Chapter 3 - Linear Regression

Libraries

Basic functions that perform least squares linear regression and other simple analyses come standard with the base distribution, but more exotic functions require additional libraries.
Here we load the MASS package, which is a very large collection of data sets and functions. We also load the ISLR package, which includes the data sets associated with this book.

 library (MASS)
 library (ISLR)
## Warning: package 'ISLR' was built under R version 3.4.3

If you receive an error message when loading any of these libraries, it likely indicates that the corresponding library has not yet been installed on your system. Some libraries, such as MASS, come with R and do not need to be separately installed on your computer. However, other packages, such as ISLR, must be downloaded the first time they are used. This can be done directly from within R.

Simple Linear Regression Model

The MASS library contains the Boston data set, which records medv (median house value) for 506 neighborhoods around Boston. We will seek to predict medv using 13 predictors such as rm (average number of rooms per house), age (average age of houses), and lstat (percent of households with low socioeconomic status).

To find out more about the data set, we can type ?Boston.
* lm() - used to fit a simple linear regression model

In this dataset, we start with medv as the response and lstat as the predictor. The basic syntax is lm(y~x,data), where y is the response, x is the predictor, and data is the data set in which these two variables are kept.

#install.packages("ISLR")
#library(ISLR)
#fix(Boston) 
names(Boston)
##  [1] "crim"    "zn"      "indus"   "chas"    "nox"     "rm"      "age"    
##  [8] "dis"     "rad"     "tax"     "ptratio" "black"   "lstat"   "medv"
lm.fit =lm(medv~lstat ,data=Boston )

# Alternate way of doing the same as above line
attach(Boston)
lm.fit =lm(medv~lstat)

Results generated from running a simple linear regression:
* lm.fit - some basic information about the model is output.
* summary(lm.fit) - Gives more detailed information. This gives us pvalues and standard errors for the coefficients, as well as the R2 statistic and F-statistic for the model.
* names() - used to find out what other pieces of information are stored in lm.fit (resultant model fit variable). Although we can extract these quantities by name — e.g. lm.fit$coefficients — it is safer to use the extractor functions like coef() to access them.
* confint() - returns a confidence interval for the coefficient estimates.

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"       "rank"         
##  [5] "fitted.values" "assign"        "qr"            "df.residual"  
##  [9] "xlevels"       "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

For instance, the 95% confidence interval associated with a lstat value of 10 is (24.47, 25.63), and the 95% prediction interval is (12.828, 37.28). As expected, the confidence and prediction intervals are centered around the same point (a predicted value of 25.05 for medv when lstat equals 10), but the latter are substantially wider.

predict (lm.fit ,data.frame(lstat=c(5 ,10 ,15) ), interval ="confidence")
##        fit      lwr      upr
## 1 29.80359 29.00741 30.59978
## 2 25.05335 24.47413 25.63256
## 3 20.30310 19.73159 20.87461
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)

There is some evidence for non-linearity in the relationship between lstat and medv. We will explore this issue later in this lab.

Some more options to modify the plot and lines are - lwd(), pch() etc. The lwd=3 command causes the width of the regression line to be increased by a factor of 3; this works for the plot() and lines() both. We can also use the pch option to create different plotting symbols. Its value varies from 1-20.

Diagnostic Plots
Four diagnostic plots are automatically produced by applying the plot() function directly to the output from lm(). In general, this command will produce one plot at a time, and hitting Enter will generate the next plot. However, it is often convenient to view all four plots together. We can achieve this by using the par() function.
* par() - tells R to split the display screen into separate panels so that multiple plots can be viewed simultaneously. For example, par(mfrow=c(2,2)) divides the plotting region into a 2 × 2 grid of panels.

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

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

Alternatively,
* residuals() - compute the residuals from a linear regression fit. * rstudent() - will return the studentized residuals, and we can use this function to plot the residuals against the fitted values.
* which.max() - identifies the index of the largest element of a vector. In this case, it tells us which observation has the largest leverage statistic.

On the basis of the residual plots, there is some evidence of non-linearity. Leverage statistics can be computed for any number of predictors using the hatvalues() function.

plot(hatvalues(lm.fit ))

which.max (hatvalues (lm.fit))
## 375 
## 375

Multiple Linear Regression

In order to fit a multiple linear regression model using least squares, we again use the lm() function. The syntax lm(y∼x1+x2+x3) is used to fit a model with three predictors, x1, x2, and x3. The summary() function now outputs the regression coefficients for all the predictors.

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

The Boston data set contains 13 variables, and so it would be cumbersome to have to type all of these in order to perform a regression using all of the predictors. Instead, we can use a . to represent all variables.
What if we would like to perform a regression using all of the variables but one? The following syntax: “~.-” or medv~.-age results in a regression using all predictors except age.

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
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
#Alternatively, the update() function can be used.
# age has a high p-value so lets remove and build a new model.
lm.fit1=update (lm.fit , ∼.-age)

We can access the individual components of a summary object by name (type ?summary.lm to see what is available). Hence
* summary(lm.fit) $r.sq - gives us the R2, and
* summary(lm.fit) $sigma - gives us the RSE.
* vif() - part of the car package,vif() can be used to compute variance inflation factors.

Most VIF’s are low to moderate for this data.

#install.packages("car")
library (car)
## Warning: package 'car' was built under R version 3.4.3
vif(lm.fit)
##     crim       zn    indus     chas      nox       rm      age      dis 
## 1.792192 2.298758 3.991596 1.073995 4.393720 1.933744 3.100826 3.955945 
##      rad      tax  ptratio    black    lstat 
## 7.484496 9.008554 1.799084 1.348521 2.941491

Interaction Terms

It is easy to include interaction terms in a linear model using the lm() function. The syntax lstat:black tells R to include an interaction term between lstat and black. The syntax lstat*age simultaneously includes lstat, age, and the interaction term lstat×age as predictors; it is a shorthand for lstat+age+lstat:age.

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

The lm() function can also accommodate non-linear transformations of the predictors. For instance, given a predictor X, we can create a predictor X2 using I(X^2). I() is needed since the ^ has a special meaning in a formula; wrapping as we do allows the standard usage in R, which is to raise X to the power 2.
We now perform a regression of medv onto lstat and lstat2.

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

The near-zero p-value associated with the quadratic term suggests that it leads to an improved model.
* anova() - used to further quantify the extent to which the quadratic fit is superior to the linear fit.

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

Here:

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

We see that when the lstat2 term is included in the model, there is little discernible pattern in the residuals.
In order to create a cubic fit, we can include a predictor of the form I(X^3). However, this approach can start to get cumbersome for higherorder polynomials. A better approach involves using the poly() function to create the polynomial within lm(). For example, the following command produces a fifth-order polynomial fit and a log transformation fit:

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

This suggests that including additional polynomial terms, up to fifth order, leads to an improvement in the model fit! However, further investigation of the data reveals that no polynomial terms beyond fifth order have significant p-values in a regression fit.

Qualitative Predictors

We will now examine the Carseats data, which is part of the ISLR library. We will attempt to predict Sales (child car seat sales) in 400 locations based on a number of predictors.

The Carseats data includes qualitative predictors such as Shelveloc, an indicator of the quality of the shelving location—that is, the space within a store in which the car seat is displayed—at each location. The predictor Shelveloc takes on three possible values, Bad, Medium, and Good.

#fix(Carseats)
names(Carseats)
##  [1] "Sales"       "CompPrice"   "Income"      "Advertising" "Population" 
##  [6] "Price"       "ShelveLoc"   "Age"         "Education"   "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 )
##        Good Medium
## Bad       0      0
## Good      1      0
## Medium    0      1

Given a qualitative variable such as Shelveloc, R generates dummy variables automatically. R has created a ShelveLocGood dummy variable that takes on a value of 1 if the shelving location is good, and 0 otherwise. It has also created a ShelveLocMedium dummy variable that equals 1 if the shelving location is medium, and 0 otherwise. A bad shelving location corresponds to a zero for each of the two dummy variables. The fact that the coefficient for ShelveLocGood in the regression output is positive indicates that a good shelving location is associated with high sales (relative to a bad location). And ShelveLocMedium has a smaller positive coefficient, indicating that a medium shelving location leads to higher sales than a bad shelving location but lower sales than a good shelving location.

Writing Functions

As we have seen, R comes with many useful functions, and still more functions are available by way of R libraries. However, we will often be interested in performing an operation for which no function is available. In this setting, we may want to write our own function. For instance, a simple function called LoadLibraries() that reads in the ISLR and MASS libraries.

The { symbol informs R that multiple commands are about to be input. Hitting Enter after typing { will cause R to print the + symbol. We can then input as many commands as we wish, hitting Enter after each one. Finally the } symbol informs R that no further commands will be entered.

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

LoadLibraries()
## [1] " The libraries have been loaded ."