden skinfold caliper. Am. J. Phys. Anthrop., 13: VANDENBERG, S. G.,1968: A factor analysis of garment measures of Dutch women. Hum. Biol., 40: WEINER, J. S. and J.A. LOURIE, 1969: IBP Handbook, No.9. Human Biology, a Guide to Field Methods. Blackwell, Oxford. (Received January )

15 On the Validity of Principal Component Analysis 437