library("ggplot2")
library("car")
library("MASS")
assurances <- read.table("Assurances.txt", header = TRUE)
head(assurances)
## cost age sex bmi children smoker region
## 1 19151.22 19 female 27.900 0 yes southwest
## 2 9146.12 18 male 33.770 1 no southeast
## 3 11774.53 28 male 33.000 3 no southeast
## 4 11539.90 33 male 22.705 0 no northwest
## 5 11100.11 32 male 28.880 0 no northwest
## 6 10027.75 31 female 25.740 0 no southeast
dim(assurances)
## [1] 2772 7
n <- dim(assurances)[1]
assurances$sex <- factor(assurances$sex)
assurances$smoker <- factor(assurances$smoker)
assurances$region <- factor(assurances$region)
(a) Statistiques descriptives
summary(assurances)
## cost age sex bmi children
## Min. : 4776 Min. :18.00 female:1366 Min. :15.96 Min. :0.000
## 1st Qu.:10863 1st Qu.:26.00 male :1406 1st Qu.:26.22 1st Qu.:0.000
## Median :12692 Median :39.00 Median :30.45 Median :1.000
## Mean :13414 Mean :39.11 Mean :30.70 Mean :1.102
## 3rd Qu.:15116 3rd Qu.:51.00 3rd Qu.:34.77 3rd Qu.:2.000
## Max. :24812 Max. :64.00 Max. :53.13 Max. :5.000
## smoker region
## no :2208 northeast:658
## yes: 564 northwest:664
## southeast:766
## southwest:684
##
##
par(mfrow = c(2,2))
hist(assurances$cost, main = "Histogramme de cost", xlab = "cost")
hist(assurances$age, main = "Histogramme de age", xlab = "age")
hist(assurances$bmi, main = "Histogramme de bmi", xlab = "bmi")
hist(assurances$children, main = "Histogramme de children", xlab = "children")
par(mfrow = c(1,1))
x <- ggplot(assurances, aes(age, cost))
x + geom_point() + theme_minimal()
x <- ggplot(assurances, aes(bmi, cost))
x + geom_point() + theme_minimal()
x <- ggplot(assurances, aes(factor(children), cost))
x + geom_boxplot()
x <- ggplot(assurances, aes(sex, cost))
x + geom_boxplot()
x <- ggplot(assurances, aes(smoker, cost))
x + geom_boxplot()
x <- ggplot(assurances, aes(region, cost))
x + geom_boxplot()
x <- ggplot(assurances, aes(x = age, y = cost, color = smoker))
x + geom_point() + theme_bw()
x <- ggplot(assurances, aes(x = bmi, y = cost, color = smoker))
x + geom_point() + theme_bw()
x <- ggplot(assurances, aes(x = smoker, y = cost, fill = sex))
x + geom_boxplot()
x <- ggplot(assurances, aes(x = bmi, y = cost, color = age))
x + geom_jitter(width = .2)
(b) Multicolinéarité
cor(assurances[, c("age", "bmi", "children")])
## age bmi children
## age 1.00000000 0.113048451 0.037574294
## bmi 0.11304845 1.000000000 -0.001492284
## children 0.03757429 -0.001492284 1.000000000
modele.additif <- lm(cost ~ age + sex + bmi + children + smoker + region, data = assurances)
vif(modele.additif)
## GVIF Df GVIF^(1/(2*Df))
## age 1.018201 1 1.009060
## sex 1.009875 1 1.004925
## bmi 1.114274 1 1.055592
## children 1.004017 1 1.002006
## smoker 1.013467 1 1.006711
## region 1.107600 3 1.017178
(c) Sélection de modèle
modele.max <- lm(cost ~ (age + sex + bmi + children + smoker + region)^2 +
I(age^2) + I(bmi^2), data = assurances)
modele.min <- lm(cost ~ 1, data = assurances)
Forward
modele.forward <- step(modele.min, direction = "forward",
scope = list(lower = modele.min, upper = modele.max))
## Start: AIC=45373.65
## cost ~ 1
##
## Df Sum of Sq RSS AIC
## + smoker 1 2.3691e+10 1.1893e+10 42338
## + age 1 3.4405e+09 3.2144e+10 45094
## + I(age^2) 1 3.2512e+09 3.2333e+10 45110
## + bmi 1 1.7766e+09 3.3807e+10 45234
## + I(bmi^2) 1 1.7368e+09 3.3847e+10 45237
## + region 3 3.8036e+08 3.5204e+10 45350
## + children 1 1.4519e+08 3.5439e+10 45364
## + sex 1 8.9018e+07 3.5495e+10 45369
## <none> 3.5584e+10 45374
##
## Step: AIC=42337.58
## cost ~ smoker
##
## Df Sum of Sq RSS AIC
## + age 1 3875877459 8.0167e+09 41246
## + I(age^2) 1 3696472854 8.1961e+09 41308
## + bmi 1 1630899773 1.0262e+10 41931
## + I(bmi^2) 1 1559573048 1.0333e+10 41950
## + children 1 122554340 1.1770e+10 42311
## + region 3 60036765 1.1833e+10 42330
## + sex 1 10548460 1.1882e+10 42337
## <none> 1.1893e+10 42338
##
## Step: AIC=41246.34
## cost ~ smoker + age
##
## Df Sum of Sq RSS AIC
## + bmi 1 1125170406 6891511946 40829
## + I(bmi^2) 1 1100047751 6916634601 40839
## + children 1 76169047 7940513305 41222
## + region 3 73475038 7943207314 41227
## + I(age^2) 1 23600684 7993081668 41240
## <none> 8016682352 41246
## + sex 1 3029821 8013652531 41247
## + age:smoker 1 21176 8016661176 41248
##
## Step: AIC=40829.12
## cost ~ smoker + age + bmi
##
## Df Sum of Sq RSS AIC
## + children 1 79656364 6811855581 40799
## + region 3 82417241 6809094705 40802
## + I(age^2) 1 33489720 6858022226 40818
## + sex 1 10622655 6880889291 40827
## <none> 6891511946 40829
## + I(bmi^2) 1 463171 6891048775 40831
## + age:smoker 1 327072 6891184874 40831
## + age:bmi 1 227909 6891284037 40831
## + bmi:smoker 1 32607 6891479338 40831
##
## Step: AIC=40798.89
## cost ~ smoker + age + bmi + children
##
## Df Sum of Sq RSS AIC
## + region 3 84229954 6727625627 40770
## + sex 1 11623528 6800232053 40796
## + I(age^2) 1 10799122 6801056459 40796
## + bmi:children 1 6616487 6805239095 40798
## <none> 6811855581 40799
## + children:smoker 1 2008017 6809847565 40800
## + I(bmi^2) 1 465880 6811389701 40801
## + age:bmi 1 376625 6811478956 40801
## + age:children 1 160540 6811695041 40801
## + age:smoker 1 142977 6811712604 40801
## + bmi:smoker 1 49536 6811806046 40801
##
## Step: AIC=40770.4
## cost ~ smoker + age + bmi + children + region
##
## Df Sum of Sq RSS AIC
## + sex 1 11678868 6715946758 40768
## + I(age^2) 1 10805099 6716820528 40768
## + age:region 3 16454396 6711171231 40770
## + bmi:children 1 6426404 6721199223 40770
## <none> 6727625627 40770
## + children:smoker 1 2055236 6725570391 40772
## + I(bmi^2) 1 560929 6727064698 40772
## + age:bmi 1 255602 6727370024 40772
## + age:children 1 218765 6727406861 40772
## + bmi:smoker 1 185664 6727439963 40772
## + age:smoker 1 76977 6727548650 40772
## + bmi:region 3 4419508 6723206119 40775
## + children:region 3 3555665 6724069962 40775
## + smoker:region 3 3307185 6724318442 40775
##
## Step: AIC=40767.58
## cost ~ smoker + age + bmi + children + region + sex
##
## Df Sum of Sq RSS AIC
## + I(age^2) 1 10706404 6705240354 40765
## + age:region 3 16458255 6699488503 40767
## + sex:children 1 6702878 6709243880 40767
## + bmi:children 1 6332963 6709613796 40767
## <none> 6715946758 40768
## + children:smoker 1 1804517 6714142242 40769
## + age:sex 1 1527456 6714419303 40769
## + sex:smoker 1 1069158 6714877600 40769
## + sex:bmi 1 702893 6715243865 40769
## + I(bmi^2) 1 569345 6715377414 40769
## + sex:region 3 10218111 6705728647 40769
## + bmi:smoker 1 400226 6715546533 40769
## + age:children 1 362374 6715584384 40769
## + age:bmi 1 172976 6715773783 40770
## + age:smoker 1 102605 6715844153 40770
## + bmi:region 3 5026808 6710919950 40772
## + children:region 3 3622084 6712324675 40772
## + smoker:region 3 3212186 6712734573 40772
##
## Step: AIC=40765.16
## cost ~ smoker + age + bmi + children + region + sex + I(age^2)
##
## Df Sum of Sq RSS AIC
## + age:region 3 16607736 6688632618 40764
## + bmi:children 1 6295927 6698944427 40765
## + sex:children 1 6251027 6698989327 40765
## <none> 6705240354 40765
## + age:sex 1 1796303 6703444051 40766
## + children:smoker 1 1713131 6703527223 40766
## + sex:smoker 1 989019 6704251335 40767
## + sex:bmi 1 597242 6704643112 40767
## + sex:region 3 10241153 6694999201 40767
## + I(bmi^2) 1 517594 6704722760 40767
## + bmi:smoker 1 444037 6704796317 40767
## + age:children 1 319850 6704920504 40767
## + age:smoker 1 137261 6705103093 40767
## + age:bmi 1 2835 6705237519 40767
## + bmi:region 3 5087974 6700152380 40769
## + children:region 3 3269768 6701970586 40770
## + smoker:region 3 3050312 6702190042 40770
##
## Step: AIC=40764.28
## cost ~ smoker + age + bmi + children + region + sex + I(age^2) +
## age:region
##
## Df Sum of Sq RSS AIC
## + sex:children 1 5458301 6683174316 40764
## + bmi:children 1 5357034 6683275584 40764
## <none> 6688632618 40764
## + age:sex 1 1960958 6686671660 40765
## + children:smoker 1 1194467 6687438150 40766
## + sex:region 3 10621897 6678010721 40766
## + age:bmi 1 827352 6687805265 40766
## + sex:smoker 1 757153 6687875464 40766
## + I(bmi^2) 1 703769 6687928849 40766
## + sex:bmi 1 603617 6688029000 40766
## + bmi:smoker 1 462102 6688170515 40766
## + age:children 1 420084 6688212534 40766
## + age:smoker 1 12118 6688620500 40766
## + bmi:region 3 5215443 6683417175 40768
## + smoker:region 3 3334816 6685297802 40769
## + children:region 3 3157028 6685475590 40769
##
## Step: AIC=40764.02
## cost ~ smoker + age + bmi + children + region + sex + I(age^2) +
## age:region + children:sex
##
## Df Sum of Sq RSS AIC
## + bmi:children 1 5466478 6677707838 40764
## <none> 6683174316 40764
## + age:sex 1 2227548 6680946769 40765
## + children:smoker 1 1599552 6681574764 40765
## + sex:region 3 10835239 6672339078 40766
## + sex:smoker 1 758652 6682415664 40766
## + age:bmi 1 715036 6682459280 40766
## + sex:bmi 1 619207 6682555110 40766
## + I(bmi^2) 1 576277 6682598040 40766
## + bmi:smoker 1 346936 6682827380 40766
## + age:children 1 238658 6682935658 40766
## + age:smoker 1 1317 6683173000 40766
## + bmi:region 3 5454294 6677720023 40768
## + smoker:region 3 3503685 6679670631 40769
## + children:region 3 3019454 6680154862 40769
##
## Step: AIC=40763.75
## cost ~ smoker + age + bmi + children + region + sex + I(age^2) +
## age:region + children:sex + bmi:children
##
## Df Sum of Sq RSS AIC
## <none> 6677707838 40764
## + age:sex 1 2034618 6675673220 40765
## + children:smoker 1 1464410 6676243428 40765
## + age:bmi 1 875496 6676832343 40765
## + sex:region 3 10432510 6667275329 40765
## + sex:bmi 1 735957 6676971882 40765
## + age:children 1 720578 6676987260 40765
## + I(bmi^2) 1 697462 6677010377 40765
## + sex:smoker 1 600846 6677106992 40766
## + bmi:smoker 1 195681 6677512158 40766
## + age:smoker 1 48 6677707790 40766
## + bmi:region 3 5713777 6671994062 40767
## + children:region 3 3446755 6674261084 40768
## + smoker:region 3 3240470 6674467369 40768
Backward
modele.backward <- step(modele.max, direction = "backward",
scope = list(lower = modele.min, upper = modele.max))
## Start: AIC=40792.52
## cost ~ (age + sex + bmi + children + smoker + region)^2 + I(age^2) +
## I(bmi^2)
##
## Df Sum of Sq RSS AIC
## - smoker:region 3 2524721 6648435354 40788
## - children:region 3 3708221 6649618854 40788
## - bmi:region 3 8322979 6654233612 40790
## - sex:region 3 9495513 6655406145 40790
## - age:smoker 1 9478 6645920111 40791
## - bmi:smoker 1 184342 6646094975 40791
## - sex:smoker 1 483846 6646394479 40791
## - age:bmi 1 545208 6646455841 40791
## - sex:bmi 1 670474 6646581107 40791
## - age:children 1 807910 6646718543 40791
## - children:smoker 1 1238051 6647148684 40791
## - age:sex 1 1868026 6647778659 40791
## - I(bmi^2) 1 3854094 6649764727 40792
## <none> 6645910633 40793
## - sex:children 1 5982153 6651892786 40793
## - age:region 3 15687128 6661597761 40793
## - bmi:children 1 6150521 6652061154 40793
## - I(age^2) 1 8951735 6654862368 40794
##
## Step: AIC=40787.58
## cost ~ age + sex + bmi + children + smoker + region + I(age^2) +
## I(bmi^2) + age:sex + age:bmi + age:children + age:smoker +
## age:region + sex:bmi + sex:children + sex:smoker + sex:region +
## bmi:children + bmi:smoker + bmi:region + children:smoker +
## children:region
##
## Df Sum of Sq RSS AIC
## - children:region 3 3687608 6652122962 40783
## - bmi:region 3 7890397 6656325751 40785
## - bmi:smoker 1 25472 6648460826 40786
## - age:smoker 1 33012 6648468366 40786
## - sex:smoker 1 444056 6648879410 40786
## - age:bmi 1 545075 6648980429 40786
## - sex:bmi 1 656111 6649091465 40786
## - sex:region 3 10460791 6658896145 40786
## - age:children 1 996207 6649431561 40786
## - children:smoker 1 1474156 6649909510 40786
## - age:sex 1 1707922 6650143276 40786
## - I(bmi^2) 1 3562028 6651997382 40787
## <none> 6648435354 40788
## - sex:children 1 5788502 6654223856 40788
## - age:region 3 15484148 6663919502 40788
## - bmi:children 1 6327563 6654762917 40788
## - I(age^2) 1 9147061 6657582415 40789
##
## Step: AIC=40783.11
## cost ~ age + sex + bmi + children + smoker + region + I(age^2) +
## I(bmi^2) + age:sex + age:bmi + age:children + age:smoker +
## age:region + sex:bmi + sex:children + sex:smoker + sex:region +
## bmi:children + bmi:smoker + bmi:region + children:smoker
##
## Df Sum of Sq RSS AIC
## - bmi:region 3 7637876 6659760838 40780
## - bmi:smoker 1 7452 6652130414 40781
## - age:smoker 1 34399 6652157361 40781
## - sex:smoker 1 398046 6652521008 40781
## - age:bmi 1 459717 6652582680 40781
## - sex:bmi 1 695382 6652818344 40781
## - age:children 1 862162 6652985124 40781
## - sex:region 3 10634529 6662757491 40782
## - children:smoker 1 1474290 6653597252 40782
## - age:sex 1 1741549 6653864511 40782
## - I(bmi^2) 1 3442122 6655565084 40783
## <none> 6652122962 40783
## - age:region 3 15356662 6667479624 40784
## - sex:children 1 5867747 6657990709 40784
## - bmi:children 1 5974080 6658097042 40784
## - I(age^2) 1 9636136 6661759099 40785
##
## Step: AIC=40780.29
## cost ~ age + sex + bmi + children + smoker + region + I(age^2) +
## I(bmi^2) + age:sex + age:bmi + age:children + age:smoker +
## age:region + sex:bmi + sex:children + sex:smoker + sex:region +
## bmi:children + bmi:smoker + children:smoker
##
## Df Sum of Sq RSS AIC
## - age:smoker 1 26400 6659787238 40778
## - bmi:smoker 1 26757 6659787595 40778
## - sex:smoker 1 359816 6660120654 40778
## - age:bmi 1 559100 6660319938 40779
## - I(bmi^2) 1 777839 6660538677 40779
## - sex:region 3 10632442 6670393280 40779
## - age:children 1 1028276 6660789113 40779
## - sex:bmi 1 1070641 6660831479 40779
## - children:smoker 1 1623907 6661384745 40779
## - age:sex 1 2103459 6661864297 40779
## <none> 6659760838 40780
## - bmi:children 1 5481030 6665241868 40781
## - sex:children 1 5729510 6665490348 40781
## - age:region 3 15373696 6675134534 40781
## - I(age^2) 1 9578883 6669339721 40782
##
## Step: AIC=40778.3
## cost ~ age + sex + bmi + children + smoker + region + I(age^2) +
## I(bmi^2) + age:sex + age:bmi + age:children + age:region +
## sex:bmi + sex:children + sex:smoker + sex:region + bmi:children +
## bmi:smoker + children:smoker
##
## Df Sum of Sq RSS AIC
## - bmi:smoker 1 31707 6659818945 40776
## - sex:smoker 1 355050 6660142288 40776
## - age:bmi 1 558882 6660346121 40777
## - I(bmi^2) 1 782063 6660569301 40777
## - age:children 1 1019028 6660806266 40777
## - sex:region 3 10664570 6670451808 40777
## - sex:bmi 1 1056104 6660843342 40777
## - children:smoker 1 1601002 6661388240 40777
## - age:sex 1 2080136 6661867374 40777
## <none> 6659787238 40778
## - bmi:children 1 5485767 6665273005 40779
## - sex:children 1 5750152 6665537390 40779
## - age:region 3 15528520 6675315759 40779
## - I(age^2) 1 9565771 6669353009 40780
##
## Step: AIC=40776.32
## cost ~ age + sex + bmi + children + smoker + region + I(age^2) +
## I(bmi^2) + age:sex + age:bmi + age:children + age:region +
## sex:bmi + sex:children + sex:smoker + sex:region + bmi:children +
## children:smoker
##
## Df Sum of Sq RSS AIC
## - sex:smoker 1 389251 6660208196 40774
## - age:bmi 1 567003 6660385948 40775
## - I(bmi^2) 1 773312 6660592257 40775
## - age:children 1 1019320 6660838265 40775
## - sex:bmi 1 1067401 6660886346 40775
## - sex:region 3 10694004 6670512949 40775
## - children:smoker 1 1611300 6661430245 40775
## - age:sex 1 2099258 6661918204 40775
## <none> 6659818945 40776
## - bmi:children 1 5551326 6665370272 40777
## - sex:children 1 5798429 6665617374 40777
## - age:region 3 15518917 6675337863 40777
## - I(age^2) 1 9550190 6669369136 40778
##
## Step: AIC=40774.48
## cost ~ age + sex + bmi + children + smoker + region + I(age^2) +
## I(bmi^2) + age:sex + age:bmi + age:children + age:region +
## sex:bmi + sex:children + sex:region + bmi:children + children:smoker
##
## Df Sum of Sq RSS AIC
## - age:bmi 1 609977 6660818173 40773
## - I(bmi^2) 1 846995 6661055191 40773
## - age:children 1 991897 6661200093 40773
## - sex:bmi 1 1047148 6661255344 40773
## - sex:region 3 10862029 6671070225 40773
## - children:smoker 1 1512221 6661720418 40773
## - age:sex 1 2140364 6662348560 40773
## <none> 6660208196 40774
## - bmi:children 1 5685307 6665893503 40775
## - sex:children 1 5791743 6665999939 40775
## - age:region 3 15762924 6675971120 40775
## - I(age^2) 1 9583625 6669791821 40776
##
## Step: AIC=40772.73
## cost ~ age + sex + bmi + children + smoker + region + I(age^2) +
## I(bmi^2) + age:sex + age:children + age:region + sex:bmi +
## sex:children + sex:region + bmi:children + children:smoker
##
## Df Sum of Sq RSS AIC
## - I(bmi^2) 1 1005174 6661823347 40771
## - age:children 1 1010046 6661828219 40771
## - sex:bmi 1 1121198 6661939371 40771
## - sex:region 3 10758118 6671576291 40771
## - children:smoker 1 1561686 6662379859 40771
## - age:sex 1 2236729 6663054902 40772
## <none> 6660818173 40773
## - age:region 3 15156008 6675974181 40773
## - bmi:children 1 5570846 6666389019 40773
## - sex:children 1 5899326 6666717499 40773
## - I(age^2) 1 10299298 6671117472 40775
##
## Step: AIC=40771.15
## cost ~ age + sex + bmi + children + smoker + region + I(age^2) +
## age:sex + age:children + age:region + sex:bmi + sex:children +
## sex:region + bmi:children + children:smoker
##
## Df Sum of Sq RSS AIC
## - age:children 1 953915 6662777262 40770
## - sex:bmi 1 972936 6662796284 40770
## - sex:region 3 10610400 6672433747 40770
## - children:smoker 1 1694625 6663517973 40770
## - age:sex 1 2159152 6663982499 40770
## <none> 6661823347 40771
## - age:region 3 14903568 6676726915 40771
## - bmi:children 1 5397290 6667220638 40771
## - sex:children 1 6089842 6667913189 40772
## - I(age^2) 1 10366685 6672190032 40773
##
## Step: AIC=40769.55
## cost ~ age + sex + bmi + children + smoker + region + I(age^2) +
## age:sex + age:region + sex:bmi + sex:children + sex:region +
## bmi:children + children:smoker
##
## Df Sum of Sq RSS AIC
## - sex:region 3 10389810 6673167072 40768
## - sex:bmi 1 1020139 6663797402 40768
## - children:smoker 1 1647399 6664424661 40768
## - age:sex 1 2144478 6664921741 40768
## <none> 6662777262 40770
## - bmi:children 1 4855934 6667633196 40770
## - age:region 3 14791249 6677568512 40770
## - sex:children 1 6451045 6669228308 40770
## - I(age^2) 1 10420483 6673197746 40772
##
## Step: AIC=40767.87
## cost ~ age + sex + bmi + children + smoker + region + I(age^2) +
## age:sex + age:region + sex:bmi + sex:children + bmi:children +
## children:smoker
##
## Df Sum of Sq RSS AIC
## - sex:bmi 1 1141367 6674308439 40766
## - children:smoker 1 1453870 6674620942 40766
## - age:sex 1 2256073 6675423145 40767
## <none> 6673167072 40768
## - age:region 3 14501199 6687668271 40768
## - bmi:children 1 5274893 6678441965 40768
## - sex:children 1 6258866 6679425938 40768
## - I(age^2) 1 10412481 6683579553 40770
##
## Step: AIC=40766.34
## cost ~ age + sex + bmi + children + smoker + region + I(age^2) +
## age:sex + age:region + sex:children + bmi:children + children:smoker
##
## Df Sum of Sq RSS AIC
## - children:smoker 1 1364781 6675673220 40765
## - age:sex 1 1934989 6676243428 40765
## <none> 6674308439 40766
## - age:region 3 14496428 6688804867 40766
## - bmi:children 1 5150037 6679458476 40766
## - sex:children 1 6198272 6680506711 40767
## - I(age^2) 1 10539606 6684848045 40769
##
## Step: AIC=40764.91
## cost ~ age + sex + bmi + children + smoker + region + I(age^2) +
## age:sex + age:region + sex:children + bmi:children
##
## Df Sum of Sq RSS AIC
## - age:sex 1 2.0346e+06 6.6777e+09 40764
## <none> 6.6757e+09 40765
## - bmi:children 1 5.2735e+06 6.6809e+09 40765
## - age:region 3 1.5055e+07 6.6907e+09 40765
## - sex:children 1 5.8227e+06 6.6815e+09 40765
## - I(age^2) 1 1.0642e+07 6.6863e+09 40767
## - smoker 1 2.3560e+10 3.0235e+10 44950
##
## Step: AIC=40763.75
## cost ~ age + sex + bmi + children + smoker + region + I(age^2) +
## age:region + sex:children + bmi:children
##
## Df Sum of Sq RSS AIC
## <none> 6.6777e+09 40764
## - age:region 3 1.4890e+07 6.6926e+09 40764
## - bmi:children 1 5.4665e+06 6.6832e+09 40764
## - sex:children 1 5.5677e+06 6.6833e+09 40764
## - I(age^2) 1 1.0372e+07 6.6881e+09 40766
## - smoker 1 2.3558e+10 3.0236e+10 44948
Stepwise
modele.stepwise <- step(modele.min, direction = "both",
scope = list(lower = modele.min, upper = modele.max))
## Start: AIC=45373.65
## cost ~ 1
##
## Df Sum of Sq RSS AIC
## + smoker 1 2.3691e+10 1.1893e+10 42338
## + age 1 3.4405e+09 3.2144e+10 45094
## + I(age^2) 1 3.2512e+09 3.2333e+10 45110
## + bmi 1 1.7766e+09 3.3807e+10 45234
## + I(bmi^2) 1 1.7368e+09 3.3847e+10 45237
## + region 3 3.8036e+08 3.5204e+10 45350
## + children 1 1.4519e+08 3.5439e+10 45364
## + sex 1 8.9018e+07 3.5495e+10 45369
## <none> 3.5584e+10 45374
##
## Step: AIC=42337.58
## cost ~ smoker
##
## Df Sum of Sq RSS AIC
## + age 1 3.8759e+09 8.0167e+09 41246
## + I(age^2) 1 3.6965e+09 8.1961e+09 41308
## + bmi 1 1.6309e+09 1.0262e+10 41931
## + I(bmi^2) 1 1.5596e+09 1.0333e+10 41950
## + children 1 1.2255e+08 1.1770e+10 42311
## + region 3 6.0037e+07 1.1833e+10 42330
## + sex 1 1.0548e+07 1.1882e+10 42337
## <none> 1.1893e+10 42338
## - smoker 1 2.3691e+10 3.5584e+10 45374
##
## Step: AIC=41246.34
## cost ~ smoker + age
##
## Df Sum of Sq RSS AIC
## + bmi 1 1.1252e+09 6.8915e+09 40829
## + I(bmi^2) 1 1.1000e+09 6.9166e+09 40839
## + children 1 7.6169e+07 7.9405e+09 41222
## + region 3 7.3475e+07 7.9432e+09 41227
## + I(age^2) 1 2.3601e+07 7.9931e+09 41240
## <none> 8.0167e+09 41246
## + sex 1 3.0298e+06 8.0137e+09 41247
## + age:smoker 1 2.1176e+04 8.0167e+09 41248
## - age 1 3.8759e+09 1.1893e+10 42338
## - smoker 1 2.4127e+10 3.2144e+10 45094
##
## Step: AIC=40829.12
## cost ~ smoker + age + bmi
##
## Df Sum of Sq RSS AIC
## + children 1 7.9656e+07 6.8119e+09 40799
## + region 3 8.2417e+07 6.8091e+09 40802
## + I(age^2) 1 3.3490e+07 6.8580e+09 40818
## + sex 1 1.0623e+07 6.8809e+09 40827
## <none> 6.8915e+09 40829
## + I(bmi^2) 1 4.6317e+05 6.8910e+09 40831
## + age:smoker 1 3.2707e+05 6.8912e+09 40831
## + age:bmi 1 2.2791e+05 6.8913e+09 40831
## + bmi:smoker 1 3.2607e+04 6.8915e+09 40831
## - bmi 1 1.1252e+09 8.0167e+09 41246
## - age 1 3.3701e+09 1.0262e+10 41931
## - smoker 1 2.3974e+10 3.0866e+10 44983
##
## Step: AIC=40798.89
## cost ~ smoker + age + bmi + children
##
## Df Sum of Sq RSS AIC
## + region 3 8.4230e+07 6.7276e+09 40770
## + sex 1 1.1624e+07 6.8002e+09 40796
## + I(age^2) 1 1.0799e+07 6.8011e+09 40796
## + bmi:children 1 6.6165e+06 6.8052e+09 40798
## <none> 6.8119e+09 40799
## + children:smoker 1 2.0080e+06 6.8098e+09 40800
## + I(bmi^2) 1 4.6588e+05 6.8114e+09 40801
## + age:bmi 1 3.7662e+05 6.8115e+09 40801
## + age:children 1 1.6054e+05 6.8117e+09 40801
## + age:smoker 1 1.4298e+05 6.8117e+09 40801
## + bmi:smoker 1 4.9536e+04 6.8118e+09 40801
## - children 1 7.9656e+07 6.8915e+09 40829
## - bmi 1 1.1287e+09 7.9405e+09 41222
## - age 1 3.3258e+09 1.0138e+10 41899
## - smoker 1 2.3953e+10 3.0764e+10 44976
##
## Step: AIC=40770.4
## cost ~ smoker + age + bmi + children + region
##
## Df Sum of Sq RSS AIC
## + sex 1 1.1679e+07 6.7159e+09 40768
## + I(age^2) 1 1.0805e+07 6.7168e+09 40768
## + age:region 3 1.6454e+07 6.7112e+09 40770
## + bmi:children 1 6.4264e+06 6.7212e+09 40770
## <none> 6.7276e+09 40770
## + children:smoker 1 2.0552e+06 6.7256e+09 40772
## + I(bmi^2) 1 5.6093e+05 6.7271e+09 40772
## + age:bmi 1 2.5560e+05 6.7274e+09 40772
## + age:children 1 2.1876e+05 6.7274e+09 40772
## + bmi:smoker 1 1.8566e+05 6.7274e+09 40772
## + age:smoker 1 7.6977e+04 6.7275e+09 40772
## + bmi:region 3 4.4195e+06 6.7232e+09 40775
## + children:region 3 3.5557e+06 6.7241e+09 40775
## + smoker:region 3 3.3072e+06 6.7243e+09 40775
## - region 3 8.4230e+07 6.8119e+09 40799
## - children 1 8.1469e+07 6.8091e+09 40802
## - bmi 1 1.1332e+09 7.8609e+09 41200
## - age 1 3.2966e+09 1.0024e+10 41874
## - smoker 1 2.3801e+10 3.0528e+10 44961
##
## Step: AIC=40767.58
## cost ~ smoker + age + bmi + children + region + sex
##
## Df Sum of Sq RSS AIC
## + I(age^2) 1 1.0706e+07 6.7052e+09 40765
## + age:region 3 1.6458e+07 6.6995e+09 40767
## + sex:children 1 6.7029e+06 6.7092e+09 40767
## + bmi:children 1 6.3330e+06 6.7096e+09 40767
## <none> 6.7159e+09 40768
## + children:smoker 1 1.8045e+06 6.7141e+09 40769
## + age:sex 1 1.5275e+06 6.7144e+09 40769
## + sex:smoker 1 1.0692e+06 6.7149e+09 40769
## + sex:bmi 1 7.0289e+05 6.7152e+09 40769
## + I(bmi^2) 1 5.6934e+05 6.7154e+09 40769
## + sex:region 3 1.0218e+07 6.7057e+09 40769
## + bmi:smoker 1 4.0023e+05 6.7155e+09 40769
## + age:children 1 3.6237e+05 6.7156e+09 40769
## + age:bmi 1 1.7298e+05 6.7158e+09 40770
## + age:smoker 1 1.0260e+05 6.7158e+09 40770
## - sex 1 1.1679e+07 6.7276e+09 40770
## + bmi:region 3 5.0268e+06 6.7109e+09 40772
## + children:region 3 3.6221e+06 6.7123e+09 40772
## + smoker:region 3 3.2122e+06 6.7127e+09 40772
## - region 3 8.4285e+07 6.8002e+09 40796
## - children 1 8.2508e+07 6.7985e+09 40799
## - bmi 1 1.1410e+09 7.8569e+09 41201
## - age 1 3.2821e+09 9.9980e+09 41869
## - smoker 1 2.3731e+10 3.0447e+10 44956
##
## Step: AIC=40765.16
## cost ~ smoker + age + bmi + children + region + sex + I(age^2)
##
## Df Sum of Sq RSS AIC
## + age:region 3 1.6608e+07 6.6886e+09 40764
## + bmi:children 1 6.2959e+06 6.6989e+09 40765
## + sex:children 1 6.2510e+06 6.6990e+09 40765
## <none> 6.7052e+09 40765
## + age:sex 1 1.7963e+06 6.7034e+09 40766
## + children:smoker 1 1.7131e+06 6.7035e+09 40766
## + sex:smoker 1 9.8902e+05 6.7043e+09 40767
## + sex:bmi 1 5.9724e+05 6.7046e+09 40767
## + sex:region 3 1.0241e+07 6.6950e+09 40767
## + I(bmi^2) 1 5.1759e+05 6.7047e+09 40767
## + bmi:smoker 1 4.4404e+05 6.7048e+09 40767
## + age:children 1 3.1985e+05 6.7049e+09 40767
## + age:smoker 1 1.3726e+05 6.7051e+09 40767
## + age:bmi 1 2.8350e+03 6.7052e+09 40767
## - I(age^2) 1 1.0706e+07 6.7159e+09 40768
## - sex 1 1.1580e+07 6.7168e+09 40768
## + bmi:region 3 5.0880e+06 6.7002e+09 40769
## + children:region 3 3.2698e+06 6.7020e+09 40770
## + smoker:region 3 3.0503e+06 6.7022e+09 40770
## - children 1 5.9336e+07 6.7646e+09 40788
## - region 3 8.4288e+07 6.7895e+09 40794
## - age 1 1.3462e+08 6.8399e+09 40818
## - bmi 1 1.1457e+09 7.8510e+09 41200
## - smoker 1 2.3721e+10 3.0427e+10 44956
##
## Step: AIC=40764.28
## cost ~ smoker + age + bmi + children + region + sex + I(age^2) +
## age:region
##
## Df Sum of Sq RSS AIC
## + sex:children 1 5.4583e+06 6.6832e+09 40764
## + bmi:children 1 5.3570e+06 6.6833e+09 40764
## <none> 6.6886e+09 40764
## - age:region 3 1.6608e+07 6.7052e+09 40765
## + age:sex 1 1.9610e+06 6.6867e+09 40765
## + children:smoker 1 1.1945e+06 6.6874e+09 40766
## + sex:region 3 1.0622e+07 6.6780e+09 40766
## + age:bmi 1 8.2735e+05 6.6878e+09 40766
## + sex:smoker 1 7.5715e+05 6.6879e+09 40766
## + I(bmi^2) 1 7.0377e+05 6.6879e+09 40766
## + sex:bmi 1 6.0362e+05 6.6880e+09 40766
## + bmi:smoker 1 4.6210e+05 6.6882e+09 40766
## + age:children 1 4.2008e+05 6.6882e+09 40766
## + age:smoker 1 1.2118e+04 6.6886e+09 40766
## - I(age^2) 1 1.0856e+07 6.6995e+09 40767
## - sex 1 1.1583e+07 6.7002e+09 40767
## + bmi:region 3 5.2154e+06 6.6834e+09 40768
## + smoker:region 3 3.3348e+06 6.6853e+09 40769
## + children:region 3 3.1570e+06 6.6855e+09 40769
## - children 1 5.6050e+07 6.7447e+09 40785
## - bmi 1 1.1581e+09 7.8467e+09 41205
## - smoker 1 2.3583e+10 3.0271e+10 44947
##
## Step: AIC=40764.02
## cost ~ smoker + age + bmi + children + region + sex + I(age^2) +
## age:region + children:sex
##
## Df Sum of Sq RSS AIC
## + bmi:children 1 5.4665e+06 6.6777e+09 40764
## <none> 6.6832e+09 40764
## - children:sex 1 5.4583e+06 6.6886e+09 40764
## - age:region 3 1.5815e+07 6.6990e+09 40765
## + age:sex 1 2.2275e+06 6.6809e+09 40765
## + children:smoker 1 1.5996e+06 6.6816e+09 40765
## + sex:region 3 1.0835e+07 6.6723e+09 40766
## + sex:smoker 1 7.5865e+05 6.6824e+09 40766
## + age:bmi 1 7.1504e+05 6.6825e+09 40766
## + sex:bmi 1 6.1921e+05 6.6826e+09 40766
## + I(bmi^2) 1 5.7628e+05 6.6826e+09 40766
## + bmi:smoker 1 3.4694e+05 6.6828e+09 40766
## + age:children 1 2.3866e+05 6.6829e+09 40766
## + age:smoker 1 1.3170e+03 6.6832e+09 40766
## - I(age^2) 1 1.0415e+07 6.6936e+09 40766
## + bmi:region 3 5.4543e+06 6.6777e+09 40768
## + smoker:region 3 3.5037e+06 6.6797e+09 40769
## + children:region 3 3.0195e+06 6.6802e+09 40769
## - bmi 1 1.1579e+09 7.8411e+09 41205
## - smoker 1 2.3555e+10 3.0239e+10 44946
##
## Step: AIC=40763.75
## cost ~ smoker + age + bmi + children + region + sex + I(age^2) +
## age:region + children:sex + bmi:children
##
## Df Sum of Sq RSS AIC
## <none> 6.6777e+09 40764
## - age:region 3 1.4890e+07 6.6926e+09 40764
## - bmi:children 1 5.4665e+06 6.6832e+09 40764
## - children:sex 1 5.5677e+06 6.6833e+09 40764
## + age:sex 1 2.0346e+06 6.6757e+09 40765
## + children:smoker 1 1.4644e+06 6.6762e+09 40765
## + age:bmi 1 8.7550e+05 6.6768e+09 40765
## + sex:region 3 1.0433e+07 6.6673e+09 40765
## + sex:bmi 1 7.3596e+05 6.6770e+09 40765
## + age:children 1 7.2058e+05 6.6770e+09 40765
## + I(bmi^2) 1 6.9746e+05 6.6770e+09 40765
## + sex:smoker 1 6.0085e+05 6.6771e+09 40766
## + bmi:smoker 1 1.9568e+05 6.6775e+09 40766
## + age:smoker 1 4.8000e+01 6.6777e+09 40766
## - I(age^2) 1 1.0372e+07 6.6881e+09 40766
## + bmi:region 3 5.7138e+06 6.6720e+09 40767
## + children:region 3 3.4468e+06 6.6743e+09 40768
## + smoker:region 3 3.2405e+06 6.6745e+09 40768
## - smoker 1 2.3558e+10 3.0236e+10 44948
formula(modele.forward)
## cost ~ smoker + age + bmi + children + region + sex + I(age^2) +
## age:region + children:sex + bmi:children
formula(modele.backward)
## cost ~ age + sex + bmi + children + smoker + region + I(age^2) +
## age:region + sex:children + bmi:children
formula(modele.stepwise)
## cost ~ smoker + age + bmi + children + region + sex + I(age^2) +
## age:region + children:sex + bmi:children
extractAIC(modele.forward, k = 2)[2]
## [1] 40763.75
extractAIC(modele.backward, k = 2)[2]
## [1] 40763.75
extractAIC(modele.stepwise, k = 2)[2]
## [1] 40763.75
modele.retenu <- modele.stepwise
summary(modele.retenu)
##
## Call:
## lm(formula = cost ~ smoker + age + bmi + children + region +
## sex + I(age^2) + age:region + children:sex + bmi:children,
## data = assurances)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5841.3 -1057.0 47.2 1081.0 6638.7
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5562.9316 361.0265 15.409 < 2e-16 ***
## smokeryes 7303.7768 74.0585 98.622 < 2e-16 ***
## age 102.1166 14.9362 6.837 9.93e-12 ***
## bmi 104.8036 6.8140 15.381 < 2e-16 ***
## children -100.5789 128.7842 -0.781 0.43488
## regionnorthwest -411.6447 254.2379 -1.619 0.10553
## regionsoutheast -805.2898 245.8442 -3.276 0.00107 **
## regionsouthwest -491.6113 254.0440 -1.935 0.05307 .
## sexmale -210.7982 80.0466 -2.633 0.00850 **
## I(age^2) -0.3753 0.1814 -2.069 0.03860 *
## age:regionnorthwest 7.7586 6.1133 1.269 0.20451
## age:regionsoutheast 12.4697 5.8789 2.121 0.03400 *
## age:regionsouthwest 0.7899 6.0883 0.130 0.89678
## children:sexmale 74.0695 48.8534 1.516 0.12959
## bmi:children 6.0949 4.0570 1.502 0.13313
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1556 on 2757 degrees of freedom
## Multiple R-squared: 0.8123, Adjusted R-squared: 0.8114
## F-statistic: 852.5 on 14 and 2757 DF, p-value: < 2.2e-16
p <- length(coef(modele.retenu)) - 1
(d) Résidus et leviers
h <- hatvalues(modele.retenu)
range(h)
## [1] 0.002380277 0.039741677
sum(h)
## [1] 15
residus.bruts <- resid(modele.retenu)
residus.standardises <- rstandard(modele.retenu)
residus.studentises <- rstudent(modele.retenu)
summary(cbind(residus.standardises, residus.studentises))
## residus.standardises residus.studentises
## Min. :-3.76129 Min. :-3.770296
## 1st Qu.:-0.68154 1st Qu.:-0.681471
## Median : 0.03038 Median : 0.030372
## Mean :-0.00002 Mean :-0.000045
## 3rd Qu.: 0.69724 3rd Qu.: 0.697178
## Max. : 4.27915 Max. : 4.292651
plot(h, xlab = "Numero d'observation", ylab = "Leviers")
abline(h = 3*(p+1)/n, lty = 2, col = "red")
plot(residus.standardises, xlab = "Numero d'observation", ylab = "Residus standardises")
abline(h = 0, col = "red")
abline(h = 3, lty = 2, col = "green")
abline(h = -3, lty = 2, col = "green")
plot(residus.studentises, xlab = "Numero d'observation", ylab = "Residus studentises")
abline(h = 0, col = "red")
abline(h = 3, lty = 2, col = "green")
abline(h = -3, lty = 2, col = "green")
plot(h, residus.standardises, xlab = "Leviers", ylab = "Residus standardises")
abline(h = 0, col = "red")
abline(v = 3*(p+1)/n, lty = 2, col = "red")
(indices.h <- which(as.vector(h) > 3*(p+1)/n))
## [1] 167 439 641 1086 1553 1825 2027 2472
which(abs(as.vector(residus.standardises)) > 3)
## [1] 84 298 540 694 709 771 1520 1800 1979 2078 2095
which(abs(as.vector(residus.studentises)) > 3)
## [1] 84 298 540 694 709 771 1520 1800 1979 2078 2095
(e) Validation des hypothèses
Linéarité
scatter.smooth(residus.standardises ~ assurances$age,
xlab = "age", ylab = "Residus standardises")
scatter.smooth(residus.standardises ~ assurances$bmi,
xlab = "bmi", ylab = "Residus standardises")
scatter.smooth(residus.standardises ~ assurances$children,
xlab = "children", ylab = "Residus standardises")
Variance constante
plot(fitted(modele.retenu), residus.standardises,
xlab = "Valeurs predites", ylab = "Residus standardises")
abline(h = 0, col = "red")
abline(h = 1.96, lty = 2, col = "green")
abline(h = -1.96, lty = 2, col = "green")
Indépendance
plot(residus.standardises, xlab = "Numero d'observation",
ylab = "Residus standardises")
abline(h = 0, col = "red")
abline(h = 1.96, lty = 2, col = "green")
abline(h = -1.96, lty = 2, col = "green")
plot(residus.standardises[2:n], residus.standardises[1:(n-1)],
xlab = "Residus standardises precedents", ylab = "Residus standardises")
abline(h = 0, col = "red")
abline(v = 0, col = "red")
Normalité
hist(residus.standardises, freq = FALSE,
xlab = "Residus standardises", ylab = "Histogramme", main = "")
axe.x <- seq(-4, 4, 0.01)
axe.densite <- dnorm(seq(-4, 4, 0.01))
lines(axe.x, axe.densite, col = "green")
qqnorm(residus.standardises, xlab = "Quantiles theoriques",
ylab = "Quantiles empiriques", main = "")
qqline(residus.standardises)
(f) Observations influentes
d1 <- cooks.distance(modele.retenu)
max(d1)
## [1] 0.01331256
qf(0.5, p+1, n-(p+1))
## [1] 0.9561561
indices.d1 <- which(as.vector(d1) > qf(0.5, p+1, n-(p+1)))
d2 <- dffits(modele.retenu)
max(abs(d2))
## [1] 0.447407
3*sqrt(p/(n-p))
## [1] 0.2137411
indices.d2 <- which(as.vector(abs(d2)) > 3*sqrt(p/(n-p)))
d3 <- dfbetas(modele.retenu)
max(abs(d3))
## [1] 0.3397856
3/sqrt(n)
## [1] 0.05698029
d3.matrice <- abs(d3) > 3/sqrt(n)
indices.d3 <- which(as.vector(rowSums(d3.matrice)) > 0)
d4 <- covratio(modele.retenu)
max(abs(1-d4))
## [1] 0.08443621
3*(p+1)/n
## [1] 0.01623377
indices.d4 <- which(as.vector(abs(1-d4)) > 3*(p+1)/n)
indices.d1
## integer(0)
length(indices.d2)
## [1] 23
length(indices.d3)
## [1] 320
length(indices.d4)
## [1] 105
(indices <- intersect(intersect(indices.d2, indices.d3), indices.d4))
## [1] 26 84 298 540 694 771 781 846 1120 1520 1722 1800 1825 1874 1979
## [16] 2285
modele.sans.obs.influentes <- lm(formula(modele.retenu), data = assurances[-indices, ])
data.frame(coef.avec = coef(modele.retenu),
coef.sans = coef(modele.sans.obs.influentes))
## coef.avec coef.sans
## (Intercept) 5562.9316257 5695.0676346
## smokeryes 7303.7767506 7308.4275189
## age 102.1166419 95.3033875
## bmi 104.8035560 103.6809585
## children -100.5789247 -123.9831189
## regionnorthwest -411.6446849 -449.6914251
## regionsoutheast -805.2898205 -792.1578668
## regionsouthwest -491.6113102 -459.3255734
## sexmale -210.7982092 -204.4617067
## I(age^2) -0.3753328 -0.2881425
## age:regionnorthwest 7.7585626 8.6378602
## age:regionsoutheast 12.4696588 13.1624352
## age:regionsouthwest 0.7899005 0.7895369
## children:sexmale 74.0694828 81.7288060
## bmi:children 6.0949284 7.1702517
data.frame(s.e.avec = sqrt(diag(vcov(modele.retenu))),
s.e.sans = sqrt(diag(vcov(modele.sans.obs.influentes))))
## s.e.avec s.e.sans
## (Intercept) 361.0265001 353.0903691
## smokeryes 74.0584597 72.3837328
## age 14.9362467 14.6139921
## bmi 6.8139667 6.6800289
## children 128.7842498 127.7140269
## regionnorthwest 254.2379041 248.3759238
## regionsoutheast 245.8441809 240.2275719
## regionsouthwest 254.0439744 248.4840398
## sexmale 80.0466014 78.3077113
## I(age^2) 0.1813751 0.1775186
## age:regionnorthwest 6.1133351 5.9689553
## age:regionsoutheast 5.8789457 5.7485976
## age:regionsouthwest 6.0883294 5.9555948
## children:sexmale 48.8534162 47.9965851
## bmi:children 4.0570478 4.0467626
