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Applied Sciences: Physical Fitness and Performance

A New Nonexercise-Based V̇O2max Equation for Aerobically Trained Females


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Medicine & Science in Sports & Exercise: October 2004 - Volume 36 - Issue 10 - p 1804-1810
doi: 10.1249/01.MSS.0000142299.42797.83
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The measure of maximal oxygen uptake (V̇O2max) is used for many purposes including diagnostic tests, quantifying training intensity for aerobic exercise prescription, monitoring the effects of aerobic training programs, and classifying individuals for health risk (2,3). Typically, direct determination of V̇O2max involves measurement of expired gas samples during an incremental exercise test to exhaustion using a treadmill or cycle ergometer (1,3). Although this method may be optimal for determining V̇O2max, often it is not practical because it requires specialized equipment and trained personnel. Therefore, a number of nonexercise- and exercise-based regression equations to estimate V̇O2max have been developed using variables such as gender, age, height, weight, walking time, frequency of sweating, and Ẇmax (power output at V̇O2max) (8,9,11,12,15,16,21,24,26–28). No studies, however, have utilized habitual physical activity indices such as the intensity, frequency, and duration of exercise training to estimate V̇O2max. These indices have been shown to affect the oxidative adaptations to endurance training and, therefore, may be particularly useful for estimating V̇O2max in aerobically trained individuals (14).

Recently, Malek et al. (18) evaluated the validity of 18 published regression equations for estimating V̇O2max in samples of aerobically trained males and females. These equations utilized various combinations of demographic information (e.g., age, height, and weight), ratings of leisure time physical activity, and/or Ẇmax to predict V̇O2max (18). The equations were selected for cross-validation because of their widespread popularity for estimating V̇O2max and/or proposed accuracy for the general populations of adult males and females. The results of the cross-validation analyses indicated that there were significant (P < 0.006) mean differences between actual and predicted V̇O2max for all 18 equations (18). Furthermore, the total prediction error values for estimating V̇O2max were greater than 10% of the mean for actual V̇O2max for all of the equations. Based on the cross-validation analyses, Malek and colleagues (18) concluded that 16 of the 18 equations were not recommended for estimating V̇O2max or prescribing exercise intensity for an aerobic training program in aerobically trained males or females. Only the equations of Storer et al. (26) that included age, body weight, and Ẇmax as predictor variables were recommended for estimating V̇O2max in aerobically trained subjects. The practicality of these equations is limited, however, because they require the subject to perform an exhaustive cycle ergometer test to estimate V̇O2max. Malek et al. (18) suggested that in an attempt to improve the prediction accuracy of the equations future studies should use the CE (constant error = mean difference between actual and predicted V̇O2max) values from their investigation to adjust the y-intercepts of the equations. This can be accomplished by adding the estimated CE for each prediction equation to the y-intercept of that equation (17). Theoretically, this procedure results in a CE value of zero for the modified equation.

The purposes of the present study were to (a) modify previously published V̇O2max equations using the CE values for aerobically trained females from Malek et al. (18), (b) cross-validate the modified equations to determine their accuracy for estimating V̇O2max in aerobically trained females, (c) derive a new nonexercise-based equation for estimating V̇O2max in aerobically trained females if the modified equations are found to be inaccurate, and (d) cross-validate the new V̇O2max equation using the predicted residual sum of squares (PRESS) statistic and determining total error in an independent sample of aerobically trained females.


Experimental approach and design.

The current study included three parts: (a) cross-validation procedures were used to determine the accuracy of nine V̇O2max prediction equations. The y-intercepts of the nine equations had been modified according to the recommendation of Malek et al. (18); (b) because the errors associated with the modified equations were too high for practical use, a new nonexercise-based equation was developed which included demographic information and habitual physical activity indices as predictor variables; and (c) the accuracy of the new nonexercise-based equation was assessed using two separate cross-validation procedures.


A total of 115 aerobically trained females participated in the present study. We operationally defined an aerobically trained female as someone who had participated in continuous aerobic exercise three or more sessions per week for a minimum of 1 h per session, for at least the past 18 months. In addition, the subjects were asked questions related to their habitual physical activity. Specifically, information was obtained regarding the mode (e.g., “What type of exercise do you perform?”); frequency (e.g., “How many sessions per week do you exercise?”); duration (e.g., “How many hours per week do you exercise?”); length of time performing habitual physical activity (e.g., “How long have you consistently, no more than 1 month without exercise, been exercising?”); and intensity of the exercise performed (e.g., “Indicate, in general, the intensity at which you perform your exercise regimen.”). With regard to intensity, subjects rated their perceived exertion using the Borg scale (5). All procedures were approved by the University Institutional Review Board (IRB) for Human Subjects, and participants signed an IRB approved informed consent.

Maximal cycle ergometer test.

Maximal exercise performance was assessed using an incremental exercise protocol on a cycle ergometer (Ergoline 800S; SensorMedics Corp., Yorba Linda, CA). Seat height was adjusted so that subject’s legs were at near full extension during each pedal revolution. The power output was continuously increased in ramp fashion by computer control. The exercise duration for the ramp phase was 8–12 min as suggested by Buchfuhrer et al. (6). After a period of stabilization at rest, the subjects performed unloaded pedaling (i.e., 0 W) for 3 min followed by the ramp increase in power output (i.e., 30 W·min−1). The subjects were asked to maintain a cycling cadence of 70 rev·min−1. The ramp power output increased until the subject reached voluntary fatigue. A cool-down period with no resistance was performed until heart rate was near that during the unloaded pedaling phase.

Minute ventilation (V̇E) was measured using a mass flow meter and expired fractional concentrations of oxygen and carbon dioxide were continuously monitored by paramagnetic oxygen analyzer and nondispersive infrared CO2 analyzer, respectively (2900; SensorMedics Corp.) (23). The subjects wore a nose clip and breathed through a mouthpiece (2700; Hans Rudolph, Kansas City, MO.). The metabolic cart and breathing valve were calibrated before each test. Oxygen uptake (V̇O2) and carbon dioxide output (V̇CO2) were calculated breath-by-breath using standard algorithms. Breath-by-breath data were presented as a five-breath rolling average. Heart rate was continuously obtained throughout the exercise (Quinton 5000; Seattle, WA). A subject’s data were used if she met two of the following three criteria during the test (3,4,10): a) 90% of age-predicted maximum heart rate, b) respiratory exchange ratio > 1.20, and c) a plateauing of oxygen uptake (≤150 mL·min−1 in V̇O2 over the last 30 s of the test). Maximum oxygen uptake was determined by taking the highest V̇O2 value in the last 30 s of the exercise test. In the present study, Ẇmax was defined as the power output at V̇O2max.

The nine V̇O2max equations for females cross-validated by Malek et al. (18) are included in Table 1 along with the CE values. The modified equations were cross-validated in the present study on a random sample of 50 females according to the recommendations of Lohman (17). For example, the original equation 3F (Table 1) was V̇O2max (mL·min−1) = (9.39*Ẇmax) + (7.70*BW) − (5.88*age) + 136.7. This equation was modified by adding the cross-validation CE of 132 mL·min−1 from Malek et al. (18) to the y-intercept of the original equation 3F (136.7 + 132 = 268.7). Thus, the modified equation 3F equation cross-validated in the present study was: V̇O2max (mL·min−1) = (9.39*Ẇmax) + (7.70*BW) − (5.88*age) + 268.7 (Table 2).

V̇O2max prediction equations for cycle ergometry that were cross-validated against maximum incremental test.
Modified V̇O2max prediction equations for cycle ergometry that were cross-validated against maximum incremental test.

Statistical analysis.

The cross-validation analyses of the nine modified equations in this study (Table 2) were based on an evaluation of the actual V̇O2max versus the predicted V̇O2max via calculation of the constant error (CE = mean difference for actual V̇O2max − predicted V̇O2max), Pearson product-moment correlation (r), standard error of estimate

For the derivation of the new nonexercise-based equation, V̇O2max, weight, height, age, duration, number of years performing habitual physical activity, and intensity at which the subject’s performed their daily exercise were examined using the Statistical Package for the Social Sciences software (v. 12.0, SPSS Inc., Chicago, IL) for screening of missing values, outliers, and distributional properties and transformed, if necessary, for parametric analyses. The positively skewed length of time that subject’s had been performing habitual physical activity was transformed with a natural logarithm to reduce skewness, and improve the normality, linearity, and homoscedasticity of residuals. No cases had missing data.

To generate a new equation for estimating V̇O2max for aerobically trained females, we computed hierarchical linear regression of the nonexercise variables onto V̇O2max (mL·min−1) for 80 subjects selected randomly from the pool of 115 subjects. Specifically, we entered the anthropometric variables (e.g., body weight and height) into the first block then entered age into the second block, and then entered the habitual physical activity indices (e.g., duration, intensity of the exercise, and the length of time subjects performed habitual physical activity) into the third, fourth, and fifth blocks, respectively. This approach was taken to compare the relative contribution, based on R2 change, of the three groups of variables (anthropometric, age, and habitual physical activity indices) to the prediction of V̇O2max in aerobically trained females.

The new nonexercise-based equation was cross-validated using the predicted residual sum of squares (PRESS) method (13). The PRESS approach to cross-validation is based on the error in prediction for each case when only that case is deleted from the model-generating process (20). This term is called the “predicted residual” in SAS (22) and the “deleted residual” in SPSS (25). PRESS is defined as the sum of squares of the predicted or deleted residuals, and the PRESS adjusted R2 (Rp2) can be calculated as 1 − (PRESS/SStotal). Also, we calculated a PRESS standard error of estimate (SEEp) using the following equation:

A second series of cross-validation analyses of the new equation based on 80 subjects was conducted on the remaining 35 aerobically trained females using the same statistical methods (e.g., CE, r, SEE, and TE) that were used to cross-validate the modified equations in the present study. A power analysis showed that with N = 80, the power exceeded 80% for a test to detect an R2 added of 0.05 for a single predictor added to a regression model that had attained R2 = 0.55 using k = 5 predictors (alpha = 0.05) (7). Because of the large range between the highest and lowest CE values reported for females by Malek et al. (18) we selected the median CE value (902 mL·min−1) to conduct a power analysis for the second cross-validation analyses. With a total of 35 subjects the power to detect a large effect size (Cohen’s d = 2.24, alpha = 0.05) between group means for a two-tailed paired t-test exceeded 80%.


Cross-validation of the modified equations (N = 50).

Table 3 includes the results of the cross-validation analyses for the nine modified equations. The mean (±SD) actual V̇O2max for the random sample of 50 females used to cross-validate the modified equations was 2597 ± 399 mL·min−1. None of the modified equations had a significant CE value at a Bonferroni corrected alpha of P ≤ 0.006 (0.05/9). The validity coefficients (r) ranged from 0.52 (equation 8F) to 0.89 (equation 3F). The SEE values ranged from 186 (equation 3F) to 344 mL·min−1 (equation 8F). The TE values ranged from 217 (equation 3F) to 374 mL·min−1 (equation 1F). These values corresponded to %TE (e.g., [TE/mean of actual V̇O2max] × 100) that were ≥ 8% of the mean actual V̇O2max.

Cross-validation of the modified equations for maximal oxygen uptake (V̇O2max) in aerobically trained females (N = 50).

Derivation of the new nonexercise equation (N = 80).

Eighty aerobically trained females were randomly selected for the derivation group from the pool of 115 subjects (Table 4). All six predictors were significantly related to actual V̇O2max (Table 5), and each predictor contributed independently (P≤ 0.01, Table 6) to the model. As shown in Table 6, the proportion of variance in V̇O2max predicted by the model (R2 adjusted) was 0.67 (SEE = 247 mL·min−1).

Characteristics of subjects (mean ± SD).
Correlation matrix of V̇O2max and independent variables.
New V̇O2max prediction equation for aerobically trained females.

First cross-validation of the new nonexercise equation using PRESS.

The cross-validation results of the PRESS method for the new nonexercise-based equation are shown in Table 6. For this model, Rp2 was nearly as large as the R2 adjusted (0.63 vs 0.67), and the SEEp value was nearly equal to the corresponding SEE value (247 vs 259 mL·min−1).

Second cross-validation of the new nonexercise equation using an independent sample (N = 35).

Table 7 includes the results of the cross-validation analyses for the new nonexercise equation (Table 6) based on the subsample of 35 aerobically trained females who were withheld from the derivation of the equation. The mean predicted V̇O2max was 2658 mL·min−1 (Table 7) compared with the actual V̇O2max of 2729 mL·min−1 (Table 4). The mean CE value of 71 mL·min−1 (the difference between predicted and actual) was not significantly different from zero, t (34) = 1.61, P > 0.05. The validity coefficient (r) and SEE values were 0.76 and 264 mL·min−1, respectively. The TE value was 268 mL·min−1, which corresponded to a %TE of 10% of mean actual V̇O2max.

Cross-validation of new maximal oxygen uptake (V̇O2max) equation in a random independent sample of aerobically trained females.


When compared with the results of Malek et al. (18) for the previously published equations, the cross-validation analyses of the modified equations in the present study (Table 3) resulted in lower CE, SEE, and TE values and higher validity coefficients (r) for all equations. Despite these improvements in accuracy, however, the %TE values for the modified nonexercise-based equations (equations 1F, 2F, 4F, 5F, 6F, 7F, 8F, and 9F) were ≥ 13%. Equation 3F had a %TE of 8%, which was an improvement from the original cross-validation value (% TE = 12%) reported by Malek et al. (18). Equation 3F, however, is limited for practical use because it requires a maximal test to determine Ẇmax. In cases where Ẇmax is available, however, the modified version of equation 3F (Table 2) is recommended for estimating V̇O2max in aerobically trained females.

To further increase the accuracy of a nonexercise-based model, a new equation using demographic variables and habitual physical activity indices was developed. The derivation of the new nonexercise-based equation in the present study resulted in a validity coefficient of R = 0.83 and a SEE of 247 mL·min−1 (Table 6). This SEE was substantially less than those for the modified equations (Table 3; 312–344 mL·min−1) cross-validated in the present study. Information about the subject’s habitual physical activity was used as predictor variables to develop the new nonexercise-based equation. In the current study, the regression equation yielded R2-adjusted = 0.33 when using only height and weight. This value increased to R2-adjusted = 0.40 when age was added to the equation. Next, the habitual physical activity indices (e.g., duration, intensity of the exercise, and the length of time subjects performed habitual physical activity) were added in the final three steps of the regression model resulting in R2-adjusted values of 0.48, 0.58, and 0.67, respectively (Table 6). The habitual physical activity indices, which have not been used in previously developed equations, substantially improved the accuracy for the estimation of V̇O2max in aerobically trained females when compared with the use of age, height, and weight alone.

To determine the generalizability of the new equation, we conducted two cross-validation analyses. For the PRESS procedure, the R2 adjusted for the new equation (R2 = 0.67) and Rp2 (Rp2 = 0.63) were similar. This was also true for the SEE (247 mL·min−1) and SEEp (259 mL·min−1) values. In the second analysis, the actual V̇O2max values for an independent cross-validation sample of aerobically trained females (N = 35) was compared with estimated V̇O2max values from the new nonexercise-based equation. The results indicated close agreement between the actual and predicted V̇O2max values as represented by the small, nonsignificant CE value (71 mL·min−1). In addition, the SEE% of 9.7% was lower than the 10–20% values reported for most field methods such as step tests, submaximal cycle ergometer tests, or walk/run tests used to estimate V̇O2max (19,28). Furthermore, the TE, which is the best single criterion for determining the accuracy of an equation because it combines the errors associated with the SEE and CE (17), was 268 mL·min−1, which corresponded to a %TE of 10%. This value was 3–4% less than the %TE values for the modified nonexercise equations (Table 3).

There are a number of benefits to using two separate cross-validation procedures. First, through the use of the PRESS statistic, we were able to determine the relative accuracy of our equation based on N − 1 cases from the sample used to derive the equation. Holiday et al. (13) stated “The PRESS statistic and associated residuals do not require the data to be split, [and] yield alternative unbiased estimates of R2 and SEE. . .” (p. 612). As a result of using the PRESS statistic, the new nonexercise-based equation was found to have high generalizability for a population of aerobically trained females (Table 6). Studies such as Neder et al. (21) and Davis et al. (9) used only the PRESS statistic to validate their equations. Thus, these studies provided only a single-level validation approach. Herein lies a major difference between the present study and previous studies, which have derived new equations for estimating V̇O2max. An essential concern in developing a new equation is how well the model will work for new cases from the same population. The present study addressed this concern by using the PRESS statistic to determine the generalizability of the new nonexercise-based equation. To further corroborate the validity of the equation, the actual and predicted V̇O2max values for an independent sample of aerobically trained females were compared (Table 7). Therefore, the second benefit of using two separate cross-validation procedures is that the investigator provides two levels of validation. Holiday et al. (13) stated, “A central tenet of cross-validation is that the custodians of the model should not release a prediction equation to the user community without some assurance that it will do a good job” (p. 616). To our knowledge, no other studies in the literature have used the two-level validation approach outlined in the present study for a nonexercise-based V̇O2max equation. Therefore, based on the results of two separate cross-validation procedures, the new nonexercise-based equation derived in the present study is recommended for estimating V̇O2max for aerobically trained females.


The results of the present study, in conjunction with those of Malek et al. (18), indicated that the original and modified versions (Table 2) of the nonexercise-based equations resulted in TE values (% TE = 13–14%) that were too large to be of practical value for estimating V̇O2max in aerobically trained females. The modified equation of Storer et al. (Tables 2 and 3) is recommended for estimating V̇O2max in aerobically trained females only when an estimate of Ẇmax from a maximal cycle ergometer test can be obtained. However, because this requires the subject to perform a maximal cycle ergometer test to determine Ẇmax for the estimation of V̇O2max the new nonexercise-based equation derived in the present study (Table 6), overcomes this limitation by using only demographic and habitual physical activity information to estimate V̇O2max. Given the cross-validation %TE of 10%, if Ẇmax is not available, the new nonexercise-based equation derived in this study is recommended for estimating V̇O2max in aerobically trained females.

We thank the various triathlete, cycling, running, and adventure racing clubs in Southern California for participating in this study.

Note: Data were collected while Mr. Malek was a research associate in the David Geffen School of Medicine at UCLA.


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