{"id":297,"date":"2014-10-09T10:05:21","date_gmt":"2014-10-09T08:05:21","guid":{"rendered":"http:\/\/wordpress.callac.online\/index.php\/la-regression-lineaire-simple\/"},"modified":"2023-10-14T20:14:36","modified_gmt":"2023-10-14T18:14:36","slug":"la-regression-lineaire-simple","status":"publish","type":"page","link":"https:\/\/wordpress.callac.online\/index.php\/logiciels-mathematiques\/le-logiciel-r\/les-regressions-avec-r\/la-regression-lineaire-simple\/","title":{"rendered":"La r\u00e9gression lin\u00e9aire simple"},"content":{"rendered":"

Un fichier de donn\u00e9es<\/h2>\n

Travaillons sur une enqu\u00eate de satisfactions dans un h\u00f4pital, r\u00e9cup\u00e9r\u00e9 lors d’un cours de FUN (France Unit\u00e9 Num\u00e9rique). Les fichiers sont disponibles ici<\/a>.<\/p>\n

Par la suite, ces donn\u00e9es seront stock\u00e9es dans la variable satis<\/em> .<\/p>\n

Dans ces donn\u00e9es comme souvent, les observations sont en lignes et les variables en colonnes.<\/p>\n

Pr\u00e9sentation<\/h2>\n

On cherche \u00e0 expliquer une variable quantitative Y \u00e0 l’aide d’une variable quantitative X en utilisant un mod\u00e8le lin\u00e9aire :<\/p>\n Y=a + b \\times X + \\textrm{bruit} <\/span>\n

En pratique, on devra v\u00e9rifier que le bruit suit une loi normale.<\/p>\n

Un exemple<\/h2>\n

\u00c9tudions si le score de relation varie en fonction de l’\u00e2ge des personnes.<\/p>\n

Calculons les param\u00e8tres avec la commande lm(Y~X)<\/em> :<\/p>\n

mod.sr.a<-lm(score.relation~age,data=satis) \nsummary(mod.sr.a)<\/pre>\n

\nCall:
\nlm(formula = score.relation ~ age, data = satis)<\/code><\/p>\n
<\/code><\/code><\/p>\n
Residuals:
\nMin 1Q Median 3Q Max
\n-22.047 -2.447 1.339 3.620 5.673<\/p>\n
<\/code><\/code><\/p>\n
Coefficients:
\nEstimate Std. Error t value Pr(>|t|)
\n(Intercept) 33.76634 0.84496 39.962 <2e-16 *** age 0.02668 0.01485 1.796 0.0734 .
\n\u2014\u00a0–
\nSignif. codes: 0 \u2018***\u2019 0.001 \u2018**\u2019 0.01 \u2018*\u2019 0.05 \u2018.\u2019 0.1 \u2018 \u2019 1<\/p>\n

\n<\/code><\/p>\n
Residual standard error: 4.609 on 347 degrees of freedom
\n(185 observations deleted due to missingness)
\nMultiple R-squared:  0.00921,\tAdjusted R-squared:  0.006355
\nF-statistic: 3.226 on 1 and 347 DF,  p-value: 0.07336
\n<\/code><\/p>\n
On y voit les coefficients avec leur p-value respectifs :
\n– a = 33.76634 avec p=<2e-16 – b = 0.02668 avec p = 0.0734 Mais on y trouve aussi la valeur du p-value (0.07336), que l’on avait dans la test de corr\u00e9lation lin\u00e9aire :<\/p>\n
cor.test(satis$score.relation,satis$age,use=\"complete.obs\")<\/pre>\n

\nPearson's product-moment correlation<\/code><\/p>\n

\n<\/code><\/p>\n
data:  satis$score.relation and satis$age
\nt = 1.796, df = 347, p-value = 0.07336
\nalternative hypothesis: true correlation is not equal to 0
\n95 percent confidence interval:
\n-0.009102243  0.198945290
\nsample estimates:
\ncor
\n0.09596955
\n<\/code><\/p>\n
Cette valeur \u00e9tant trop \u00e9lev\u00e9e (sup\u00e9rieure \u00e0 0.05), on peut pas affirmer que les deux variables sont li\u00e9s.<\/p>\n
Test de normalit\u00e9 du bruit<\/h2>\n
Pour juger de la validit\u00e9 d’un mod\u00e8le de r\u00e9gression, on peut tracer un histogramme :<\/p>\n
hist(resid(mod.sr.a),col=\"brown\")<\/pre>\n
\"histo_resid_01.png\"<\/p>\n
On voit que le bruit ne suit pas une loi normale. Le mod\u00e8le n’est pas valide.<\/p>\n
Repr\u00e9sentation graphique<\/h2>\n
plot(satis$age,satis$score.relation,las=1,xlab=\"\u00e2ge\",ylab=\"Score de relation\")\nabline(mod.sr.a,lwd=2,col=\"red\")<\/pre>\n
\"reglin_01.png\"<\/p>\n
Cas particulier : la variable X est binaire<\/h2>\n
Cherchons \u00e0 expliquer le score de relation par le fait de recommander l’\u00e9tablissement.<\/p>\n
Pr\u00e9parons la variable binaire :<\/p>\n
satis$recommander.b<-ifelse(satis$recommander==2,1,0)<\/pre>\n
Calculons les coefficients :<\/p>\n
mod.sr.r<-lm(score.relation~recommander.b,data=satis) summary(mod.sr.r)<\/pre>\n

\nCall:
\nlm(formula = score.relation ~ recommander.b, data = satis)<\/code><\/p>\n
<\/code><\/code><\/p>\n
Residuals:
\nMin 1Q Median 3Q Max
\n-19.1091 -1.6826 0.8909 2.9975 7.8909<\/p>\n
<\/code><\/code><\/p>\n
Coefficients:
\nEstimate Std. Error t value Pr(>|t|)
\n(Intercept) 32.1091 0.3918 81.959 <2e-16 *** recommander.b 4.5735 0.4763 9.602 <2e-16 ***
\n\u2014\u00a0–
\nSignif. codes: 0 \u2018***\u2019 0.001 \u2018**\u2019 0.01 \u2018*\u2019 0.05 \u2018.\u2019 0.1 \u2018 \u2019 1<\/p>\n

\n<\/code><\/p>\n
Residual standard error: 4.109 on 338 degrees of freedom
\n(194 observations deleted due to missingness)
\nMultiple R-squared:  0.2143,\tAdjusted R-squared:  0.212
\nF-statistic: 92.19 on 1 and 338 DF,  p-value: < 2.2e-16
\n<\/code><\/p>\n
On voit que les valeurs de p sont faibles.<\/p>\n
On lit aussi b=4.5735.<\/p>\n
Si la variable recommande augmente de 1 (passe de 0 \u00e0 1), le score de relation augmente de 4.5735.<\/p>\n
Comparons avec le test de Student :<\/p>\n
t.test(score.relation~recommander.b,data=satis,var.equal=TRUE)<\/pre>\n

\nTwo Sample t-test<\/code><\/p>\n

\n<\/code><\/p>\n
data:  score.relation by recommander.b
\nt = -9.6016, df = 338, p-value < 2.2e-16
\nalternative hypothesis: true difference in means is not equal to 0
\n95 percent confidence interval:
\n-5.510459 -3.636576
\nsample estimates:
\nmean in group 0 mean in group 1
\n32.10909        36.68261
\n<\/code><\/p>\n
On y retrouve la p-value tr\u00e8s faible.<\/p>\n
On voit aussi l’\u00e9cart des moyennes entre les deux sous-groupes :<\/p>\n
36.68261-32.10909<\/p>\n

\n[1] 4.57352
\n<\/code><\/p>\n
Une autre fa\u00e7on de trouver ces valeurs est de r\u00e9aliser deux pr\u00e9dictions :<\/p>\n
predict(mod.sr.r,data=satis,data.frame(recommander.b=0))<\/pre>\n

\n1
\n32.10909
\n<\/code><\/p>\n
predict(mod.sr.r,data=satis,data.frame(recommander.b=1))<\/pre>\n

\n1
\n36.68261
\n<\/code><\/p>\n
36.68261-32.10909<\/p>\n

\n[1] 4.57352
\n<\/code><\/p>\n","protected":false},"excerpt":{"rendered":"
Un fichier de donn\u00e9es Travaillons sur une enqu\u00eate de satisfactions dans un h\u00f4pital, r\u00e9cup\u00e9r\u00e9 lors d’un cours de FUN (France Unit\u00e9 Num\u00e9rique). Les fichiers sont disponibles ici. Par la suite,…<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":2076,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-297","page","type-page","status-publish","hentry"],"yoast_head":"\nLa r\u00e9gression lin\u00e9aire simple - Maths & Num\u00e9rique<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/wordpress.callac.online\/index.php\/logiciels-mathematiques\/le-logiciel-r\/les-regressions-avec-r\/la-regression-lineaire-simple\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"La r\u00e9gression lin\u00e9aire simple - Maths & Num\u00e9rique\" \/>\n<meta property=\"og:description\" content=\"Un fichier de donn\u00e9es Travaillons sur une enqu\u00eate de satisfactions dans un h\u00f4pital, r\u00e9cup\u00e9r\u00e9 lors d’un cours de FUN (France Unit\u00e9 Num\u00e9rique). 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