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PHYS THER
Vol. 86, No. 9, September 2006, pp. 1263-1270
DOI: 10.2522/ptj.20060034
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Research Reports

Allometric Analysis of Physical Performance Measures in Older Adults

Yong-Hao Pua

YH Pua, BSc (Hons), is Senior Physiotherapist, Physiotherapy Department, Rehabilitative Services, Alexandra Hospital, 378 Alexandra Rd, Singapore 159964

Address all correspondence to Mr Pua at: puayonghao{at}gmail.com


Submitted January 31, 2006; Accepted April 26, 2006


   Abstract
 
Background and Purpose. Body mass is a confounding variable in human performance, and adjusting physical performance measures for body mass differences would allow meaningful individual and group comparisons. The purpose of this study was to allometrically determine the relationship between body mass and handgrip and ankle dorsiflexor performance on the Timed "Up & Go" Test (TUGT). Subjects. One hundred thirty-one subjects (33 male and 98 female) participated. Methods. All physical performance measures were adjusted for the influence of body mass, sex, and age using an allometric scaling procedure. Results. For handgrip force, the body mass exponent from allometric analysis was 0.63. For the TUGT, the body mass exponent was 0.073. For ankle dorsiflexor force and torque, the body mass exponents were 0.82 and 0.91, respectively. Discussion and Conclusion. The body mass exponents for handgrip force and the TUGT agree with previous clinical data and theoretical expectations. Studies investigating normalized handgrip force in older adults should consider an allometric scaling approach.

Key Words: Aging • Body mass • Log-linear modeling


   Introduction
Top
 Abstract
 Introduction
Method
 Results
 Discussion
 Conclusion
 References
 
Improving and evaluating balance and muscle performance are important concerns in geriatric rehabilitation and aging research.1 Body mass is a known confounding variable in human performance,2 and adjusting physical performance measures for differences in body mass would allow meaningful individual and group comparisons. Furthermore, the extent to which body mass is related to physical performance measures would add valuable insight into the construction of normative reference standards. To date, however, the relationship between body mass and physical performance measures has seldom been determined empirically within the geriatric population. Most authors have used non-normalized results,3–5 or they have arbitrarily chosen the ratio standard approach of scaling.6–9

It is known that most physiological functions vary allometrically with body mass,10 and allometric modeling has attracted renewed attention in the arena of exercise science. Specifically, a body mass–adjusted index of performance parameter, Pnorm, may be modeled using the formula, Pnorm = P·M-b, where P denotes the measured physical performance, M is body mass, and b (b value) is the derived mass exponent. Under the presumption of geometric similarity, muscular force is proportional to M2/3, and muscular torque is the product of muscular force and moment arm (because muscular force {propto} muscle cross-sectional area {propto} height2 and M {propto} height3, so muscular strength {propto} M2/3). Inasmuch as moment arm, a linear dimension, is proportional to M1/3, it follows that the b values for modeling muscular force and torque output are 0.67 and 1.0, respectively (M2/3·M1/3 = M).11

Jaric12,13 was among the first investigators to articulate the concept that the magnitude of the b values is dependent on the groups of physical tests studied. Accordingly, Jaric13 suggested that b=0 for tests of rapid movements (eg, jumping, running) and b=0.67 for tests of exertion of external forces (eg, gripping, weight lifting). Despite mounting evidence in young adult and athletic populations11,14,15 demonstrating preliminary support for Jaric's13 postulates, little is known regarding these associations in older adults. Given that Timed "Up & Go" Test (TUGT) handgrip and ankle dorsiflexor performance are important predictors of falls and function in older adults,3–5,7,16–18 the purpose of this study was to allometrically determine their relationship to body mass. Based on the aforementioned considerations, 3 hypotheses were examined. The first hypothesis was that the mass exponent would be 0.67 for handgrip force. The second hypothesis was that, when modeling ankle dorsiflexor force and torque, the mass exponents would be 0.67 and 1.0, respectively. Given that the TUGT requires participants to walk at a comfortable fast and secure pace,18 the decision was made to classify the TUGT as a "test of rapid movement"13 in this study. Consequently, the third hypothesis was that the mass exponent would be 0 for TUGT measurements.


   Method
Top
 Abstract
 Introduction
 Method
Results
 Discussion
 Conclusion
 References
 
Subjects

This study was a cross-sectional survey of community-dwellers who participated in a 1-day health screening. This research was carried out with local ethical approval, and all participants gave informed consent. All subjects in the study walked unaided and did not participate in any forms of resistance training. They did not have any orthopedic pathology, any diagnosed neurological deficit, or any other medical conditions that may have affected their ability to participate in the study. In allometric scaling, a fundamental assumption is that the physiological variable is influenced by muscle mass and body mass is used as a proxy in the absence of muscle mass estimates.19 In order to secure a more homogenous sample, only subjects with a body mass index (BMI) of 25 kg/m2 or less were recruited. Consequently, only 131 subjects (33 male and 98 female) satisfied all of the selection criteria. The subjects had a mean age of 62.8 years (SD=6.89, range=50–84). Of the 131 subjects, ankle dorsiflexor data were available for only 121 subjects.

Measurements

For all subjects, height (to nearest 0.1 cm) and weight (to the nearest 0.1 kg) were measured by one physical therapist who was not involved in assessing the physical performance measures. All measurements were taken with subjects in light indoor clothes without shoes. Weight was measured using calibrated digital scales,* and height was measured with a portable stadiometer.* All subjects also were questioned regarding their dominant upper and lower extremities (preferred for throwing a ball and kicking a ball, respectively).20

Maximum handgrip force was measured using a Baseline hydraulic handgrip dynamometer.{dagger} Subjects were seated, and the dynamometer was adjusted until the proximal interphalangeal joints were flexed to 90 degrees with the elbow flexed. After one practice trial, the subjects were instructed to increase the handgrip force to their maximum and to sustain the contractions for 5 seconds. For each subject, handgrip force was measured bilaterally, and the peak force recorded represented the maximum handgrip force. Reliability of handgrip force measurements was not assessed in this study, although excellent interrater reliability was previously demonstrated in a study of 30 community-dwelling women.21

Isometric ankle dorsiflexor force was measured using a Nicholas handheld dynamometer.{ddagger} During testing, the subjects were positioned in long sitting with the hip flexed between 70 and 80 degrees. After one practice trial, the subjects were instructed to dorsiflex their foot as hard as they could, for a 5-second isometric contraction, against the end piece of the dynamometer. The end piece was placed just proximal to the metatarsophalangeal joints. Measurements were obtained bilaterally, and all tests were "make" tests. For each subject, the peak force represented the maximum ankle dorsiflexor force. Additionally, with the ankle at 0 degrees of dorsiflexion, the perpendicular distance (to nearest 0.001 m) from the lateral ankle malleolus to the base of the fifth metatarsal bone was measured using a measuring tape. Subsequently, ankle dorsiflexor torque was calculated by multiplying the measured force by its perpendicular distance. Sixteen subjects underwent a retest session to determine the intratester reliability of the ankle dorsiflexor force and torque measurements. Intraclass correlation coefficients (ICC[3,1]) for measurements of force and torque were .63 and .73, respectively.

Timed "Up & Go" Test

The TUGT was performed as described by Podsiadlo and Richardson.22 To perform the test, subjects sat on a standard-height chair (46 cm high) with armrests. On the command "go," subjects stood up, walked 3 m at a normal and safe pace, turned around, returned to the chair, and sat down. The TUGT was measured with a stopwatch. A practice trial was given, followed by 2 timed trials. Test-retest reliability was calculated for the 2 timed trials, and excellent intratester reliability (ICC[3,1]=.90) was demonstrated.

Data Analysis

For handgrip and ankle dorsiflexor performance, data from the dominant side were used for analysis. For the TUGT, the average timing of the 2 trials was used. The influence of body mass (M), age, and sex on physical performance measures (P) was modeled using the general allometric equation, P = Mß1·(ß0 + ß2age + ß3sex). Age was incorporated within the exponential term because of the expected exponential decline in the subjects studied.2 Sex was coded as 1 for female subjects and 0 for male subjects. The allometric equation was linearized by taking natural logarithms of both sides, yielding a corresponding linear equation of the form, lnP = ß1lnM + ß0 + ß2age + ß3sex. Next, the lnM/sex/age interaction terms were entered into the regression models to determine the viability of a common mass exponent for both sexes across the age range. If the interaction terms were not significant, the regression analyses were repeated, excluding the interaction terms, to generate the mass exponents.23

After the determination of regression equations, regression diagnostics were performed by the analysis of the residuals. Linearity and homoscedasticity were investigated by plotting the residuals against the fitted values. Visual inspection of the residual distribution was performed to identify possible influential points.

A Pearson correlation (r) was performed between the independent and dependent variables to further assist in the interpretation of results. Significance was at the .05 level. All statistical analyses were performed using SPSS for Windows (Version 11.0).§


   Results
Top
 Abstract
 Introduction
 Method
 Results
Discussion
 Conclusion
 References
 
Table 1 summarizes the subjects' descriptive characteristics. The correlations between the physical performance measures and independent variables are reported in Table 2. The correlation between age and TUGT scores was moderately high (r=.528) at P<.01, whereas the correlation between body mass and TUGT scores was weak (r=–.117, P=.18). The correlations between body mass and handgrip force and between sex and handgrip force were moderately high (r=.621 and –.594, respectively) at P<.01. Body mass was correlated to ankle dorsiflexor force (r=.227, P<.05) and torque (r=.286, P<.01). The comparison between ankle dorsiflexor force and torque was statistically significant at P<.001.


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  		Table 1. Descriptive Statistics of the Sample (N=131)

 

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  		Table 2. Matrix of Simple Correlations (r) Among Variablesa

 
For all measurements, the initial multivariate analyses to determine commonality of slopes did not produce any significant interaction terms. Consequently, regression analyses were repeated without the interaction terms to generate the mass exponents (ß1). Table 3 shows the regression coefficients for the independent variables of the allometric models. For handgrip force, the mass exponent (0.63) approximated the theoretical 0.67, and a mass exponent of 1.0 could be excluded on a 95% confidence interval (95% CI) basis (95% CI=0.34–0.91). Although the mass exponent (0.91) for ankle dorsiflexor torque was greater than that for ankle dorsiflexor force (0.82), the mass exponent of the latter measurement was not significantly different from 1.0 and 0 (95% CI=–0.04–1.70, P=.06). For the TUGT, the mass exponent (0.073) was not statistically different from the predicted value of 0 (P=.56).


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  		Table 3. Allometric Analysis Examining the Relationship Between Physical Performance Measures and Body Mass, Controlling for Age and Sexa

 
All multiple regression analyses were examined for collinearity, heteroscedasticity, and linearity. For all regression models, the tolerance among the independent variables was greater than 0.5, indicating relatively low collinearity. Examination of plots of predicted values versus residuals indicated that the residuals were evenly dispersed about zero, thus meeting the assumption of homoscedasticity. For handgrip force and ankle dorsiflexor torque, the linear assumption was evaluated by examining plots of lnM versus the unstandardized residuals. The Figure shows the scatterplot for handgrip force, and no systematic variations were apparent. Finally, using the data from Table 3, a power function ratio was constructed for handgrip force and ankle dorsiflexor torque. The correlation between the adjusted handgrip force and body mass was 0.02, indicating that the allometric process produced a physical performance measure that was body-mass independent.


Figure 1
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  		Figure. Plot of raw residuals from the allometric regression analyses of handgrip force against ln (body mass).

 

   Discussion
Top
 Abstract
 Introduction
 Method
 Results
 Discussion
Conclusion
 References
 
In this study, an allometric approach was used to determine the relationship between body mass and 2 groups of physical performance tests in older adults. This study also examined whether allometric mass exponents were different between muscle force and torque.

In my sample, the mass exponent for handgrip force was 0.63. This result is close to the predicted 0.67, and a value of 1.0 can be excluded on a 95% CI basis (95% CI=0.31–0.91). The present findings are consistent with those of Foley and colleagues,24 who reported a mass exponent of 0.4 (95% CI=0.026–0.78) in 104 older adults. These findings are also in agreement with reports from Markovic and Jaric14 and Vanderburgh et al,15 who found allometric exponents of 0.26 and 0.51, respectively, in college-aged men and women. Because the relationship between body mass and handgrip force is nonlinear, the results of the present study and other studies raise questions about the use of ratio-standards scaling to normalize handgrip force.

The ability to dorsiflex the ankle is an important requirement for safe ambulation, and cross-sectional7 and longitudinal17 studies have suggested that older adults with lower ankle dorsiflexor force are at greater risk for falling. Given that measured forces vary as a function of distance of the dynamometer to the ankle joint, measurements of ankle dorsiflexor torque are more meaningful than measurements of ankle dorsiflexor force because they allow a more accurate comparison between subjects. In this study, the correlation analysis indicated that when a 3-dimensional variable (body mass) is paired with a 2-dimensional variable (ankle dorsiflexor force), the resulting correlation (r=.227) is lower (P<.001) compared with the correlation calculated when two 3-dimensional variables (body mass and ankle dorsiflexor torque) are paired (r=.286). Moreover, the mass exponent for ankle dorsiflexor torque (0.91) was closer to 1.0 than 0.67. Although a point estimate represents the single most plausible value in light of the observed data, the standard errors that surround the mass exponents were large. Specifically, for ankle dorsiflexor force, the 95% CI of the mass exponent included 1.0 and 0 (P=.06). Collectively, the results of the study fail to reject the second null hypothesis that mass exponents are no different for measurements of ankle dorsiflexor torque and force.

Comparing the results of this study with those of other studies is difficult because the present study is the first, to date, to allometrically examine ankle dorsiflexor force and torque as measured by a handheld dynamometer. In contrast, Owings et al25 examined ankle dorsiflexor torque in 79 older adults using an isokinetic device. Owings et al25 adjusted torque values for differences in body size (body mass x height) and reported a 95% CI of 0.71–1.25 for the body size exponent. Although the results of the study by Owings et al25 cannot be directly related to those of the present study, it is noteworthy that a linear relationship between torque and body weight would argue for a body size (body mass x height) exponent of 0.75 (because body size {propto} M·M1/3=M4/3 and muscular torque {propto} M, so muscular torque {propto} body size3/4) and indirectly supports the hypothesis of this study.

In this study, the correlation between age and the ankle dorsiflexor data was not statistically significant (r=–.155 and –.137 for force and torque measurements, respectively). These findings are in contrast to previous reports of an inverse association between age and ankle muscle force and torque.6,10 In the present study, measurements were obtained by 4 physical therapists with at least 4 years of clinical experience. However, the examiners did not use the handheld dynamometer routinely in their practice. The ICCs (3,1) for ankle dorsiflexor force and torque were .63 and .73, respectively. In the study by Andrews et al,6 the authors did not state the ICC values for ankle dorsiflexion. However, they reported that the interrater reliability was disconcerting despite considerable experience of their testers. Furthermore, Gunter et al5 reported poor reliability of handheld dynamometry measurements obtained in a sample of older people. Although other authors26,27 have found excellent reliability of measurements obtained using handheld dynamometry, it must be acknowledged that many factors such as testing position, study population, and the ability of the examiners to stabilize the dynamometer influence the repeatability of measurements.6,26,27 Thus, it seems possible that the relatively low reliability of ankle measurements in this study may obscure their relationship with age and body mass and partially explains the discrepant results found in this and other studies.

As may be predicted, allometric analysis of the TUGT scores generated a mass exponent that was close to 0. This finding is in agreement with that of Payette et al,21 who found no correlation between TUGT scores and fat-free mass in 30 frail elderly women. The results of the present study extend Payette and colleagues'21 findings to include a more heterogeneous group of subjects (male and female) with a considerably wider age range. In contrast, the results of this study do not agree with those of Bischoff and colleagues,18 who found a weak bivariate association (r=.18, P<.0001) between body mass and TUGT scores in 413 community-dwelling older women. Although the authors studied a bigger sample size with a large body mass range (43–115 kg), the BMI range of their subjects was concomitantly larger (17.9–42.4 kg/m2). Ostensibly, the true relationship between body mass and TUGT scores cannot be readily discerned from their data due to the heterogeneity of body composition in the sample studied, which indicates that greater confidence can be placed on the findings of the present study.

Several potential limitations exist in this study. First, in order to secure a homogenous sample on which allometric analysis is performed, this study was necessarily limited to a sample of free-living community dwellers who did not fall in the past year. Second, as mandated by the inclusion criteria of the study, the BMI of the subjects did not exceed 25 kg/m2. Given that the relationship between BMI and body fat is age- and sex-dependent,28 it is possible that BMI may be unable to reliably differentiate between lean and fat mass. Thus, it is reasonable to question whether homogeneity of body composition can be completely assumed in our sample. Third, although some studies4,6,29 have demonstrated an association between height and physical performance measures, height was not included in the allometric analyses. In this study, similar to the findings of previous studies,9,29 height was correlated with age (r=–.22, P<.01) and body mass (r=.57, P<.000). In contrast, the correlation between body mass and age was considerably lower in the present study (r=–.17, P=.048). Consequently, modeling height along with body mass resulted in problems of collinearity (tolerance for ln[height]=0.34). In the presence of collinearity, the mass exponents generated may be numerically inaccurate.23


   Conclusion
Top
 Abstract
 Introduction
 Method
 Results
 Discussion
 Conclusion
References
 
The mass exponents for handgrip force and TUGT scores agree with previous clinical data and theoretical expectations. Future studies reporting body mass–adjusted handgrip force should consider using an allometric scaling approach. For ankle dorsiflexor torque and force, future research should examine whether their mass exponents differ by using direct estimates of muscle mass and obtaining more reliable ankle dorsiflexor measurements.30 Ultimately, the value of allometric scaling resides in its ability to reveal relationships that may otherwise remain obscure.24,25


   Footnotes
 
The author thanks Mrs Marguerita Dass for the direction of the study and his fellow physical therapists and physical therapist assistants for their help with data collection. The author also acknowledges the cooperation of the study participants.

This research was carried out with local ethical approval from the Domain-Specific Review Board of the National Healthcare Group (NHG).

* Seca, Hamburg, Germany. Back

{dagger} Fabrication Enterprises Inc, PO Box 1500, White Plains, NY 10602. Back

{ddagger} Lafayette Instrument Co, 3700 Sagamore Pkwy N, PO Box 5729, Lafayette, IN 47903. Back

§ SPSS Inc, 233 S Wacker Dr, Chicago, IL 60606. Back


   References
Top
 Abstract
 Introduction
 Method
 Results
 Discussion
 Conclusion
 References
 

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