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<div id="content">
<h1 class="title">Analyse du risque de défaillance des joints toriques de la navette Challenger</h1>
<p>
<b>Préambule:</b> Les explications données dans ce document sur le contexte
de l'étude sont largement reprises de l'excellent livre d'Edward
R. Tufte intitulé <i>Visual Explanations: Images and Quantities, Evidence
and Narrative</i>, publié en 1997 par <i>Graphics Press</i> et réédité en 2005,
ainsi que de l'article de Dalal et al. intitulé <i>Risk Analysis of the
Space Shuttle: Pre-Challenger Prediction of Failure</i> et publié en 1989
dans <i>Journal of the American Statistical Association</i>.
</p>
<div id="outline-container-org95fbd2d" class="outline-2">
<h2 id="org95fbd2d"><span class="section-number-2">1</span> Contexte</h2>
<div class="outline-text-2" id="text-1">
<p>
Dans cette étude, nous vous proposons de revenir sur <a href="https://fr.wikipedia.org/wiki/Accident_de_la_navette_spatiale_Challenger">l'accident de la
navette spatiale Challenger</a>. Le 28 Janvier 1986, 73 secondes après son
lancement, la navette Challenger se désintègre (voir Figure <a href="#org7a7557d">1</a>)
et entraîne avec elle, les sept astronautes à son bord. Cette
explosion est due à la défaillance des deux joints toriques
assurant l'étanchéité entre les parties hautes et basses des
propulseurs (voir Figure <a href="#orge54c304">2</a>). Ces joints ont perdu de leur
efficacité en raison du froid particulier qui régnait au moment du
lancement. En effet, la température ce matin là était juste en dessous
de 0°C alors que l'ensemble des vols précédents avaient été effectués
à une température d'au moins 7 à 10°C de plus.
</p>
<div id="org7a7557d" class="figure">
<p><img src="challenger5.jpg" alt="challenger5.jpg" />
</p>
<p><span class="figure-number">Figure&nbsp;1&nbsp;: </span>Photos de la catastrophe de Challenger.</p>
</div>
<div id="orge54c304" class="figure">
<p><img src="o-ring.png" alt="o-ring.png" />
</p>
<p><span class="figure-number">Figure&nbsp;2&nbsp;: </span>Schéma des propulseurs de la navette challenger. Les joints toriques (un joint principale et un joint secondaire) en caoutchouc de plus de 11 mètres de circonférence assurent l'étanchéité entre la partie haute et la partie basse du propulseur.</p>
</div>
<p>
Le plus étonnant est que la cause précise de cet accident avait été
débattue intensément plusieurs jours auparavant et était encore
discutée la veille même du décollage, pendant trois heures de
télé-conférence entre les ingénieurs de la Morton Thiokol
(constructeur des moteurs) et de la NASA. Si la cause immédiate de
l'accident (la défaillance des joints toriques) a rapidement été
identifiée, les raisons plus profondes qui ont conduit à ce désastre
servent régulièrement de cas d'étude, que ce soit dans des cours de
management (organisation du travail, décision technique malgré des
pressions politiques, problèmes de communication), de statistiques
(évaluation du risque, modélisation, visualisation de données), ou de
sociologie (symptôme d'un historique, de la bureaucratie et du
conformisme à des normes organisationnelles).
</p>
<p>
Dans l'étude que nous vous proposons, nous nous intéressons
principalement à l'aspect statistique mais ce n'est donc qu'une
facette (extrêmement limitée) du problème et nous vous invitons à lire
par vous même les documents donnés en référence dans le
préambule. L'étude qui suit reprend donc une partie des analyses
effectuées cette nuit là et dont l'objectif était d'évaluer
l'influence potentielle de la température et de la pression à laquelle
sont soumis les joints toriques sur leur probabilité de
dysfonctionnement. Pour cela, nous disposons des résultats des
expériences réalisées par les ingénieurs de la NASA durant les 6
années précédant le lancement de la navette Challenger.
</p>
<p>
Dans le répertoire <code>module2/exo5/</code> de votre espace <code>gitlab</code>, vous
trouverez les données d'origine ainsi qu'une analyse pour chacun des
différents parcours proposés. Cette analyse comporte quatre étapes:
</p>
<ol class="org-ol">
<li>Chargement des données</li>
<li>Inspection graphique des données</li>
<li>Estimation de l'influence de la température</li>
<li>Estimation de la probabilité de dysfonctionnement des joints
toriques</li>
</ol>
<p>
Les deux premières étapes ne supposent que des compétences de base en
R ou en python. La troisième étape suppose une familiarité avec la
régression logistique (généralement abordée en L3 ou M1 de stats,
économétrie, bio-statistique, &#x2026;) et la quatrième étape des bases de
probabilités (niveau lycée). Nous vous présentons donc dans la
prochaine section une introduction à la régression logistique qui ne
s'attarde pas sur les détails du calcul, mais juste sur le sens donné
aux résultats de cette régression.
</p>
</div>
</div>
<div id="outline-container-org1c79137" class="outline-2">
<h2 id="org1c79137"><span class="section-number-2">2</span> Introduction à la régression logistique</h2>
<div class="outline-text-2" id="text-2">
<p>
Imaginons que l'on dispose des données suivantes qui indiquent pour
une cohorte d'individus s'ils ont déclaré une maladie particulière ou
pas. Je montre ici l'analyse avec R mais le code python n'est pas forcément
très éloigné. Les données sont stockées dans une data frame dont voici
un bref résumé:
</p>
<div class="org-src-container">
<pre class="src src-R">summary(df)
str(df)
</pre>
</div>
<pre class="example">
Age Malade
Min. :22.01 Min. :0.000
1st Qu.:35.85 1st Qu.:0.000
Median :50.37 Median :1.000
Mean :50.83 Mean :0.515
3rd Qu.:65.37 3rd Qu.:1.000
Max. :79.80 Max. :1.000
'data.frame': 400 obs. of 2 variables:
$ Age : num 75.1 76.4 38.6 70.2 59.2 ...
$ Malade: int 1 1 0 1 1 1 0 0 1 1 ...
</pre>
<p>
Voici une représentation graphique des données qui permet de mieux
percevoir le lien qu'il peut y avoir entre l'âge et le fait de
contracter cette maladie ou pas:
</p>
<div class="org-src-container">
<pre class="src src-R">ggplot(df,aes(x=Age,y=Malade)) + geom_point(alpha=.3,size=3) + theme_bw()
</pre>
</div>
<div class="figure">
<p><object type="image/svg+xml" data="fig1.svg" class="org-svg">
Sorry, your browser does not support SVG.</object>
</p>
</div>
<p>
Il apparaît clairement sur ces données que plus l'on est âgé, plus la
probabilité de développer cette maladie est importante. Mais comment
estimer cette probabilité à partir uniquement de ces valeurs binaires
(malade/pas malade)? Pour chaque tranche d'âge (par exemple de 5 ans),
on pourrait regarder la fréquence de la maladie (le code qui suit est
un peu compliqué car le calcul de l'intervalle de confiance pour ce
type de données nécessite un traitement particulier via la fonction
<code>binconf</code>).
</p>
<div class="org-src-container">
<pre class="src src-R">age_range=5
df_grouped = df <span class="org-ess-XopX">%&gt;%</span> mutate(Age=age_range*(floor(Age/age_range)+.5)) <span class="org-ess-XopX">%&gt;%</span>
group_by(Age) <span class="org-ess-XopX">%&gt;%</span> summarize(Malade=sum(Malade),N=n()) <span class="org-ess-XopX">%&gt;%</span>
rowwise() <span class="org-ess-XopX">%&gt;%</span>
do(data.frame(Age=.$Age, binconf(.$Malade, .$N, alpha=0.05))) <span class="org-ess-XopX">%&gt;%</span>
as.data.frame()
ggplot(df_grouped,aes(x=Age)) + geom_point(data=df,aes(y=Malade),alpha=.3,size=3) +
geom_errorbar(data=df_grouped,
aes(x=Age,ymin=Lower, ymax=Upper, y=PointEst), color=<span class="org-string">"darkred"</span>) +
geom_point(data=df_grouped, aes(x=Age, y=PointEst), size=3, shape=21, color=<span class="org-string">"darkred"</span>) +
theme_bw()
</pre>
</div>
<div class="figure">
<p><object type="image/svg+xml" data="fig1bis.svg" class="org-svg">
Sorry, your browser does not support SVG.</object>
</p>
</div>
<p>
L'inconvénient de cette approche est que ce calcul est effectué
indépendemment pour chaque tranches d'âges, que la tranche d'âge est
arbitraire, et qu'on n'a pas grande idée de la façon dont ça
évolue. Pour modéliser cette évolution de façon plus continue, on
pourrait tenter une régression linéaire (le modèle le plus simple
possible pour rendre compte de l'influence d'un paramètre) et ainsi
estimer l'effet de l'âge sur la probabilité d'être malade:
</p>
<div class="org-src-container">
<pre class="src src-R">ggplot(df,aes(x=Age,y=Malade)) + geom_point(alpha=.3,size=3) +
theme_bw() + geom_smooth(method=<span class="org-string">"lm"</span>)
</pre>
</div>
<div class="figure">
<p><object type="image/svg+xml" data="fig2.svg" class="org-svg">
Sorry, your browser does not support SVG.</object>
</p>
</div>
<p>
La ligne bleue est la régression linéaire au sens des moindres carrés
et la zone grise dans laquelle est la zone de confiance à 95% de cette
estimation (avec les données dont l'on dispose et cette hypothèse de
linéarité, la ligne bleue est la plus probable et il y a 95% de chance
que la vraie ligne soit dans cette zone grise).
</p>
<p>
Mais on voit clairement dans cette représentation graphique que cette
estimation n'a aucun sens. Une probabilité doit être comprise entre 0
et 1 et avec une régression linéaire on arrivera forcément pour des
valeurs un peu extrêmes (jeune ou âgé) à des prédictions aberrantes
(négative ou supérieures à 1). C'est tout simplement dû au fait qu'une
régression linéaire fait l'hypothèse que \(\textsf{Malade} =
\alpha.\textsf{Age} + \beta + \epsilon\), où \(\alpha\) et \(\beta\) sont des nombres réels et \(\epsilon\)
est un bruit (une variable aléatoire de moyenne nulle), et estime \(\alpha\)
et \(\beta\) à partir des données.
</p>
<p>
Cette technique n'a pas de sens pour estimer une probabilité et il
convient donc d'utiliser ce que l'on appelle une <a href="https://fr.wikipedia.org/wiki/R%C3%A9gression_logistique">régression
logistique</a>:
</p>
<div class="org-src-container">
<pre class="src src-R">ggplot(df,aes(x=Age,y=Malade)) + geom_point(alpha=.3,size=3) +
theme_bw() +
geom_smooth(method = <span class="org-string">"glm"</span>,
method.args = list(family = <span class="org-string">"binomial"</span>)) + xlim(20,80)
</pre>
</div>
<div class="figure">
<p><object type="image/svg+xml" data="fig3.svg" class="org-svg">
Sorry, your browser does not support SVG.</object>
</p>
</div>
<p>
Ici, la bibliothèque <code>ggplot</code> fait tous les calculs de régression
logistique pour nous et nous montre uniquement le résultat "graphique"
mais dans l'analyse que nous vous proposerons pour Challenger, nous
réalisons la régression et la prédiction à la main (en <code>R</code> ou en <code>python</code>
selon le parcours que vous choisirez) de façon à pouvoir effectuer si
besoin une inspection plus fine. Comme avant, la courbe bleue indique
l'estimation de la probabilité d'être malade en fonction de l'âge et
la zone grise nous donne des indications sur l'incertitude de cette
estimation, i.e., "sous ces hypothèses et étant donné le peu de
données qu'on a et leur variabilité, il y a 95% de chances pour que la
vraie courbe se trouve quelque part (n'importe où) dans la zone
grise".
</p>
<p>
Dans ce modèle, on suppose que \(P[\textsf{Malade}] = \pi(\textsf{Age})\) avec
\(\displaystyle\pi(x)=\frac{e^{\alpha.x + \beta}}{1+e^{\alpha.x + \beta}}\). Cette
formule (étrange au premier abord) a la bonne propriété nous donner
systématiquement une valeur comprise entre 0 et 1 et de bien tendre
rapidement vers \(0\) quand l'âge tend vers \(-\infty\) et vers \(1\) quand l'âge
tend vers \(+\infty\) (mais ce n'est pas bien sûr pas la seule motivation).
</p>
<p>
En conclusion, lorsque l'on dispose de données événementielles
(binaires) et que l'on souhaite estimer l'influence d'un paramètre sur
la probabilité d'occurrence de l'évènement (maladie, défaillance,
&#x2026;), le modèle le plus naturel et le plus simple est celui de la
régression logistique. Notez, que même en se restreignant à une petite
partie des données (par exemple, uniquement les patients de moins de
50 ans), il est possible d'obtenir une estimation assez raisonnable,
même si, comme on pouvait s'y attendre, l'incertitude augmente
singulièrement.
</p>
<div class="org-src-container">
<pre class="src src-R">ggplot(df[df$Age&lt;50,],aes(x=Age,y=Malade)) + geom_point(alpha=.3,size=3) +
theme_bw() +
geom_smooth(method = <span class="org-string">"glm"</span>,
method.args = list(family = <span class="org-string">"binomial"</span>),fullrange = <span class="org-type">TRUE</span>) + xlim(20,80)
</pre>
</div>
<div class="figure">
<p><object type="image/svg+xml" data="fig4.svg" class="org-svg">
Sorry, your browser does not support SVG.</object>
</p>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="date">Date: Juin 2018</p>
<p class="author">Auteur: Konrad Hinsen, Arnaud Legrand, Christophe Pouzat</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>
</html>
module2/exo5/challenger.org 0 → 100644
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#+TITLE: Analyse du risque de défaillance des joints toriques de la navette Challenger
#+AUTHOR: Konrad Hinsen, Arnaud Legrand, Christophe Pouzat
#+DATE: Juin 2018
#+LANGUAGE: fr
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#+LATEX_HEADER: \usepackage{palatino}
#+LATEX_HEADER: \usepackage{svg}
#+LATEX_HEADER: \let\epsilon=\varepsilon
*Préambule:* Les explications données dans ce document sur le contexte
de l'étude sont largement reprises de l'excellent livre d'Edward
R. Tufte intitulé /Visual Explanations: Images and Quantities, Evidence
and Narrative/, publié en 1997 par /Graphics Press/ et réédité en 2005,
ainsi que de l'article de Dalal et al. intitulé /Risk Analysis of the
Space Shuttle: Pre-Challenger Prediction of Failure/ et publié en 1989
dans /Journal of the American Statistical Association/.
* Contexte
Dans cette étude, nous vous proposons de revenir sur [[https://fr.wikipedia.org/wiki/Accident_de_la_navette_spatiale_Challenger][l'accident de la
navette spatiale Challenger]]. Le 28 Janvier 1986, 73 secondes après son
lancement, la navette Challenger se désintègre (voir Figure [[fig:photo]])
et entraîne avec elle, les sept astronautes à son bord. Cette
explosion est due à la défaillance des deux joints toriques
assurant l'étanchéité entre les parties hautes et basses des
propulseurs (voir Figure [[fig:oring]]). Ces joints ont perdu de leur
efficacité en raison du froid particulier qui régnait au moment du
lancement. En effet, la température ce matin là était juste en dessous
de 0°C alors que l'ensemble des vols précédents avaient été effectués
à une température d'au moins 7 à 10°C de plus.
#+NAME: fig:photo
#+ATTR_LATEX: :width 10cm
#+CAPTION: Photos de la catastrophe de Challenger.
file:challenger5.jpg
# #+NAME: fig:photo
# #+ATTR_LATEX: :width 6cm
# #+CAPTION: Photo montage de la catastrophe de Challenger. De gauche à droite, de haut en bas : traînée de fumée après la désintégration de la navette spatiale américaine Challenger 73 secondes après son lancement ; débris d'un propulseur d'appoint à poudre ; joints toriques brûlés, les fautifs de l'accident ; décollage final de Challenger ; cérémonie funéraire tenue par le président Ronald Reagan en hommage aux astronautes ; explosion en vol de Challenger
# file:Challenger_Photo_Montage.jpg
# # From https://upload.wikimedia.org/wikipedia/commons/a/a8/Challenger_Photo_Montage.jpg
#+NAME: fig:oring
#+ATTR_LATEX: :width 10cm
#+CAPTION: Schéma des propulseurs de la navette challenger. Les joints toriques (un joint principale et un joint secondaire) en caoutchouc de plus de 11 mètres de circonférence assurent l'étanchéité entre la partie haute et la partie basse du propulseur.
file:o-ring.png
# From https://upload.wikimedia.org/wikipedia/commons/a/a8/Challenger_Photo_Montage.jpg
# https://i0.wp.com/www.kylehailey.com/wp-content/uploads/2014/01/Screen-Shot-2013-12-30-at-12.05.04-PM-1024x679.png?zoom=2&resize=594%2C393
Le plus étonnant est que la cause précise de cet accident avait été
débattue intensément plusieurs jours auparavant et était encore
discutée la veille même du décollage, pendant trois heures de
télé-conférence entre les ingénieurs de la Morton Thiokol
(constructeur des moteurs) et de la NASA. Si la cause immédiate de
l'accident (la défaillance des joints toriques) a rapidement été
identifiée, les raisons plus profondes qui ont conduit à ce désastre
servent régulièrement de cas d'étude, que ce soit dans des cours de
management (organisation du travail, décision technique malgré des
pressions politiques, problèmes de communication), de statistiques
(évaluation du risque, modélisation, visualisation de données), ou de
sociologie (symptôme d'un historique, de la bureaucratie et du
conformisme à des normes organisationnelles).
Dans l'étude que nous vous proposons, nous nous intéressons
principalement à l'aspect statistique mais ce n'est donc qu'une
facette (extrêmement limitée) du problème et nous vous invitons à lire
par vous même les documents donnés en référence dans le
préambule. L'étude qui suit reprend donc une partie des analyses
effectuées cette nuit là et dont l'objectif était d'évaluer
l'influence potentielle de la température et de la pression à laquelle
sont soumis les joints toriques sur leur probabilité de
dysfonctionnement. Pour cela, nous disposons des résultats des
expériences réalisées par les ingénieurs de la NASA durant les 6
années précédant le lancement de la navette Challenger.
Dans le répertoire ~module2/exo5/~ de votre espace =gitlab=, vous
trouverez les données d'origine ainsi qu'une analyse pour chacun des
différents parcours proposés. Cette analyse comporte quatre étapes:
1. Chargement des données
2. Inspection graphique des données
3. Estimation de l'influence de la température
4. Estimation de la probabilité de dysfonctionnement des joints
toriques
Les deux premières étapes ne supposent que des compétences de base en
R ou en python. La troisième étape suppose une familiarité avec la
régression logistique (généralement abordée en L3 ou M1 de stats,
économétrie, bio-statistique, ...) et la quatrième étape des bases de
probabilités (niveau lycée). Nous vous présentons donc dans la
prochaine section une introduction à la régression logistique qui ne
s'attarde pas sur les détails du calcul, mais juste sur le sens donné
aux résultats de cette régression.
* Introduction à la régression logistique
Imaginons que l'on dispose des données suivantes qui indiquent pour
une cohorte d'individus s'ils ont déclaré une maladie particulière ou
pas. Je montre ici l'analyse avec R mais le code python n'est pas forcément
très éloigné. Les données sont stockées dans une data frame dont voici
un bref résumé:
#+begin_src R :results output :session *R* :exports none
library(Hmisc) # pour calculer un intervalle de confiance sur des données binomiales
library(ggplot2)
library(dplyr)
set.seed(42)
proba = function(age) {
val=(age-50)/4
return(exp(val)/(1+exp(val)))
}
df = data.frame(Age = runif(400,min=22,max=80))
df$Malade = sapply(df$Age, function(x) rbinom(n=1,size=1,prob=proba(x)))
#+end_src
#+RESULTS:
#+begin_src R :results output :session *R* :exports both
summary(df)
str(df)
#+end_src
#+RESULTS:
#+begin_example
Age Malade
Min. :22.01 Min. :0.00
1st Qu.:37.06 1st Qu.:0.00
Median :53.30 Median :1.00
Mean :52.42 Mean :0.53
3rd Qu.:66.29 3rd Qu.:1.00
Max. :79.36 Max. :1.00
'data.frame': 100 obs. of 2 variables:
$ Age : num 75.1 76.4 38.6 70.2 59.2 ...
$ Malade: int 1 1 0 1 0 0 1 0 1 1 ...
#+end_example
Voici une représentation graphique des données qui permet de mieux
percevoir le lien qu'il peut y avoir entre l'âge et le fait de
contracter cette maladie ou pas:
#+begin_src R :results output graphics :file fig1.svg :exports both :width 4 :height 3 :session *R*
ggplot(df,aes(x=Age,y=Malade)) + geom_point(alpha=.3,size=3) + theme_bw()
#+end_src
#+ATTR_LATEX: :width 8cm
#+RESULTS:
[[file:fig1.svg]]
Il apparaît clairement sur ces données que plus l'on est âgé, plus la
probabilité de développer cette maladie est importante. Mais comment
estimer cette probabilité à partir uniquement de ces valeurs binaires
(malade/pas malade)? Pour chaque tranche d'âge (par exemple de 5 ans),
on pourrait regarder la fréquence de la maladie (le code qui suit est
un peu compliqué car le calcul de l'intervalle de confiance pour ce
type de données nécessite un traitement particulier via la fonction
=binconf=).
#+begin_src R :results output graphics :file fig1bis.svg :exports both :width 4 :height 3 :session *R*
age_range=5
df_grouped = df %>% mutate(Age=age_range*(floor(Age/age_range)+.5)) %>%
group_by(Age) %>% summarize(Malade=sum(Malade),N=n()) %>%
rowwise() %>%
do(data.frame(Age=.$Age, binconf(.$Malade, .$N, alpha=0.05))) %>%
as.data.frame()
ggplot(df_grouped,aes(x=Age)) + geom_point(data=df,aes(y=Malade),alpha=.3,size=3) +
geom_errorbar(data=df_grouped,
aes(x=Age,ymin=Lower, ymax=Upper, y=PointEst), color="darkred") +
geom_point(data=df_grouped, aes(x=Age, y=PointEst), size=3, shape=21, color="darkred") +
theme_bw()
#+end_src
#+ATTR_LATEX: :width 8cm
#+RESULTS:
[[file:fig1bis.svg]]
L'inconvénient de cette approche est que ce calcul est effectué
indépendemment pour chaque tranches d'âges, que la tranche d'âge est
arbitraire, et qu'on n'a pas grande idée de la façon dont ça
évolue. Pour modéliser cette évolution de façon plus continue, on
pourrait tenter une régression linéaire (le modèle le plus simple
possible pour rendre compte de l'influence d'un paramètre) et ainsi
estimer l'effet de l'âge sur la probabilité d'être malade:
#+begin_src R :results output graphics :file fig2.svg :exports both :width 4 :height 3 :session *R*
ggplot(df,aes(x=Age,y=Malade)) + geom_point(alpha=.3,size=3) +
theme_bw() + geom_smooth(method="lm")
#+end_src
#+ATTR_LATEX: :width 8cm
#+RESULTS:
[[file:fig2.svg]]
La ligne bleue est la régression linéaire au sens des moindres carrés
et la zone grise dans laquelle est la zone de confiance à 95% de cette
estimation (avec les données dont l'on dispose et cette hypothèse de
linéarité, la ligne bleue est la plus probable et il y a 95% de chance
que la vraie ligne soit dans cette zone grise).
Mais on voit clairement dans cette représentation graphique que cette
estimation n'a aucun sens. Une probabilité doit être comprise entre 0
et 1 et avec une régression linéaire on arrivera forcément pour des
valeurs un peu extrêmes (jeune ou âgé) à des prédictions aberrantes
(négative ou supérieures à 1). C'est tout simplement dû au fait qu'une
régression linéaire fait l'hypothèse que $\textsf{Malade} =
\alpha.\textsf{Age} + \beta + \epsilon$, où $\alpha$ et $\beta$ sont des nombres réels et $\epsilon$
est un bruit (une variable aléatoire de moyenne nulle), et estime $\alpha$
et $\beta$ à partir des données.
Cette technique n'a pas de sens pour estimer une probabilité et il
convient donc d'utiliser ce que l'on appelle une [[https://fr.wikipedia.org/wiki/R%25C3%25A9gression_logistique][régression
logistique]]:
#+begin_src R :results output graphics :file fig3.svg :exports both :width 4 :height 3 :session *R*
ggplot(df,aes(x=Age,y=Malade)) + geom_point(alpha=.3,size=3) +
theme_bw() +
geom_smooth(method = "glm",
method.args = list(family = "binomial")) + xlim(20,80)
#+end_src
#+ATTR_LATEX: :width 8cm
#+RESULTS:
[[file:fig3.svg]]
Ici, la bibliothèque =ggplot= fait tous les calculs de régression
logistique pour nous et nous montre uniquement le résultat "graphique"
mais dans l'analyse que nous vous proposerons pour Challenger, nous
réalisons la régression et la prédiction à la main (en =R= ou en =python=
selon le parcours que vous choisirez) de façon à pouvoir effectuer si
besoin une inspection plus fine. Comme avant, la courbe bleue indique
l'estimation de la probabilité d'être malade en fonction de l'âge et
la zone grise nous donne des indications sur l'incertitude de cette
estimation, i.e., "sous ces hypothèses et étant donné le peu de
données qu'on a et leur variabilité, il y a 95% de chances pour que la
vraie courbe se trouve quelque part (n'importe où) dans la zone
grise".
Dans ce modèle, on suppose que $P[\textsf{Malade}] = \pi(\textsf{Age})$ avec
$\displaystyle\pi(x)=\frac{e^{\alpha.x + \beta}}{1+e^{\alpha.x + \beta}}$. Cette
formule (étrange au premier abord) a la bonne propriété nous donner
systématiquement une valeur comprise entre 0 et 1 et de bien tendre
rapidement vers $0$ quand l'âge tend vers $-\infty$ et vers $1$ quand l'âge
tend vers $+\infty$ (mais ce n'est pas bien sûr pas la seule motivation).
En conclusion, lorsque l'on dispose de données événementielles
(binaires) et que l'on souhaite estimer l'influence d'un paramètre sur
la probabilité d'occurrence de l'évènement (maladie, défaillance,
...), le modèle le plus naturel et le plus simple est celui de la
régression logistique. Notez, que même en se restreignant à une petite
partie des données (par exemple, uniquement les patients de moins de
50 ans), il est possible d'obtenir une estimation assez raisonnable,
même si, comme on pouvait s'y attendre, l'incertitude augmente
singulièrement.
#+begin_src R :results output graphics :file fig4.svg :exports both :width 4 :height 3 :session *R*
ggplot(df[df$Age<50,],aes(x=Age,y=Malade)) + geom_point(alpha=.3,size=3) +
theme_bw() +
geom_smooth(method = "glm",
method.args = list(family = "binomial"),fullrange = TRUE) + xlim(20,80)
#+end_src
#+ATTR_LATEX: :width 8cm
#+RESULTS:
[[file:fig4.svg]]
* Emacs Setup :noexport:
This document has local variables in its postembule, which should
allow Org-mode (9) to work seamlessly without any setup. If you're
uncomfortable using such variables, you can safely ignore them at
startup. Exporting may require that you copy them in your .emacs.
# Local Variables:
# eval: (add-to-list 'org-latex-packages-alist '("" "minted"))
# eval: (setq org-latex-listings 'minted)
# eval: (setq org-latex-minted-options '(("style" "Tango") ("bgcolor" "Moccasin") ("frame" "lines") ("linenos" "true") ("fontsize" "\\small")))
# eval: (setq org-latex-pdf-process '("pdflatex -shell-escape -interaction nonstopmode -output-directory %o %f" "pdflatex -shell-escape -interaction nonstopmode -output-directory %o %f" "pdflatex -shell-escape -interaction nonstopmode -output-directory %o %f"))
# End:
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