#chapter9 선형 회귀
#1 단순 선형 회귀(Simple Linear Regression)
#1.1 모델 생성
data(cars)
head(cars)
## speed dist
## 1 4 2
## 2 4 10
## 3 7 4
## 4 7 22
## 5 8 16
## 6 9 10
m <- lm(dist~speed, cars)
m
##
## Call:
## lm(formula = dist ~ speed, data = cars)
##
## Coefficients:
## (Intercept) speed
## -17.579 3.932
#1.2 선형회귀 결과 추출
coef(m)
## (Intercept) speed
## -17.579095 3.932409
fitted(m)[1:4]
## 1 2 3 4
## -1.849460 -1.849460 9.947766 9.947766
residuals(m)[1:4]
## 1 2 3 4
## 3.849460 11.849460 -5.947766 12.052234
fitted(m)[1:4]+residuals(m)[1:4]
## 1 2 3 4
## 2 10 4 22
confint(m)
## 2.5 % 97.5 %
## (Intercept) -31.167850 -3.990340
## speed 3.096964 4.767853
deviance(m)
## [1] 11353.52
sum((cars$dist-predict(m,newdata=cars))^2)
## [1] 11353.52
#1.3 예측과 신뢰구간
m
##
## Call:
## lm(formula = dist ~ speed, data = cars)
##
## Coefficients:
## (Intercept) speed
## -17.579 3.932
predict(m, newdata=data.frame(speed=3))
## 1
## -5.781869
coef(m)
## (Intercept) speed
## -17.579095 3.932409
-17.579095+3.932409*3
## [1] -5.781868
predict(m,newdata=data.frame(speed=c(3)),interval="confidence")
## fit lwr upr
## 1 -5.781869 -17.02659 5.462853
predict(m, newdata=data.frame(speed=c(3)),interval="prediction")
## fit lwr upr
## 1 -5.781869 -38.68565 27.12192
#1.4 모형 평가
summary(m)
##
## Call:
## lm(formula = dist ~ speed, data = cars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -29.069 -9.525 -2.272 9.215 43.201
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -17.5791 6.7584 -2.601 0.0123 *
## speed 3.9324 0.4155 9.464 1.49e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15.38 on 48 degrees of freedom
## Multiple R-squared: 0.6511, Adjusted R-squared: 0.6438
## F-statistic: 89.57 on 1 and 48 DF, p-value: 1.49e-12
#1.5 ANOVA 및 모델간의 비교
anova(m)
## Analysis of Variance Table
##
## Response: dist
## Df Sum Sq Mean Sq F value Pr(>F)
## speed 1 21186 21185.5 89.567 1.49e-12 ***
## Residuals 48 11354 236.5
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
full <- lm(dist~speed, data=cars)
reduced <- lm(dist~1, data=cars)
full
##
## Call:
## lm(formula = dist ~ speed, data = cars)
##
## Coefficients:
## (Intercept) speed
## -17.579 3.932
reduced
##
## Call:
## lm(formula = dist ~ 1, data = cars)
##
## Coefficients:
## (Intercept)
## 42.98
anova(reduced, full)
## Analysis of Variance Table
##
## Model 1: dist ~ 1
## Model 2: dist ~ speed
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 49 32539
## 2 48 11354 1 21186 89.567 1.49e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#1.6 모델 평가 차트
plot(m)
par(mfrow=c(1,2))
plot(m, which=c(4,6))
#1.7 회귀 직선의 시각화
plot(cars$speed, cars$dist)
abline(coef(m))
summary(cars$speed)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 4.0 12.0 15.0 15.4 19.0 25.0
predict(m, newdata=data.frame(speed=seq(4.0,25.0,.2)),
interval="confidence")
## fit lwr upr
## 1 -1.8494599 -12.3295433 8.630624
## 2 -1.0629781 -11.3914503 9.265494
## 3 -0.2764964 -10.4538419 9.900849
## 4 0.5099854 -9.5167401 10.536711
## 5 1.2964672 -8.5801681 11.173102
## 6 2.0829489 -7.6441504 11.810048
## 7 2.8694307 -6.7087129 12.447574
## 8 3.6559124 -5.7738833 13.085708
## 9 4.4423942 -4.8396905 13.724479
## 10 5.2288759 -3.9061655 14.363917
## 11 6.0153577 -2.9733411 15.004056
## 12 6.8018394 -2.0412521 15.644931
## 13 7.5883212 -1.1099353 16.286578
## 14 8.3748029 -0.1794301 16.929036
## 15 9.1612847 0.7502220 17.572347
## 16 9.9477664 1.6789765 18.216556
## 17 10.7342482 2.6067864 18.861710
## 18 11.5207299 3.5336015 19.507858
## 19 12.3072117 4.4593685 20.155055
## 20 13.0936934 5.3840304 20.803356
## 21 13.8801752 6.3075269 21.452823
## 22 14.6666569 7.2297936 22.103520
## 23 15.4531387 8.1507617 22.755516
## 24 16.2396204 9.0703583 23.408883
## 25 17.0261022 9.9885055 24.063699
## 26 17.8125839 10.9051205 24.720047
## 27 18.5990657 11.8201149 25.378016
## 28 19.3855474 12.7333949 26.037700
## 29 20.1720292 13.6448606 26.699198
## 30 20.9585109 14.5544057 27.362616
## 31 21.7449927 15.4619173 28.028068
## 32 22.5314745 16.3672758 28.695673
## 33 23.3179562 17.2703541 29.365558
## 34 24.1044380 18.1710177 30.037858
## 35 24.8909197 19.0691247 30.712715
## 36 25.6774015 19.9645252 31.390278
## 37 26.4638832 20.8570613 32.070705
## 38 27.2503650 21.7465676 32.754162
## 39 28.0368467 22.6328708 33.440823
## 40 28.8233285 23.5157900 34.130867
## 41 29.6098102 24.3951377 34.824483
## 42 30.3962920 25.2707195 35.521864
## 43 31.1827737 26.1423359 36.223212
## 44 31.9692555 27.0097826 36.928728
## 45 32.7557372 27.8728522 37.638622
## 46 33.5422190 28.7313356 38.353102
## 47 34.3287007 29.5850240 39.072377
## 48 35.1151825 30.4337109 39.796654
## 49 35.9016642 31.2771944 40.526134
## 50 36.6881460 32.1152800 41.261012
## 51 37.4746277 32.9477831 42.001472
## 52 38.2611095 33.7745318 42.747687
## 53 39.0475912 34.5953700 43.499812
## 54 39.8340730 35.4101600 44.257986
## 55 40.6205547 36.2187852 45.022324
## 56 41.4070365 37.0211522 45.792921
## 57 42.1935182 37.8171927 46.569844
## 58 42.9800000 38.6068653 47.353135
## 59 43.7664818 39.3901562 48.142807
## 60 44.5529635 40.1670792 48.938848
## 61 45.3394453 40.9376757 49.741215
## 62 46.1259270 41.7020141 50.549840
## 63 46.9124088 42.4601875 51.364630
## 64 47.6988905 43.2123128 52.185468
## 65 48.4853723 43.9585276 53.012217
## 66 49.2718540 44.6989880 53.844720
## 67 50.0583358 45.4338660 54.682806
## 68 50.8448175 46.1633459 55.526289
## 69 51.6312993 46.8876225 56.374976
## 70 52.4177810 47.6068976 57.228664
## 71 53.2042628 48.3213777 58.087148
## 72 53.9907445 49.0312717 58.950217
## 73 54.7772263 49.7367885 59.817664
## 74 55.5637080 50.4381356 60.689280
## 75 56.3501898 51.1355172 61.564862
## 76 57.1366715 51.8291331 62.444210
## 77 57.9231533 52.5191774 63.327129
## 78 58.7096350 53.2058377 64.213432
## 79 59.4961168 53.8892949 65.102939
## 80 60.2825985 54.5697223 65.995475
## 81 61.0690803 55.2472853 66.890875
## 82 61.8555620 55.9221418 67.788982
## 83 62.6420438 56.5944417 68.689646
## 84 63.4285255 57.2643269 69.592724
## 85 64.2150073 57.9319319 70.498083
## 86 65.0014891 58.5973838 71.405594
## 87 65.7879708 59.2608022 72.315139
## 88 66.5744526 59.9223000 73.226605
## 89 67.3609343 60.5819835 74.139885
## 90 68.1474161 61.2399526 75.054880
## 91 68.9338978 61.8963012 75.971494
## 92 69.7203796 62.5511175 76.889642
## 93 70.5068613 63.2044844 77.809238
## 94 71.2933431 63.8564797 78.730206
## 95 72.0798248 64.5071766 79.652473
## 96 72.8663066 65.1566436 80.575970
## 97 73.6527883 65.8049451 81.500632
## 98 74.4392701 66.4521417 82.426398
## 99 75.2257518 67.0982901 83.353214
## 100 76.0122336 67.7434437 84.281023
## 101 76.7987153 68.3876526 85.209778
## 102 77.5851971 69.0309641 86.139430
## 103 78.3716788 69.6734223 87.069935
## 104 79.1581606 70.3150691 88.001252
## 105 79.9446423 70.9559435 88.933341
## 106 80.7311241 71.5960827 89.866166
speed <- seq(min(cars$speed),max(cars$speed), .1)
ys <- predict(m, newdata=data.frame(speed=speed), interval="confidence")
matplot(speed, ys, type='n')
matlines(speed,ys)
#2
#2.1 모델 생성 및 평가
m <- lm(Sepal.Length ~Sepal.Width + Petal.Length + Petal.Width, data=iris)
m
##
## Call:
## lm(formula = Sepal.Length ~ Sepal.Width + Petal.Length + Petal.Width,
## data = iris)
##
## Coefficients:
## (Intercept) Sepal.Width Petal.Length Petal.Width
## 1.8560 0.6508 0.7091 -0.5565
summary(m)
##
## Call:
## lm(formula = Sepal.Length ~ Sepal.Width + Petal.Length + Petal.Width,
## data = iris)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.82816 -0.21989 0.01875 0.19709 0.84570
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.85600 0.25078 7.401 9.85e-12 ***
## Sepal.Width 0.65084 0.06665 9.765 < 2e-16 ***
## Petal.Length 0.70913 0.05672 12.502 < 2e-16 ***
## Petal.Width -0.55648 0.12755 -4.363 2.41e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3145 on 146 degrees of freedom
## Multiple R-squared: 0.8586, Adjusted R-squared: 0.8557
## F-statistic: 295.5 on 3 and 146 DF, p-value: < 2.2e-16
#2.3 중선형회귀모형의 시각화
with(iris, plot(Sepal.Width, Sepal.Length, cex=.7, pch=as.numeric(Species)))
as.numeric(iris$Species)
## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [36] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## [71] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3
## [106] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## [141] 3 3 3 3 3 3 3 3 3 3
legend("topright", levels(iris$Species), pch=1:3, bg="white")
levels(iris$Species)
## [1] "setosa" "versicolor" "virginica"
m <- lm(Sepal.Length~Sepal.Width+Species, data=iris)
coef(m)
## (Intercept) Sepal.Width Speciesversicolor Speciesvirginica
## 2.2513932 0.8035609 1.4587431 1.9468166
abline(2.25, 0.80, lty=1)
abline(2.25 + 1.45, 0.80, lty=2)
abline(2.25 + 1.94, 0.80, lty=3)
with(iris, plot(Sepal.Width, Sepal.Length, cex=.7, pch=as.numeric(Species)))
m <- lm(Sepal.Length~Sepal.Width + Species, data=iris)
coef(m)
## (Intercept) Sepal.Width Speciesversicolor Speciesvirginica
## 2.2513932 0.8035609 1.4587431 1.9468166
abline(2.25, 0.80, lty=1)
abline(2.25 + 1.45, 0.80, lty=2)
abline(2.25 + 1.94, 0.80, lty=3)
legend("topright",levels(iris$Species),pch=1:3,bg="white")
#2.4 표현식을 위한 I()의 사용
x <- 1:1000
y <- x^2 + 3*x + 5 + rnorm(1000)
lm(y~I(x*2)+x)
##
## Call:
## lm(formula = y ~ I(x * 2) + x)
##
## Coefficients:
## (Intercept) I(x * 2) x
## -167162 502 NA
lm(y~x^2)
##
## Call:
## lm(formula = y ~ x^2)
##
## Coefficients:
## (Intercept) x
## -167162 1004
x1 <- 1:1000
x2 <- 3 * x1
y <- 3* (x1+x2) + rnorm(1000)
lm(y~I(x1 + x2))
##
## Call:
## lm(formula = y ~ I(x1 + x2))
##
## Coefficients:
## (Intercept) I(x1 + x2)
## 0.04739 3.00000
lm(y~x1+x2)
##
## Call:
## lm(formula = y ~ x1 + x2)
##
## Coefficients:
## (Intercept) x1 x2
## 0.04739 12.00000 NA
#2.5 변수의 변환
x <- 101:200
y <- exp(3*x +rnorm(100))
lm(log(y)~x)
##
## Call:
## lm(formula = log(y) ~ x)
##
## Coefficients:
## (Intercept) x
## -0.5382 3.0037
x <- 101:200
y <- log(x) + rnorm(100)
lm(y~log(x))
##
## Call:
## lm(formula = y ~ log(x))
##
## Coefficients:
## (Intercept) log(x)
## 0.5847 0.8880
#2.6 상호 작용
data(Orange)
Orange
## Tree age circumference
## 1 1 118 30
## 2 1 484 58
## 3 1 664 87
## 4 1 1004 115
## 5 1 1231 120
## 6 1 1372 142
## 7 1 1582 145
## 8 2 118 33
## 9 2 484 69
## 10 2 664 111
## 11 2 1004 156
## 12 2 1231 172
## 13 2 1372 203
## 14 2 1582 203
## 15 3 118 30
## 16 3 484 51
## 17 3 664 75
## 18 3 1004 108
## 19 3 1231 115
## 20 3 1372 139
## 21 3 1582 140
## 22 4 118 32
## 23 4 484 62
## 24 4 664 112
## 25 4 1004 167
## 26 4 1231 179
## 27 4 1372 209
## 28 4 1582 214
## 29 5 118 30
## 30 5 484 49
## 31 5 664 81
## 32 5 1004 125
## 33 5 1231 142
## 34 5 1372 174
## 35 5 1582 177
with(Orange, plot(Tree, circumference, xlab="tree", ylab="circumference"))
with(Orange, interaction.plot(age, Tree, circumference))
Orange[,"fTree"] <- factor(Orange[,"Tree"],ordered=FALSE)
m <- lm(circumference ~ fTree * age, data=Orange)
anova(m)
## Analysis of Variance Table
##
## Response: circumference
## Df Sum Sq Mean Sq F value Pr(>F)
## fTree 4 11841 2960 27.2983 8.428e-09 ***
## age 1 93772 93772 864.7348 < 2.2e-16 ***
## fTree:age 4 4043 1011 9.3206 9.402e-05 ***
## Residuals 25 2711 108
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
head(model.matrix(m))
## (Intercept) fTree1 fTree5 fTree2 fTree4 age fTree1:age fTree5:age
## 1 1 1 0 0 0 118 118 0
## 2 1 1 0 0 0 484 484 0
## 3 1 1 0 0 0 664 664 0
## 4 1 1 0 0 0 1004 1004 0
## 5 1 1 0 0 0 1231 1231 0
## 6 1 1 0 0 0 1372 1372 0
## fTree2:age fTree4:age
## 1 0 0
## 2 0 0
## 3 0 0
## 4 0 0
## 5 0 0
## 6 0 0
#3 이상치
data(Orange)
m <- lm(circumference~age + I(age), data=Orange)
rstudent(m)
## 1 2 3 4 5
## 6.372161e-05 -4.735583e-01 -5.474343e-02 -4.051405e-01 -1.250387e+00
## 6 7 8 9 10
## -9.461338e-01 -1.884773e+00 1.318100e-01 -3.258711e-03 9.737615e-01
## 11 12 13 14 15
## 1.359966e+00 9.960118e-01 1.744963e+00 7.283842e-01 6.372161e-05
## 16 17 18 19 20
## -7.773767e-01 -5.647282e-01 -7.042510e-01 -1.480911e+00 -1.080244e+00
## 21 22 23 24 25
## -2.143664e+00 8.788132e-02 -3.019190e-01 1.018008e+00 1.881656e+00
## 26 27 28 29 30
## 1.311188e+00 2.045052e+00 1.226404e+00 6.372161e-05 -8.652966e-01
## 31 32 33 34 35
## -3.087970e-01 1.696690e-02 -2.897485e-01 4.323381e-01 -4.040268e-01
data(Orange)
Orange <- rbind(Orange, data.frame(Tree=as.factor(c(6,6,6)),
age=c(118,484,664),
circumference=c(177,50,30)))
tail(Orange)
## Tree age circumference
## 33 5 1231 142
## 34 5 1372 174
## 35 5 1582 177
## 36 6 118 177
## 37 6 484 50
## 38 6 664 30
#4 변수 선택
#4.1 단계적 변수 선택
library(mlbench)
## Warning: package 'mlbench' was built under R version 3.3.3
data(BostonHousing)
m <- lm(medv~., data=BostonHousing)
m2 <- step(m, direction="both")
## Start: AIC=1589.64
## medv ~ crim + zn + indus + chas + nox + rm + age + dis + rad +
## tax + ptratio + b + lstat
##
## Df Sum of Sq RSS AIC
## - age 1 0.06 11079 1587.7
## - indus 1 2.52 11081 1587.8
## <none> 11079 1589.6
## - chas 1 218.97 11298 1597.5
## - tax 1 242.26 11321 1598.6
## - crim 1 243.22 11322 1598.6
## - zn 1 257.49 11336 1599.3
## - b 1 270.63 11349 1599.8
## - rad 1 479.15 11558 1609.1
## - nox 1 487.16 11566 1609.4
## - ptratio 1 1194.23 12273 1639.4
## - dis 1 1232.41 12311 1641.0
## - rm 1 1871.32 12950 1666.6
## - lstat 1 2410.84 13490 1687.3
##
## Step: AIC=1587.65
## medv ~ crim + zn + indus + chas + nox + rm + dis + rad + tax +
## ptratio + b + lstat
##
## Df Sum of Sq RSS AIC
## - indus 1 2.52 11081 1585.8
## <none> 11079 1587.7
## + age 1 0.06 11079 1589.6
## - chas 1 219.91 11299 1595.6
## - tax 1 242.24 11321 1596.6
## - crim 1 243.20 11322 1596.6
## - zn 1 260.32 11339 1597.4
## - b 1 272.26 11351 1597.9
## - rad 1 481.09 11560 1607.2
## - nox 1 520.87 11600 1608.9
## - ptratio 1 1200.23 12279 1637.7
## - dis 1 1352.26 12431 1643.9
## - rm 1 1959.55 13038 1668.0
## - lstat 1 2718.88 13798 1696.7
##
## Step: AIC=1585.76
## medv ~ crim + zn + chas + nox + rm + dis + rad + tax + ptratio +
## b + lstat
##
## Df Sum of Sq RSS AIC
## <none> 11081 1585.8
## + indus 1 2.52 11079 1587.7
## + age 1 0.06 11081 1587.8
## - chas 1 227.21 11309 1594.0
## - crim 1 245.37 11327 1594.8
## - zn 1 257.82 11339 1595.4
## - b 1 270.82 11352 1596.0
## - tax 1 273.62 11355 1596.1
## - rad 1 500.92 11582 1606.1
## - nox 1 541.91 11623 1607.9
## - ptratio 1 1206.45 12288 1636.0
## - dis 1 1448.94 12530 1645.9
## - rm 1 1963.66 13045 1666.3
## - lstat 1 2723.48 13805 1695.0
formula(m2)
## medv ~ crim + zn + chas + nox + rm + dis + rad + tax + ptratio +
## b + lstat
step(m,direction="both")
## Start: AIC=1589.64
## medv ~ crim + zn + indus + chas + nox + rm + age + dis + rad +
## tax + ptratio + b + lstat
##
## Df Sum of Sq RSS AIC
## - age 1 0.06 11079 1587.7
## - indus 1 2.52 11081 1587.8
## <none> 11079 1589.6
## - chas 1 218.97 11298 1597.5
## - tax 1 242.26 11321 1598.6
## - crim 1 243.22 11322 1598.6
## - zn 1 257.49 11336 1599.3
## - b 1 270.63 11349 1599.8
## - rad 1 479.15 11558 1609.1
## - nox 1 487.16 11566 1609.4
## - ptratio 1 1194.23 12273 1639.4
## - dis 1 1232.41 12311 1641.0
## - rm 1 1871.32 12950 1666.6
## - lstat 1 2410.84 13490 1687.3
##
## Step: AIC=1587.65
## medv ~ crim + zn + indus + chas + nox + rm + dis + rad + tax +
## ptratio + b + lstat
##
## Df Sum of Sq RSS AIC
## - indus 1 2.52 11081 1585.8
## <none> 11079 1587.7
## + age 1 0.06 11079 1589.6
## - chas 1 219.91 11299 1595.6
## - tax 1 242.24 11321 1596.6
## - crim 1 243.20 11322 1596.6
## - zn 1 260.32 11339 1597.4
## - b 1 272.26 11351 1597.9
## - rad 1 481.09 11560 1607.2
## - nox 1 520.87 11600 1608.9
## - ptratio 1 1200.23 12279 1637.7
## - dis 1 1352.26 12431 1643.9
## - rm 1 1959.55 13038 1668.0
## - lstat 1 2718.88 13798 1696.7
##
## Step: AIC=1585.76
## medv ~ crim + zn + chas + nox + rm + dis + rad + tax + ptratio +
## b + lstat
##
## Df Sum of Sq RSS AIC
## <none> 11081 1585.8
## + indus 1 2.52 11079 1587.7
## + age 1 0.06 11081 1587.8
## - chas 1 227.21 11309 1594.0
## - crim 1 245.37 11327 1594.8
## - zn 1 257.82 11339 1595.4
## - b 1 270.82 11352 1596.0
## - tax 1 273.62 11355 1596.1
## - rad 1 500.92 11582 1606.1
## - nox 1 541.91 11623 1607.9
## - ptratio 1 1206.45 12288 1636.0
## - dis 1 1448.94 12530 1645.9
## - rm 1 1963.66 13045 1666.3
## - lstat 1 2723.48 13805 1695.0
##
## Call:
## lm(formula = medv ~ crim + zn + chas + nox + rm + dis + rad +
## tax + ptratio + b + lstat, data = BostonHousing)
##
## Coefficients:
## (Intercept) crim zn chas1 nox
## 36.341145 -0.108413 0.045845 2.718716 -17.376023
## rm dis rad tax ptratio
## 3.801579 -1.492711 0.299608 -0.011778 -0.946525
## b lstat
## 0.009291 -0.522553
m2 <- step(m, direction="both")
## Start: AIC=1589.64
## medv ~ crim + zn + indus + chas + nox + rm + age + dis + rad +
## tax + ptratio + b + lstat
##
## Df Sum of Sq RSS AIC
## - age 1 0.06 11079 1587.7
## - indus 1 2.52 11081 1587.8
## <none> 11079 1589.6
## - chas 1 218.97 11298 1597.5
## - tax 1 242.26 11321 1598.6
## - crim 1 243.22 11322 1598.6
## - zn 1 257.49 11336 1599.3
## - b 1 270.63 11349 1599.8
## - rad 1 479.15 11558 1609.1
## - nox 1 487.16 11566 1609.4
## - ptratio 1 1194.23 12273 1639.4
## - dis 1 1232.41 12311 1641.0
## - rm 1 1871.32 12950 1666.6
## - lstat 1 2410.84 13490 1687.3
##
## Step: AIC=1587.65
## medv ~ crim + zn + indus + chas + nox + rm + dis + rad + tax +
## ptratio + b + lstat
##
## Df Sum of Sq RSS AIC
## - indus 1 2.52 11081 1585.8
## <none> 11079 1587.7
## + age 1 0.06 11079 1589.6
## - chas 1 219.91 11299 1595.6
## - tax 1 242.24 11321 1596.6
## - crim 1 243.20 11322 1596.6
## - zn 1 260.32 11339 1597.4
## - b 1 272.26 11351 1597.9
## - rad 1 481.09 11560 1607.2
## - nox 1 520.87 11600 1608.9
## - ptratio 1 1200.23 12279 1637.7
## - dis 1 1352.26 12431 1643.9
## - rm 1 1959.55 13038 1668.0
## - lstat 1 2718.88 13798 1696.7
##
## Step: AIC=1585.76
## medv ~ crim + zn + chas + nox + rm + dis + rad + tax + ptratio +
## b + lstat
##
## Df Sum of Sq RSS AIC
## <none> 11081 1585.8
## + indus 1 2.52 11079 1587.7
## + age 1 0.06 11081 1587.8
## - chas 1 227.21 11309 1594.0
## - crim 1 245.37 11327 1594.8
## - zn 1 257.82 11339 1595.4
## - b 1 270.82 11352 1596.0
## - tax 1 273.62 11355 1596.1
## - rad 1 500.92 11582 1606.1
## - nox 1 541.91 11623 1607.9
## - ptratio 1 1206.45 12288 1636.0
## - dis 1 1448.94 12530 1645.9
## - rm 1 1963.66 13045 1666.3
## - lstat 1 2723.48 13805 1695.0
formula(m2)
## medv ~ crim + zn + chas + nox + rm + dis + rad + tax + ptratio +
## b + lstat