Bull. Natl. Mus. Nat. Sci., Ser. D, 43, pp. 7–15, December 22, 2017

Allometric relationship between hand bone lengths andstature in a modern Japanese population
Megumi Kondo1, Naomichi Ogihara2, Shuji Matsuʼura1, Sayaka Anada1, Kohta Ito2,
Mitsunori Murata3, Toshiaki Tanaka4, Shun Matsuura5,
Shozo Takai6 and Ken-ichi Shinoda7
1

Laboratory of Physical Anthropology, Ochanomizu University, Bunkyo Ward, Tokyo 112–8610, Japan
E-mail: kondo.megumi@ocha.ac.jp
2
Department of Mechanical Engineering, Keio University, Yokohama City, Kanagawa 223–8522, Japan
3
Wayo Womenʼs University, Ichikawa City, Chiba 272–8533, Japan
4
Tanaka Growth Clinic, Yoga 2–36–7, Setagaya Ward, Tokyo 158–0097, Japan
5
Department of Administration Engineering, Keio University,
Yokohama City, Kanagawa 223–8522, Japan
6
University of Tsukuba, Tsukuba City, Ibaraki 305–8577, Japan
7
Department of Anthropology, National Museum of Nature and Science,
Tsukuba City, Ibaraki 305–0005, Japan

Abstract Human proximal limb elements are known to scale with body size with smaller allometry coefﬁcients than distal limb elements. In the present study, we tested if this allometric tendency is applicable to the human hand. Speciﬁcally, the allometric relationships between handbone lengths and stature in a modern Japanese population were examined based on a total of 197
anteroposterior X-ray images of the adult hand. The lengths of 19 metacarpal and phalangeal elements were measured by digitizing the images. We regressed each log-transformed bone length
against log-transformed stature using reduced major axis (RMA) regression, and compared theallometry coefﬁcients. Our results demonstrated that, in all the examined hand bone elements,
allometry coefﬁcients were signiﬁcantly larger than one, indicating positive allometry. It was also
found that the allometry coefﬁcients of the distal hand bones were generally larger than those of
the proximal bones, indicating that the general allometric relationship previously reported for themain limb bones is also applicable to the hand bones. No statistical differences were observed in
the allometric relationship between males and females for all the examined hand bones except the
5th middle phalanx, possibly owing to larger variabilities in female 5th middle phalanx length.
These results contribute to the understanding of normal growth and developmental patterns as wellas to secular change of human hand proportions. Our results also provide some new insights into
grasping biomechanics of the hand.
Key words: Allometric relationships, Hand bone lengths and stature, Metacarpals and phalanges,
Planar radiography, Modern Japanese

Introduction
Human limb elements are known to scale allometrically with body size. Previous studies
investigated the allometric relationships of thelengths of the humerus, radius, femur and tibia(Holliday and Ruff, 2001; Sylvester et al., 2008;
Auerbach and Sylvester, 2011). These studies
demonstrated that the scaling relationships ofsome of the limb elements were non-isometricand that the proximal limb elements scaled withcomparatively smaller allometry coefﬁcients thanthe distal elements. Therefore, as a general tendency in humans, if stature is higher, the lengthsof the distal limb segments (radius and tibia) relative to the proximal limb segments (humerus
and femur) tend to be greater. However, suchallometric relationships have not been investi-

Kondo et al.

8

gated in the other parts of the postcranium.
The present study aimed to investigate if thisallometric tendency is also applicable to thehuman hand. Relationships between metacarpal
length and stature have been analyzed for the
purpose of stature estimation (Musgrave andHarneja, 1978; Kimura, 1992; Meadows andJantz, 1992; Wilbur, 1998: Takai et al., 2005).
However, no studies have previously compared
the allometric coefﬁcients of all the metacarpals
and phalanges of the hand to investigate theirscaling relationships. Such information may contribute to our understanding of normal growthand development patterns of the human hand,secular changes of hand proportions, and aspectsof grasping biomechanics.
Materials and Methods
In the present study, we used anteroposterior
X-ray images of the right hands of a total of 197
adult participants (61 males and 136 females)
who participated in the Ogi Growth Study performed by Saga University School of Medicine
from 1979 to 1988 (Takai, 1990; Takai and Shinoda, 1991). This study is a longitudinal growth
study of Japanese students who resided in Ogi
city, Saga prefecture, Japan. Along with standard
anthropometric measurements, an anteroposterior
X-ray image of the right hand was taken every
year from an age of 6 to a maximum age of 18
years (all the X-ray ﬁlms are currently housed in
the Department of Anthropology, National
Museum of Nature and Science). However, in thepresent study, we selected only the X-ray images(anonymized in an unlinkable fashion) of thehand after epiphyseal fusion, since our focus wason the adult hand.
The X-ray ﬁlms were scanned with a resolution of 150 dpi (a pixel size of 0.169 mm) using aﬁlm scanner (EPSON GT-X980). On each digital
image, the most proximal and distal ends of themetacarpals (MC), proximal phalanges (PP),middle phalanges (MP), and distal phalanges(DP) along the shaft axis were manually digitized. This was done using a custom-made soft-

ware named Dicom&16Tiff Viewer Ver.2.2 (Ogihara Lab., Dept. Mech. Eng., Keio University),
and the distances between the proximal and distal ends of the bones were calculated (Figure 1).
The measurement was conducted by one of theauthors (S.A.), twice on a different day. The
mean of the two values was used as the length ofeach MC or phalanx. The intra-observer error of
the length measurement was approximately
0.3 mm for all the bones, corresponding to ＜1%of the length of the MCs and PPs, and ＜2% of
the MPs and DPs, indicating that the measurements were made in a highly reproducible fashion.
To explore the scaling relationships of the 19
metacarpal and phalangeal bones with stature,
we regressed each of the log-transformed bonelength against log-transformed stature using
reduced major axis (RMA) regression (Hofman,
1988; Sjøvold, 1990), a model II linear regression (R version 3.2.4, The R Project for Statistical Computing) with custom packages. We used
RMA regression because both bone lengths andFig. 1. Anteroposterior X-ray image of the hand(left) and enlargement of the MP3 part (right)
showing two digitized points connected by aline segment representing the MP3 length.

Allometry between hand bone lengths and stature

stature were measured with errors. All regression
analyses were performed separately for males
and females. The allometry coefﬁcients (slopes)
and intercepts of the male and female regression
lines were computed and statistically tested byStudentʼs t-test. Using the same software, we also
tested if the regression line signiﬁcantly deviatesfrom one, i.e., geometric similarity.
The present study was approved by the Ethics
Committee of the National Museum of Natureand Science.
Results and Discussion
Bivariate plots with RMA regression lines ofthe log-transformed 19 hand bones relative to
log-transformed stature are shown in Figure 2.
Table 1 lists the allometry coefﬁcients and intercepts of the RMA regression lines of all 19 handbones. For all hand bones, the allometry coefﬁcients and intercepts of the male regression lines
were not statistically different from those of thefemale regression lines except for the regression
lines for the MP5 (p＜0.001). All regression
lines signiﬁcantly deviated from one, indicatingthat all the hand bones scale with positive allometry.
Figure 3 compares the allometry coefﬁcientsof the hand bones from the 1st to 5th rays. As theﬁgure shows, the allometry coefﬁcients for theMCs and PPs were smaller than those for theMPs and DPs in both males and females, indicating that the two proximal elements scale withcomparatively smaller allometry coefﬁcients thanthe distal bones. We also observed that the allometry coefﬁcients tended to increase with distancefrom the 3rd ray, particularly in females.
In the present study, we found that the proximal bones scaled with comparatively smaller
allometry coefﬁcients than the distal bones in thehuman hand, as previously demonstrated for thefour major limb bones (Holliday and Ruff, 2001;
Sylvester et al., 2008; Auerbach and Sylverster,2011). Furthermore, Auerbach and Sylvester(2011) reported that the allometry coefﬁcients ofhumerus, radius, femur and tibia are 1.03, 1.25,

9

1.12 and 1.28, respectively, for males and 1.03,
1.21, 1.17, and 1.35, respectively, for females,
showing that the allometry coefﬁcients of theseproximal bones are smaller than those of handbones. Therefore, there seems to exist a generaltendency that distal elements grow faster in relation to proximal element in human limbs. The
mechanism underlying the emergence of thisallometric relationship is beyond the scope of thepresent study, but it should be related to differential skeletal growth deﬁned by the expression ofHox genes (Reno et al., 2008) or differences intiming of epiphyseal fusion between proximaland distal elements. In addition, it was noted thatthe allometry coefﬁcients tended to increase with
distance from the 3rd ray, particularly in females.
The functional signiﬁcances of these extracted
allometric patterns are currently obscure. However, it might be related to change in mechanicalrequirement of the human hand with increasing
size. How the human hand proportion emerges indevelopment and how changes in skeletal proportion of the hand affect manipulative capability
should be investigated in future studies. It must
be noted, however, Auerbach and Sylvester(2011) used a model I regression analysis to calculate the allometry coefﬁcients and their samples were of a mixed population from different
geographical origins, indicating that direct comparisons of the allometry coefﬁcients were actually impossible. To conﬁdently conclude that
there exists a general tendency that distal elements grow faster in relation to proximal elementin human forelimbs, the allometric relationshipsof the humerus, radius, metacarpals and phalanges should be more rigorously investigated using
the same method and samples from the same
population.
The present study also found that there is basically no statistical difference in the allometric
relationship between males and females except
for the MP5. This is owing to the fact that thevariance of the MP5 length was much larger infemales than in males. There were a certain number of female participants who had shorter-thanexpected MP5 as shown in Figure 2. The same

Kondo et al.

10

Fig. 2. Bivariate plots with RMA regression lines of log-transformed hand bones relative to log-transformed stature. Red circle, female; blue triangle, male. Red line, female regression line; blue line, male regression line.

Allometry between hand bone lengths and stature

Fig. 2.

Continued.

11

12

Kondo et al.

Fig. 2.

tendency, although not signiﬁcant, can beobserved in DP1. We can offer no explanations
as to why this occurs, but distal hand bones distant from the 3rd ray, i.e. the center of the hand,
may be developmentally more variable thanother hand bones.
As mentioned before, the length of metacarpals has been used to estimate the stature (Musgrave and Harneja, 1978; Kimura, 1992; Meadows and Jantz, 1992; Wilbur, 1998; Takai et al.,2005). The regression equations derived in thepresent study can also be used for stature estimation. In the previous studies, however, the lengthof metacarpals was directly regressed with stature without logarithmic transformation, assuming that the allometric relationship between theContinued.

metacarpal length and the stature was relatively
minor. Furthermore, the previous studies usedordinary least square regression (model I)
whereas the present study used RMA regression(geometric mean regression of model II). Whenthe two variables both contain error as in thepresent case, a model I (Y on X) regression tends
to underestimates the slope, thus resulting inoverestimation of the shortest individuals andunderestimation of the tallest (Sjøvold, 1990).
Therefore, the present approach might have provided theoretically better equations for stature
estimation based on the length of metacarpals.
However, this must also be conﬁrmed in futurestudies.
Here we have focused on the adult hand using

Allometry between hand bone lengths and stature

Fig. 2.

Continued.

13

Kondo et al.

14
Table 1.

Allometry coefﬁcients and intercepts of the RMA regression lines
Slope

MC1
MC2
MC3
MC4
MC5
PP1
PP2
PP3
PP4
PP5
MP2
MP3
MP4
MP5
DP1
DP2
DP3
DP4
DP5

Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined
Male
FemaleCombined

1.79
1.83
1.60
1.81
1.67
1.49
1.76
1.71
1.48
1.72
1.76
1.49
1.73
1.82
1.51
2.03
2.01
1.72
1.73
1.69
1.41
1.68
1.62
1.39
1.65
1.62
1.43
1.72
1.75
1.48
1.96
2.13
1.68
1.82
1.83
1.50
1.89
1.92
1.60
2.63
4.27
2.89
2.33
2.71
2.13
1.98
2.11
1.80
1.89
2.07
1.79
2.00
2.03
1.85
2.40
2.28
1.97

95% conﬁdence interval
1.45
1.59
1.46
1.45
1.46
1.36
1.41
1.49
1.35
1.37
1.53
1.35
1.38
1.58
1.37
1.65
1.74
1.56
1.38
1.47
1.28
1.35
1.40
1.26
1.32
1.41
1.30
1.37
1.51
1.34
1.55
1.84
1.51
1.45
1.58
1.35
1.51
1.66
1.45
2.08
3.63
2.57
1.81
2.30
1.89
1.56
1.81
1.62
1.48
1.78
1.61
1.58
1.75
1.68
1.89
1.95
1.77

2.22
2.10
1.75
2.26
1.91
1.64
2.20
1.96
1.63
2.15
2.03
1.64
2.16
2.10
1.67
2.49
2.32
1.89
2.16
1.95
1.56
2.09
1.87
1.53
2.07
1.87
1.57
2.16
2.03
1.64
2.47
2.46
1.86
2.28
2.13
1.67
2.37
2.23
1.77
3.32
5.02
3.26
3.00
3.19
2.39
2.52
2.47
2.01
2.42
2.41
1.99
2.53
2.37
2.05
3.04
2.67
2.19

Intercept

95% conﬁdence interval

−4.11
−4.20
−3.46
−4.00
−3.53
−2.96
−3.85
−3.69
−2.96
−3.77
−3.89
−3.02
−3.84
−4.12
−3.14
−5.03
−4.96
−4.03
−3.94
−3.81
−2.92
−3.74
−3.53
−2.80
−3.68
−3.57
−2.95
−4.01
−4.10
−3.23
−4.95
−5.47
−4.03
−4.42
−4.44
−3.39
−4.67
−4.76
−3.73
−7.21
−12.44
−8.06
−6.18
−7.36
−5.51
−5.16
−5.56
−4.57
−4.82
−5.39
−4.50
−5.17
−5.25
−4.69
−6.49
−6.11
−5.10

−5.48
−5.07
−3.95
−5.45
−4.29
−3.42
−5.27
−4.49
−3.43
−5.17
−4.75
−3.51
−5.24
−5.02
−3.64
−6.54
−5.94
−4.58
−5.35
−4.63
−3.39
−5.08
−4.33
−3.26
−5.04
−4.36
−3.41
−5.42
−5.00
−3.74
−6.59
−6.54
−4.62
−5.92
−5.38
−3.91
−6.22
−5.74
−4.28
−9.45
−14.82
−9.24
−8.33
−8.89
−6.36
−6.90
−6.71
−5.22
−6.53
−6.47
−5.13
−6.88
−6.32
−5.31
−8.57
−7.35
−5.83

−3.00
−3.45
−3.02
−2.85
−2.86
−2.55
−2.72
−2.99
−2.54
−2.66
−3.14
−2.58
−2.72
−3.35
−2.69
−3.81
−4.11
−3.54
−2.82
−3.10
−2.49
−2.67
−2.83
−2.38
−2.60
−2.89
−2.53
−2.89
−3.32
−2.78
−3.64
−4.55
−3.50
−3.23
−3.63
−2.91
−3.44
−3.91
−3.23
−5.44
−10.41
−7.01
−4.51
−6.06
−4.76
−3.78
−4.57
−3.98
−3.48
−4.46
−3.93
−3.81
−4.34
−4.12
−4.86
−5.05
−4.45

Bold text at MP5 indicates a statistically signiﬁcant difference between male and female with a p-value less than 0.05.

Allometry between hand bone lengths and stature

Fig. 3.

15

Allometry coefﬁcients of the metacarpals and phalanges from the 1st to 5th rays. A, male; B, female.

the X-ray images of hands taken after epiphyseal
fusion, and thus did not study the ontogenetic
changes of the hand bones. However, longitudinal-growth radiographs of the same participants
are available, along with standard anthropometricmeasurements. Continued investigations ofdevelopmental change of hand bone lengths arenecessary to further clarify some of the quesitons
raised above.
Acknowledgement
We would like to extend our appreciation to
the Editor of this journal, Dr. Kazuhiro Sakaue
for his generous help. The manuscript was
improved with valuable suggestions and comments from anonymous reviewers.
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