14. Régression linéaire multiple
239
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4
6
8
10
12
−5
0
5
10
valeur prédite
résidu
(a) premier modèle
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1.8
2.0
2.2
2.4
2.6
2.8
−0.5
0.0
0.5
valeur prédite
résidu
(b) second modèle
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−3
−2
−1
0
1
2
3
−2 −1 0
1
2
3
4
résidu standardisé
(c) premier modèle
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−3
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−3 −2 −1 0
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2
3
résidu standardisé
(d) second modèle
Figure 14.9 – Analyse des résidus pour deux modèles.
pas au milieu des données à chaque âge, d’autre part la variabilité des observations diminue visiblement avec l’âge). Le graphique (a) de la figure 14.9, où l’on
voit les résidus en fonction des valeurs prédites pour ce premier modèle, nous
montre une information équivalente (les grandes valeurs prédites correspondant
aux jeunes âges). Le qq-plot des résidus sur le graphique (c) de cette figure nous
montre en outre que l’hypothèse de normalité n’est pas non plus satisfaite.
Afin de « stabiliser la variance » et de nous approcher de la normalité,
on a ici log-transformé la variable réponse, c’est-à-dire que l’on a modélisé
Z = log(Y ) au lieu de Y . Comme la relation entre Z et X 1 n’est pas non
plus linéaire, on a en outre considéré un modèle quadratique, qui est un cas
particulier de modèle de régression linéaire multiple, obtenu en introduisant
dans le modèle un second prédicteur égal au carré du premier (on a donc X 2 =
X
2
1 ). On a ainsi calculé l’hyperplan de régression suivant :
z
∗ = 3.745 − 0.239x 1 + 0.00706x
2
1 .
Une analyse graphique des résidus pour ce second modèle nous est présentée
dans les graphiques (b) et (d) de la figure 14.9, où l’on voit que les hypothèses
de linéarité, d’homoscédasticité et de normalité sont à présent raisonnablement
satisfaites. Si on désire représenter les résultats de ce second modèle sur l’échelle
239
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12
−5
0
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valeur prédite
résidu
(a) premier modèle
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0.5
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résidu
(b) second modèle
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4
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(c) premier modèle
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résidu standardisé
(d) second modèle
Figure 14.9 – Analyse des résidus pour deux modèles.
pas au milieu des données à chaque âge, d’autre part la variabilité des observations diminue visiblement avec l’âge). Le graphique (a) de la figure 14.9, où l’on
voit les résidus en fonction des valeurs prédites pour ce premier modèle, nous
montre une information équivalente (les grandes valeurs prédites correspondant
aux jeunes âges). Le qq-plot des résidus sur le graphique (c) de cette figure nous
montre en outre que l’hypothèse de normalité n’est pas non plus satisfaite.
Afin de « stabiliser la variance » et de nous approcher de la normalité,
on a ici log-transformé la variable réponse, c’est-à-dire que l’on a modélisé
Z = log(Y ) au lieu de Y . Comme la relation entre Z et X 1 n’est pas non
plus linéaire, on a en outre considéré un modèle quadratique, qui est un cas
particulier de modèle de régression linéaire multiple, obtenu en introduisant
dans le modèle un second prédicteur égal au carré du premier (on a donc X 2 =
X
2
1 ). On a ainsi calculé l’hyperplan de régression suivant :
z
∗ = 3.745 − 0.239x 1 + 0.00706x
2
1 .
Une analyse graphique des résidus pour ce second modèle nous est présentée
dans les graphiques (b) et (d) de la figure 14.9, où l’on voit que les hypothèses
de linéarité, d’homoscédasticité et de normalité sont à présent raisonnablement
satisfaites. Si on désire représenter les résultats de ce second modèle sur l’échelle