Abstract
Studies assessing changes in maximal aerobic capacity (V˙o _{2 max}) associated with aging have traditionally employed the ratio ofV˙o _{2 max} to body weight. Loglinear, ordinary leastsquares, and weighted leastsquares models may avoid some of the inherent weaknesses associated with the use of ratios. In this study we used four different methods to examine the ageassociated decline inV˙o _{2 max} in a crosssectional 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 leastsquares model that included age, fatfree mass, and an indicator variable denoting exercise training status. The model accounted for 66% of the variance inV˙o _{2 max} and satisfied all the important general linear model assumptions. The other approaches failed to satisfy one or more of these assumptions. The results indicated thatV˙o _{2 max} declines at the same rate in athletic and sedentary men (0.24 l/min or 9%/decade) and that 35% of this decline (0.08 l ⋅ min^{−1} ⋅ decade^{−1}) is due to the ageassociated loss of fatfree mass.
 exercise
 heteroscedasticity
 weighted least squares
 loglinear model
 ratio
there is an ageassociated decline in maximal aerobic capacity (V˙o _{2 max}) in trained and untrained men (6, 8, 10, 13, 22). This ageassociated decline in V˙o _{2 max} is due to many factors, including decreases in maximum heart rate, stroke volume, arteriovenous O_{2}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 ageassociated decline inV˙o _{2 max} (3, 12, 17, 22,27). There is conflicting evidence on whether regular aerobic exercise training attenuates the ageassociated decline inV˙o _{2 max} (13,25, 28).
Different statistical models have been used to analyze the relationships among the physiological variables associated with the ageassociated decline inV˙o _{2 max}. The statistical approach employed to adjust theV˙o _{2 max} for body size and composition must be considered carefully; the approach must be statistically valid. All other factors being equal, a larger individual would be expected to have a higherV˙o _{2 max} than a smaller individual. To account for differences in body habitus, the simple ratio of V˙o _{2 max} to a measure of body size [e.g.,V˙o _{2 max}ml ⋅ kg body wt^{−1} ⋅ min^{−1}or V˙o _{2 max}ml ⋅ kg fatfree mass (FFM)^{−1} ⋅ min^{−1}] is often used as the dependent variable in studies of the agerelated decline in V˙o _{2 max} (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, andZ are uncorrelated,X/Z andY/Z are correlated under minimally restrictive assumptions (23). In the case ofV˙o _{2 max}, theV˙o _{2 max}/body size ratio may not make V˙o _{2 max}totally independent of body size. Given the physiological and functional importance of theV˙o _{2 max}age relationship, 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 agerelated change inV˙o _{2 max}: ordinary leastsquares (OLS) regression ofV˙o _{2 max} on a measure of body size, such as FFM or weight (31), and loglinear (LL) or allometric models (20, 21). OLS and LL models are not without problems. OLS residuals inV˙o _{2 max} models are often heteroscedastic; 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 BoxCox transformation (4), in which the dependent variable is raised to the power (i.e.,V˙o _{2 max} ^{γ}) that maximizes the likelihood that the model residuals follow a normal distribution. This method has not been widely used in studies that have examined the ageassociated decline inV˙o _{2 max}.
In this paper an alternative method to modeling V˙o _{2 max} is proposed: weighted least squares (WLS). WLS 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 ofV˙o _{2 max}, the potential errors caused by the failure of the RS to remove the effect of body size on V˙o _{2 max}, and the potential problems caused by the heteroscedasticity inherent in OLS models of V˙o _{2 max}, we believe that further analysis of the agerelated decline inV˙o _{2 max} is warranted. Therefore, we compare the results obtained from RS, OLS, LL, and WLS models of the ageassociated decline inV˙o _{2 max}. In each case we examine how well the model satisfies the general linear model (GLM) assumptions of normality and homoscedasticity.
METHODS AND MATERIALS
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 8yr 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 BaltimoreWashington 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.
Measurement of Body Composition
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} × (weight in kg)^{0.427} × 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.
Measurement ofV˙o_{2 max}
An exercise treadmill test to ≥85% of the predicted ageadjusted heart rate (220 − age) was performed according to the protocol of Bruce and Horsten (5) to exclude subjects with previously undiagnosed heart disease. On a subsequent visit theV˙o _{2 max} was determined using a modified Balke protocol, as previously described (14). The grade of the treadmill was increased every 2 min until the subject was exhausted and could not continue (14). TheV˙o _{2 max} tests fulfilled at least two of the three following criteria:1) the heart rate at maximal exercise was ≥85% of the ageadjusted maximal heart rate,2) the respiratory exchange quotient was >1.10, and 3) there was a plateau in O_{2} consumption defined as a change in O_{2} consumption of <0.2 l/min during the final two collection periods. AbsoluteV˙o _{2 max}, measured in liters per minute, is 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.
Statistical models ofV˙o_{2 max}.
Four statistical models were used to determine the ageassociated decline in V˙o _{2 max}.
1) The form of the RS model is
2) The form of the OLS regression is
3) The univariate form of the LL model is
When the univariate form of the LL model is used, it is usual practice to verify that the variableV˙o
_{2 max}/F
4) WLS regression is identical to OLS regression, except each subject ireceives a weightw_{i}
. The goal of WLS is to minimize the sums of the squares
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 ttests 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 formw_{i}
= 1/F
It is possible to demonstrate that using the weights 1/F
Equation 10
shows that the variance of the transformed residuals is constant as long as the weightsw_{i}
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
It is possible to extend Eq. 9
to include multiple regressors, e.g.
Factors related to the ageassociated decline inV˙o_{2 max}.
We 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 = sedentary, 1 = 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 agerelated decline inV˙o _{2 max} differed between athletes and sedentary subjects, 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 ageassociated decline and effect of ET onV˙o _{2 max}. Because the LL model is log linear, it does not permit a test of the hypothesis of parallel linear declines inV˙o _{2 max}.
Contribution of skeletal muscle loss to the ageassociated decline in V˙o_{2 max}.
We estimated the contribution of the loss of skeletal muscle to the decline in V˙o
_{2 max} as
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 variancecovariance 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 WilksShapiro test were obtained from SAS proc reg and proc univariate, respectively. SAS proc IML was used to obtain the ML estimate of α and to compute the likelihood ratio test for the hypothesis α = 0. An ML grid search algorithm for estimating α is available in SPSS Professional Statistics 6.1 (WLS procedure).
RESULTS
The correlations among selected measures of body size, age, andV˙o _{2 max} are shown in Table 2. All the body size variables correlated with V˙o _{2 max}(P < 0.01). Regression results for the unadjusted and adjusted RS, OLS, LL, and WLS models are shown in Table 3. Only variables significant atP < 0.05 in the regression are listed.
Effect of Age Unadjusted for Body Habitus
All the parameter estimates of the OLS model
All the parameters of the LL model
To obtain the WLS model, the OLS model
Effect of Age Adjusted for Body Habitus
The RS model was
For the OLS model
For the LL model
WLS results were based on the OLS model (Eq.21 ) using weights 1/FFM^{1.30}. The WLS and OLS estimates and standard errors were almost identical (Table 3). The WLS model explained 66% of the variance inV˙o _{2 max}compared with 64% for OLS. The residuals met all the important GLM assumptions. As in the RS and OLS models, the rates of decline inV˙o _{2 max} were the 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 agerelated declines inV˙o _{2 max} that were similar in athletes and sedentary men, as in the OLS model. A plot of the decline in V˙o _{2 max}with age based on the coefficients from the WLS model is shown in Fig.2.
Effect of Age Adjusted for FFM and Height
Nevill (19) suggested that height may be an important covariate in LL models of V˙o _{2 max}. In this study, height was not a significant predictor ofV˙o _{2 max} in any of the models, and its inclusion had little effect on the estimates of the ageassociated 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 inEq. 10 with FFM as a predictor of withinsubject variance [Harvey’s test χ^{2} (1) = 2.64,P > 0.05].
Effect of Age Adjusted for FFM and Fat Mass
Toth et al. (30) suggested that the ratio of FFM to fat mass was an important factor contributing to the ageassociated decline inV˙o _{2 max}. In this study we found 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 AgeAssociated Decline inV˙o_{2 max}
The OLS, WLS, and LL models suggested that ∼35% of the ageassociated decline inV˙o _{2 max} was due to a loss of skeletal muscle. Given thatV˙o _{2 max} declines 0.24 l ⋅ min^{−1} ⋅ decade^{−1}, the loss of skeletal muscle accounts for 0.08 l ⋅ min^{−1} ⋅ decade^{−1}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 FFM to the ageassociated decline inV˙o _{2 max}.
DISCUSSION
The WLS regression incorporating age, FFM, and a dichotomous indicator of physical conditioning status yielded a model ofV˙o _{2 max} that accounted for the largest proportion of variance and met all the important GLM assumptions. AbsoluteR ^{2} 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 ageassociated decline inV˙o _{2 max}.
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 ageassociated 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 inV˙o _{2 max}). The OLS estimate was nearly identical to the WLS estimate, despite the failure of the OLS model to satisfy the assumption of homoscedasticity. This estimate (0.24 l ⋅ min^{−1} ⋅ decade^{−1}) suggests a 9% decline inV˙o _{2 max}between 60 and 70 yr of age, which is consistent with other studies that indicate that the average healthy sedentary man >25 yr of age is expected to lose 9–11% of hisV˙o _{2 max} per decade (10,13, 30).
Because the WLS models satisfied all the important GLM assumptions, the estimates produced by the WLS analysis were BLUE, and thettests 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 modelR ^{2} increased slightly compared with the (homoscedastic) RS model. We also found that the estimates were virtually unchanged over the range α = 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 = 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 logtransformed 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 ageassociated change inV˙o _{2 max} in healthy men are multifactorial. Factors implicated in the ageassociated decline inV˙o _{2 max}include decreases in maximum heart rate, stroke volume, arteriovenous O_{2} 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 larger estimates of the percent decline inV˙o _{2 max} in the 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 ∼35% of the decline in V˙o _{2 max}is due to the ageassociated 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 to be 33%. Recent findings suggest that the decrease inV˙o _{2 max} that is associated with the loss of skeletal 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 O_{2} 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 decreased theirV˙o _{2 max} at the same absolute rate, ∼0.24 l ⋅ min^{−1} ⋅ decade^{−1}. However, the V˙o _{2 max} was 0.96 l/min higher in the athletes than in the sedentary men at all ages. Although some crosssectional studies report similar absolute declines in V˙o _{2 max} over time in athletes and in sedentary individuals (25), other longitudinal studies suggest thatV˙o _{2 max} declines at a slower rate in athletes than in sedentary men (27). Hagberg (10) estimated that the 5.5% decline inV˙o _{2 max} per decade in master athletes is ∼50% of the rate of decline in agematched sedentary men. It is noteworthy that there is a correlation between the change in training intensity and longitudinal changes inV˙o _{2 max} (17). Possibly, the attenuation of the rate of the ageassociated decrease inV˙o _{2 max} may be achievable only for a select group of elite older athletes who are able to continue highintensity training over an extended period of time and also preserve their FFM. The athletes in the present study were heterogeneous with respect to theirV˙o _{2 max} per kilogram of FFM and the intensity and duration of their training. This heterogeneity may have had an impact on the ageassociated decline inV˙o _{2 max}.
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 atlarge 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 subjectselection 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 ageassociated decline inV˙o _{2 max} (18). More direct measures of skeletal muscle mass, such as that provided by magnetic resonance imaging or dual Xray absorptiometry, may provide a more accurate quantification of muscle mass and prediction of the ageassociated decline inV˙o _{2 max}. Third, younger men were not included 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 of decline in V˙o _{2 max} in sedentary males.
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 ageassociated decline inV˙o _{2 max}, which the RS model, without an explicit FFM term, could not. Another advantage of OLS and WLS was their suitability for testing the hypothesis of parallel linear declines inV˙o _{2 max} in athletic and sedentary individuals; the nonlinear 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.
Acknowledgments
We acknowledge the contributions of Drs. E. Bleecker, P. Coon, J. Fleg, R. Pratley, M. J. BusbyWhitehead, and D. Drinkwater and the research exercise physiologists who assisted in measuringV˙o _{2 max} 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.
Footnotes

This work was supported by National Institute on Aging (NIA) Grant R01AG07660, The Johns Hopkins Academic Teaching Nursing Home Award P01AG04402, Grant K07AG00608, the Department of Veterans Affairs Geriatric Research, Education, and Clinical Center (GRECC), The Johns Hopkins Bayview GCRC (Grant M01RR02719), the Claude D. Pepper Older Americans Independence Center NIA Grant P60AG12583, 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.
 Copyright © 1998 the American Physiological Society