机器学习——线性回归（一元、多元、高阶、交互项）R语言

线性回归

一元线性回归

fix(women)

A data.frame: 6 × 2

	height	weight
1	58	115
2	59	117
3	60	120
4	61	123
5	62	126
6	63	129

?women

# 查看数据的列明
names(women)

1. 'height'
2. 'weight'

``` # 查看数据类型 class(women) ``` 'data.frame'

# 线性拟合
fit.lm<-lm(weight~height,data=women)
fit.lm

Call:
lm(formula = weight ~ height, data = women)
Coefficients:
(Intercept) height
-87.52 3.45

summary(fit.lm)
# Residuals—残差统计量、intercept-表示截距、Estimate-包含由普通最小二乘法计算出来的估计回归系数、Std.error-估计的回归系数的标准误差、
# Multiple R-squared-拟合优度越大越好、F-statistic-判断方程的显著性检验

Call:
lm(formula = weight ~ height, data = women)

Residuals:
Min      1Q  Median      3Q     Max
-1.7333 -1.1333 -0.3833  0.7417  3.1167
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -87.51667    5.93694  -14.74 1.71e-09 ***
height        3.45000    0.09114   37.85 1.09e-14 ***
-----------------------------------------------------

Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.525 on 13 degrees of freedom
Multiple R-squared:  0.991,	Adjusted R-squared:  0.9903
F-statistic:  1433 on 1 and 13 DF,  p-value: 1.091e-14

names(fit.lm)

1. 'coefficients'
2. 'residuals'
3. 'effects'
4. 'rank'
5. 'fitted.values'
6. 'assign'
7. 'qr'
8. 'df.residual'
9. 'xlevels'
10. 'call'
11. 'terms'
12. 'model'

#回归诊断图的理解
#残差与拟合图，本质上残差服从正态分布与估计值无关的假设，与估计值无关，残差应该在y=0上下随机波动
#QQ图用来检测残差是否服从正态分布
#方差相同，红线应该是水平波动不可以存在上下波动
#检查是否存在特别极端的点cook的内部即可
#使用par()函数在同一窗口中创建多个图，mfrow： 决定了网格的行值和列
par(mfrow=c(2,2))
plot(fit.lm)

#绘制散点图
plot(women$height,women$weight)
#添加拟合直线abline(a,b,h,v)a,b指定线的截距和斜率、h为水平线指定y、v为垂直线指定x
abline(fit.lm)

# 预测
#构造要预测数据，newdata的类型必须是dataframe结构，
#而且必须是与原来的名称相同
newdata <- data.frame(height= seq(50, 60, 0.5))
#predict.lm函数进行预测：predict(object,newdata，interval)object代表的是模型对象、interval代表的是置信区间的类型
#confidence是对均值做区间估计、prediction是对随机变量做区间预测
pred=predict(fit.lm, newdata, interval = "prediction")
pred

A matrix: 21 × 3 of type dbl

	fit	lwr	upr
1	84.98333	80.47778	89.48888
2	86.70833	82.26669	91.14998
3	88.43333	84.05432	92.81235
4	90.15833	85.84061	94.47606
5	91.88333	87.62550	96.14116
6	93.60833	89.40894	97.80772
7	95.33333	91.19087	99.47580
8	97.05833	92.97122	101.14545
9	98.78333	94.74992	102.81674
10	100.50833	96.52692	104.48975
11	102.23333	98.30213	106.16453
12	103.95833	100.07550	107.84116
13	105.68333	101.84696	109.51971
14	107.40833	103.61642	111.20025
15	109.13333	105.38383	112.88284
16	110.85833	107.14911	114.56756
17	112.58333	108.91219	116.25448
18	114.30833	110.67300	117.94367
19	116.03333	112.43148	119.63519
20	117.75833	114.18755	121.32911
21	119.48333	115.94117	123.02550

多项式回归

#对于多项式拟合存在两种方式
fit2.lm <- lm(weight~height+I(height^2),data=women)
fit3.lm <- lm(weight~poly(height,2,raw=TRUE),data=women)
summary(fit2.lm)
summary(fit3.lm)
#对于训练集的数据进行拟合预测用fitted、对于新的样本用predict
fitted(fit2.lm)
fitted.values(fit2.lm)

Call:
lm(formula = weight ~ height + I(height^2), data = women)

Residuals:
Min       1Q   Median       3Q      Max
-0.50941 -0.29611 -0.00941  0.28615  0.59706

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 261.87818   25.19677  10.393 2.36e-07 ***
height       -7.34832    0.77769  -9.449 6.58e-07 ***
I(height^2)   0.08306    0.00598  13.891 9.32e-09 ***
-----------------------------------------------------

Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.3841 on 12 degrees of freedom
Multiple R-squared:  0.9995,	Adjusted R-squared:  0.9994
F-statistic: 1.139e+04 on 2 and 12 DF,  p-value: < 2.2e-16

Call:
lm(formula = weight ~ poly(height, 2, raw = TRUE), data = women)

Residuals:
Min       1Q   Median       3Q      Max
-0.50941 -0.29611 -0.00941  0.28615  0.59706

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)                  261.87818   25.19677  10.393 2.36e-07 ***
poly(height, 2, raw = TRUE)1  -7.34832    0.77769  -9.449 6.58e-07 ***
poly(height, 2, raw = TRUE)2   0.08306    0.00598  13.891 9.32e-09 ***
----------------------------------------------------------------------

Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.3841 on 12 degrees of freedom
Multiple R-squared:  0.9995,	Adjusted R-squared:  0.9994
F-statistic: 1.139e+04 on 2 and 12 DF,  p-value: < 2.2e-16

1

115.102941176471

2

117.473109243697

3

120.009405300582

4

122.711829347123

5

125.580381383323

6

128.615061409179

7

131.815869424693

8

135.182805429864

9

138.715869424693

10

142.415061409179

11

146.280381383323

12

150.311829347124

13

154.509405300582

14

158.873109243697

15

163.402941176471

1

115.102941176471

2

117.473109243697

3

120.009405300582

4

122.711829347123

5

125.580381383323

6

128.615061409179

7

131.815869424693

8

135.182805429864

9

138.715869424693

10

142.415061409179

11

146.280381383323

12

150.311829347124

13

154.509405300582

14

158.873109243697

15

163.402941176471

coef(fit2.lm)#查看回归系数

(Intercept)

261.878183581103

height

-7.34831932773043

I(height^2)

0.0830639948286958

plot(women$height,women$weight)
#lines与abline的区别是前者做的是一般连线图，其输入的是点向量
#后者输入的是回归模型对象，之可以添加直线绘图，可以达到相同的目的
lines(women$height,fitted(fit2.lm))

#plot(women$height,women$weight)
#fit4.lm <- lm(height~weight+I(weight^2),data=women)
#abline(fit2.lm)

par(mfrow=c(2,2))
plot(fit2.lm)  #残差

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# 尝试通过添加高阶项，比如（x＾3）来提高回归的预测精度。
fit22.lm<-lm(weight~poly(height,3,raw=T),data=women)  #多项式的方法，包含所有指定高阶项以下的项,而不仅仅只是给出指定的阶数
summary(fit22.lm)

par(mfrow=c(2,2))
plot(fit22.lm)  #残差

plot(women$height,women$weight)
lines(women$height,fitted(fit22.lm))

Call:
lm(formula = weight ~ poly(height, 3, raw = T), data = women)

Residuals:
Min       1Q   Median       3Q      Max
-0.40677 -0.17391  0.03091  0.12051  0.42191

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)               -8.967e+02  2.946e+02  -3.044  0.01116 *
poly(height, 3, raw = T)1  4.641e+01  1.366e+01   3.399  0.00594 **
poly(height, 3, raw = T)2 -7.462e-01  2.105e-01  -3.544  0.00460 **
poly(height, 3, raw = T)3  4.253e-03  1.079e-03   3.940  0.00231 **
-------------------------------------------------------------------

Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.2583 on 11 degrees of freedom
Multiple R-squared:  0.9998,	Adjusted R-squared:  0.9997
F-statistic: 1.679e+04 on 3 and 11 DF,  p-value: < 2.2e-16

](https://upload-images.jianshu.io/upload_images/28957475-d875e23e671762ed.png?imageMogr2/auto-orient/strip%7CimageView2/2/w/1240))

多元线性回归：高阶的、交互项的

library(MASS)
#data()#因为前面有加载报的操作，所以后边data()中什么命令也没有，本质上是data(,package="MASS")
#看Boston数据集
data(Boston)
head(Boston)
tail(Boston,3)
# fix(Boston)#在R中修改数据
names(Boston)  #住宅结构与居住环境信息
?Boston

A data.frame: 6 × 14

	crim	zn	indus	chas	nox	rm	age	dis	rad	tax	ptratio	black	lstat	medv
1	0.00632	18	2.31	0	0.538	6.575	65.2	4.0900	1	296	15.3	396.90	4.98	24.0
2	0.02731	0	7.07	0	0.469	6.421	78.9	4.9671	2	242	17.8	396.90	9.14	21.6
3	0.02729	0	7.07	0	0.469	7.185	61.1	4.9671	2	242	17.8	392.83	4.03	34.7
4	0.03237	0	2.18	0	0.458	6.998	45.8	6.0622	3	222	18.7	394.63	2.94	33.4
5	0.06905	0	2.18	0	0.458	7.147	54.2	6.0622	3	222	18.7	396.90	5.33	36.2
6	0.02985	0	2.18	0	0.458	6.430	58.7	6.0622	3	222	18.7	394.12	5.21	28.7

A data.frame: 3 × 14

	crim	zn	indus	chas	nox	rm	age	dis	rad	tax	ptratio	black	lstat	medv
504	0.06076	0	11.93	0	0.573	6.976	91.0	2.1675	1	273	21	396.90	5.64	23.9
505	0.10959	0	11.93	0	0.573	6.794	89.3	2.3889	1	273	21	393.45	6.48	22.0
506	0.04741	0	11.93	0	0.573	6.030	80.8	2.5050	1	273	21	396.90	7.88	11.9

1. 'crim'
2. 'zn'
3. 'indus'
4. 'chas'
5. 'nox'
6. 'rm'
7. 'age'
8. 'dis'
9. 'rad'
10. 'tax'
11. 'ptratio'
12. 'black'
13. 'lstat'
14. 'medv'

#拟合多元线性回归 
lm.fit=lm(medv~lstat+rm,data=Boston)
lm.fit
summary(lm.fit)
names(lm.fit)

Call:
lm(formula = medv ~ lstat + rm, data = Boston)

Coefficients:
(Intercept)        lstat           rm
-1.3583      -0.6424       5.0948

Call:
lm(formula = medv ~ lstat + rm, data = Boston)

Residuals:
Min      1Q  Median      3Q     Max
-18.076  -3.516  -1.010   1.909  28.131

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.35827    3.17283  -0.428    0.669
lstat       -0.64236    0.04373 -14.689   <2e-16 ***
rm           5.09479    0.44447  11.463   <2e-16 ***
----------------------------------------------------

Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 5.54 on 503 degrees of freedom
Multiple R-squared:  0.6386,	Adjusted R-squared:  0.6371
F-statistic: 444.3 on 2 and 503 DF,  p-value: < 2.2e-16

1. 'coefficients'
2. 'residuals'
3. 'effects'
4. 'rank'
5. 'fitted.values'
6. 'assign'
7. 'qr'
8. 'df.residual'
9. 'xlevels'
10. 'call'
11. 'terms'
12. 'model'

#提取回归系数
coef(lm.fit)
#拟合样本数据
fitted(lm.fit)

(Intercept)

-1.35827281187449

lstat

-0.642358334244129

rm

5.09478798433654

1

28.941013680603

2

25.4842056605593

3

32.6590747685798

4

32.4065199998349

5

31.6304069906576

6

28.0545270059976

7

21.2870784553023

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17.7855965266756

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8.1046933839978

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18.2465067305075

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17.9949622289472

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20.7322130905842

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18.5534841969081

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23.6447410660871

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23.1089582312963

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22.9239451976971

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24.6525760358365

18

19.73611045094

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18.9297215033518

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20.5737759641471

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13.5173240750685

22

20.1483217520967

23

17.9089669708705

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15.4876460563005

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18.3528103591559

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16.5621090140552

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18.7444028109183

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18.34995811367

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23.5101884680665

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13.2309525887229

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11.1559662530231

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33.155648488302

42

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25.5063093520635

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23.4274733730326

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21.0318339224932

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19.0308000359423

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17.2869620498852

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6.35742723749689

50

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28.4222397493317

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23.7851847604981

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19.132935493086

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32.4841016988669

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27.455351303557

58

30.8304866690991

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25.5426211789588

60

22.9159917295573

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19.4438929108913

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19.7615779561921

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27.2106068255392

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26.99027936289

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18.7335005772485

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25.6152963101065

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24.9248809495225

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22.9428716808724

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23.3267053192647

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22.4657440619383

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22.7230509664968

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29.516290370606

82

27.7263016832065

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26.4324330592939

84

25.2371735973556

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25.0128404446232

86

28.2255716016626

87

21.0261487355303

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24.405420099229

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30.807935756029

90

31.0462888240524

91

25.6758047589223

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26.0065058869992

93

26.2207073757894

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26.2964101031837

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23.6764825425992

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28.1230146616317

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22.7565620252632

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37.0472428465693

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36.1897499723857

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32.4484767909942

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28.2625960862591

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24.445575134786

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21.2751450357578

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22.1410064262989

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17.8716899194778

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16.3885033477181

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26.326867827025

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14.7868207262998

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12.0152158401217

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16.6199276988068

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4.76981118873819

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6.87170964910109

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31.0664154091936

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35.6805173060786

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31.7854674340077

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36.8553783120707

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33.1466252262699

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29.3196801349634

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31.4488028134934

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31.2487193747197

201

⋯

202

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28.6989718359882

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29.5293386267301

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22.6634884381694

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15.8586984893853

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21.8921983461714

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23.079912953518

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#提取系数的置信区间，就是置信区间
confint(lm.fit,level = 0.95)

A matrix: 3 × 2 of type dbl

	2.5 %	97.5 %
(Intercept)	-7.5919003	4.8753547
lstat	-0.7282772	-0.5564395
rm	4.2215504	5.9680255

#回归诊断
par(mfrow=c(2,2))
plot(lm.fit)

带有交互项

#带交互项的回归
#注意：变量x和y在lm拟合中，x*y和x：y的意思是不一样的。
summary(lm(medv~lstat*age,data=Boston)) #单变量+交互
summary(lm(medv~age*lstat,data=Boston)) #单变量+交互

Call:
lm(formula = medv ~ lstat * age, data = Boston)

Residuals:
Min      1Q  Median      3Q     Max
-15.806  -4.045  -1.333   2.085  27.552

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 36.0885359  1.4698355  24.553  < 2e-16 ***
lstat       -1.3921168  0.1674555  -8.313 8.78e-16 ***
age         -0.0007209  0.0198792  -0.036   0.9711
lstat:age    0.0041560  0.0018518   2.244   0.0252 *
------------------------------------------------------------------------------------------------------

Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 6.149 on 502 degrees of freedom
Multiple R-squared:  0.5557,	Adjusted R-squared:  0.5531
F-statistic: 209.3 on 3 and 502 DF,  p-value: < 2.2e-16

Call:
lm(formula = medv ~ age * lstat, data = Boston)

Residuals:
Min      1Q  Median      3Q     Max
-15.806  -4.045  -1.333   2.085  27.552

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 36.0885359  1.4698355  24.553  < 2e-16 ***
age         -0.0007209  0.0198792  -0.036   0.9711
lstat       -1.3921168  0.1674555  -8.313 8.78e-16 ***
age:lstat    0.0041560  0.0018518   2.244   0.0252 *
----------------------------------------------------

Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 6.149 on 502 degrees of freedom
Multiple R-squared:  0.5557,	Adjusted R-squared:  0.5531
F-statistic: 209.3 on 3 and 502 DF,  p-value: < 2.2e-16

多元线性回归高阶

## 多元线性回归高阶情况
summary(lm(medv~lstat+age+poly(rm,3,raw=T),data=Boston))
summary(lm(medv~lstat+I(rm^3),data=Boston))#这种高阶的情况也是可以的只不过只能输出指定阶数的项

Call:
lm(formula = medv ~ lstat + age + poly(rm, 3, raw = T), data = Boston)

Residuals:
Min       1Q   Median       3Q      Max
-31.1634  -2.6180  -0.4345   1.9570  27.7091

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)           160.858226  37.336812   4.308 1.98e-05 ***
lstat                  -0.682214   0.047720 -14.296  < 2e-16 ***
age                    -0.004674   0.009731  -0.480   0.6313
poly(rm, 3, raw = T)1 -52.259779  18.264834  -2.861   0.0044 **
poly(rm, 3, raw = T)2   6.003959   2.936142   2.045   0.0414 *
poly(rm, 3, raw = T)3  -0.159260   0.154874  -1.028   0.3043
--------------------------------------------------------------------------------------------------------------------------

Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4.781 on 500 degrees of freedom
Multiple R-squared:  0.7324,	Adjusted R-squared:  0.7298
F-statistic: 273.8 on 5 and 500 DF,  p-value: < 2.2e-16

Call:
lm(formula = medv ~ lstat + I(rm^3), data = Boston)

Residuals:
Min       1Q   Median       3Q      Max
-24.5407  -2.9667  -0.6511   2.0786  27.9885

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 18.312269   1.195934   15.31   <2e-16 ***
lstat       -0.609726   0.039741  -15.34   <2e-16 ***
I(rm^3)      0.046324   0.003136   14.77   <2e-16 ***
-----------------------------------------------------

Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 5.196 on 503 degrees of freedom
Multiple R-squared:  0.6821,	Adjusted R-squared:  0.6808
F-statistic: 539.5 on 2 and 503 DF,  p-value: < 2.2e-16