The New Power-Duration Model in WKO4November 5, 2013 | Andrew R. Coggan, Ph.D.

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The power-duration relationship

Linear time scale Log time scale

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The new power-duration model in WKO4

• A heart-warming tale in which the mean maximal power chart and the power profiling table meet, fall in love, get married, and have a child that turns out to be somewhat of a child prodigy.

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The new power-duration model in WKO4

• The most advanced mathematical model of the exercise intensity-duration relationship ever developed.

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 5

Webinar series topics

• Part I: A primer on modeling• Part II: Limitations of current models• Part III: Introduction of a superior approach• Part IV: Present and possible future applications

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The $64,000 question

Why?

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The power-duration relationship

Linear time scale Log time scale

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 8

The $64,000 answer

Because having a robust mathematical description of an individual’s power-duration relationship provides quantitative insight re. their unique abilities and paves the way for other analyses.

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 9

Importance of new power-duration model

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 10

Evolution of rider phenotyping

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Uses for the power-duration model

1. Auto FTP2. Pmax3. FRC4. dFRC5. Phenotyping6. Individual adaptations scores7. Auto match finding8. Identification of outliers9. Optimal pacing strategy10. Application to other sports

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 12

The new power-duration model in WKO4

Part I: A primer on modeling

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 13

Some mathematical models

Coggan AR, Ruby BC, Zderic TW. FASEB J 1999; 13: A1055

Coggan AR, et al. J Nucl Cardiol 2009; 16: 562-570.

Herrero P, et al. J Nucl Med 2007; 48: 955-964.

Herrero P, et al. J Nucl Med 2007; 48: 2046-2055.

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 14

Caveats re. mathematical modeling

• Mathematical models are theoretical constructs that may, or may not, have a basis in reality

• All models are wrong, but some are useful

• The usefulness of any model depends in part upon its domain of validity

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 15

Example of a limited domain of validity

Model structure Model validation

Coggan AR, et al. J Nucl Cardiol 2009; 16: 562-570.

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 16

How do you evaluate mathematical models?

1. Statistical criteriaa. F testb. Runs testc. AIC

2. Residualsa. Distributionb. Magnitudec. Bias

3. Parameter estimatesa. Numberb. Independencec. Precision

4. External validation

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 17

Most important take-home message?

R2 is not an adequate indicator of the goodness-of-fit of a multi-parameter, non-linear model!

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 18

R2 as a measure of goodness-of-fit

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 19

R2 as a measure of goodness-of-fit

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 20

R2 as a measure of goodness-of-fit

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 21

How do you evaluate mathematical models?

1. Statistical criteriaa. F testb. Runs testc. AIC

2. Residualsa. Magnitudeb. Distributionc. Bias

3. Parameter estimatesa. Numberb. Independencec. Precision

4. External validation

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 22

What is a “normalized residual”?

x 100%

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Residuals of linear fit

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 24

How do you evaluate mathematical models?

1. Statistical criteriaa. F testb. Runs testc. AIC

2. Residualsa. Magnitudeb. Distributionc. Bias

3. Parameter estimatesa. Numberb. Independencec. Precision

4. External validation

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 25

The critical power model

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 26

Distribution of normalized residuals

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 27

Normalized residuals vs. duration

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Independence of parameter estimates

W’ CP

W’ 1.00 0.07

CP 1.00

Correlation matrix for parameters of critical power model

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 29

External validation

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 30

Model vs. reality

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 31

Distribution of normalized residuals

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 32

Normalized residuals vs. duration

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 33

Limitations of the critical power model

“The critical power concept is not without its limitations...in particular, it tends to greatly overestimate the maximal power that can be generated for only a few seconds, and it predicts that there should be a power output below which fatigue will never occur. In addition, the exact values obtained for W' and critical power depend in part on the testing protocol, e.g., the exact combination of powers and durations used to define the curve...”

- Yours truly, post to the wattage list on 2/11/2002

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 34

Summary

• Modeling the power-duration relationship provides many potential benefits

• Such benefits, however, will only be realizable with a sufficiently robust model

• Standard criteria and approaches (which do not include R2) exist for evaluating mathematical models

• The classic critical power model fails to adequately describe the entire power-duration relationship

Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved. 35

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