Aerobic fitness is often defined in terms of maximal oxygen uptake (V̇O_{2max} ). The measurement of V̇O_{2max} is used for many purposes including quantifying training intensity for aerobic exercise prescription, monitoring the effects of aerobic training programs, and classifying individuals for health risk (^{1,3} ). Whereas highly accurate laboratory techniques are preferred when assessing V̇O_{2max} it is often necessary to utilize “field” methods because of the time, cost, equipment, and personnel requirements associated with laboratory-based V̇O_{2max} testing. Consequently, a number of equations have been developed for estimating V̇O_{2max} based on readily available measures of gender, age, height, and weight (^{2,6,7,9,10,13,14,20,21,24,26} ).

These equations, however, may not be accurate if they are applied to populations that differ from those used to develop the prediction model (^{15,27} ). The decision regarding which equation to use should be based on its accuracy for the population being examined. Existing equations have not been scrutinized for their sensitivity, specificity, and strength of predictive power in aerobically trained subjects; and, thus, it is possible that the level of aerobic fitness may influence the accuracy with which these equations can be used to estimate V̇O_{2max} (^{1} ).

Cross-validation analyses provide a way to compare the accuracy of various equations for a particular population (^{12,16} ). Although this procedure is commonly used to validate body composition equations (^{11,16,25} ), we are not aware of any studies that have cross-validated the V̇O_{2max} prediction equations examined in the present study. Therefore, the purpose of this investigation was to cross-validate existing V̇O_{2max} prediction equations on samples of aerobically trained males and females.

METHODS
Approach to the problem and experimental design.
The present study used a cross-validation design to determine the concurrent validity with which 18 equations from the literature could be used to estimate V̇O_{2max} in samples of aerobically trained males and females. The predicted V̇O_{2max} values from the equations were statistically compared with actual V̇O_{2max} values derived from maximal cycle ergometer tests.

Subjects.
A total of 142 aerobically trained subjects (93 males and 49 females) volunteered for this investigation (Table 1 ). The subjects included triathletes (N = 66), cyclists (N = 35), marathon runners (N = 26), adventure racers (N = 8), rowers (N = 4), and swimmers (N = 3). We perationally defined an aerobically trained individual 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. All procedures were approved by the University Institutional Review Board (IRB) for Human Subjects, and participants signed an IRB approved informed consent.

TABLE 1: Characteristics of subjects (mean ± SD).

V̇O_{2max} equations.
The equations in this study were selected for cross-validation because they were frequently cited in the literature and were derived using cycle ergometry to determine V̇O_{2max} (Tables 2 and 3 ). Also, the equations included readily available variables such as gender, age, height, and body weight. In addition, four equations (equation 2M, equation 2F, equation 6M, and equation 6F) used a leisure time score (lei) to quantify the subject's level of physical activity (^{14,18} ), and equations 3M and 3F included maximal work rate (Ẇ_{max} ) which was defined as the highest power output attained at V̇O_{2max} (^{24).}

TABLE 2: Characteristics of existing V̇O2max prediction equations for cycle ergometry.

TABLE 3: V̇O2max prediction equations for cycle ergometry which were cross-validated against maximum incremental test.

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. (^{5} ). After a period of stabilization at rest, 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.

Minute ventilation (V̇_{E} ) 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 CO_{2} analyzer, respectively (2900, SensorMedics Corp.). 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 prior to each test. Oxygen uptake (V̇O_{2} ) and carbon dioxide output (V̇CO_{2} ) 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 they met two of the following three criteria (^{3,4,8} ): a) 90% of age-predicted heart rate, b) respiratory exchange ratio >1.20, and c) a plateauing of oxygen uptake (≤150 mL·min^{−1} in V̇O_{2} over the last 30 s of the test). Maximum oxygen uptake was determined by taking the highest V̇O_{2} value in the last 30 s of the exercise test. In the present study, Ẇ_{max} was defined as the power output at V̇O_{2max} .

Statistical analysis.
The cross-validation analyses of the 18 equations in this study (Table 3 ) were based on an evaluation of the actual V̇O_{2max} versus the predicted V̇O_{2max} via calculation of the constant error (CE = mean difference for actual V̇O_{2max} − predicted V̇O_{2max} ), Pearson product-moment correlation (r), standard error of estimateEQUATION

, and total errorEQUATION

. All analyses were performed using Statistical Package for the Social Sciences software (version 11.1; SPSS Inc., Chicago, IL).

Post hoc power analyses were conducted based on the lowest and highest CE values for the males (equations 3M and 7M) and females (equations 3F and 7F). The results indicated that the power to detect a significant difference between the means for the actual and predicted V̇O_{2max} values ranged from 0.87 and 1.0.

RESULTS
Descriptive characteristics of the subjects are presented in Table 1 . The mean V̇O_{2max} determined by the incremental test was 4098 ± 648 mL·min^{−1} and 2714 ± 444 mL·min^{−1} for males and females, respectively.

Table 4 presents the results of the cross-validation analyses. The mean difference (CE) between the actual and predicted V̇O_{2max} was analyzed using a dependent t -test. For each sample (males and females), the Bonferroni correction was used to adjust the significance level (0.05/9 = P < 0.006) for each comparison. The CE values ranged from −216 (equation 3M) to 1415 mL·min^{−} ^{1} (equation 7M) for the males, and 132 (equation 3F) to 1037 mL·min^{−} ^{1} (equation 7F) for the females. All equations resulted in CE values that differed significantly from zero (P < 0.006).

TABLE 4: Cross-validation of maximal oxygen uptake (V̇O2max) in aerobically trained subjects.

The validity coefficients ranged from r = 0.36 (equation 8M) to r = 0.88 (equation 3M) for males, and r = 0.41 (equation 8F) to r = 0.80 (equation 3F) for females. The SEE values ranged from 316 (equation 3M) to 609 mL·min^{−1} (equation 8M) for males, and 266 (equation 3F) to 409 mL·min^{−1} (equation 8F) for females. TE, which accounts for the errors associated with both the CE and SEE ranged from 413 (equation 3M) to 1535 mL·min^{−1} (equation 7M) for males, and 317 (equation 3F) to 1110 mL·min^{−1} (equation 7F) for females.

DISCUSSION
The following criteria were used in the present study to evaluate the results of the cross-validation analyses (^{22,23} ): a) the mean values for actual and predicted V̇O_{2max} should be comparable; b) the TE should be calculated because it reflects the true difference between the actual and predicted V̇O_{2max} , whereas the SEE reflects only the error associated with the regression between the variables; c) there should be close similarity between TE and SEE because it reflects the relationship between the regression line for actual versus the predicted V̇O_{2max} and the line of identity; d) a low SEE is desirable and is preferred over correlation coefficients because it is not sensitive to differences in means and they are affected by differences between samples in the variability of V̇O_{2max} values; and e) there should be no relationship between the CE and actual V̇O_{2max.}

Seventeen of the 18 equations (all equations except equation 3M) cross-validated in the present study significantly (P < 0.006) underestimated actual V̇O_{2max} (Table 4 ). For the males, the CE values ranged from −216 to 1415 mL·min^{−1} , which corresponded to 2.9–18.4 mL·kg^{−1} ·min^{−1} . These CE values represented 5.4 and 34.4% of the mean of actual V̇O_{2max} . For the females, the CE values ranged from 132 to 1037 mL·min^{−1} , which corresponded to 2.3–16.7 mL·kg^{−1} ·min^{−1} or 5.2–38.2% of the mean of actual V̇O_{2max} . These findings indicated that, in almost all cases, the currently available equations (equations 1, 4, 5, 7, 8, 9, and 10) that utilized demographic information such as gender, age, height, and body weight, as well as equations (equation 2, 3F, and 6) that also included a leisure time score or Ẇ_{max} systematically underestimate actual V̇O_{2max} in aerobically trained males and females. Equation 3M, however, which utilized Ẇ_{max} along with age and body weight overestimated actual V̇O_{2max} by 216 mL·min^{−1} .

There were significant (P < 0.05), positive correlations (r = 0.54 to 0.92) between the CE values and actual V̇O_{2max} for all equations in the present study except equation 3M (r = 0.03) and equation 3F (r = 0.20). The differences between actual and predicted V̇O_{2max} were greatest at the upper end of the actual V̇O_{2max} distribution. Thus, the equations better predicted V̇O_{2max} in the subjects with the lowest levels of aerobic fitness. This may have been due to the fact that the equations were derived on untrained subjects but cross-validated on aerobically trained samples. These findings suggested that 16 of the 18 equations used to estimate V̇O_{2max} in the present study exhibited some population specificity as it relates to the subjects' level of aerobic fitness.

As shown in Table 4 , the validity coefficients ranged from r = 0.36 (equation 8M) to r = 0.88 (equation 3M) for the males, and r = 0.41 (equation 8F) to r = 0.80 (equation 3F) for the females. These values were lower than those from the original derivation studies presented in Table 3 . This shrinkage of the multiple correlation is typical of cross-validation analysis (^{19} ). Furthermore, the SEE values ranged from 316 (equation 3M) to 609 mL·min^{−1} (equation 8M) for the males, and 266 (equation 3F) to 409 mL·min^{−1} (equation 8F) for the females. The SEE values expressed as a percentage of the mean of actual V̇O_{2max} (SEE%) ranged from 7.7 to (equation 3M) to 14.9% (equation 8M) for the males, and 9.8 (equation 3F) to 15.1% (equation 8F) for the females. Typically, SEE values for estimating V̇O_{2max} from various field methods such as step tests, submaximal cycle ergometer tests, or walk/run tests represent approximately 10–20% of the actual V̇O_{2max} (^{17,28} ). Therefore, the SEE values for the equations in the present study were similar to those associated with other indirect methods for estimating V̇O_{2max} .

Although the SEE provides important information concerning the error associated with the regression for actual versus predicted V̇O_{2max} , the TE is the best single criterion for determining the accuracy of an equation because it combines the errors associated with the SEE and CE (^{16} ). In the present study, equation 3M (413 mL·min^{−1} ; 10% of actual V̇O_{2max} ); and equation 3F (317 mL·min^{−1} ; 12% of actual V̇O_{2max} ) exhibited the lowest TE values for the male and female samples, respectively (Table 4 ). In addition, the SEE and TE will be equal only when the means for actual and predicted V̇O_{2max} values are identical (CE = 0) (^{16} ). Therefore, valid equations exhibit close agreement between the SEE and TE values (^{16} ). In the present investigation, there were large differences (≥244 mL·min^{−1} for the males and ≥ 163 mL·min^{−1} for the females) between the SEE and TE for all equations except equation 3M (97 mL·min^{−1} ) and equation 3F (51 mL·min^{−1} ) (Table 4 ).

It should be noted that the magnitude of the differences between the SEE and TE values for equations 1, 2, 4, 5, 6, 7, 8, and 9 were due mostly to the large CE values. Therefore, it may be possible to improve the accuracy of the equations in the present study by correcting for the systematic errors (i.e., CE values). To do so, future studies should use the procedure recommended by Lohman (^{16} ) to modify the original equations by correcting the y -intercepts using the CE values from the present study such that the CE = 0 and then cross-validate the modified equations on independent samples of aerobically trained males and females. For example, equation 4M (Table 3 ) is as follows: V̇O_{2max} (mL·min^{−1} ) = [(nwt + awt)/2]* (50.72 − (0.372 * age)). For aerobically trained males, this equation would be modified by adding the CE of 1219 mL·min^{−} ^{1} , resulting in the following equation: V̇O_{2max} (mL·min^{−1} ) = [(nwt + awt)/2]* (50.72 − (0.372 * age)) + 1219.

Based on the low SEE and TE values, the nonsignificant relationships between the CE values and actual V̇O_{2max} high validity coefficients, and small differences between the SEE and TE values, equations 3M and 3F (Table 4 ) of Storer et al. (^{24} ) are recommended for estimating V̇O_{2max} in aerobically trained males and females. These equations (equations 3M and 3F), however, have practical limitations compared to the nonexercise equations in the present study because they require the completion of a maximal cycle ergometer test for the determination of Ẇ_{max} . The salient feature of the nonexercise equations is that they utilize easily attainable demographic and/or survey information for the estimation of V̇O_{2max} without having the subject perform an exhaustive exercise bout. It is possible that modifying the y -intercepts of the nonexercise equations in the present study using the procedure of Lohman (^{16} ) may sufficiently reduce the error for one or more of the equations such that V̇O_{2max} could be accurately estimated in aerobically trained subjects using only demographic data. An independent cross-validation study would be necessary to test this hypothesis. If, however, modifying the nonexercise equations in this way does not result in a valid model, it may be possible to improve the practicality of estimating V̇O_{2max.} by deriving new equations from only demographic data in large samples of aerobically trained males and females.

SUMMARY
The results of the present study indicated that the equations (equations 3M and 3F) of Storer et al. (^{24} ), which included age, body weight, and Ẇ_{max} , most accurately estimated V̇O_{2max} in aerobically trained males (TE = 413 mLmin^{−1} ) and females (317 mL·min^{−1} ). The applicability of these equations, however, is limited because they require a maximal exercise bout to determine Ẇ_{max} in order to estimate V̇O_{2max.} The TE values for the nonexercise equations were ≥ 564 mL·min^{−1} and, therefore, are not recommended for estimating V̇O_{2max} in aerobically trained males and females. Future studies should modify the nonexercise equations using the CE values from the present investigation and then cross-validate them on samples of aerobically trained males and females to determine their accuracy for estimating V̇O_{2max} . If, however, modifying the nonexercise equations does not improve their accuracy, new equations should be derived using only demographic data to estimate V̇O_{2max} in aerobically trained samples of males and 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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