Hormonal fluctuations

Cycling

Estimation of the phases

• Athletes have been asked to report the beginning and the end of their bleeding for each period.

• We estimate their ovulation day using a robust linear regression model based on their cycle length (Soumpasis et al., 2020).

• The menstrual cycle is then split in three phases:

  • Menstruations
  • Follicular
  • Luteal

Power Data

Mean Maximal Power Curves

• Consider some generic exercise which last $T$ seconds.

• Let $Z=\{z_t,t=1,2,\dots ,T\}$ a sequence of observation of the power output.

• Assume that, given $t_1,t_2$ two timestamps such that $t_2>t_1$, $t_2-t_1$ is constant.

• An MMP curve is defined as $$X\left(t\right)=\underset{t_2-t_1=t}{max}\frac{z_{t_1}+\cdots +z_{t_2}}{t_2-t_1}.$$

Mean Maximal Power Curves

Mean and standard deviation per phase

Model

• MMP curves consist of random realizations from a stochastic process $X=\left\{X\right(t):t\in [1,T\left]\right\}$ with continuous trajectories.

• We consider the following model $$X_{jklmn}\left(t\right)={\mu }_k\left(t\right)+B_{jk}\left(t\right)+C_{lk}\left(t\right)+D_{mk}\left(t\right)+E_{jklmn}\left(t\right)$$ where

  • $X_{jklmn}\left(t\right)$: MMP output for a particular observation.
  • ${\mu }_k\left(t\right)$: fixed effect for the phase of the menstrual cycle.
  • $B_{jk}\left(t\right)$: phase-specific functional random intercept accounting for athlete.
  • $C_{lk}\left(t\right)$: phase-specific functional random intercept accounting for training intensity.
  • $D_{mk}\left(t\right)$: phase-specific functional random intercept accounting for bike type.
  • $E_{ijl}\left(t\right)$: smooth error term accounting for observation-specific variability.

Mean comparison

• We are interested in testing the following hypothesis:

$$\left(H_0\right):{\mu }_k={\mu }_{k^{\prime }}\text{v.s}\left(H_1\right):{\mu }_k\ne {\mu }_{k^{\prime }}$$

• We consider the test statistic:

$$S_N=\frac{N_k{N}_{k^{\prime }}}{NT}{\int }_T\left\{{\mu }_k\right(t)-{\mu }_{k^{\prime }}(t){\}}^{2}dt$$

• The sampled bootstrap statistics, under the assumption of equality of the mean curves, are compared to $S_N$ computed on the observed data.

Mean comparison

Variance decomposition

Following Cederbaum (2017):

• Standardize the curves per phase.

• Estimate the covariance of each random effects.

• Perform an eigendecomposition of the covariances.

• Estimate the variability induced by each random effects.

Variance decomposition

Takeaway ideas

• Power output data exhibits the variable nature of performance in women's professional cycling.

• We have not proven that there is no variation between phases, we have failed to find evidence of variation between phases.

• The athletes are likely to achieve their peak performance in each phase.

• These results may be helpful for coaches who use these curves for training planning or the comprehension of their athletes.

Thank you for your attention!

References

Cederbaum, J. (2017). “Functional linear mixed models for complex correlation structures and general sampling grids”. Text.PhDThesis.

McNulty, K. L., K. J. Elliott-Sale, E. Dolan, et al. (2020). “The Effects of Menstrual Cycle Phase on Exercise Performance in Eumenorrheic Women: A Systematic Review and Meta-Analysis”. In: Sports Medicine 50.10, pp. 1813–1827. ISSN: 1179-2035. DOI: 10.1007/s40279-020-01319-3.

Soumpasis, I., B. Grace, and S. Johnson (2020). “Real-Life Insights on Menstrual Cycles and Ovulation Using Big Data”. In: Human Reproduction Open 2020.2. DOI: 10.1093/hropen/hoaa011. pmid: pmid. URL: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7164578/ (visited on Jul. 18, 2023).