The appropriate use of regression equations for the estimation of segmental inertia parameters

.I. Biomechonics Vol. 22. No. 617, pp. 683-689. Printed m Great Britain

OoZl-9290/89 f3.00+ .I0 CQ 1989 Pergamon Press plc

1989.

THE APPROPRIATE

USE OF REGRESSION

FOR THE ESTIMATION OF SEGMENTAL PARAMETERS

EQUATIONS INERTIA

M. R. YEADON and M. MORLOCK Biomechanics Laboratory, Faculty of Physical Education, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N lN4 Abstract-Linear regression equations are commonly used in conjunction with experimental data to provide linear relationships between quantities which are dimensionally distinct. In many cases theoretical relationships between such quantities are known and can be used as a basis for non-linear regression equations. This study compares linear and non-linear approaches for estimating the segmental moments of inertia from anthropometric measurements using the data of Chandler et al. [Chandler et al. (1975) Investigation of inertial properties of the human body. AMRL Technical Report 74-137, Wright Patterson Air Force Base, OH.] Right limb data were used to derive the equations while left limb data were used as a cross-validation sample to evaluate the inertia estimates calculated from the equations. For the limb segments the standard error estimates had average values of 21% for the linear equations and 13% for the non-linear equations. Data on a 10 yr-old boy was used to compare the two approaches outside the sample range. The mean txrcentaae residuals were 286% for the linear equations and 20% for the non-linear equations. A set oi non-linear equations is provided.

INTRODUCTION Quantitative mechanical analyses of human movement require estimates of body segment mass centres,

masses and moments of inertia. Attempts have been made to obtain estimates of such segmental inertia parameters from anthropometric measurements of an individual by using linear regression equations based upon cadaver data (Barter, 1957; Clauser et al., 1969; Hinrichs, 1985). Hinrichs (1985) expressed the moments of inertia of body segments as linear multivariable functions of length and perimeter measurements. However, such use of linear equations to express relationships between quantities which are dimensionally distinct may be inappropriate and requires justification. When quantities are dimensionally commensurate, consideration must be given to the most appropriate form of the linear equation to be used. Barter (1957) used the data of three dissection studies, involving 12 cadavers, to express segmental mass as a linear function of total body mass. He believed that the equations would provide better estimates of segment masses than estimates obtained using mean ratio values of segmental mass to total body mass. It should be noted that the use of a mean ratio value corresponds to a regression line constrained to pass through the origin whereas Barter’s regression line is not constrained in this way. For a given segment the regression line of Barter will indeed provide a closer fit to the data than the constrained regression line since there is an additional degree of freedom. A close fitting regression equation, however, may not provide good estimates for data other than the sample data upon which the equation is based. Receiurd in jinnlform

18 July 1988.

Dapena (1978) developed a method of scaling the segmental moment of inertia values given by Whitsett (1963) using body mass and standing height of a subject. The method assumed that segment mass was proportional to body mass, segment length was proportional to standing height and that segmental moments of inertia were proportional to mass times length squared. Such a non-linear approach based upon theoretical considerations is intuitively more appealing than the routine use of linear regression. Forwood et al. (1985) used a modified scaling method based upon body mass and segment length but found that this did not improve the inertia estimates significantly. A non-linear scaling procedure which uses only segmental measurements, such as length and perimeter, may be more successful. This paper compares the estimates of segmental moments of inertia provided by linear regression equations and by non-linear equations based on theoretical considerations. THEORETICALCONSIDERATIONS In the study of Chandler et al. (1975) anthropometric measurements of six cadavers were taken prior to dissection. The cadavers were adult males with

body masses between 50.6 and 89.2 kg and heights between 1.64 and 1.82 m. Each cadaver was dissected into 14 segments and moments of inertia were determined for each segment about six axes using a pendulum technique. The directions of the principal axes and the corresponding principal moments of inertia were then calculated from these six moment of inertia values. Thus for each body segment there are length, width and perimeter measurements in addition to principal moment of inertia values. These data provide an opportunity to relate segmental principal

683

684

M. R. YEADON and M. MORLOCK

moment of inertia values to standard anthropometric measurements. Linear approach

Suppose that the moment of inertia I of a body segment about some axis is an unknown functionfof n anthropometric measurements xi (i= 1, n). Thus: I=f(x,,

x2, . . . x,).

If a, is a particular value of the measurement (i= 1, n) a Taylor series expansion yields:

/

d

(1) xi h

I=f(a,,

a*, . . .

1n

a2c

(2)

+12$(X,-ai)(Xj-Uj)

1

II

where third and higher order terms have been neglected. If second order terms are neglected, equation (2) takes the form: I=k,+i

1

kixi

(3)

where k,=f(a,,

n af

a,, . ..a,)-I---ai I

axi

4

af and ki=--. axi

b W

Fig. 1. A torso segment of depth d, width w and height h.

Equation (3) represents the n-dimensional tangent plane to the surface defined by equation (1). If the differences (xi - ai) are small enough then equation (3) will give a close approximation to the actual value of I. If only a certain number of points on the surface I=j-(Xi, x2, . . . x,) are known then the surface may again be approximated by a plane of the form given by equation (3). Linear regression determines the constants ki (i=O, n) by minimizing the mean squared distance of the data points from the plane. The linear regression equation will provide a close approximation to the functionfproviding the curvature of the surface I =j(x,, x2, . . . x,) and the range of values of each anthropometric measurement are small.

the expressions: (5)

IX= M [b2w2 + b,h2]

(6)

1,=M[b,d2+b,h2]

(7)

I,=M[b,d2+b2w2]

(8)

where the positive constants b,, b,, b,, b, are characteristic of the density distribution p. For example the class of rectangular blocks with a uniform density of 1 has constants b,= 1, b, = b, = b, = l/12. Substituting equation (5) in equations (6), (7) and (8) gives: I,=dwh [c2w2 -tc,h2]

Non-linear approach

Consider a torso segment with a given density distribution p(x, y, z) such that the principal inertia axes x, y, z are aligned with the anterior-posterior, medial-lateral and longitudinal anatomical axes. Another torso segment will be said to have a similar density distribution D(x, y, z) providing there exist constants a,, a2, a3 such that: D(x, Y, z)=p(a,x,

M=b,,dwh

a,y, a&.

(4)

If a number of torso segments have similar density distributions then their moments of inertia will be related to anthropometric measurements in the following way. Let the dimensions of a torso segment be defined by depth, width and height measurements d, w and h (Fig. 1). The mass M and principal moments of inertia I,, I,, I, about the mass centre will be given by

(9)

I,,=dwh[c,d2+cSh2]

(10)

I,=dwh[c,d2+c2w2]

(11)

where ci, c2 and cJ are positive constants which are characteristic of the density distribution of a torso segment. If d, w, h, I,, I,, and I, are known for n torso segments then c , , c2, c3 may be determined by minimizing E where: E=f:[dwh(c2w2+c,h2)-IX-J2 +i

[dwh(c,d2+c3h2)-IJ2

1 [dwh(c,d2+c2w2)-I,]‘.

+i 1

685

Estimation of segmental inertia parameters For E to be a minimum aE aE -=-=-= ac, ac,

aE

a+

0.

Thus: 2S,,c, +S21~t+S31~g=R1 S,,c,f2S,,c,+S,,c,=R2 S13c1 +S23~2+2S,,~3=R3

where S, 1=I

(d3wh)’

S,, = 1 (d2w2h)2 S, 1=I

(dZwh2)2

s,,=s,*

S22 = 1 (dw3h)’ S,, = 1 (dw’h’)’ s,3=s31 s23=s32

S,, =I

(dwh3)2

R, =c

d3wh(I,+I,)

R2=x

dw3h(I,+I,)

R3=~dwh3(I,+I,).

Inequality (19) is similar to the triangle inequality which states that the length of one side of a triangle must be smaller than the sum of the lengths of the other two sides. Any non-planar mass distribution must satisfy inequality (19). In the study of Chandler et al. (1975) moments of (12) inertia were determined about the three anatomical (13) axes of each torso segment and also about three other (14) axes. The principal moments of inertia and directions of the principal axes were calculated using these six moment of inertia values. Of the six cadaver torso segments only one has principal moments of inertia which satisfy inequality (19). In addition the X and Y principal axes, rather than lying close to the anatomical X and Y axes, lie approximately midway between the posterior -anterior and medial -lateral anatomical axes. Such a systematic error must have arisen during the calculation of the three principal inertias from the six experimental values. Because of these errors in the calculated principal inertias and principal axes, the original moment of inertia values about the anatomical axes were reconstructed from the principal inertias and the directional angles of the principal axes in the following way. If the principal axes make angles z,, ct2,a3 with an anatomical axis then the moment ofinertia I about the anatomical axis is given by:

are then obtained by solving equaCl, c2, c3 tions (12), (13) and (14) for these three unknowns. For the remaining body segments it will be assumed that I,= I,,. This assumption is necessary since depth and width measurements are not available in the data of Chandler et al. (1975). The assumption is also reasonable since I, and I, are generally close in value for segments other than the torso. Assuming that depth d and width w are proportional to perimeter p, equations (9), (lo), (11) give: I,=k,p4h

(15)

I,=+IZ+k2p2h3

(16)

where I, = I, = I, will be called the transverse moment of inertia. If the variables are transformed using the substitutions u=p4h and u=p2h3 equations (15) and (16) take the forms: I==k,u (17) I,-+I;=k,c.

(18)

The constants k, and k, may then be determined by using linear regression. Thus linear regression may be used to determine the constants in the non-linear equations (15) and (16). Available data

From equations (9), (lo), (11) it can be seen that: 1,<1,+1; since the constants

c1 ,

c2, c3 are positive.

(19)

1 = I,, co? !I, + I, co? a2 + I,, co? !x3

(20)

where I,,, I,,, I:, are the principal moments of inertia. This procedure was followed for all body segments to produce a pseudo data set of segmental moment of inertia values about the anatomical axes.

METHOD

Data on anthropometric dimensions and segmental moments of inertia were taken from the study of Chandler et al. (1975). The segmental moments of inertia about anatomical axes were derived from the published data as described above. For segments other than the torso, I, and I, values differed by less than 5% in general and were averaged to give I,, the mean transverse moment of inertia. Linear multivariable and non-linear regression equations were determined for I, and I,, the transverse and longitudinal moments of inertia of the head and right limb segments. For the torso segments linear and nonlinear equations were determined for the anterior-posterior, medial-lateral and longitudinal moments of inertia I,, I, and I,. For each segment a single length h and up to three perimeters p, , p2, pj were available from the data of Chandler et al. (1975). For each non-torso segment linear regression equations for I, and I, were based upon h and the available perimeters. One perimeter was used for the head, two perimeters for the hand and three perimeters for the remaining segments.

M. R. YEADON

686

In addition to the length h and perimeters pl, pz and p3 of the torso segment, three width measurements wl,

w2 and wj were available. The linear regression equations for I,, I, and I, were based upon h, p 1,p2, p3, w1, w2 and wj. Linear multi-variable regression equations were derived using the procedure in SPSS Update 7-9 (Hull and Nie, 1981). A forward stepwise inclusion model was used with an F-to-enter of a = 0.05. In the first step the program selected the anthropometric variable which explained the highest amount of variance of the inertia variable, providing it was significantly correlated wilh the inertia variable at the a=0.05 level. Further steps were performed only when the F change due to the inclusion of another anthropometric variable was significant at the a = 0.05 level. The non-linear equations for the non-torso segments were based on the length h and a mean perimeter p. When there were three perimeters the mean perimeter was calculated as p = (PI + 2p, + p,)/4, the endpoint perimeters p1 and p3 having half the weighting of the mid-perimeter pz. The SPSS linear regression package was used with the longitudinal and transverse moments of inertia I, and I, in the forms: I,= k,p4h

(15)

Z,=fZ,+k,p2h3

(16)

where k, and k, are constants. In a first step the coefficient k, was determined by obtaining the linear regression line for I, against p4h which was forced to pass through the origin. The constant k, was calculated only if the correlation between p4h and I, was significant at the a =0.05 level. The same procedure was then used to determine k, for the equation: [I,-+k,p4h]

= k2p2h3

obtained from equations (15) and (16). For the torso segment a mean depth d was calculated from the mean perimeters p and mean width w using the formula d =(p-2w)/(n-2) for a stadium (Yeadon, 1989). The non-linear equations were based upon the mean depth d, the mean width w and the length h, as described in the theory section. The linear and non-linear equations were compared in three ways. (1) for the torso, head and right limb segments unbiased standard error of fit estimates sr were determined for each equation using the formula: sy= ;(Ji-Z,)‘/(6-k) 1

1

and M. MORLOCK

tion (21) with k=O. Since the limb equations were based upon the right limb data, the left limb data comprise an independent sample for the evaluation of the equations. (3) In order to compare the linear and non-linear equations outside the anthropometric range of the sample, segmental moments of inertia were determined from anthropometric measurements of a 10 yrold boy using the inertia model of Yeadon (1989). These moments of inertia were used as criteria data values to compare the inertia estimates provided by the linear and non-linear equations. RESULTS AND DISCUSSION

Table 1 lists the percentage standard error of fit estimates for the linear and non-linear equations. In two of the 16 cases linear fits were rejected since there was no significant correlation at the 5% level between any anthropometric variable and the moment of inertia. For the non-linear fits there was always significant correlation between the transformed anthropometric variable and the moment of inertia. This suggests that the non-linear equations were more appropriate than the linear equations. On the other hand the standard error of fit estimates have average values of 11% for the linear equations and 15% for the nonlinear equations. This suggests that the linear equations fit the data better than the non-linear equations. Even if the linear equations do fit the right limb data better than the non-linear equations, it does not necessarily follow that the linear equations will provide better estimates of inertias for segments other than those upon which the equations are based. Table 2 lists the standard error estimates of the left limb inertias. Since the equations are based upon the right limb data, the left limb data comprise an independent sample for the evaluation of the equations. The standard error estimates have average values of Table 1. Standard errors of fit ~~(1)and +(I) of the linear and non-linear equations expressed as a percentage of the mean of the six inertia values Linear

Head

112(21) 1

where Ji are the inertia estimates given by the equation, Ii are the inertia data values and k is the number of constants used in the equation (Spiegel and Boxer, 1961). (2) Standard error estimates for the limb equations were also determined using the left limb data as a cross-validation sample in conjunction with equa-

Torso Right Right Right Right Right Right

upper arm forearm hand thigh calf foot

Mean value

Non-linear

8

9

5 * 7 17 11 * 5

13 9 11 11 13 34 8

15 15 12 9 17 5 7 7

13 15 17 23 28 17 33 11

9

13

11

20

Note 1: the standard error values for the torso inertias I, and I, have been averaged to give a mean value ~(1,). Note 2: * indicates that there was no significant correlation at the 5% level between any anthropometric variable and the moment of inertia.

Estimation of segmental inertia parameters Table 2. Standard errors sL(Z) and +,(I) of the inertia estimates given by the linear and non-linear equations using the left limb data as a cross-validation sample Linear

Non-linear

sL.(lz)

%(I,)

%(lt)

Left Left Left Left Left Left

upper arm forearm hand thigh calf foot

Mean value

SN(lz) W)

W) * 19 14 29 * 10

22 25 39 32 16 14

10 11 15 8 5 7

12 15 38 13 14 8

18

25

9

17

Note: * indicates that there was no equation based upon the right limb data due to the lack of significant correlation between each anthropometric variable and the moment of inertia. 21% for the linear equations and 13% for the nonlinear equations. Moreover, 11 of the 12 mean standard errors for the non-linear equations were smaller

than the corresponding mean errors for the linear equations. This indicates that the non-linear equations provide better estimates of the segmental inertias than the linear equations for limb segments which lie within the anthropometric range of the right limb samples. Although the right limb data are independent in the sense that they are not used in the formulation of the linear and non-linear equations, it should be recognized that both left and right limbs came from the same six cadavers. This may have the effect of reducing the standard error estimates. It is possible to consider the results presented in Table 2 in greater detail. For the non-linear equations the standard error estimates of the right hand inertias are larger than those of the other segments. It may be speculated that this arises from different amounts of hand flexion which would affect the inertia values. In view of the small sample size, however, such detailed consideration of the results should be undertaken with some caution. It may be observed from Table 1 and Table 2 that the standard error estimates are of similar magnitude for the non-linear equations whereas for the linear equations the standard error estimates in Table 2 are larger than those in Table 1. This may be edplained in the following way. The choice of anthropometric variables in a non-linear equation was based upon theoretical considerations whereas the decision to include an anthropometric variable in a linear equation was made statistically. A consequence of this statistical decision making is that the linear equations may reflect correlations between errors in the anthropometric and inertia values of the right limbs. Because the number of degrees of freedom in the linear equations was only 2, 3 or 4, such data errors could have produced large errors in the linear equations. These errors in the linear equations are not reflected in the standard error estimates in Table 1 but are apparent

687

in the standard error estimates in Table 2 since the latter are based upon an independent data set. Outside the anthropometric range of the particular segments upon which the equations are based it is to be expected that the linear equations will not perform well. It is of interest, however, to determine whether the non-linear equations provide reasonable estimates of the segmental moments of inertia outside the sample range. Since the non-linear equations model the segments in a geometrical way they should be valid for a different population providing the segmental mass distribution is similar. This means that segmental densities and, to some extent, segmental shape should be comparable if the non-linear equations are to be applied to a different population. There is some evidence that the mass distributions of human body segments do not change greatly. In the study of Jensen (1986) it was found that the ratio of radius of gyration to segment length remained relatively constant for boys between the ages of 4 and 15 yr. In addition, the use of more than one perimeter in the non-linear equations permits some change in this ratio. Thus, there can be some expectation that the non-linear equations will perform reasonably well on another population. Table 3 lists the percentage residuals of the segmental inertia estimates given by the linear and nonlinear equations for 10 yr-old boy. The mean percentage residuals were 286% for the linear equations and 20% for the non-linear equations. Since the criterion values are based on a purely theoretical model and the non-linear estimates are based upon a simple model incorporating experimental data, this level of agreement (20%) may be regarded as quite good. This positive result with one subject indicates that it may be worthwhile to test the non-linear equations for a wide range of subjects using values calculated from an inertia model as criteria. Table 3. Percentage residuals rL(I) and rN(I) of the inertia estimates given by the linear and non-linear equations for a subject lying outside the sample range Linear TL(l,) rL(Iz) W)

Non-linear r&J rN(li) (%I

Head Torso Upper arm Forearm Hand Thigh Calf Foot

29 44 * 335 121 348 * 23

2 439 534 783 411 217 36 94

13 30 20 22 19 19 11 32

17 18 14 26 6 37 18 19

Mean value

150

422

21

20

Note 1: percentage residuals for right and left limbs have been averaged. Note 2: percentage residuals for the torso inertias I, and I, have been averaged to give a mean value r(l,). Note 3: * indicates that there was no equation based on the original right limb data set.

688

M. R.

YEADON and

It may be concluded that the non-linear equations are superior to the linear equations and that the nonlinear equations can provide reasonable estimates of segmental moments of inertia even when the anthropometric measurements lie outside the sample range of Chandler et at. (1975). Non-linear regression equations expressing segmental moments of inertia as functions of anthropometric variables are presented in the Appendix. For the limb segments these equations are based upon both the right and left limb data of Chandler et al. (1975) rather than just the right limb data.

M.

MORLOCK

Table 4. (Contd.) Segment

Torso

h PI? WI P2,

h P1 P2 P3

Forearm

h Pl

Acknowledgement-This study was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

P2 P3

h

Hand

PI PZ

REFERENCES

Barter, J. T. (1957) Estimation of the mass of body segments. WADC Technical Report 57-260, Wright-Patterson Air Force Base, OH. Chandler, R. F., Clauser, C. E., McConville, J. T.;Reynolds, H. M. and Young, J. W. (1975) Investigation of inertial properties of the human body. AMRL Technical Report 74-137, Wright-Patterson Air Force Base, OH. Clauser, C. E., McConville, J. T. and Young, J. W. (1969) Weight, volume and center of mass of segments of the human body. AMRL Technical Report 69-70, WrightPatterson Air Force Base, OH. Dapena, J. (1978) A method to determine the angular momentum of a human body about three orthogonal axes passing through its centre of gravity. J. Biomechanics 11, 251-256. Forwood, M. R., Neal, R.4. and Wilson, B. D. (1985) Scaling segmental moments of inertia for individual subjects. J. Biomechanics 18, 755-761. Hinrichs, R. N. (1985) Regression equations to predict segmental moments of inertia from anthropometric measurements: an extension of the data of Chandler et al. (1975). J. Biomechanics 18, 621-624. Hull, C. H. and Nie, N. H. (1981) SPSS Update 7-9, New Procedures and Facilities. McGraw-Hill, New York. Jensen, R. K. (1986) Body segment mass, radius and radius of gyration proportions of children. J. Biomechanics 19,

Length: trochanter to acromion Perimeter, width: nipple Perimeter, width: umbilicus Perimeter, width: hip Length: shoulder centre to elbow centre Perimeter: below axilla Perimeter: maximum Perimeter: elbow Length: elbow centre to wrist centre Perimeter: elbow Perimeter: maximum Perimeter: wrist Length: wrist centre to tip of finger III Perimeter: wrist Perimeter: metacarpal-phalangeal joints

Note: the hand is in a flexed/relaxed orientation Thigh

h PI P2

P3 h

Calf

P1 Pl P3

h

Foot

PI P2 Ps

Length: hip centre to knee centre Perimeter: below gluteal furrow Perimeter: mid-thigh Perimeter: knee Length: knee centre to ankle centre Perimeter: knee Perimeter: maximum Perimeter: minimum near ankle Length: heel to toe Perimeter: minimum near ankle Perimeter: arch Perimeter: ball

For segments with three perimeters the mean perimeter p is calculated as p =(pl +2p, + p,)/4, for the hand p=(p, +p2)/2, for the head p=pl, for the torso the mean

width w =(wl + 2w, + wq)/4. For the non-torso segments the segmental moments of inertia I, and I, about the longitudinal and transverse axes are given by:

359-368.

Spiegel, M. R. and Boxer, R. W. (1961) Theory and Problems of Stntistics, pp. 243-244. McGraw-Hill, New York. Whitsett, C. (1963) Some dynamic response characteristics of weightless man. AMRL Technical Report 63-18. WrightPatterson Air Force Base, OH. Yeadon, M. R. (1989) The simulation of aerial movement, Part II: a mathematical inertia model of the human body. J. Biomechanics (submitted).

w2

P3r w3

Upper arm

Definition

Variable

l,=k,p4h

(15)

l,=~l,+k,p=h=

(16)

where linear measurements are in m and moments of inertia are in kgm2. Values for k, and k, are given to three significant figures.

Table 5 Segment APPENDIX: NON-LINEAR

EQUATIONS

FOR SEGMENTAL MOMENTS OF INERTIA Table 4.

Segment Head

Variable h PI

Definition Length: chin to vertex Perimeter: above ear

Head Upper arm Forearm Hand Thigh Calf Foot

k,

k,

0.701 0.979 0.810 1.309 1.593 0.853 1.001

2.33 6.11 4.98 7.68 8.12 5.73 3.72

689

Estimation of segmental inertia parameters For the torso segment the segmental moments of inertia I,, lY, I, about anterior-posterior, medial-lateral and longitudinal axes are given by: I,=dwh

[c,w*+c$~]

I,=dwh[c,d*+c,h’]

Iz=dwh[c,d2+c2wZ]

where the mean depth d = (p- Zw)/(n - 2). Table 6

(22) (23)

Segment Torso

(24)