Predictors of age-associated decline in maximal aerobic capacity: a comparison of four statistical models Mitchell J. Rosen, John D. Sorkin, Andrew P. Goldberg, James M. Hagberg and Leslie I. Katzel Journal of Applied Physiology 84:2163-2170, 1998. You might find this additional information useful... This article cites 30 articles, 12 of which you can access free at: http://jap.physiology.org/cgi/content/full/84/6/2163#BIBL This article has been cited by 4 other HighWire hosted articles: Test of the principle of initial value in rat genetic models of exercise capacity L. G. Koch, C. L. Green, A. D. Lee, J. E. Hornyak, G. T. Cicila and S. L. Britton Am J Physiol Regulatory Integrative Comp Physiol, February 1, 2005; 288 (2): R466-R472. [Abstract] [Full Text] [PDF]

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Journal of Applied Physiology publishes original papers that deal with diverse areas of research in applied physiology, especially those papers emphasizing adaptive and integrative mechanisms. It is published 12 times a year (monthly) by the American Physiological Society, 9650 Rockville Pike, Bethesda MD 20814-3991. Copyright © 2005 by the American Physiological Society. ISSN: 8750-7587, ESSN: 1522-1601. Visit our website at http://www.the-aps.org/.

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Greater rate of decline in maximal aerobic capacity with age in endurance-trained than in sedentary men A. E. Pimentel, C. L. Gentile, H. Tanaka, D. R. Seals and P. E. Gates J Appl Physiol, June 1, 2003; 94 (6): 2406-2413. [Abstract] [Full Text] [PDF]

modeling in physiology Predictors of age-associated decline in maximal aerobic capacity: a comparison of four statistical models MITCHELL J. ROSEN,1 JOHN D. SORKIN,1,2 ANDREW P. GOLDBERG,1 JAMES M. HAGBERG,1,3 AND LESLIE I. KATZEL1 1Division of Gerontology, Department of Medicine, University of Maryland, and Geriatric Research, Education, and Clinical Center, Geriatrics Service, Baltimore Veterans Affairs Medical Center, Baltimore 21201; 2Laboratory of Clinical Investigation, Metabolism Section, National Institute on Aging, Baltimore 21224; and 3Department of Kinesiology, University of Maryland, College Park, Maryland 20742

exercise; heteroscedasticity; weighted least squares; loglinear model; ratio

THERE IS AN AGE-ASSOCIATED decline in maximal aerobic ˙ O2 max) in trained and untrained men (6, 8, capacity (V ˙ O2 max is due 10, 13, 22). This age-associated decline in V to many factors, including decreases in maximum heart rate, stroke volume, arteriovenous O2 difference, and skeletal muscle mass and an increase in adiposity (10, 13, 22). Changes in physical activity habits, leading to a sedentary lifestyle, also contribute to the age-associ˙ O2 max (3, 12, 17, 22, 27). There is ated decline in V conflicting evidence on whether regular aerobic exercise training attenuates the age-associated decline in ˙ O2 max (13, 25, 28). V Different statistical models have been used to analyze the relationships among the physiological variables associated with the age-associated decline in ˙ O2 max. The statistical approach employed to adjust the V

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˙ O2 max for body size and composition must be considV ered carefully; the approach must be statistically valid. All other factors being equal, a larger individual would ˙ O2 max than a smaller be expected to have a higher V individual. To account for differences in body habitus, ˙ O2 max to a measure of body size [e.g., the simple ratio of V ˙ O2 max ml · kg body wt21 · min21 or V˙ O2 max ml · kg fat-free V mass (FFM)21 · min21] is often used as the dependent ˙ O2 max variable in studies of the age-related decline in V (3, 12, 27). We will refer to this model as the ratio standard (RS) model. The use of ratios as dependent variables in regression models has been criticized for a variety of reasons (2, 9, 20, 28, 30, 31, 33), but the main drawback is that the ratio Y/X is correlated with X (21, 32, 33). In a classic paper, Pearson (23) showed that if X, Y, and Z are three random variables and X, Y, and Z are uncorrelated, X/Z and Y/Z are correlated under minimally restrictive assumptions (23). In the case of ˙ O2 max, the V˙ O2 max/body size ratio may not make V˙ O2 max V totally independent of body size. Given the physiologi˙ O2max-age relationcal and functional importance of the V ship, it is important that this relationship is assessed in valid statistical and mathematical terms to ensure appropriate interpretation of the results. Two alternatives to the RS model have been used in the statistical analysis of the age-related change in ˙ O2 max: ordinary least-squares (OLS) regression of V ˙ O2 max on a measure of body size, such as FFM or V weight (31), and log-linear (LL) or allometric models (20, 21). OLS and LL models are not without problems. ˙ O2 max models are often heteroscedasOLS residuals in V tic; i.e., they increase in variability as the measure of body habitus increases (20, 33). Heteroscedasticity can affect the standard errors of the parameter estimates and thus adversely affect tests of significance (24). LL models have been criticized for their tendency to overfit biological data, modeling what is sometimes a linear phenomenon with a nonlinear model (1). Additionally, LL models assume a zero intercept, which usually involves extrapolation well beyond the range of observed data (1). A variant of LL is the Box-Cox transformation (4), in which the dependent variable is raised to 2163

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Rosen, Mitchell J., John D. Sorkin, Andrew P. Goldberg, James M. Hagberg, and Leslie I. Katzel. Predictors of age-associated decline in maximal aerobic capacity: a comparison of four statistical models. J. Appl. Physiol. 84(6): 2163–2170, 1998.—Studies assessing changes in maximal ˙ O2max) associated with aging have traditionaerobic capacity (V ˙ O2 max to body weight. Log-linear, ally employed the ratio of V ordinary least-squares, and weighted least-squares models may avoid some of the inherent weaknesses associated with the use of ratios. In this study we used four different methods ˙ O2 max in a crossto examine the age-associated decline in V sectional sample of 276 healthy men, aged 45–80 yr. Sixtyone of the men were aerobically trained athletes, and the remainder were sedentary. The model that accounted for the largest proportion of variance was a weighted least-squares model that included age, fat-free mass, and an indicator variable denoting exercise training status. The model ac˙ O2 max and satisfied all the counted for 66% of the variance in V important general linear model assumptions. The other approaches failed to satisfy one or more of these assumptions. ˙ O2 max declines at the same rate in The results indicated that V athletic and sedentary men (0.24 l/min or 9%/decade) and that 35% of this decline (0.08 l · min21 · decade21 ) is due to the age-associated loss of fat-free mass.

˙ O2 max STATISTICAL MODELS OF AGE-ASSOCIATED DECLINE IN V

2164

˙ O2 maxg ) that maximizes the likelihood the power (i.e., V that the model residuals follow a normal distribution. This method has not been widely used in studies that ˙ O2 max. have examined the age-associated decline in V In this paper an alternative method to modeling ˙ O2 max is proposed: weighted least squares (WLS). WLS V models retain the advantages of OLS models, producing ‘‘best linear unbiased estimates’’ (BLUE) and overcoming the problem of heteroscedasticity (16, 24). Because of the widespread use of the RS models in studies ˙ O2 max, the potential errors caused by the failure of of V ˙ O2 max, and the RS to remove the effect of body size on V the potential problems caused by the heteroscedasticity ˙ O2 max, we believe that inherent in OLS models of V ˙ O2 max is further analysis of the age-related decline in V warranted. Therefore, we compare the results obtained from RS, OLS, LL, and WLS models of the age˙ O2 max. In each case we examine associated decline in V how well the model satisfies the general linear model (GLM) assumptions of normality and homoscedasticity. Subjects This study was approved by the Institutional Review Boards of the University of Maryland School of Medicine and Johns Hopkins University Bayview Medical Center. All subjects provided informed consent before participation. Over an 8-yr period, healthy nonsmoking men, ages 45–80 yr, with no prior history of cardiovascular disease and a wide range of body mass index (BMI) and physical conditioning status, were recruited to participate in exercise training and weight loss studies (14, 35). Characteristics of the subjects are summarized in Table 1. All subjects underwent a history and physical examination and evaluation of fasting blood chemistries. Subjects were healthy and on no medications. Exclusion criteria included history of coronary artery disease (by clinical history and electrocardiogram), pulmonary disease, hypertension (blood pressure $160/90 mmHg), hyperlipidemia, diabetes mellitus (fasting plasma glucose $140 mg/dl), or any other significant medical problems that would interfere with their ability to undergo maximal exercise treadmill testing. The study sample included a cohort of 61 healthy athletes recruited from participants of the Maryland Senior Olympics and athletic clubs in the Baltimore-Washington metropolitan area and 215 sedentary subjects (35). The sedentary subjects exercised ,20 min twice per week. The athletes exercised vigorously at least four times per week and had ,25% body fat by hydrodensitometry.

Table 1. Subject characteristics

Height and weight were measured, and BMI was computed as the ratio of body weight in kilograms to height in meters squared. Body surface area in meters squared was calculated as (height in cm)0.718 3 (weight in kg)0.427 3 0.007449 (7). Body density was determined by hydrostatic weighing, with percent body fat calculated after correction for residual lung volume with use of the Siri model (29). FFM was calculated as body weight minus fat mass.

˙ O2 max Measurement of V An exercise treadmill test to $85% of the predicted ageadjusted heart rate (220 2 age) was performed according to the protocol of Bruce and Horsten (5) to exclude subjects with previously undiagnosed heart disease. On a subsequent visit ˙ O2 max was determined using a modified Balke protocol, the V as previously described (14). The grade of the treadmill was increased every 2 min until the subject was exhausted and ˙ O2 max tests fulfilled at least two could not continue (14). The V of the three following criteria: 1) the heart rate at maximal exercise was $85% of the age-adjusted maximal heart rate, 2) the respiratory exchange quotient was .1.10, and 3) there was a plateau in O2 consumption defined as a change in O2 consumption of ,0.2 l/min during the final two collection ˙ O2 max, measured in liters per minute, is periods. Absolute V used in the statistical modeling. Statistical Analysis All statistical analyses were performed using SAS version 6.11 run on the Windows 3.1 operating system. ˙ O2 max. Four statistical models were Statistical models of V ˙ O2 max. used to determine the age-associated decline in V 1) The form of the RS model is

˙ O2 max/FFM 5 b0 1 b1 X1 1 b2 X2 1, . . ., 1e V

where X1, X2, etc., are independent variables, b0, b1, etc., are parameters, and e is a normally and independently distributed error term having equal variance for all levels of the independent variable(s) in the target population. 2) The form of the OLS regression is

˙ O2 max 5 b0 1 b1FFM 1 b2 X2 1, . . ., 1e V

Characteristic

Athletes (n 5 61)

Sedentary (n 5 215)

Age, yr Weight, kg BMI, kg/m2 Height, cm %Body fat FFM, kg ˙ O2 max , l/min V

62.4 6 6.1 69.8 6 6.9 23.1 6 2.2 174 6 6 15.1 6 5.2 59.2 6 5.8 3.38 6 0.48

60.0 6 7.9* 91.2 6 13.2* 29.5 6 3.5* 175 6 7 28.4 6 5.9* 64.0 6 7.5* 2.66 6 0.51*

Values are means 6 SD. BMI, body mass index; FFM, fat-free ˙ O2 max , maximal aerobic capacity. * P , 0.01 compared with mass; V athletes.

(2)

where FFM (5 X1 ), X2, etc., are independent variables, b0, b1, etc., are parameters, and e is an error term with the same properties as in Eq. 1. OLS estimation ensures that the residuals of the model are uncorrelated with FFM, X2, etc. (24). 3) The univariate form of the LL model is

˙ O2 max 5 b0FFMb1d V

Group

(1)

(3)

where b0 and b1 are parameters and d is an error term. Equation 3 is referred to as an allometric model. The multivariable form is

˙ O2 max 5 b0FFMb1Xb22, . . . , Xbpp V exp(bp11 W1 1 bp12 W2 1, . . ., bp1k Wk)d

(4)

where FFM (5 X1 ), X2, etc., are variables related in an LL ˙ O2 max, W1, W2, etc., are variables related linearly manner to V ˙ O2 max, b0, b1, etc., are parameters, and d is an error term. to V

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METHODS AND MATERIALS

Measurement of Body Composition

˙ O2 max STATISTICAL MODELS OF AGE-ASSOCIATED DECLINE IN V For example, when age is used in the multivariable model, Eq. 4 becomes Y 5 b0FFMb1 exp(b2 AGE)d

(5)

One approach to estimating the parameters of the univariate or multivariable models is to use OLS on the logarithmic transformation of the model, e.g.

˙ O2 max) 5 log(b0) 1 b1 log(FFM) 1 log(d) log(V

(6)

or

˙ O2 max) 5 log(b0) 1 b1 log(FFM) 1 b2 AGE 1 log(d) log(V

(7)

n

o w 3 V˙ O i

i51

2 maxi (obs)

4

˙ O2 maxi (pred) 2V

2

(8)

where the first and second terms in parentheses are the ˙ O2 max values, respectively. In OLS, observed and predicted V wi 5 1 (for all i). For example, if the error variance increases along with the values of a regressor (e.g., FFM), observations that have higher values on the regressor receive lower weights than observations that have lower values. The model then reflects the greater uncertainty in the predicted values at higher values of the regressor. If the weights are chosen to be proportional to the reciprocal of the error variance and the other GLM assumptions are met, the WLS model has the desirable property of producing parameter estimates that are BLUE. Moreover, the estimated variances of the parameters will be unbiased and so will the t-tests on which these variances are based. If heteroscedasticity exists, the estimates of the variance of the parameters produced by OLS are not minimum variance estimates, so OLS estimates will not, in general, be BLUE (24). In this study, WLS weights took the form wi 5 1/FFMai . When this form is used, the following assumption, known as multiplicative heteroscedasticity, is implied s2i 5 s2 FFMai

(9)

In Eq. 9, s2i is the error variance at the level of FFM that equals FFMi, s2 is the error variance from the OLS model, and a is the parameter related to the degree of heteroscedasticity (homoscedasticity implies a 5 0). Equation 9 suggests that the variation of the residuals of model shown in Eq. 2 increases as FFM increases. This assumption is usually examined by plotting the fitted residuals eˆ against FFM. Harvey (11) provided a method for obtaining the maximum likelihood (ML) estimator of a. Other ways of obtaining the ML estimates of s2 and a have been proposed (16). It is possible to demonstrate that using the weights 1/FFMai is equivalent to transforming the linear model by dividing (24). For example, if FFM is the only each term by FFMa/2 i

regressor, the transformed model is

˙ O2 max/FFMa/2 5 (b0/FFMa/2) 1 b1FFM12a/2 V 1 (e/FFMa/2)

(10)

The model can be estimated by fitting the regression on the transformed variables. Equation 10 shows that the variance of the transformed residuals is constant as long as the weights wi are proportional to the reciprocal of the error variance. Given the usual regression assumption that the independent variables are fixed, we can write the variance of the error term as a var(ei/FFMa/2 i ) 5 (1/FFMi ) var(ei)

(11)

Now, apply Eq. 9, setting s2i 5 var(ei ) (1/FFMai ) var(ei) 5 (1/FFMai )s2FFMai 5 s2

(12)

There is another interesting consequence of Eq. 10. If a 5 2, Eq. 10 becomes

˙ O2 max/FFM 5 b0/FFM 1 b1 1 e/FFM V

(13)

Then, if b0 5 0 (zero intercept), Eq. 13 is structurally identical to the RS model (Eq. 1) with only an intercept term. This suggests that if a 5 2 and b0 5 0, the RS model (Eq. 13) will be homoscedastic. Moreover, if the transformation does not greatly perturb the correlation matrix, multivariable RS models should also have this property, given the previous assumptions. It is possible to extend Eq. 9 to include multiple regressors, e.g. s2i 5 s2 FFMai 1 HEIGHTai 2

(14)

The weights for WLS derived from Eq. 10 are 1/(FFMai 1)HEIGHTai 2. Note that the regressors in Eq. 9 or 14 need not necessarily be regressors in the OLS model (Eq. 2). ˙ O2 max. We Factors related to the age-associated decline in V initially examined models that included age, FFM, body weight, fat mass, height, BMI, body surface area, and a dichotomous exercise training variable (ET) that denoted whether the individual was an athlete (0 5 sedentary, 1 5 athlete). Body weight, fat mass, BMI, height, and body surface area were not significant predictors in any of the models at the 0.05 level when FFM was included and were not included in the analyses. To determine whether the slopes of the age-related decline ˙ O2 max differed between athletes and sedentary subjects, in V analyses were also performed with RS, OLS, and WLS models that included interactions between age and ET. For each model type we obtained estimates of the age-associated ˙ O2 max. Because the LL model is decline and effect of ET on V log linear, it does not permit a test of the hypothesis of ˙ O2 max. parallel linear declines in V Contribution of skeletal muscle loss to the age-associated ˙ O2 max. We estimated the contribution of the loss of decline in V ˙ O2 max as skeletal muscle to the decline in V %contribution 5 100 ·

bAGE (adjusted) 2 bAGE (unadjusted) bAGE (unadjusted)

(15)

where bAGE(adjusted) and bAGE(unadjusted) are the regression coefficients for age in the models adjusted and unadjusted for FFM,

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An alternative approach is to estimate the coefficients in the original scale by use of nonlinear regression methods. In this study the former method was used. When the univariate form of the LL model is used, it is ˙ O2 max/FFMb1, which usual practice to verify that the variable V is proportional to the residuals of Eq. 3, has a correlation near zero with FFM (32, 33). 4) WLS regression is identical to OLS regression, except each subject i receives a weight wi. The goal of WLS is to minimize the sums of the squares

2165

˙ O2 max STATISTICAL MODELS OF AGE-ASSOCIATED DECLINE IN V

2166

Effect of Age Unadjusted for Body Habitus

respectively

All the parameter estimates of the OLS model

Linear unadjusted adjusted

˙ O2 max 5 b0 1 b1 AGE 1 b2 ET 1 e V

(16)

˙ O2 max 5 b0 1 b1 AGE 1 b2 ET V 1 b3 FFM 1 e (17) Log-linear

unadjusted adjusted

˙ O2 max 5 b0 exp(b2 AGE 1 b3 ET)d V

(18)

˙ O2 max 5 b0 FFMb1 exp(b2 AGE 1 b3 ET)d V

(19)

RESULTS

The correlations among selected measures of body ˙ O2 max are shown in Table 2. All the body size, age, and V ˙ O2 max (P , 0.01). Regressize variables correlated with V sion results for the unadjusted and adjusted RS, OLS, LL, and WLS models are shown in Table 3. Only variables significant at P , 0.05 in the regression are listed.

Table 2. Correlations between age, measures of body ˙ O2 max habitus, and V

FFM Fat mass Height ˙ O2 max V

Age

FFM

Fat Mass

Height

20.40 20.29 20.24 20.38

0.43 0.65 0.39

0.25 20.19

0.34

All correlations are significant at P , 0.01.

(20)

were significant at P , 0.01 (Table 3). This model ˙ O2 max. There was accounted for 48% of the variance in V no evidence of lack of normality (Wilks-Shapiro test, P . 0.10) or heteroscedasticity due to lack of fit (White’s test, P . 0.7). However, it is of interest that multiplicative heteroscedasticity with FFM was present [Harvey’s test, x2 (1) 5 4.22, P , 0.05]. There was no significant interaction between age and ET. Therefore, ˙ O2 max, the absolute age-associated rate of decline in V 0.36 l/min, was the same in athletes and sedentary men. The OLS estimate of the effect of exercise training (the coefficient of ET) was 0.78 l/min (Table 3). All the parameters of the LL model

˙ O2 max 5 b0 exp(b2 AGE 1 b3 ET)d V

(21)

were significant at P , 0.01 (Table 3). This model ˙ O2 max. The accounted for 46% of the variance in V residuals were normally distributed, but there was evidence of heteroscedasticity due to lack of fit (White’s test, P , 0.05). Unlike the OLS and WLS models, in the ˙ O2 max per decade is LL model the percent decline of V independent of the baseline value. The age-associated decline was proportional to exp(20.014 AGE) 5 0.986 exp(AGE). To obtain the WLS model, the OLS model

˙ O2 max 5 b0 1 b1 AGE 1 b2 ET 1 e V

(22)

was adjusted for multiplicative heteroscedasticity by using weights 1/FFM0.87. All the parameter estimates were significant at P , 0.01 (Table 3). This model ˙ O2 max. The WLS accounted for 49% of the variance in V model met all the GLM assumptions and yielded estimates of the training effect and of the age-associated decline that were nearly identical to the OLS model. Effect of Age Adjusted for Body Habitus The RS model was

˙ O2 max/FFM 5 b0 1 b1 AGE 1 b2 ET 1 e V

(23)

˙ O2 max/FFM did not completely remove the The ratio V effects of FFM; the correlation between the ratio and FFM was 20.17 (P , 0.01). All parameter estimates for the model (Eq. 23) were significant at P , 0.01 (Table 3). The model accounted for 60% of the variance in ˙ O2 max. There was no evidence of heteroscedasticity due V to lack of fit and no correlation between the absolute residuals and FFM (r 5 0.01). However, the residuals were not normally distributed (P , 0.02). There was no significant interaction between age and ET. Thus the age-associated rate of decline relative to FFM was the same in athletes and sedentary men, ,0.0034 l · kg FFM21 · min21 · decade21. The estimate of the ET effect

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Tests of assumptions. A key issue in the selection of an appropriate mathematical model for describing a set of data is whether the model fulfills the appropriate underlying statistical assumptions. It was therefore important to examine the distribution of the residuals in the various models for heteroscedasticity. For OLS models that contained FFM we used the likelihood ratio test provided by Harvey (11) to determine the presence of multiplicative heteroscedasticity (Eqs. 9 and 14). For LL models we assessed multiplicative heteroscedasticity by computing the correlation between the absolute value of the residuals and FFM (24). Even if multiplicative heteroscedasticity is not present, other forms of heteroscedasticity may exist. We used the method of White (34) to test for the possibility of other forms of heteroscedasticity. This test is sensitive to heteroscedasticity if it causes the variance-covariance matrix of the OLS estimators to vary from its asymptotic (large sample) form (15). Variation from the asymptotic form can result if one or more of the linear model assumptions are not satisfied. We used the WilksShapiro test to test for violations of normality. White’s test and the Wilks-Shapiro test were obtained from SAS proc reg and proc univariate, respectively. SAS proc IML was used to obtain the ML estimate of a and to compute the likelihood ratio test for the hypothesis a 5 0. An ML grid search algorithm for estimating a is available in SPSS Professional Statistics 6.1 (WLS procedure).

˙ O2 max 5 b0 1 b1 AGE 1 b2 ET 1 e V

˙ O2 max STATISTICAL MODELS OF AGE-ASSOCIATED DECLINE IN V

2167

Table 3. Regression results for variables significant at P , 0.05 in models unadjusted and adjusted for body habitus Intercept Method

R2

Coefficient

SE

Age, yr t

Coefficient

SE

ET t

Coefficient

SE

FFM, kg t

Coefficient

SE

t

0.035 0.83 0.036

0.0032 0.076 0.0032

10.9 10.9 11.2

Unadjusted for body habitus OLS LL WLS

0.48 0.46 0.49

4.83 1.79 4.82

0.20 0.077 0.2

23.8 23.3 23.7

20.036 20.014 20.036

RS OLS LL WLS

0.60 0.64 0.63 0.66

0.062 1.84 21.96 1.77

0.0027 0.33 0.35 0.32

22.6 5.6 25.6 5.4

20.00034 20.024 20.009 20.024

0.0034 0.0013 0.0033

210.8 210.9 210.9

0.78 0.28 0.79

0.062 0.023 0.06

12.7 11.9 13.1

0.016 0.96 0.33 0.96

0.00083 0.054 0.02 0.051

19.4 17.9 16.4 18.6

Adjusted for body habitus 0.000045 0.0031 0.0011 0.003

27.5 27.8 27.9 27.8

ET, exercise training status (0 5 sedentary, 1 5 athlete); OLS, ordinary least squares; WLS, weighted least squares; LL, log linear; RS, ratio standard. Estimates for LL models are in log units.

˙ O2 max 5 b0 1 b1 FFM 1 b2 AGE 1 b3 ET 1 e V

(24)

all parameter estimates were significant at P , 0.01. The OLS model accounted for 64% of the variance in ˙ O2 max. There was no evidence of lack of normality or V heteroscedasticity due to lack of fit. The absolute residuals correlated with FFM (r 5 0.18, P , 0.01). Harvey’s test for multiplicative heteroscedasticity confirmed this finding: x2 (1) 5 7.52, P , 0.01. A plot of the residuals vs. FFM, in which the heteroscedasticity is depicted, is shown in Fig. 1. As in the RS model, the rates of decline were not significantly different in athletes and sedentary men. The calculated decline in ˙ O2 max was 0.24 l · min21 · decade21, and the estimate of V the exercise conditioning effect was 0.96 l/min.

For the LL model

˙ O2 max 5 b0 FFMb1 exp(b2 AGE 1 b3 ET)d V

(25)

all parameter estimates were significant at P , 0.01. ˙ O2 max. The model accounted for 63% of the variance in V However, there was evidence of lack of normality and of heteroscedasticity due to lack of fit (Wilks-Shapiro and White’s tests, both P , 0.01). There was no evidence of multiplicative heteroscedasticity; the absolute residuals were uncorrelated with FFM (r 5 0.01). For this model the age-associated decline was proportional to exp(20.009 AGE) 5 0.99 exp(AGE), suggesting an 8.4% ˙ O2 max per decade in athletes and sedentary decline in V men. WLS results were based on the OLS model (Eq. 21) using weights 1/FFM1.30. The WLS and OLS estimates and standard errors were almost identical (Table 3). The WLS model explained 66% of the variance in ˙ O2 max compared with 64% for OLS. The residuals met V all the important GLM assumptions. As in the RS and ˙ O2 max were the OLS models, the rates of decline in V same in trained and untrained men (i.e., there was no significant interaction between age and ET). The WLS model yielded estimates of the training effect and of the ˙ O2 max that were similar in age-related declines in V athletes and sedentary men, as in the OLS model. A ˙ O2 max with age based on the plot of the decline in V coefficients from the WLS model is shown in Fig. 2. Effect of Age Adjusted for FFM and Height

Fig. 1. Diagnostic scatter plot of ordinary least-squares (OLS) residuals vs. fat-free mass (FFM). Residuals are derived from OLS model ˙ O2 max) 5 1.84 2 0.024 AGE 1 0.96 ET 1 maximal aerobic capacity (V 0.035 FFM, where ET is exercise training.

Nevill (19) suggested that height may be an im˙ O2 max. In this portant covariate in LL models of V ˙ O2 max in study, height was not a significant predictor of V any of the models, and its inclusion had little effect on the estimates of the age-associated decline. This may be due in part to the relatively small variation in height (coefficient of variation of 4%) in these subjects. Height also did not contribute when used in the model in Eq. 10 with FFM as a predictor of within-subject variance [Harvey’s test x2 (1) 5 2.64, P . 0.05].

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due to the difference in intercept was 0.016 l/min (Table 3). For the OLS model

2168

˙ O2 max STATISTICAL MODELS OF AGE-ASSOCIATED DECLINE IN V

˙ O2 max vs. age in athletes (s) and sedentary Fig. 2. Scatter plot of V men (1). Dashed and solid lines, best-fit weighted least-squares ˙ O2 max 5 1.77 2 0.024 AGE 1 0.96 ET 1 regression for each group, V 0.036 FFM, at mean of FFM for entire sample.

Toth et al. (30) suggested that the ratio of FFM to fat mass was an important factor contributing to the ˙ O2 max. In this study we found age-associated decline in V that neither fat mass nor the ratio of FFM to fat mass contributed significantly to any of the models (P . 0.05) when FFM was already included. Contribution of Loss of FFM to the Age-Associated ˙ O2 max Decline in V The OLS, WLS, and LL models suggested that ,35% ˙ O2 max was due to a of the age-associated decline in V ˙ O2 max declines 0.24 loss of skeletal muscle. Given that V l · min21 · decade21, the loss of skeletal muscle accounts for 0.08 l · min21 · decade21 of this quantity. Because FFM was not in the RS model as a predictor, the RS model cannot provide an estimate of the contribution of ˙ O2 max. FFM to the age-associated decline in V DISCUSSION

The WLS regression incorporating age, FFM, and a dichotomous indicator of physical conditioning status ˙ O2 max that accounted for the largest yielded a model of V proportion of variance and met all the important GLM assumptions. Absolute R2 for the WLS model was ,6% higher than for the RS model, 3% higher than for the LL model, and 2% higher than for the OLS model (Table 3). These results support the use of WLS models to examine the physiological factors underlying the ˙ O2 max. age-associated decline in V Despite their failure to satisfy the underlying GLM assumptions, the RS and OLS models incorporating age, FFM, and ET provided estimates similar to the WLS model of the age-associated decline in athletes and sedentary men. The RS model incorporating age, FFM, and ET produced estimates that were ,10% lower than those produced by the OLS and WLS models (the LL model suggested an exponential decay in

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Effect of Age Adjusted for FFM and Fat Mass

˙ O2 max). The OLS estimate was nearly identical to the V WLS estimate, despite the failure of the OLS model to satisfy the assumption of homoscedasticity. This estimate (0.24 l · min21 · decade21 ) suggests a 9% decline in ˙ O2 max between 60 and 70 yr of age, which is consistent V with other studies that indicate that the average healthy sedentary man .25 yr of age is expected to lose ˙ O2 max per decade (10, 13, 30). 9–11% of his V Because the WLS models satisfied all the important GLM assumptions, the estimates produced by the WLS analysis were BLUE, and the t-tests of the regression coefficients were correct. All the other methods failed to satisfy one or more of the assumptions. Some investigators recommend verifying the assumption of homoscedasticity when scaling for differences in body habitus (20, 33). In this study the findings were generally robust to violation of this assumption. Multiplicative heteroscedasticity did not significantly affect the OLS standard errors, and the OLS model R2 increased slightly compared with the (homoscedastic) RS model. We also found that the estimates were virtually unchanged over the range a 5 0–2 (data not shown). This insensitivity of the WLS estimates may have been due to the fact that the correlation between the absolute residuals and FFM was only r 5 0.18. It is possible that WLS may have a greater impact on the estimates and standard errors when the correlation is higher, e.g., r . 0.30. Higher correlations have been reported elsewhere (33). In the log-transformed LL models the residuals displayed lack of normality and heteroscedasticity due to lack of fit. A post hoc analysis (data not shown) determined that both problems were due to the presence of an indicator variable in the analysis. When separate analyses were carried out within the athlete and sedentary groups, the residuals were normally distributed and homoscedastic. Heteroscedasticity due to lack of fit is diagnostic of model misspecification and suggests that the logarithmic transformation was not the appropriate metric when the dichotomous training variable was included in LL models. The mechanisms underlying the age-associated ˙ O2 max in healthy men are multifactorial. change in V Factors implicated in the age-associated decline in ˙ O2max include decreases in maximum heart rate, stroke V volume, arteriovenous O2 difference, and skeletal muscle mass, an increase in adiposity, and a decline in daily, regular physical activity (3, 10, 12, 13, 17, 22, 27, 30). In the present study the models adjusted for FFM accounted for more variance than the unadjusted models (Table 3). The unadjusted models yielded lar˙ O2 max in the ger estimates of the percent decline in V athletes and sedentary men than did models that included FFM. This difference between unadjusted and adjusted models, which has been noted in RS models that do not adjust for muscle mass (8), suggests that ˙ O2 max is due to the age,35% of the decline in V associated decrease in skeletal muscle mass (OLS, WLS, and LL estimate). This finding is consistent with that of Toth et al. (30), who estimated the contribution

˙ O2 max STATISTICAL MODELS OF AGE-ASSOCIATED DECLINE IN V

etry, may provide a more accurate quantification of muscle mass and prediction of the age-associated de˙ O2 max. Third, younger men were not included cline in V in this study, which has affected the estimates of the intercept. Finally, training intensity was treated as a dichotomous variable. This may not adequately account for the heterogeneity in effects of exercise training and leisure time activities in the sedentary and athletic populations. Toth et al. (30) concluded that controlling for leisure time activity reduced the degree ˙ O2 max in sedentary males. of decline in V In this study the OLS and WLS models possessed certain advantages over the corresponding RS and LL models. First, OLS and WLS were able to provide estimates of the contribution of the loss of FFM to the ˙ O2 max, which the RS model, age-associated decline in V without an explicit FFM term, could not. Another advantage of OLS and WLS was their suitability for testing the hypothesis of parallel linear declines in ˙ O2max in athletic and sedentary individuals; the nonlinV ear LL model was not appropriate for addressing this question. OLS or WLS should prove useful in studies where these issues (or similar ones) comprise part of the investigation, with WLS providing a slightly more efficient analysis. We acknowledge the contributions of Drs. E. Bleecker, P. Coon, J. Fleg, R. Pratley, M. J. Busby-Whitehead, and D. Drinkwater and the ˙ O2 max research exercise physiologists who assisted in measuring V and body composition in the subjects in this study. In addition, we thank Drs. A. Gardner, J. R. Hebel, and A. Yataco for helpful comments and suggestions. This work was supported by National Institute on Aging (NIA) Grant R01-AG-07660, The Johns Hopkins Academic Teaching Nursing Home Award P01-AG-04402, Grant K07-AG-00608, the Department of Veterans Affairs Geriatric Research, Education, and Clinical Center (GRECC), The Johns Hopkins Bayview GCRC (Grant M01-RR02719), the Claude D. Pepper Older Americans Independence Center NIA Grant P60-AG-12583, and National Institutes of Health Intramural Funds from the Gerontology Research Center, Laboratory of Clinical Investigation, Metabolism Section, NIA. Present address and address for reprint requests: M. J. Rosen, 42 Haymarket Ln., Bryn Mawr, PA 19010. Received 29 May 1997; accepted in final form 5 February 1998. REFERENCES 1. Albrecht, G. H., B. R. Gelvin, and S. E. Hartman. Ratios as a size adjustment in morphometrics. Am. J. Phys. Anthropol. 91: 441–468, 1993. 2. Allison, D. B., F. Paultre, M. I. Goran, E. T. Poehlman, and S. B. Heymsfield. Statistical considerations regarding the use of ratios to adjust data. Int. J. Obes. Relat. Metab. Disord. 19: 644–652, 1995. 3. Badenhop, D. T., P. A. Cleary, S. F. Schaal, E. L. Fox, and R. L. Bartels. Physiological adjustments to higher- and lowerintensity exercise in elders. Med. Sci. Sports Exerc. 15: 496–502, 1983. 4. Box, G. E. P., and D. R. Cox. An analysis of transformations. J. R. Stat. Soc. B 26: 211–252, 1964. 5. Bruce, R. A., and T. R. Horsten. Exercise testing in the evaluation of patients with ischemic heart disease. Prog. Cardiovasc. Dis. 11: 371–390, 1969. 6. Buskirk, E. R., and J. L. Hodgson. Age and aerobic power: the rate of change in men and women. Federation Proc. 46: 1824– 1829, 1987.

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to be 33%. Recent findings suggest that the decrease in ˙ O2 max that is associated with the loss of skeletal V muscle may be due, at least in trained subjects, to reduced aerobic capacity per kilogram of active muscle (26). Reduced aerobic capacity would result from ageassociated changes in maximal O2 delivery and be independent of any actual loss of muscle fibers. In the present study the OLS and WLS models indicated that the athletes and sedentary men de˙ O2 max at the same absolute rate, ,0.24 creased their V ˙ O2 max was 0.96 l/min l · min21 · decade21. However, the V higher in the athletes than in the sedentary men at all ages. Although some cross-sectional studies report simi˙ O2 max over time in athletes and lar absolute declines in V in sedentary individuals (25), other longitudinal stud˙ O2 max declines at a slower rate in ies suggest that V athletes than in sedentary men (27). Hagberg (10) ˙ O2 max per decade in estimated that the 5.5% decline in V master athletes is ,50% of the rate of decline in age-matched sedentary men. It is noteworthy that there is a correlation between the change in training ˙ O2 max (17). intensity and longitudinal changes in V Possibly, the attenuation of the rate of the age˙ O2 max may be achievable only associated decrease in V for a select group of elite older athletes who are able to continue high-intensity training over an extended period of time and also preserve their FFM. The athletes in the present study were heterogeneous with respect ˙ O2 max per kilogram of FFM and the intensity to their V and duration of their training. This heterogeneity may have had an impact on the age-associated decline in ˙ O2 max. V A strength of this study is that the athletes and untrained men underwent a vigorous medical evaluation, and only those men with no evidence of cardiovascular or other disease were enrolled. Also, the athletes enrolled in this study were selected only if they were still competitive in their age class. This reduced the potentially confounding effects of disease on the ageassociated declines in cardiovascular fitness. Second, individuals with a wide range of obesity and fitness levels were included in the study population. Nevertheless, this study has several limitations. First, the subjects were not randomly selected from the at-large older population, limiting the generalizability of the study findings. The fact that several other studies found similar rates of loss of fitness with age suggests that any bias introduced by our subject-selection process is probably small. Another limitation was the use of hydrodensitometry as the method of assessing FFM. Hydrodensitometry does not yield a direct measurement of skeletal muscle mass; muscle mass must be calculated from the observed density. This calculation depends on several assumptions. In an aging population, changes in bone density and residual lung volume may confound the hydrodensitometric estimate of FFM and, hence, the contribution of FFM and fat mass to the ˙ O2 max (18). More direct meaage-associated decline in V sures of skeletal muscle mass, such as that provided by magnetic resonance imaging or dual X-ray absorptiom-

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23.

24. 25.

26.

27.

28.

29. 30.

31.

32.

33.

34.

35.

aging, sex, and physical training on cardiovascular responses to exercise. Circulation 86: 494–503, 1992. Pearson, K. Mathematical contributions to the theory of evolution on a form of spurious correlation which may arise when indices are used in the measurement of organs. Proc. R. Soc. Lond. B Biol. Sci. 60: 489–497, 1987. Pindyck, R. S., and D. L. Rubinfeld. Econometric Models and Economic Forecasts. New York: McGraw-Hill, 1976, p. 139–142. Pollock, M. L., H. S. Miller, and J. Wilmore. Physiological characteristics of champion American track athletes 40 to 75 years of age. J. Gerontol. A Biol. Sci. Med. Sci. 29: 645–649, 1987. Proctor, D. N., and M. J. Joyner. Skeletal muscle mass and the ˙ O2 max in trained older subjects. J. Appl. Physiol. 82: reduction of V 1411–1415, 1997. Rogers, M. A., J. M. Hagberg, W. H. Martin III, A. A. Ehsani, ˙ O2 max with aging in master and J. O. Holloszy. Decline in V athletes and sedentary men. J. Appl. Physiol. 68: 2195–2199, 1990. Rogers, D. M., B. L. Olson, and J. H. Wilmore. Scaling for the ˙ O2-to-body size relationship among children and adults. J. Appl. V Physiol. 79: 958–967, 1995. Siri, W. E. The gross composition of the body. Adv. Biol. Med. Phys. 4: 239–240, 1956. Toth, M. J., A. W. Gardner, P. A. Ades, and E. T. Poehlman. Contribution of body composition and physical activity age˙ O2 in men and women. J. Appl. Physiol. related decline in peak V 77: 647–652, 1994. Toth, M. J., M. I. Goran, P. A. Ades, D. B. Howard, and E. T. Poehlman. Examination of data normalization procedures for ˙ O2 data. J. Appl. Physiol. 75: 2288–2292, 1993. expressing peak V Vanderburgh, P. M., M. T. Mahar, and C. H. Chou. Allometric scaling of grip strength by body mass in college-age men and women. Res. Q. Exerc. Sport 66: 8–84, 1995. Welsman, J. R., N. Armstrong, A. M. Nevill, E. M. Win˙ O2 for differences in body ter, and B. J. Kirby. Scaling peak V size. Med. Sci. Sports Exerc. 28: 259–265, 1996. White, H. A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrics 48: 817–838, 1980. Yataco, A. R., J. Busby-Whitehead, D. T. Drinkwater, and L. I. Katzel. Relationship of body composition and cardiovascular fitness to lipoprotein lipid profiles in master athletes and sedentary men. Aging Clin. Exp. Res. 9: 88–94, 1997.

Downloaded from jap.physiology.org on April 5, 2005

7. Ewald, G. A., and C. R. McKenzie (Editors). Manual of Medical Therapeutics (28th ed.). New York: Little, Brown, 1995, p. 577. 8. Fleg, J. L., and E. G. Lakatta. Role of muscle loss in the ˙ O2 max. J. Appl. Physiol. 65: 1147– age-associated reduction in V 1151, 1988. 9. Goran, M. I., D. B. Allison, and E. T. Poehlman. Issues relating to normalization of body fat content in men and women. Int. J. Obes. Relat. Metab. Disord. 19: 638–643, 1985. ˙ O2 max with 10. Hagberg, J. M. Effect of training on the decline of V aging. Federation Proc. 46: 1830–1833, 1987. 11. Harvey, A. C. Estimating regression models with multiplicative heteroscedasticity. Econometrica 44: 461–465, 1976. 12. Heath, G. W., J. M. Hagberg, A. A. Ehsani, and J. O. Holloszy. A physiological comparison of young and older endurance athletes. J. Appl. Physiol. 51: 634–640, 1981. 13. Holloszy, J. O., and W. M. Kohrt. Exercise. In: Handbook of Physiology. Aging. Bethesda, MD: Am. Physiol. Soc., 1995, chapt. 24, p. 633–666. 14. Katzel, L. I., E. R. Bleecker, E. G. Colman, E. M. Rogus, J. D. Sorkin, and A. P. Goldberg. Effects of weight loss vs. aerobic exercise training on risk factors for coronary disease in healthy, obese, middle-aged and older men. JAMA 274: 1915–1921, 1995. 15. Kennedy, P. A Guide to Econometrics (3rd ed.). Cambridge, MA: MIT Press, 1992, p. 90, 118–119, 130, 131. 16. Kmenta, J. Elements of Econometrics. New York: Macmillan, 1971, p. 263–264. 17. Marti, B., and H. Howald. Long-term effects of physical training on aerobic capacity: controlled study of former elite athletes. J. Appl. Physiol. 69: 1451–1459, 1990. 18. Martin, A., and D. Drinkwater. Variability in the measures of body fat. Assumptions or techniques? Sports Med. 11: 277–288, 1991. 19. Nevill, A. M. The need to scale for differences in body size and mass: an explanation of Kleiber’s 0.75 mass exponent. J. Appl. Physiol. 77: 2870–2873, 1994. 20. Nevill, A. M., and R. L. Holder. Scaling, normalizing, and per ratio standards: an allometric modeling approach. J. Appl. Physiol. 79: 1027–1031, 1995. 21. Nevill, A. M., R. Ramsbottom, and C. Williams. Scaling physiological measurements for individuals of different body size. Eur. J. Appl. Physiol. 65: 110–117, 1992. 22. Ogawa, T., R. Spina, W. H. Martin III, W. M. Kohrt, K. B. Schechtman, J. O. Holloszy, and A. A. Ehsani. Effects of