Functional multilevel modelling of the influence of the menstrual cycle on the performance of female cyclists

class: center, middle, inverse, title-slide

.title[
# Functional multilevel modelling of the influence of the menstrual cycle on the performance of female cyclists
]
.subtitle[
## <br>
]
.author[
### <strong>S. Golovkine</strong> · T. Chassard · A. Meigné · E. Brunet · J.-F. Toussaint · J. Antero
]
.institute[
### 37th International Workshop on Statistical Modelling
]
.date[
### July 18th, 2023
]

---

# Hormonal fluctuations

<br><br><br>
<figure>
    <center>
    <img src="data:image/png;base64,#./img/menstrual_cycle.svg" alt="cycle" width="120%"/>
    <figcaption>Schema of a menstrual cycle (adapted from <a href='https://doi.org/10.1007/s40279-020-01319-3'>McNulty et al., 2020</a>).<figcaption>
    </center>
</figure>

---
# Cycling

<div class="row">
  <div class="column">
    <center>
    <img src="data:image/png;base64,#./img/route.jpg" alt="route" width="90%"/>
    <figcaption>Road<figcaption>
    </center>
  </div>
  <div class="column">
    <center>
    <img src="data:image/png;base64,#./img/piste.jpg" alt="track" width="90%"/>
    <figcaption>Track<figcaption>
    </center>
  </div>
  <div class="column">
    <center>
    <img src="data:image/png;base64,#./img/crosscountry.jpg" alt="crosscountry" width="90%"/>
    <figcaption>Cross-country<figcaption>
    </center>
  </div>
</div>

---
# 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 (<a href='https://doi.org/10.1093/hropen/hoaa011'>Soumpasis et al., 2020</a>).

* The menstrual cycle is then split in three phases:

- Menstruations
    - Follicular
    - Luteal

---
# Power Data

<figure>
    <center>
    <img src="data:image/png;base64,#./img/1.svg" alt="observation" width="80%"/>
    <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 timestamps 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

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

---
# Mean and standard deviation per phase

<figure>
    <center>
    <img src="data:image/png;base64,#./img/golovkine_ppr_mean.svg" alt="mean_ppr" width="47.5%"/>
    <img src="data:image/png;base64,#./img/golovkine_ppr_std.svg" alt="std_ppr" width="47.5%"/>
    <figcaption>Mean and standard deviation per phase.<figcaption>
    </center>
</figure>

---
# Model

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

* We consider the following model
`$$X_{jklmn}(t) = \mu_k(t) + B_{jk}(t) + C_{lk}(t) + D_{mk}(t) + E_{jklmn}(t)$$`
where 
    * `$X_{jklmn}(t)$`: MMP output for a particular observation.
    * `$\mu_k(t)$`: fixed effect for the phase of the menstrual cycle.
    * `$B_{jk}(t)$`: phase-specific functional random intercept accounting for athlete.
    * `$C_{lk}(t)$`: phase-specific functional random intercept accounting for training intensity.
    * `$D_{mk}(t)$`: phase-specific functional random intercept accounting for bike type.
    * `$E_{ijl}(t)$`: smooth error term accounting for observation-specific variability.

---
# Mean comparison

* We are interested in testing the following hypothesis:

`$$(H_0): \mu_k = \mu_{k^\prime} \quad\text{v.s}\quad (H_1): \mu_k \neq \mu_{k^\prime}$$`

* We consider the test statistic:

`$$S_N = \frac{N_k N_{k^\prime}}{NT} \int_{T} \{\mu_k(t) - \mu_{k^\prime}(t)\}^2 \mathrm{d}t$$`
* 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 <a href='https://edoc.ub.uni-muenchen.de/22121/1/Cederbaum_Jona.pdf'>Cederbaum (2017)</a>:

* 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

<table>
  <tr>
    <th>Variability source</th>
    <th>Variance explained (in `\%`)</th> 
  </tr>
  <tr>
    <td>Phase</td>
    <td>`2.41 \times 10^{-3}`</td>
  </tr>
  <tr>
    <td>Athlete</td>
    <td>`22.0`</td>
  </tr>
  <tr>
    <td>RPE</td>
    <td>`11.5`</td>
  </tr>
  <tr>
    <td>Bike type</td>
    <td>`16.6`</td>
  </tr>
  <tr>
    <td>Observation</td>
    <td>`49.8`</td>
  </tr>
  <tr>
    <td>Error variance</td>
    <td>`6.60 \times 10^{-11}`</td>
  </tr>
  <caption>Full variance decomposition using a functional random intercept for phase with variance explained of `99.999%`.</caption>
</table>

---
# 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.

<br>
<h2 style="color:#005844;"><center>Thank you for your attention!</center></h2>

---
# References

<p><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></p>

<p><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: <em>Sports Medicine</em> 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></p>

<p><cite><a id='bib-soumpasisReallifeInsightsMenstrual2020'></a><a href="#cite-soumpasisReallifeInsightsMenstrual2020">Soumpasis, I., B. Grace, and S. Johnson</a>
(2020).
&ldquo;Real-Life Insights on Menstrual Cycles and Ovulation Using Big Data&rdquo;.
In: <em>Human Reproduction Open</em> 2020.2.
DOI: <a href="https://doi.org/10.1093/hropen/hoaa011">10.1093/hropen/hoaa011</a>.
pmid: pmid.
URL: <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7164578/">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7164578/</a> (visited on Jul. 18, 2023).</cite></p>