Lab05
Dasha Djukic-Min
3/8/2022
R Programming Lab (Linear Regression)
Adapted from ISLR Chapter 3 Lab
Load datasets from MASS and ISLR packages
# install.packages("arm") # For coefficient plots
library(arm)## Loading required package: MASS## Loading required package: Matrix## Loading required package: lme4##
## arm (Version 1.12-2, built: 2021-10-15)## Working directory is /Users/dasha/Desktoplibrary(MASS)
library(ISLR)
attach(Boston)
names(Boston)## [1] "crim" "zn" "indus" "chas" "nox" "rm" "age"
## [8] "dis" "rad" "tax" "ptratio" "black" "lstat" "medv"What is the Boston dataset?
?BostonSimple linear regression
plot(medv~lstat,Boston)
fit1=lm(medv~lstat,data=Boston)
fit1##
## Call:
## lm(formula = medv ~ lstat, data = Boston)
##
## Coefficients:
## (Intercept) lstat
## 34.55 -0.95summary(fit1)##
## Call:
## lm(formula = medv ~ lstat, data = Boston)
##
## 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-16abline(fit1,col="red")
names(fit1)## [1] "coefficients" "residuals" "effects" "rank"
## [5] "fitted.values" "assign" "qr" "df.residual"
## [9] "xlevels" "call" "terms" "model"confint(fit1)## 2.5 % 97.5 %
## (Intercept) 33.448457 35.6592247
## lstat -1.026148 -0.8739505predict(fit1,data.frame(lstat=c(0,5,10,15)),interval="confidence")## fit lwr upr
## 1 34.55384 33.44846 35.65922
## 2 29.80359 29.00741 30.59978
## 3 25.05335 24.47413 25.63256
## 4 20.30310 19.73159 20.87461Multiple linear regression
fit2=lm(medv~lstat+age,data=Boston)
summary(fit2)##
## 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-16fit3=lm(medv~.,Boston)
summary(fit3)##
## 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-16par(mfrow=c(2,2))
plot(fit3)
Update function to respecify the model, i.e. remove age and indus variables
fit4=update(fit3,~.-age-indus)
summary(fit4)##
## Call:
## lm(formula = medv ~ crim + zn + chas + nox + rm + dis + rad +
## tax + ptratio + black + lstat, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.5984 -2.7386 -0.5046 1.7273 26.2373
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 36.341145 5.067492 7.171 2.73e-12 ***
## crim -0.108413 0.032779 -3.307 0.001010 **
## zn 0.045845 0.013523 3.390 0.000754 ***
## chas 2.718716 0.854240 3.183 0.001551 **
## nox -17.376023 3.535243 -4.915 1.21e-06 ***
## rm 3.801579 0.406316 9.356 < 2e-16 ***
## dis -1.492711 0.185731 -8.037 6.84e-15 ***
## rad 0.299608 0.063402 4.726 3.00e-06 ***
## tax -0.011778 0.003372 -3.493 0.000521 ***
## ptratio -0.946525 0.129066 -7.334 9.24e-13 ***
## black 0.009291 0.002674 3.475 0.000557 ***
## lstat -0.522553 0.047424 -11.019 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.736 on 494 degrees of freedom
## Multiple R-squared: 0.7406, Adjusted R-squared: 0.7348
## F-statistic: 128.2 on 11 and 494 DF, p-value: < 2.2e-16Set the next plot configuration
par(mfrow=c(2,2))
plot(fit4)
par(mfrow=c(1,1))
coefplot(fit4)
Nonlinear terms and Interactions
fit5=lm(medv~lstat*age,Boston)
summary(fit5)##
## 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-16I() identity function for squared term to interpret as-is
Combine two command lines with semicolon
fit6=lm(medv~lstat +I(lstat^2),Boston); summary(fit6)##
## Call:
## lm(formula = medv ~ lstat + I(lstat^2), data = Boston)
##
## 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-16attach(Boston)## The following objects are masked from Boston (pos = 3):
##
## age, black, chas, crim, dis, indus, lstat, medv, nox, ptratio, rad,
## rm, tax, znpar(mfrow=c(1,1))
plot(medv~lstat)
points(lstat,fitted(fit6),col="red",pch=20)
fit7=lm(medv~poly(lstat,4))
points(lstat,fitted(fit7),col="blue",pch=20)
Show all plotting characters (pch) in once chart, double in size (cex=2)
plot(1:20,1:20,pch=1:20,cex=2)
Qualitative predictors
fix(Carseats)
names(Carseats)## [1] "Sales" "CompPrice" "Income" "Advertising" "Population"
## [6] "Price" "ShelveLoc" "Age" "Education" "Urban"
## [11] "US"summary(Carseats)## Sales CompPrice Income Advertising
## Min. : 0.000 Min. : 77 Min. : 21.00 Min. : 0.000
## 1st Qu.: 5.390 1st Qu.:115 1st Qu.: 42.75 1st Qu.: 0.000
## Median : 7.490 Median :125 Median : 69.00 Median : 5.000
## Mean : 7.496 Mean :125 Mean : 68.66 Mean : 6.635
## 3rd Qu.: 9.320 3rd Qu.:135 3rd Qu.: 91.00 3rd Qu.:12.000
## Max. :16.270 Max. :175 Max. :120.00 Max. :29.000
## Population Price ShelveLoc Age Education
## Min. : 10.0 Min. : 24.0 Bad : 96 Min. :25.00 Min. :10.0
## 1st Qu.:139.0 1st Qu.:100.0 Good : 85 1st Qu.:39.75 1st Qu.:12.0
## Median :272.0 Median :117.0 Medium:219 Median :54.50 Median :14.0
## Mean :264.8 Mean :115.8 Mean :53.32 Mean :13.9
## 3rd Qu.:398.5 3rd Qu.:131.0 3rd Qu.:66.00 3rd Qu.:16.0
## Max. :509.0 Max. :191.0 Max. :80.00 Max. :18.0
## Urban US
## No :118 No :142
## Yes:282 Yes:258
##
##
##
## fit1=lm(Sales~.+Income:Advertising+Age:Price,Carseats)
summary(fit1)##
## Call:
## lm(formula = Sales ~ . + Income:Advertising + Age:Price, 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-16contrasts(Carseats$ShelveLoc) # what is contrasts function/## Good Medium
## Bad 0 0
## Good 1 0
## Medium 0 1Writing R functions
Combine the lm, plot and abline functions to create a one step regression fit plot function
regplot=function(x,y){
fit=lm(y~x)
plot(x,y)
abline(fit,col="red")
}
attach(Carseats)
regplot(Price,Sales)
Allow extra room for additional arguments/specifications
regplot=function(x,y,...){
fit=lm(y~x)
plot(x,y,...)
abline(fit,col="red")
}
regplot(Price,Sales,xlab="Price",ylab="Sales",col="blue",pch=20)
Additional note: try out the coefplot2 package to finetune the coefplots
install.packages(“coefplot2”,
repos="http://www.math.mcmaster.ca/bolker/R", type="source")library(coefplot2)
I can’t install “coefplot2”. I get this error message:
> install.packages(“coefplot2”,
Warning in install.packages : dependency ‘reshape’ is not available trying URL ‘http://www.math.mcmaster.ca/bolker/R/src/contrib/coefplot2_0.1.3.2.tar.gz’ Content type ‘application/x-gzip’ length 1357869 bytes (1.3 MB) ================================================== downloaded 1.3 MBrepos="http://www.math.mcmaster.ca/bolker/R", type="source")
ERROR: dependency ‘reshape’ is not available for package ‘coefplot2’ * removing ‘/Library/Frameworks/R.framework/Versions/4.1/Resources/library/coefplot2’ Warning in install.packages : installation of package ‘coefplot2’ had non-zero exit status
The downloaded source packages are in ‘/private/var/folders/8v/4k1yxpt54rx9k6mdf6m14lsr0000gn/T/Rtmp7ZPnSc/downloaded_packages’ #### > library(coefplot2) Error in library(coefplot2) : there is no package called ‘coefplot2’ #### Downloaded the xcode app on Mac, it didn’t help.
Exercise
Try other combination of interactive terms
What are other variables?
crim — per capita crime rate by town
zn — proportion of residential land zoned for lots over 25,000 sq.ft
indus — proportion of non-retail business acres per town
chas — Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)
nox — nitric oxides concentration (parts per 10 million)
rm — average number of rooms per dwelling
age — proportion of owner-occupied units built prior to 1940
dis — weighted distances to five Boston employment centres
rad — index of accessibility to radial highways
tax — full-value property-tax rate per USD 10,000
ptratio — pupil-teacher ratio by town
black — proportion of blacks by town
lstat — percentage of lower status of the population
medv — median value of owner-occupied homes in USD 1000’s
Interacting lstat and rm
fit7=lm(medv~lstat*rm,Boston)
summary(fit7)##
## Call:
## lm(formula = medv ~ lstat * rm, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -23.2349 -2.6897 -0.6158 1.9663 31.6141
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -29.12452 3.34250 -8.713 <2e-16 ***
## lstat 2.19398 0.20570 10.666 <2e-16 ***
## rm 9.70126 0.50023 19.393 <2e-16 ***
## lstat:rm -0.48494 0.03459 -14.018 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.701 on 502 degrees of freedom
## Multiple R-squared: 0.7402, Adjusted R-squared: 0.7387
## F-statistic: 476.9 on 3 and 502 DF, p-value: < 2.2e-16