Relating Training to the Performance Measure

In this section, we consider , as a performance measure, which was defined in the previous section. So through this we have:

( ) (6.1) and

̂ ( ) (6.2) where ( ) is the variance in the estimate ̂ , which must be calculated. The estimate ̂ and its variance are obtained as described in section 6.2. Finally, through (6.1) and (6.2), we conclude the model of training–performance to be

̂ ( )

The parameters can again be estimated using the method of maximum likelihood. This is done using R programming. The parameters , , are fixed for each individual rider, but

varies session by session. Table 6-1 shows the estimates for each rider.

Table ‎6-1 Estimated parameters with standard errors of the training and performance model for each rider with the t statistic and p value for the test of β=0

R ider ̂ ( ̂ ) ̂ ( ̂ ) ̂ ( ̂ ) ̂ ( ̂ ) ̂ ( ̂ ) ̂ ( ̂) t 1 88.2 6.1 31.2 15.8 1.9 0.8 2.2 5.8 215 22.1 0.0081 0.0046 1.76 0.04 2 51.5 4.3 84.5 26.5 1.4 0.7 12.7 1.7 264 31.6 0.0098 0.0052 1.88 0.03 3 36.6 4.3 36.6 17.6 6.5 0.8 1.3 0.2 233 17.9 0.0319 0.0190 1.67 0.05 4 61.5 3.7 60.6 28.4 3.3 1.4 5.2 1.5 207 13.1 0.0113 0.0018 6.28 0.00 5 60.6 8.7 57.6 13.3 0.8 0.7 5.1 3.4 250 14.7 0.0062 0.0012 5.17 0.00 6 53.2 5.4 85.4 10.1 1.8 0.9 3.7 1.1 295 89.7 0.0093 0.0019 4.89 0.00 7 61.8 3.5 74.1 19.5 3.9 3.4 2.8 2.6 197 26.8 0.0264 0.0050 5.28 0.00 8 51.3 1.5 46.7 13.3 1.1 0.8 1.8 0.9 214 21.8 0.0095 0.0053 1.79 0.04 9 41.9 2.2 58.2 18.3 3.4 5.8 12.5 8.5 213 11.7 0.0020 0.0006 3.33 0.00 10 44.8 3.2 93.8 24.6 4.8 3.5 5.1 0.6 203 21.1 0.0090 0.0020 4.50 0.00

6.3.1 Statistical Discussion of the Training Effect

In this section, we statistically study the effect of training on performance. The relationship between the accumulated training effect and our proposed performance is linearly positive. Therefore, we would like to reject the hypothesis in favour of . Through the Table 6-1, we see that there is a statistically significant relationship at 5% level between the accumulated training effect and the performance measure for all riders. Since all estimates of are positive and ̂ ( ̂) 1.64 in all cases.

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6.3.2 Practical Discussion of the Training Effect

Since there is a statistically significant increase in performance as a result of the training process, is important to determine the practical significance of that increase. This can be calculated using the change in power output from the beginning of training until the point at which a rider has completed the optimal training, as follows:

̂

The values of ̂ are shown in Table 6-1. is defined as the change between the maximum and initial accumulated training effect ( ). The change in power output is presented in Table 6-2 for each rider. This change ranges between 4% and 35%.

Table ‎6-2 Performance gain and the ATE change when the performance measure is with percentile of power output for each rider

Rider / 1 3227 26 291 0.09 2 4734 43 307 0.13 3 2457 74 291 0.25 4 4472 45 246 0.18 5 7795 47 280 0.16 6 5448 49 384 0.12 7 5794 116 323 0.35 8 4242 38 274 0.13 9 3873 08 214 0.04 10 7068 64 260 0.24 6.3.3 Discussion of Results

In this section, we discuss the results for each rider in terms of the impact of training on‎performance.‎The‎reason‎for‎this‎is‎that‎each‎rider‟s‎training‎program‎differs‎from‎those of the others as well as his individual capacities.

For rider (1), there is a statistically significant relationship that training has an impact on his performance measure , as seen in Table 6-1. Also, as noted in Table 6-2, the practical impact of the training for this rider is significant with the measure increasing by about 9%. In addition, a slight improvement in the performance measure of this rider is seen in Figure 6.3.

Rider (2) has a statistically significant relationship between the accumulated training effect and the performance measure, as highlighted in Table 6-1. In Figure 6.3, there is a clear improvement in performance over time for this rider. Practically, for this rider, the effect of training is significant with the measure increasing by about 13%, as presented in Table 6-2. Furthermore, the accumulated training effect has clearly improved, Figure 6.3 illustrates this.

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The effect of training for rider (3) is practically significant with the measure increasing by about 25%, as seen in Table 6-2. The relationship between the accumulated training effect and the performance measure for this rider is also statistically significant, as seen in Table 6-1. Moreover, the training effect over time is stable, as presented in Figure 6.3.

Rider (4) shows a statistically and practically significant relationship between the accumulated training effect and the performance measure, as presented in Tables 6-1 and 6-2. The impact of training on this rider‟s‎performance over time is increasing, as seen in Figure 6.3.

The results for Rider (5) appear statistically and practically significant in the relationship between the accumulated training effect and the performance measure, as seen in Tables 6-1 and 6-2. However, the parameter is less than 1 in this case. It is supposed to be greater than one. It seems probable that the reason for this issue is that this rider has a lot of gaps in his training programme.

Rider (6) has statistically and practically significant results in terms of training effect on performance, this is demonstrated in Tables 6-1 and 6-2. The performance measure of this rider has obviously improved by 12%, as shown in Figure 6.3. Also, the accumulated effect of training is improved.

Rider (7) reveals a significant practical improvement of 35% performance after training, this is explained in Table 6-2. Furthermore, the relationship between the accumulated training effects and the performance measure for this rider is highly statistically significant, Table 6-1 proves this. After initially rising, the performance measure declines slightly after 100 days. This is noted in Figure 6.3.

For rider (8), a statistically and practically significant training effect is illustrated in Tables 6-1 and 6-2. Although the effect of the training on performance for this rider is clearly improved, the performance is fairly stable, as shown in Figure 6.3.

Rider (9) presents a statistically significant relationship between the accumulated training effect and his performance, this is presented in Table 6-1. However, this rider expresses only a very small practical effect from training on his performance by about 4%, as can be visualised in Table 6-2. Furthermore, the accumulation of training effect for this rider is fluctuates rather a lot, this is evident in Figure 6.3.

Finally, rider (10) indicates a statistically and practically significant training effect increased by about 24%, as can be seen in Tables 6-1 and 6-2. The performance measure for this rider initially increases and after 150 days fluctuates, this is noted in Figure 6.3.

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Figure ‎6-3 Two plots for each rider: left , (symbols) vs time in days and ATE (line) vs time in days; right , vs ATE (all sessions)

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