Chapitre 3: Description de la relation entre deux variables
1. Diagramme de dispersion 2. Covariance et corr´elation 3. Moyenne mobile
4. R´egression lin´eaire
5. Ajustement
1. Diagramme de dispersion
Comme dans le chapitre pr´ec´edent, nous allons nous concentrer sur les variables quantitatives avec un grand nombre de modalit´es.
Pour visualiser l’association entre deux telles variables, le moyen le plus simple est de construire un diagramme de dispersion ou scatter plot. Voici par exemple le diagramme de dispersion des poids et tailles des ´etudiant(e)s de premi`ere ann´ee:
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160 165 170 175 180 185 190
50 55 60 65 70 75
Tailles et poids
Taille
P oids
N.B.: Pour simplifier la suite, l’´ etudiant dont le poids ´ etait particuli` erement ´ elev´ e a ´ et´ e retir´ e de l’´ echantillon.
2. Covariance et corr´elation
Le graphique semble indiquer une association entre les variables poids et taille: une plus grande taille semble correspondre en moyenne ` a un plus grand poids.
Une fa¸con de quantifier cette association est le coefficient de covariance. Pour deux variables X et Y mesur´ ees sur les mˆ emes unit´ es d’observation, le coefficient de covariance (ou simplement covariance), not´e v (X, Y ), est d´efini par:
v(X, Y ) = m (X − m(X )) (Y − m(Y )) . Exemple de calcul:
x i y i x i − m(X ) y i − m(Y ) (x i − m(X ))(y i − m(Y ))
-9 4 -7 3 -21
-5 3 -3 2 -6
3 -1 5 -2 -10
7 -3 9 -4 -36
-1 0 1 -1 -1
-7 3 -5 2 -10
Moyenne -2 1 0 0 -14
On a donc v (X, Y ) = −14.
Propri´ et´ es de la covariance
Soient X , Y et Z des variables et soient a, b, c et d des constantes.
1. Si les grandes valeurs de X sont g´en´eralement associ´ees aux grandes valeurs de Y et les petites valeurs aux petites valeurs, alors v(X, Y ) > 0.
2. Si les grandes valeurs de X sont g´en´eralement associ´ees aux petites valeurs de Y et les petites valeurs aux grandes valeurs, alors v (X, Y ) < 0.
3. v(X, X ) = s 2 (X )
4. Sym´etrie: v (X, Y ) = v (Y, X ) 5. v(X, c) = 0
6. v(aX + bY, Z ) = a v(X, Z ) + b v (Y, Z ) 7. v(aX + b, cY + d) = ac v(X, Y )
8. s 2 (X + Y ) = s 2 (X ) + s 2 (Y ) + 2v (X, Y ) 9. v(X, Y ) = m(X, Y ) − m(X )m(Y )
La propri´et´e 9. est pratique pour faire le calcul ` a la main car elle ´evite de calculer tous
les ´ecarts (x i − m(X )) et (y i − m(Y )).
L’inconv´enient de la covariance comme mesure de l’association entre deux variables est qu’elle d´epend des unit´es de mesures. Par exemple, la covariance entre les tailles et les poids des ´etudiant(e)s vaut v(T , P ) = 41.82 cm kg. Si on d´ecidait de mesurer la taille en m`etres (T m ) et le poids en grammes (P g ), on obtiendrait v(T m , P g ) = 418.2 m g.
Il est donc difficile d’interpr´eter la covariance entre deux variables.
Pour rem´edier ` a cet inconv´enient, on d´efinit le coefficient de corr´elation (ou simplement corr´elation), not´e r(X, Y ), entre les variables X et Y comme
r (X, Y ) = v (X, Y ) s(X )s(Y ) . Pour les poids et tailles, on obtient
r(T , P ) = r(T m , P g ) = 0.64.
La corr´elation est une mesure sans unit´e. Elle est donc interpr´etable mˆeme dans des cas
o` u les unit´es des variables ne nous sont pas famili`eres.
Propri´ et´ es de la corr´ elation
Soient X et Y des variables et soient a, b, c et d des constantes.
1. Si les grandes valeurs de X sont g´en´eralement associ´ees aux grandes valeurs de Y et les petites valeurs aux petites valeurs, alors r(X, Y ) > 0.
2. Si les grandes valeurs de X sont g´en´eralement associ´ees aux petites valeurs de Y et les petites valeurs aux grandes valeurs, alors r(X, Y ) < 0.
3. r(X, X ) = 1
4. Sym´etrie: r (X, Y ) = r(Y, X ) 5. r(X, c) = v(X,c)
s(X)s(c) = 0 0 est ind´ efini 6. r(aX + b, cY + d) = signe(ac) r(X, Y ) 7. r(aX + b, X ) = signe(a) r(X, X ) = ± 1 8. −1 ≤ r(X, Y ) ≤ 1
La corr´ elation entre deux variables est donc toujours comprise entre -1 et 1,
et ces bornes maximale et minimale sont atteintes lorsqu’il a y une relation
lin´ eaire parfaite entre les variables.
La corr´elation est une mesure de l’association lin´eaire entre deux variables.
Une autre formulation des propri´et´es 1. et 2. est la suivante: Si une valeur de X sup´erieure
`
a la moyenne de X est g´en´eralement associ´ee ` a une valeur de Y sup´erieure ` a la moyenne de Y , et de mˆeme pour les valeurs inf´erieures ` a la moyenne, r(X, Y ) aura tendance ` a ˆetre positif. Une association renvers´ee conduira r(X, Y ) ` a ˆetre n´egatif.
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7 8 9 10 11 12 13
17 18 19 20 21 22 23
r(X,Y) = 0.79
X
Y m(Y)
m(X)
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47 48 49 50 51 52
−1 0 1 2 3 4 5
r(X,Y) = −0.58
X
Y m(Y)
m(X)
Dans le cas des tailles et des poids, la corr´elation est donc positive:
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160 165 170 175 180 185 190
50 55 60 65 70 75
r(Taille,Poids) = 0.64
Taille P oids m(Poids)
m(Taille)
Voici quelques exemples de diagrammes de dispersion correspondant ` a diff´erentes valeurs positives de la corr´elation:
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r = 0.01
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r = 0.22
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