On the influence of the menstrual cycle on female athlete performance

class: center, middle, inverse, title-slide

.title[
# On the influence of the menstrual cycle on female athlete performance
]
.subtitle[
## A case study of cycling
]
.author[
### Juliana Antero · Tom Chassard · Steven Golovkine · Alice Meigné
]
.institute[
### SIM Talk
]
.date[
### September 19, 2023
]

---

# Presentation of the project

<figure>
 <center>
 <img src="data:image/png;base64,#./img/menstrual_cycle.svg" alt="cycle" width="100%"/>
 <figcaption>Schema of a menstrual cycle (adapted from <a id='cite-mcnultyEffectsMenstrualCycle2020'></a>(<a href='https://doi.org/10.1007/s40279-020-01319-3'>McNulty, Elliott-Sale, Dolan, Swinton, Ansdell, Goodall, Thomas, and Hicks, 2020</a>)).<figcaption>
 </center>
</figure>

---
# Cycling

---
# Power Data

<figure>
 <center>
 <img src="data:image/png;base64,#./img/1.svg" alt="observation" width="100%"/>
 <figcaption>Examples of data recorded from training.<figcaption>
 </center>
</figure>

---
# 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 time stamps such that `$t_2 > t_1$`, `$t_2 - t_1$` is constant.

* An MMP curve is defined as
`$$X(t) = \max_{t_2 - t_1 = t} \frac{z_{t_1} + \cdots + z_{t_2}}{t_2 - t_1}.$$`

---
# Mean Maximal Power Curves

<figure>
 <center>
 <img src="data:image/png;base64,#./img/ppr_per_phase.svg" alt="ppr" width="100%"/>
 <figcaption>MMP per phase.<figcaption>
 </center>
</figure>

---
# Mean per phase

<figure>
 <center>
 <img src="data:image/png;base64,#./img/mean_ppr_per_phase.svg" alt="mean_ppr" width="65%"/>
 <figcaption>Mean per phase.<figcaption>
 </center>
</figure>

---
# Standard deviation per phase

<figure>
 <center>
 <img src="data:image/png;base64,#./img/sd_ppr_per_phase.svg" alt="mean_ppr" width="65%"/>
 <figcaption>Standard deviation per phase.<figcaption>
 </center>
</figure>

---
# Tests for the difference of the means

<figure>
 <center>
 <img src="data:image/png;base64,#./img/follicular-luteal.svg" alt="follicular_luteal" width="32%"/>
 <img src="data:image/png;base64,#./img/follicular-menses.svg" alt="follicular_menses" width="32%"/>
 <img src="data:image/png;base64,#./img/menses-luteal.svg" alt="menses_luteal" width="32%"/>
 <figcaption>Confidence bands for the difference of the means <a id='cite-lieblFastFairSimultaneous2022'></a>(<a href='https://arxiv.org/abs/1910.00131'>Liebl and Reimherr, 2022</a>).<figcaption>
 </center>
</figure>

---
# Model

* MMP curves consist of random realizations from a stochastic process `$X = \{X(t) : t \in [1, 7200]\}$` with continuous trajectories.

* We consider the following model
`$$X_{ijl}(t) = \mu_j(t) + B_{jl}(t) + E_{ijl}(t)$$`
where 
    * `$X_{ijl}(t)$`: MMP output for a particular observation.
    * `$\mu_j(t)$`: fixed effect for the phase of the menstrual cycle.
    * `$B_{jl}(t)$`: phase-specific functional random intercept for training.
    * `$E_{ijl}(t)$`: smooth error term accounting for observation-specific variability.

---
# Eigenfunctions

<figure>
 <center>
 <img src="data:image/png;base64,#./img/follicular.svg" alt="follicular" width="48%"/>
 <img src="data:image/png;base64,#./img/luteal.svg" alt="luteal" width="48%"/>
 <figcaption>Phase specific means with the functional random intercept for trainings <a id='cite-cederbaumFunctionalLinearMixed2017'></a>(<a href='#bib-cederbaumFunctionalLinearMixed2017'>Cederbaum, 2017</a>).<figcaption>
 </center>
</figure>

---
# Takeaway ideas

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

* May help the athletes to have a better understanding of their performance regarding their menstrual cycle.

* Hard to give a definitive conclusion of the influence of menstrual cycle on the performance here!

* Add more menstrual cycles, determine a better classification of the trainings and include a random effect for athletes.

---
class: middle, center, inverse

# Thanks!

---
# References

<cite><a id='bib-cederbaumFunctionalLinearMixed2017'></a><a href="#cite-cederbaumFunctionalLinearMixed2017">Cederbaum, J.</a>
(2017).
&ldquo;Functional linear mixed models for complex correlation structures and general sampling grids&rdquo;.
Text.PhDThesis.</cite>

<cite><a id='bib-lieblFastFairSimultaneous2022'></a><a href="#cite-lieblFastFairSimultaneous2022">Liebl, D. and M. Reimherr</a>
(2022).
Fast and Fair Simultaneous Confidence Bands for Functional Parameters.
DOI: <a href="https://doi.org/10.48550/arXiv.1910.00131">10.48550/arXiv.1910.00131</a>.
arXiv: <a href="https://arxiv.org/abs/1910.00131">1910.00131 [math, stat]</a>.</cite>

<cite><a id='bib-mcnultyEffectsMenstrualCycle2020'></a><a href="#cite-mcnultyEffectsMenstrualCycle2020">McNulty, K. L., K. J. Elliott-Sale, E. Dolan, et al.</a>
(2020).
&ldquo;The Effects of Menstrual Cycle Phase on Exercise Performance in Eumenorrheic Women: A Systematic Review and Meta-Analysis&rdquo;.
In: Sports Medicine 50.10, pp. 1813&ndash;1827.
ISSN: 1179-2035.
DOI: <a href="https://doi.org/10.1007/s40279-020-01319-3">10.1007/s40279-020-01319-3</a>.</cite>