2. Regression and Correlation. Simple Linear Regression Software: R

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1 2. Regression and Correlation Simple Linear Regression Software: R Create txt file from SAS data set data _null_; file 'C:\Documents and Settings\sphlab\Desktop\slr1.txt'; set temp; put input day:date7. calls fhigh flow high low rain snow weekday year sunday subzero; run; ##### You need to delete the dot signs at the beginning of each line######## 1.Read in data from text file data<-read.table("c:/documents and Settings/liyuan/Desktop/640TA/slr2.txt",header=T) attach(data) 2. Partical listing of output list(data) [[1]] day calls fhigh flow high low rain snow weekday year sunday subzero Plot of calls over time par(mfrow=c(2,2)) plot(day,calls, xlim=c(12000,12500), ylim=c(1000,9000), xlab= Day,ylab= Calls, main= Calls to NY Auto Club ,col= black ) \R_howto\simple linear regression ny auto club.doc Page 1 of 8

Read in data from text file data<-read.table("c:/documents and Settings/liyuan/Desktop/640TA/slr2.txt",header=T) attach(data) 2.

2 4. Tests of Assumption of Normality on Y=calls > mean(calls) [1] > length(calls) [1] 28 >sum(calls) [1] >var(calls) [1] > sum(calls^2) ##uncorrected ss## [1] > sum(((calls-mean(calls))^2) ) ##corrected ss## [1] > 100*sd(calls)/mean(calls) ##Coefficient of variation## [1] > sd(calls)/sqrt(length(calls)) ##standard error mean## [1] \R_howto\simple linear regression ny auto club.doc Page 2 of 8

717992159 > sum(((calls-mean(calls))^2) ) ##corrected ss## [1] 195747315 > 100*sd(calls)/mean(calls)

3 ##########the package fbasic should be installed first for the following function####### > skewness(calls) [1] attr(,"method") [1] "moment" > kurtosis(calls) [1] attr(,"method") [1] "excess ######the packages nortest and stats should be installed first for the following function####### >shapiro.test(calls) Shapiro-Wilk normality test data: calls W = 0.829, p-value = > cvm.test(calls) Cramer-von Mises normality test data: calls W = , p-value = > ad.test(calls) Anderson-Darling normality test data: calls A = , p-value = 6.68e Graphical Assessments of Normality of Y=calls Histogram with overlay normal hist(calls,col='lightblue', main='histogram of calls', breaks=5, include.lowest = TRUE, right = TRUE,freq=F) points(calls,dnorm(calls,mean=mean(calls),sd=sqrt(var(calls))),col='red',lty=6) \R_howto\simple linear regression ny auto club.doc Page 3 of 8

829, p-value = 0.0003628 > cvm.test(calls) Cramer-von Mises normality test data: calls W = 0.3112, p-value = 0.0002141 > ad.test(calls) Anderson-Darling normality test data: calls A = 1.

4 Quantile Quantile Plot qqnorm(calls,datax=true, main= Simple Normal QQplot for Y=calls, ylab= Calls, xlab= Normal quantiles ) qqline(calls,datax=true) \R_howto\simple linear regression ny auto club.doc Page 4 of 8

$\R_howto\simple linear regression ny auto club.$

5 qqnorm(calls,datax=true, main= Simple Normal QQplot for Y=calls ) qqline(calls,datax=true) Simple Normal QQplot for Y=calls Theoretical Quantiles Sample Quantiles 6.Scatterplot of Y=Calls vs X=low calls0<-calls[year==0] calls1<-calls[year==1] low0<-low[year==0] low1<-low[year==1] plot(low0,calls0, main="calls to NY Auto Club ",xlim=c(-10,50),ylim=c(1000,9000), xlab="low", ylab="calls", col= green ) points (low1,calls1, col="red") legend (35,9000, c( "1993:green","1994:red"), col=c("red","green") ) \R_howto\simple linear regression ny auto club.doc Page 5 of 8

Scatterplot of Y=Calls vs X=low calls0<-calls[year==0] calls1<-calls[year==1] low0<-low[year==0] low1<-low[year==1] plot(low0,calls0, main="calls to NY

6 7. Least Squares Estimation and Analysis of Variance Table lm1<-lm(calls~low) summary(lm1) coef(lm1) nova(lm1) Call: lm(formula = calls ~ low) Residuals: Min 1Q Median 3Q Max Parameter Estimates Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-11 *** low e-05 *** Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 1917 on 26 degrees of freedom Multiple R-Squared: , Adjusted R-squared: F-statistic: on 1 and 26 DF, p-value: 1.865e-05 Analysis of Variance Response: calls Df Sum Sq Mean Sq F value Pr(>F) low e-05 *** Residuals \R_howto\simple linear regression ny auto club.doc Page 6 of 8

001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 1917 on 26 degrees of freedom Multiple R-Squared: 0.5121, Adjusted R-squared: 0.4933 F-statistic: 27.28 on 1 and 26 DF, p-value: 1.

7 8. Overlay of straight line fit onto scatterplot of Y=calls vs X=low abline(lm1) 9. Residuals analysis-assessment of Normality of Residuals qqnorm(lm1$residuals, main="normality of Residuals Y=CALLS v X=LOW") 10. Residuals Analysis Detection of Outliers Using Cook s Distance \R_howto\simple linear regression ny auto club.doc Page 7 of 8

$Residuals Analysis Detection of Outliers Using Cook s Distance \R_howto\simple$

8 plot.lm(lm1,which=4, main= Cook s Distance Values for Straight Line Y=Calls v X=Low ) 10. Residuals Analysis Detection of Outliers Using Cook s Distance Diag<- ls.diag(lm1) plot(lm1$fitted,diag$stud.res,ylim=c(-2.0,2.5),xlab="predicted Value",ylab="Studentized Residual",main="Jacknife Residuals versus Predicted") abline(h=0,lty=c(3)) Jacknife Residuals versus Predicted Studentized Residual Predicted Value \R_howto\simple linear regression ny auto club.doc Page 8 of 8

$Residuals versus Predicted Studentized Residual -2-1 0 1 2 2000 3000 4000 5000 6000 7000 8000 Predicted Value \R_howto\simple$

Statistical Models in R

Statistical Models in R Some Examples Steven Buechler Department of Mathematics 276B Hurley Hall; 1-6233 Fall, 2007 Outline Statistical Models Linear Models in R Regression Regression analysis is the appropriate