Modeling the Relationship between Velocity and Time to Fatigue in Rowing

Abstract

HILL, D. W., C. ALAIN, and M. D. KENNEDY. Modeling the Relationship between Velocity and Time to Fatigue in Rowing. Med. Sci. Sports Exerc., Vol. 35, No. 12, pp. 2098–2105, 2003.

Introduction 

Several mathematical models describe the relationship between velocity and time to fatigue.

Purpose 

The purposes of this study were to evaluate different critical velocity (V_critical) models applied to rowing ergometry and to investigate prediction of performance time in a 2000-m race based on results from shorter trials.

Methods 

Sixteen men performed seven rowing ergometer tests. Velocity and time to fatigue data from the 200-m (∼0.5 min) to 1200-m (∼3 min) trials and from the 200-m to 2000-m (∼6.5 min) trials were fit to a two-parameter hyperbolic model, a three-parameter hyperbolic model, and a three-parameter exponential model.

Results 

Including data from the 2000-m trial generally resulted in higher R² and smaller SEE. V_critical from the three versions of the two-parameter model were 4.71 ± 0.28 m·s⁻¹, 4.80 ± 0.27 m·s⁻¹, and 5.04 ± 0.24 m·s⁻¹ (P < 0.001). The two three-parameter models had high R² (0.991 and 0.990, respectively) and generated parameter estimates that appeared reasonable. Time for a 2000-m race was predicted better using the two-parameter model (r > 0.974) than using the three-parameter models (r = 0.820–0.870).

Conclusion 

It is necessary to include the relatively long 2000-m predicting trial to describe accurately the velocity–time relationship in rowing. The two-parameter model may be useful in predicting time for a 2000-m race but is not otherwise appropriate for modeling when predicting trials of <1 min duration are included. Choice of model and duration of trials must be considered when using mathematical modeling to derive V_critical and other parameters in rowing.

Monod and Scherrer (²⁶) reported that there was a hyperbolic relationship between power and time to fatigue for individuals during bouts of repetitive lifting exercises performed using isolated muscle groups. They transformed the two-parameter hyperbolic relationship into a linear relationship between total work performed and time to fatigue. Whipp and colleagues (³⁶) presented a third version of the two-parameter model, regressing power against the inverse of time to fatigue measured during a series of predicting trials. The two parameters that define each of these three relationships are critical power (P_critical) and anaerobic work capacity (AWC). The critical power concept has been extended to rowing ergometer data by Kennedy and Bell (²⁴). For rowing, power is replaced by velocity, work is replaced by distance, and P_critical is replaced by critical velocity (V_critical).

The three versions of the two-parameter critical velocity model are EQUATIONEQUATIONEQUATION

The relationships are presented in Figures 1, 2, and 3, respectively.

FIGURE 1: The two-parameter hyperbolic velocity–time relationship (Equation 1), with data from a representative participant. Using predicting trials of 200 m to 1200 m (solid curve), the parameter estimates were 5.08 m·s⁻¹ for V_critical (vertical asymptote, solid line) and 55 m for AWC (the area bounded by the two asymptotes and any point on the curve). When the 2000-m predicting trial was included (dashed curve), the values were 4.99 m·s⁻¹ and 71 m, respectively.

FIGURE 2: The two-parameter linear distance–time relationship (Equation 2), with data from the same participant as in Figure 1. Using predicting trials of 200 m to 1200 m (solid line), the parameter estimates were 5.19 m·s⁻¹ for V_critical (slope) and 39 m for AWC (y-intercept). When the 2000-m predicting trial was included (dashed line), the values were 5.07 m·s⁻¹ and 53 m, respectively.

FIGURE 3: The linear velocity–(1/time) relationship (Equation 3), with data from the same participant as in Figure 1. Using predicting trials of 200 m to 1200 m (solid line), the parameter estimates were 5.32 m·s⁻¹ for V_critical (slope) and 25 m for AWC (y-intercept). When the 2000-m predicting trial was included (dashed line), the values were 5.26 m·s⁻¹ and 28 m, respectively.

P_critical or V_critical from the two-parameter model has been found to relate to measures of sustainable aerobic power (^6,20,27) and AWC to measures of anaerobic capacity (^21,23,27). Correspondingly, P_critical has been shown to respond to endurance training (^9,22) whereas AWC has been demonstrated to respond to high-intensity training (²³). In addition, the critical power concept has been used to predict performance in long events from results in shorter time trials (^14,24,34).

The two-parameter model presupposes that, regardless of exercise intensity or duration, a fixed percentage of V̇O_2max is immediately available at the onset of exercise and is sustained throughout the exercise and the anaerobic work capacity can be completely expressed. Thus, extrapolation of the relationship to extremes of intensity or duration requires that some velocity (i.e., V_critical) can be sustained for an infinite time and that an infinitely high velocity can be sustained for a very short time. Clearly, neither of these requirements is met in nature, and the model may break down when time to fatigue is less than ∼1 min (^16,28) or much greater than 30 min (³⁰). Therefore, Morton (²⁸) introduced a model that includes a third parameter, maximal power or maximal velocity (V_maximal), which is the highest possible instantaneous velocity that an individual can attain. The model can be re-parameterized such that the third parameter is k, which defines the location of the horizontal asymptote of the velocity–time curve below the x-axis. The three-parameter relationships are presented in Figure 4.

FIGURE 4: The three-parameter hyperbolic velocity–time relationship (Equations 4 and 5), with data from the same participant as in Figure 1. Using predicting trials of 200 m to 1200 m (solid curve), parameter estimates were 4.53 m·s⁻¹ for V_critical (vertical asymptote, solid line), −190 s for k (horizontal asymptote, solid line), 6.26 m·s⁻¹ for V_maximal (the point at which the solid velocity–time curve crosses the x-axis), and 328 m for A (the area bounded by the two asymptotes and any point on the curve). When the 2000-m predicting trial was included (dashed curve), values were 4.83 m·s⁻¹, −103 s, 165 m, and 6.44 m·s⁻¹, respectively.

The three-parameter hyperbolic model may be presented as

or as

Although the three-parameter model has previously been presented with parameters of P_critical and AWC (^4,5,8,28), or V_critical and AWC (¹⁷), it is presented here with parameters of V_critical and A, as it is not clear that AWC and A should have a similar physiological meaning. This model has never been applied to rowing data.

Hopkins and colleagues (¹⁶) proposed an exponential model that includes V_maximal, and they reported that it provided a better description of the intensity–time relationship than did the two-parameter hyperbolic model (Equation 3) for high-speed treadmill running at various inclines that resulted in fatigue after ∼10 s to ∼3 min. This exponential model regresses incline (or velocity, in rowing) against time to fatigue and is presented in Figure 5. The model is EQUATION

FIGURE 5: The three-parameter exponential velocity–time relationship (Equation 6), with data from the same participant as in Figure 1. Using predicting trials of 200 m to 1200 m (solid curve), parameter estimates were 5.06 m·s⁻¹ for V_critical (horizontal asymptote, solid line), 6.23 m·s⁻¹ for V_maximal (y-intercept) and 151 s for tau (time constant of the response curve). When the 2000-m predicting trial was included (dashed curve), values were 5.09 m·s⁻¹, 6.23 m·s⁻¹, and 144 s, respectively.

Although it is possible to generate versions of the exponential model with different dependent variables, this is the only version that has appeared in published form (^4,8,16,17). This model has never before been applied to rowing data.

P_critical and AWC estimates obtained using the two- and three-parameter hyperbolic models and the exponential model have been compared (⁸). Others have evaluated the effect of using the three models on only P_critical or V_critical, and not on AWC (^4,17) or the effect of using only two of the three models (²⁸). The three-parameter hyperbolic model (Equation 4) is favored over the two-parameter models for three reasons. First, in contrast to the exponential model and the two linear versions of the two-parameter models, it correctly identifies time to fatigue as the dependent variable (^8,28). Second, it produces lower estimates for P_critical (^5,8,28). Third, it overcomes the assumption that there is no upper limit to power production (^8,28). However, it is noted that P_critical derived using the three-parameter model cannot be sustained for an infinite time (^4,8). In addition, the model does not address the delayed response of the aerobic pathways at the onset of exercise (³⁷). It also assumes that the portion of the anaerobic capacity that is used is a linear function of exercise intensity and the anaerobic capacity is fully tapped only during exercise at P_critical (^4,8,28).

The purpose of this study was to evaluate different ways of modeling the velocity–time relationship in rowing. In that sense, the study was a replication of the investigation by Gaesser et al. (⁸), with some noteworthy differences in methodology. This is the first time that the three-parameter models have been applied to rowing ergometer data. More importantly, the range of exercise durations in the predicting trials was clearly outside that which is often recommended for use with the two-parameter model (³¹) and clearly within the range stipulated for use with the three-parameter models by the researchers who first introduced them (^16,28). In addition, this study was similar to studies that have attempted to predict performance in a long event based on results of shorter trials (^14,34). Prediction of performance assumes that the velocity–time relationship extends past the range of predicting trials (200 m to 1200 m in this study) to the performance distance (2000 m in this study). Therefore, another purpose of this study was to compare parameter estimates derived using the results from 200-m to 1200-m trials to parameter estimates derived using trials of 200 m to 2000 m. The first hypothesis was that including the 2000-m trial would have no effect on parameter estimates. The second hypothesis was that there would be differences in parameter estimates between the three models, with a better fit for the three-parameter models. The third hypothesis was that the prediction of 2000-m performance would be better using estimates of V_critical and AWC derived from three-parameter models, compared with estimates from the two-parameter model.

METHODS

Participants.

Participants were 16 men with at least 1 year’s experience rowing at the provincial or national level. They provided voluntary written informed consent to participate in the study, which had received ethical approval by the Faculty of Physical Education and Recreation at the University of Alberta. The men were of mean (± SD) age 23 ± 4 yr, height 186.2 ± 6.5 cm, and mass 83.7 ± 8.4 kg.

Determination of V̇O_2max.

As in other rowing studies performed at the University of Alberta (^1,11), V̇O_2max was determined using a continuous incremental test, performed on a Concept II Model C rowing machine (Concept II Inc., Morrisville, VT). The initial work rate was 100 W for 2 min. The work rate was increased 50 W every 2 min until the V_e/V̇O₂ ratio reached a minimum and began to increase. Thereafter, the work rate was increased 50 W every 1 min. When the respiratory exchange ratio reached ∼1.10, the participant performed at maximal stroke rate until volitional exhaustion. This last stage always lasted < 2 min. The mean V̇O_2max was 5.01 ± 0.50 L·min⁻¹.

Fixed distance predicting trials.

Each participant performed seven exhaustive tests on a Concept II Model C rowing machine (Concept II Inc.). These predicting trials were over distances of 200 m, 400 m, 600 m, 800 m, 1000 m, 1200 m, and 2000 m administered in randomized order, with the exception that the 2000-m trial was always performed last. At least 24 h of rest (no training) separated each trial. Each test was preceded by a 1000-m warm-up including three maximal 5-s sprints. Before each test, the rower was reminded to provide an all-out effort and to treat the trial as a race. During the tests, the rowers received no verbal encouragement but had visual reports of cumulative distance, pace, stroke rate, and elapsed time. Time to complete each trial was recorded to the nearest 0.1 s.

The participants did not perform practice trials per se. However, each used the ergometer for training and testing on a regular basis before the study.

Strategies for modeling the velocity–time relationship.

For each participant, data from the 200-m, 400-m, 600-m, 800-m, 1000-m, and 1200-m tests were fitted using three different modeling strategies: 1) traditional critical power concept (²⁶), based on a two-parameter hyperbolic relationship (Equations 1, 2, and 3); 2) three-parameter hyperbolic relationship introduced by Morton (²⁸) (Equations 4 and 5); and 3) three-parameter exponential relationship introduced by Hopkins et al. (¹⁶). These regression analyses were then repeated using data from all seven tests, i.e., with the addition of the 2000-m trial. Regression was performed using SPSS (Chicago, IL).

Oxygen equivalent of AWC and A.

To assess the significance of AWC and A, which are the parameters that are purported to reflect anaerobic work capacity, the oxygen equivalent of these two parameters were calculated as follows. It was assumed that the peak velocity attained in an incremental test is similar to the velocity in a 2000-m race (V₂₀₀₀), as the peak incremental test velocity can be sustained ∼5 min (¹⁵) to ∼7 min (²) and performance time for the 2000-m distance is ∼6.5 min. The oxygen demand at the peak incremental test velocity is ∼104% V̇O_2max (¹²). Therefore, on average for the participants in this study, the oxygen demand in the 2000 m would be 5.21 L·min⁻¹ (1.04·5.01 L·min⁻¹). Dividing this oxygen demand by the mean V₂₀₀₀ (4.93 m·s⁻¹) gives a relative oxygen demand of ∼18 mL·m⁻¹. The oxygen equivalent of AWC and A (in mL·kg⁻¹) was calculated by multiplying each value by this oxygen demand factor and then dividing this result by the mean body mass (83.7 kg).

Statistical analyses.

V_critical and AWC values from the two-parameter model were compared using a two-way ANOVA, with repeated measures across equation (Equation 1, Equation 2, Equation 3) and trials distances (200 m to 1200 m, 200 m to 2000 m). Significance for all comparisons was set at P < 0.05, and the results of post hoc comparisons were interpreted using a Bonferroni correction.

For comparisons between models, based on the explanation by Gaesser et al. (⁸), the “true” hyperbolic velocity–time version (Equation 1) was selected as the criterion version for the traditional two-parameter hyperbolic model. The two parameterizations of the three-parameter hyperbolic model generated identical values for V_critical and AWC. Values obtained using Equation 1, Equation 4, and Equation 6 were compared using a two-way ANOVA (model by trials distances). This permitted evaluation of the effects of using different models and potential interaction between these effects and the distances used in the predicting trials.

Expected times and velocities for the 2000-m trial were calculated using the six sets of V_critical and AWC that were obtained using data from the 200-m to 1200-m trials and Equations 1–6, solving each equation for time to fatigue after setting distance to 2000 m. Calculated and actual times and velocities were compared using linear regression.

RESULTS

Results obtained using the two-parameter model are in Table 1. The three versions of the model produced markedly different V_critical (F_2,30 = 514.39, P < 0.001) and AWC (F_2,30 = 493.17, P < 0.001). Addition of the 2000-m predicting trial always reduced the magnitude of the V_critical estimate (F_1,15 = 76.21, P < 0.001) and increased the magnitude of the AWC estimate (F_1,15 = 45.57, P < 0.001). There was a strong interaction effect on both V_critical (F_2,30 = 9.20, P = 0.001) and AWC (F_2,30 = 14.39, P < 0.001), and quantitative differences in the effect of adding the 2000-m trial. Specifically, addition of the 2000-m predicting trial had a smaller effect on estimates obtained using the velocity–(1/time) relationship in Equation 3 than on the estimates from Equations 1 and 2.

TABLE 1: Estimates of V_critical (m·s⁻¹) and AWC (m) derived using the three versions of the two-parameter hyperbolic model and the results from predicting trials of 200 to 1200 m or from predicting trials of 200 to 2000 m. There were significant interaction effects for each variable.

The criterion values generated using each model are presented in Table 2. V_critical from Equation 4 or 5 was less than V_critical from Equation 1, which was less than V_critical from Equation 6 (F_2,30 = 53.19, P < 0.001). There was a significant model by trials distances interaction effect (F_2,30 = 15.43, P < 0.001). Results of post hoc comparisons revealed that addition of the 2000-m predicting trial reduced V_critical from the two-parameter model, V_critical from the three-parameter hyperbolic model, and had no significant effect on V_critical from the exponential model.

TABLE 2: Criterion estimates of V_critical (m·s⁻¹), AWC (m), and V_maximal (m·s⁻¹) derived using the three different models and the results of predicting trials of 200 to 1200 m or 200 to 2000 m. There were significant interaction effects for all variables except SEE of V_maximal.

Parameter A from Equation 4 or 5 was higher than AWC from Equation 1 (F_1,15 = 29.46, P < 0.001). There was a significant interaction effect (F_1,15 = 8.81, P = 0.010). Results of post hoc comparisons revealed that addition of the 2000-m trial increased the estimate of AWC but decreased the estimate of A. The oxygen equivalents of AWC and A were 19 mL·kg⁻¹ and 42 mL·kg⁻¹, respectively.

V_maximal from Equation 4 was higher than the value generated by Equation 6 (F_1,15 = 14.32, P = 0.002). Results of post hoc comparisons revealed that addition of the 2000-m predicting trial increased the estimate of V_maximal from Equation 4 but had no effect on the exponential estimate.

As previously reported by Kennedy and Bell (²⁴), and shown in Table 3, V_critical estimates from the two-parameter model were highly correlated with. V_critical estimates from the two three-parameter models were not. For all models, there was a significant correlation between V₂₀₀₀ and the V₂₀₀₀ that were calculated based on the V_critical and AWC (and V_maximal or tau for the three-parameter models).

TABLE 3: Estimation of 2000-m velocity (V₂₀₀₀, m·s⁻¹); actual V₂₀₀₀ was 4.93 ± 0.26 m·s⁻¹.

DISCUSSION

Although the critical power concept was first described for exercise performed using isolated muscle groups (²⁶) and then extended to cycle ergometry (²⁷), it is generally assumed that it can be applied to such activities such as running (^17,18,29), swimming (^14,34,35), outdoor cycling (³³), and rowing (²⁴), simply by substituting velocity for power and distance for work. However, when fluid resistance is a determining factor in an activity, velocity does not track linearly with power output; V̇O₂, energy cost, and power output may be a function of velocity (^2,7). Even in running, at higher velocities, there is a disproportionate increase in oxygen cost (²⁵). Therefore, it is somewhat surprising that applications of critical power models to running, swimming, outdoor cycling, and rowing have been successful.

Durations of the predicting trials.

With an average duration of ∼6.5 min, the 2000-m trial lies within the range of exercise intensities and durations commonly used in the estimation of V_critical and AWC (e.g., ^4,8,17,31), although it is somewhat longer than the range suggested by Hopkins et al. (¹⁶) for use with the exponential model. Either its inclusion as a predicting trial is necessary to accurately describe the velocity–time relationship and to derive valid parameter estimates or its inclusion is not necessary. If its inclusion is not necessary, then whether or not it is used should have no effect on the value of parameter estimates.

In the present study, it was hypothesized that including the 2000-m distance would not affect the values of parameter estimates. This hypothesis was rejected. Inclusion of the 2000-m trial did affect parameter estimates. Values for V_critical from the two-parameter hyperbolic model were lower when the 2000-m test was included as a predicting trial (see Results and Table 1). This finding was consistent with the results of previous studies that have shown that using longer predicting trials lowers the estimate of V_critical derived using the two-parameter model (^3,24). Adding the 2000-m predicting trial affected the three-parameter hyperbolic model in a different and unexpected fashion. This estimate of V_criticalincreased 13%. Estimates of V_critical derived using the three-parameter exponential model were unchanged. As shown in the Results and in Table 2, depending on the model used, other parameter estimates were affected to some degree by addition of the 2000-m predicting trial, and SEE were generally smaller, and R² were generally higher, when the 2000-m trial was used. Thus, it was concluded that the 2000-m trial must be included for the velocity–time relationship to be described accurately.

Comparison of the three models.

In a letter to the editor of this journal, Poole (³¹) suggested that the duration of predicting trials for use with the two-parameter model should be between 1 and 10 min. In introducing the three-parameter hyperbolic model, Morton (²⁸) suggested that at least one trial be “about 1 min or a little less” (p. 617) in duration. In presenting the exponential model, Hopkins et al. (¹⁶) suggested that it was for use with very short exercise durations, and they included a predicting trial of ∼10-s duration. Thus, the second hypothesis was that, because the 200-m predicting trials for each participant lasted ∼0.5 min, the three-parameter models would fit the data better than the two-parameter model. This hypothesis was accepted for five reasons. First, the utility of the two-parameter model was challenged because the three versions of the model gave different values for both V_critical and AWC (¹³). Second, the effect of adding the 2000 m predicting trial on Equations 1, 2, and 3 was different. If they were providing valid parameter estimates, the effect of an intervention (such as including the 2000-m trial) should be quantitatively similar across the three versions of the model. Third, the anaerobic parameter A appeared more meaningful than AWC, as explained in the next paragraph. Fourth, the V_maximal values from the three-parameter models seemed reasonable. V₂₀₀₀ was 69% of the hyperbolic V_maximal and 78% of the exponential V_maximal. Hartmann et al. (¹⁰) reported that world-class rowers could sustain 65–75% of their maximal rowing power for 6 min, and Ingham et al. (¹⁹) found that elite rowers can sustain 77% of their maximal rowing power in a 2000-m race. Fifth, R² for the three-parameter models was greater than the R² from the “true” two-parameter hyperbolic model (Equation 1). Figure 6 shows how the three-parameter hyperbolic model appears to provide a much more accurate description of the velocity–time relationship than the two-parameter model.

FIGURE 6: Comparison of the fits of the two- and three-parameter hyperbolic velocity–time relationship (Equations 1 and 4, respectively). Data are from the results of predicting trials over 200 m to 2000 m, from the same representative participant. With the three-parameter model (dashed curve), V_critical appeared lower (4.83 m·s⁻¹ vs 4.99 m·s⁻¹); A (the larger dark rectangle that encloses the smaller light rectangle) appeared to be larger than AWC (the smaller light rectangle) (165 m vs 71 m); and the three-parameter curve appeared to fit the data points throughout the range of test durations when it was allowed to project below the x-axis, to k = −103 s.

As in previous studies using cycling (^4,8) and treadmill running (¹⁷), the three-parameter A was greater than the two-parameter AWC. The oxygen equivalent of AWC from Equation 1 was only 19 mL·kg⁻¹. The oxygen equivalent of A from Equation 4 was 42 mL·kg⁻¹. Secher (³²) has reported values of ∼90 mL·kg⁻¹ for competitive rowers. An AWC of 90 m would suggest that the 2000-m race was 5% anaerobic (90 m/2000 m), whereas the A of 195 m from Equation 4 would suggest that the 3.5 min 1200 trial was ∼16% anaerobic and the 6.5 min 2000 m race was ∼10% anaerobic. These latter values seem more reasonable, although Secher (³²) has suggested that the 2000-m race was between 21% and 30% anaerobic.

Predicting performance over 2000 m.

The third hypothesis was that, because the three-parameter modeling strategies were expected to be superior to the two-parameter strategy, they would provide better estimates of V₂₀₀₀, using the results of predicting trials of 200 m to 1200 m. From a practical standpoint, the physiological or statistical correctness of a model is irrelevant. So, although only Equations 1 and 4 correctly designate time to fatigue as the dependent variable, all six models were examined.

As shown in Figure 5, and previously reported by Kennedy and Bell (²⁴), the two-parameter V_critical were quite highly correlated with, and quantitatively similar to, actual V₂₀₀₀. In fact, these V_critical explained 88% to almost 98% of the total variance in V₂₀₀₀. When the two-parameter V_critical and AWC were combined to produce an expected or “calculated V₂₀₀₀,” there was little improvement in the relationship, with the calculated V₂₀₀₀ explaining 95% to almost 98% of the total variance in V₂₀₀₀. Performance in an event that is 6.5 min in duration should be determined by a combination of factors including such physiologic indices as maximal sustainable aerobic power, anaerobic work capacity, and a combination of neuromuscular and skill factors that determine the maximal velocity that can be achieved (^19,32). Although V_critical should provide some information about performance potential, the prediction of performance should improve quantitatively, if not statistically, as more factors are considered (¹⁹). This was not the case with the two-parameter V_critical. As calculated in this study, the two-parameter V_critical is determined by virtually the same factors that determine V₂₀₀₀, and it is a performance measure rather than a physiological measure. Thus, although the two-parameter model provides an excellent estimate of potential 2000-m performance, it provides no insight into the athlete’s strengths and weaknesses, i.e., with respect to sustainable aerobic power, anaerobic work capacity, or V_maximal.

In contrast, the V_critical that were determined using the three-parameter models explained less than 20% of the variability in V₂₀₀₀ (in fact, the correlations were not statistically significant). However, when all the parameters derived for each model were used to produce a calculated V₂₀₀₀, these calculated velocities explained 75% (Equation 4) or 66% (Equation 6) of the variance in actual V₂₀₀₀. Although this provides little direct information about the physiological significance of the parameters, it does at least show that V_critical derived using the three-parameter models is not simply a performance measure.

CONCLUSION

The critical velocity concept can be used to predict performance in a 2000-m race, based on results from shorter trials. From a coaching standpoint, it is concluded that only the two-parameter model provides an excellent prediction of 2000-m performance. From a mathematical modeling standpoint, it is concluded that, in order to accurately describe the velocity–time relationship for competitive rowers, one must use predicting trials over distances up to and including race distance (i.e., 2000 m). Finally, from a physiological standpoint, it is concluded that the three-parameter models are superior to the two-parameter hyperbolic models for rowing ergometry and the test durations used in the present study. Two-parameter models simply are not appropriate when the 200-m and 400-m predicting trials are used, whereas the three-parameter models accurately describe the velocity–time relationship and may provide distinct information about aerobic and anaerobic abilities.

Some results reported in this paper have been published previously: Kennedy and Bell (²⁴) compared performance in a simulated 2000-m race with parameter estimates that were derived by applying Equations 1, 2, and 3 to data from predicting trials over combinations of distances from 200 m to 1200 m.