Lab: Linear Regression#

Libraries#

library(MASS)
library(ISLR)

Simple Linear Regression#

library(MASS)
data(Boston)
names(Boston)
  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'
lm.fit=lm(medv~lstat)

Error in eval(predvars, data, env): object 'medv' not found
Traceback:

1. lm(medv ~ lstat)
2. eval(mf, parent.frame())
3. eval(mf, parent.frame())
4. stats::model.frame(formula = medv ~ lstat, drop.unused.levels = TRUE)
5. model.frame.default(formula = medv ~ lstat, drop.unused.levels = TRUE)
6. eval(predvars, data, env)
7. eval(predvars, data, env)

lm.fit = lm(medv ~ lstat, data=Boston)
attach(Boston)
lm.fit=lm(medv ~ lstat)

lm.fit
summary(lm.fit)

Call:
lm(formula = medv ~ lstat)

Coefficients:
(Intercept)        lstat  
      34.55        -0.95  

Call:
lm(formula = medv ~ lstat)

Residuals:
    Min      1Q  Median      3Q     Max 
-15.168  -3.990  -1.318   2.034  24.500 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 34.55384    0.56263   61.41   <2e-16 ***
lstat       -0.95005    0.03873  -24.53   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.216 on 504 degrees of freedom
Multiple R-squared:  0.5441,	Adjusted R-squared:  0.5432 
F-statistic: 601.6 on 1 and 504 DF,  p-value: < 2.2e-16

names(lm.fit)
coef(lm.fit)
  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'
(Intercept)
34.5538408793831
lstat
-0.95004935375799
confint(lm.fit)
A matrix: 2 × 2 of type dbl
2.5 %97.5 %
(Intercept)33.44845735.6592247
lstat-1.026148-0.8739505
 
predict(lm.fit, data.frame(lstat=(c(5,10,15))),
        interval="confidence")
predict(lm.fit,data.frame(lstat=(c(5,10,15))),
        interval="prediction")
A matrix: 3 × 3 of type dbl
fitlwrupr
29.8035929.0074130.59978
25.0533524.4741325.63256
20.3031019.7315920.87461
A matrix: 3 × 3 of type dbl
fitlwrupr
29.8035917.56567542.04151
25.0533512.82762637.27907
20.30310 8.07774232.52846
 
plot(lstat, medv)
abline(lm.fit)
../../_images/Ch3-linreg-lab_10_0.png 
plot(lstat, medv)
abline(lm.fit, lwd=3)
abline(lm.fit, lwd=3, col="red")
plot(lstat, medv, col="red")
plot(lstat, medv, pch=20)
plot(lstat, medv, pch="+")
plot(1:20, 1:20, pch=1:20)
../../_images/Ch3-linreg-lab_11_0.png ../../_images/Ch3-linreg-lab_11_1.png ../../_images/Ch3-linreg-lab_11_2.png ../../_images/Ch3-linreg-lab_11_3.png ../../_images/Ch3-linreg-lab_11_4.png 
par(mfrow=c(2, 2))
plot(lm.fit)
../../_images/Ch3-linreg-lab_12_0.png 
plot(predict(lm.fit), residuals(lm.fit))
 plot(predict(lm.fit), rstudent(lm.fit))
../../_images/Ch3-linreg-lab_13_0.png ../../_images/Ch3-linreg-lab_13_1.png 
plot(hatvalues(lm.fit))
which.max(hatvalues(lm.fit))
375: 375
../../_images/Ch3-linreg-lab_14_1.png

Multiple Linear Regression#

lm.fit=lm(medv ~ lstat + age, data=Boston)
summary(lm.fit)

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

Residuals:
    Min      1Q  Median      3Q     Max 
-15.981  -3.978  -1.283   1.968  23.158 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 33.22276    0.73085  45.458  < 2e-16 ***
lstat       -1.03207    0.04819 -21.416  < 2e-16 ***
age          0.03454    0.01223   2.826  0.00491 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.173 on 503 degrees of freedom
Multiple R-squared:  0.5513,	Adjusted R-squared:  0.5495 
F-statistic:   309 on 2 and 503 DF,  p-value: < 2.2e-16

lm.fit=lm(medv ~ ., data=Boston)
summary(lm.fit)

Call:
lm(formula = medv ~ ., data = Boston)

Residuals:
    Min      1Q  Median      3Q     Max 
-15.595  -2.730  -0.518   1.777  26.199 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  3.646e+01  5.103e+00   7.144 3.28e-12 ***
crim        -1.080e-01  3.286e-02  -3.287 0.001087 ** 
zn           4.642e-02  1.373e-02   3.382 0.000778 ***
indus        2.056e-02  6.150e-02   0.334 0.738288    
chas         2.687e+00  8.616e-01   3.118 0.001925 ** 
nox         -1.777e+01  3.820e+00  -4.651 4.25e-06 ***
rm           3.810e+00  4.179e-01   9.116  < 2e-16 ***
age          6.922e-04  1.321e-02   0.052 0.958229    
dis         -1.476e+00  1.995e-01  -7.398 6.01e-13 ***
rad          3.060e-01  6.635e-02   4.613 5.07e-06 ***
tax         -1.233e-02  3.760e-03  -3.280 0.001112 ** 
ptratio     -9.527e-01  1.308e-01  -7.283 1.31e-12 ***
black        9.312e-03  2.686e-03   3.467 0.000573 ***
lstat       -5.248e-01  5.072e-02 -10.347  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.745 on 492 degrees of freedom
Multiple R-squared:  0.7406,	Adjusted R-squared:  0.7338 
F-statistic: 108.1 on 13 and 492 DF,  p-value: < 2.2e-16

install.packages('car', repos='http://cloud.r-project.org')
library(car)
vif(lm.fit)

The downloaded binary packages are in
	/var/folders/r1/yzxk5h4j5r327ymmt5msy3n00000gr/T//RtmpDG6QqP/downloaded_packages

Warning message:
“package ‘car’ was built under R version 3.6.2”Loading required package: carData
crim
1.79219154743324
zn
2.29875817874944
indus
3.99159641834602
chas
1.07399532755379
nox
4.39371984757748
rm
1.93374443578326
age
3.10082551281533
dis
3.95594490637272
rad
7.48449633527445
tax
9.00855394759706
ptratio
1.79908404924889
black
1.34852107640637
lstat
2.94149107809193
lm.fit1=lm(medv ~ . - age, data=Boston)
summary(lm.fit1) 

Call:
lm(formula = medv ~ . - age, data = Boston)

Residuals:
     Min       1Q   Median       3Q      Max 
-15.6054  -2.7313  -0.5188   1.7601  26.2243 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  36.436927   5.080119   7.172 2.72e-12 ***
crim         -0.108006   0.032832  -3.290 0.001075 ** 
zn            0.046334   0.013613   3.404 0.000719 ***
indus         0.020562   0.061433   0.335 0.737989    
chas          2.689026   0.859598   3.128 0.001863 ** 
nox         -17.713540   3.679308  -4.814 1.97e-06 ***
rm            3.814394   0.408480   9.338  < 2e-16 ***
dis          -1.478612   0.190611  -7.757 5.03e-14 ***
rad           0.305786   0.066089   4.627 4.75e-06 ***
tax          -0.012329   0.003755  -3.283 0.001099 ** 
ptratio      -0.952211   0.130294  -7.308 1.10e-12 ***
black         0.009321   0.002678   3.481 0.000544 ***
lstat        -0.523852   0.047625 -10.999  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.74 on 493 degrees of freedom
Multiple R-squared:  0.7406,	Adjusted R-squared:  0.7343 
F-statistic: 117.3 on 12 and 493 DF,  p-value: < 2.2e-16

lm.fit1 = update(lm.fit, ~ . -age)

Interaction Terms#

summary(lm(medv ~ lstat*age, data=Boston))

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

Non-linear Transformations of the Predictors#

lm.fit2 = lm(medv ~ lstat + I(lstat^2))
summary(lm.fit2)

Call:
lm(formula = medv ~ lstat + I(lstat^2))

Residuals:
     Min       1Q   Median       3Q      Max 
-15.2834  -3.8313  -0.5295   2.3095  25.4148 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 42.862007   0.872084   49.15   <2e-16 ***
lstat       -2.332821   0.123803  -18.84   <2e-16 ***
I(lstat^2)   0.043547   0.003745   11.63   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.524 on 503 degrees of freedom
Multiple R-squared:  0.6407,	Adjusted R-squared:  0.6393 
F-statistic: 448.5 on 2 and 503 DF,  p-value: < 2.2e-16

lm.fit = lm(medv ~ lstat)
anova(lm.fit, lm.fit2)
A anova: 2 × 6
Res.DfRSSDfSum of SqFPr(>F)
<dbl><dbl><dbl><dbl><dbl><dbl>
50419472.38NA NA NA NA
50315347.24 14125.138135.19987.630116e-28
 
par(mfrow=c(2,2))
plot(lm.fit2)
../../_images/Ch3-linreg-lab_26_0.png 
lm.fit5 = lm(medv~poly(lstat,5))
summary(lm.fit5)

Call:
lm(formula = medv ~ poly(lstat, 5))

Residuals:
     Min       1Q   Median       3Q      Max 
-13.5433  -3.1039  -0.7052   2.0844  27.1153 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)       22.5328     0.2318  97.197  < 2e-16 ***
poly(lstat, 5)1 -152.4595     5.2148 -29.236  < 2e-16 ***
poly(lstat, 5)2   64.2272     5.2148  12.316  < 2e-16 ***
poly(lstat, 5)3  -27.0511     5.2148  -5.187 3.10e-07 ***
poly(lstat, 5)4   25.4517     5.2148   4.881 1.42e-06 ***
poly(lstat, 5)5  -19.2524     5.2148  -3.692 0.000247 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.215 on 500 degrees of freedom
Multiple R-squared:  0.6817,	Adjusted R-squared:  0.6785 
F-statistic: 214.2 on 5 and 500 DF,  p-value: < 2.2e-16

summary(lm(medv ~ log(rm), data=Boston))

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

Residuals:
    Min      1Q  Median      3Q     Max 
-19.487  -2.875  -0.104   2.837  39.816 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -76.488      5.028  -15.21   <2e-16 ***
log(rm)       54.055      2.739   19.73   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.915 on 504 degrees of freedom
Multiple R-squared:  0.4358,	Adjusted R-squared:  0.4347 
F-statistic: 389.3 on 1 and 504 DF,  p-value: < 2.2e-16

Qualitative Predictors#

#COMMENT fix(Carseats)
names(Carseats)
  1. 'Sales'
  2. 'CompPrice'
  3. 'Income'
  4. 'Advertising'
  5. 'Population'
  6. 'Price'
  7. 'ShelveLoc'
  8. 'Age'
  9. 'Education'
  10. 'Urban'
  11. 'US'
lm.fit = lm(Sales ~ . + Income:Advertising + Price:Age, data=Carseats)
summary(lm.fit)

Call:
lm(formula = Sales ~ . + Income:Advertising + Price:Age, data = Carseats)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.9208 -0.7503  0.0177  0.6754  3.3413 

Coefficients:
                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)         6.5755654  1.0087470   6.519 2.22e-10 ***
CompPrice           0.0929371  0.0041183  22.567  < 2e-16 ***
Income              0.0108940  0.0026044   4.183 3.57e-05 ***
Advertising         0.0702462  0.0226091   3.107 0.002030 ** 
Population          0.0001592  0.0003679   0.433 0.665330    
Price              -0.1008064  0.0074399 -13.549  < 2e-16 ***
ShelveLocGood       4.8486762  0.1528378  31.724  < 2e-16 ***
ShelveLocMedium     1.9532620  0.1257682  15.531  < 2e-16 ***
Age                -0.0579466  0.0159506  -3.633 0.000318 ***
Education          -0.0208525  0.0196131  -1.063 0.288361    
UrbanYes            0.1401597  0.1124019   1.247 0.213171    
USYes              -0.1575571  0.1489234  -1.058 0.290729    
Income:Advertising  0.0007510  0.0002784   2.698 0.007290 ** 
Price:Age           0.0001068  0.0001333   0.801 0.423812    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.011 on 386 degrees of freedom
Multiple R-squared:  0.8761,	Adjusted R-squared:  0.8719 
F-statistic:   210 on 13 and 386 DF,  p-value: < 2.2e-16

attach(Carseats)
contrasts(ShelveLoc)
A matrix: 3 × 2 of type dbl
GoodMedium
Bad00
Good10
Medium01

Writing Functions#

LoadLibraries
LoadLibraries()

Error in eval(expr, envir, enclos): object 'LoadLibraries' not found
Traceback:

LoadLibraries = function() {
library(ISLR)
library(MASS)
print("The libraries have been loaded.")
}

LoadLibraries

function () 
{
    library(ISLR)
    library(MASS)
    print("The libraries have been loaded.")
}
LoadLibraries()

[1] "The libraries have been loaded."