Laboratorio #1

Fiabilidad

Brayan Ivan Cruz Corona

13001595

3.6.1 Libraries

library(MASS)
library(ISLR)

3.6.2 Simple linear regression

fix(Boston)
names(Boston)

 [1] "crim"    "zn"      "indus"   "chas"    "nox"    
 [6] "rm"      "age"     "dis"     "rad"     "tax"    
[11] "ptratio" "black"   "lstat"   "medv"   

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

lm.fit

Call:
lm(formula = medv ~ lstat)

Coefficients:
(Intercept)        lstat  
      34.55        -0.95  

summary(lm.fit)

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)

 [1] "coefficients"  "residuals"     "effects"      
 [4] "rank"          "fitted.values" "assign"       
 [7] "qr"            "df.residual"   "xlevels"      
[10] "call"          "terms"         "model"        

coef(lm.fit)

(Intercept)       lstat 
 34.5538409  -0.9500494 

confint(lm.fit)

                2.5 %     97.5 %
(Intercept) 33.448457 35.6592247
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")

       fit       lwr      upr
1 29.80359 17.565675 42.04151
2 25.05335 12.827626 37.27907
3 20.30310  8.077742 32.52846

plot(lstat,medv)
abline(lm.fit)

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)

par(mfrow=c(2,2))
plot(lm.fit)

plot(predict(lm.fit),residuals(lm.fit))

plot(predict(lm.fit),rstudent(lm.fit))

plot(hatvalues(lm.fit))

which.max(hatvalues(lm.fit))

375 
375 

3.6.3 Multiple Linear Regresion

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

library(car)
vif(lm.fit)

    crim       zn    indus     chas      nox       rm 
1.792192 2.298758 3.991596 1.073995 4.393720 1.933744 
     age      dis      rad      tax  ptratio    black 
3.100826 3.955945 7.484496 9.008554 1.799084 1.348521 
   lstat 
2.941491 

lm.fit1=lm(medv~.-age, data=Boston)
summary(lm.fit1)

3.6.4 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

3.6.5 Non-linear Transformation 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)

Analysis of Variance Table

Model 1: medv ~ lstat
Model 2: medv ~ lstat + I(lstat^2)
  Res.Df   RSS Df Sum of Sq     F    Pr(>F)    
1    504 19472                                 
2    503 15347  1    4125.1 135.2 < 2.2e-16 ***
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

par(mfrow=c(2,2))
plot(lm.fit2)

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
                   
(Intercept)     ***
poly(lstat, 5)1 ***
poly(lstat, 5)2 ***
poly(lstat, 5)3 ***
poly(lstat, 5)4 ***
poly(lstat, 5)5 ***
---
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

3.6.6 Qualitative Predictors

fix(Carseats)
names(Carseats)

 [1] "Sales"       "CompPrice"   "Income"     
 [4] "Advertising" "Population"  "Price"      
 [7] "ShelveLoc"   "Age"         "Education"  
[10] "Urban"       "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
(Intercept)         6.5755654  1.0087470   6.519
CompPrice           0.0929371  0.0041183  22.567
Income              0.0108940  0.0026044   4.183
Advertising         0.0702462  0.0226091   3.107
Population          0.0001592  0.0003679   0.433
Price              -0.1008064  0.0074399 -13.549
ShelveLocGood       4.8486762  0.1528378  31.724
ShelveLocMedium     1.9532620  0.1257682  15.531
Age                -0.0579466  0.0159506  -3.633
Education          -0.0208525  0.0196131  -1.063
UrbanYes            0.1401597  0.1124019   1.247
USYes              -0.1575571  0.1489234  -1.058
Income:Advertising  0.0007510  0.0002784   2.698
Price:Age           0.0001068  0.0001333   0.801
                   Pr(>|t|)    
(Intercept)        2.22e-10 ***
CompPrice           < 2e-16 ***
Income             3.57e-05 ***
Advertising        0.002030 ** 
Population         0.665330    
Price               < 2e-16 ***
ShelveLocGood       < 2e-16 ***
ShelveLocMedium     < 2e-16 ***
Age                0.000318 ***
Education          0.288361    
UrbanYes           0.213171    
USYes              0.290729    
Income:Advertising 0.007290 ** 
Price:Age          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)

       Good Medium
Bad       0      0
Good      1      0
Medium    0      1

3.6.7 Writing Functions

LoadLibraries= function(){
  library(ISLR)
  library(MASS)
  print("Las librerias han sido cargadas.")
}

LoadLibraries()

[1] "Las librerias han sido cargadas."