delivery<-read.csv(file="d:/chilo/regression 4/delivery.csv", header=t) delivery

Transcription

1 Regression Analysis lab 4 1 Model Adequacy Checking 1.1 Import data delivery<-read.csv(file="d:/chilo/regression 4/delivery.csv", header=t) delivery observation time cases distance Fit a multiple linear regression attach(delivery) dfit <- lm(time ~ cases + distance, data=delivery) summary(dfit) 1

2 Call: lm(formula = time ~ cases + distance, data = delivery) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) * cases e-09 *** distance *** --- Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 3.26 on 22 degrees of freedom Multiple R-squared: 0.96,Adjusted R-squared: F-statistic: 261 on 2 and 22 DF, p-value: 4.69e-16 names(dfit) [1] "coefficients" "residuals" "effects" "rank" [5] "fitted.values" "assign" "qr" "df.residual" [9] "xlevels" "call" "terms" "model" dfit$fit dfit$res # residuals names(summary(dfit)) 2

3 [1] "call" "terms" "residuals" "coefficients" [5] "aliased" "sigma" "df" "r.squared" [9] "adj.r.squared" "fstatistic" "cov.unscaled" sigmahat<-summary(dfit)$sigma sigmahat [1] sigmahat2<-sigmahat^2 sigmahat2 [1] Compute residuals e<-dfit$res e ## ## ## ## ## ## ## 25 ## Compute standardized residuals MSE<-sigmahat2 d<-e/sqrt(mse) d ## ## ## ## ## ## ## 25 ##

4 1.5 Compute hat matrix n<-length(delivery$time) n [1] 25 delivery[,-c(1,2)] cases distance X<-cbind(1,delivery[,-c(1,2)]) X 1 cases distance

5 X <- as.matrix(x) t(x) %*% X 1 cases distance cases distance XtXi <- solve(t(x) %*% X) XtXi 1 cases distance e e e-05 cases e e e-05 distance e e e-06 H<-X %*% XtXi %*% t(x) H [,1] [,2] [,3] [,4] [,5] [,6] [1,] e [2,] e [3,] e [4,] e [5,] e [6,] e [7,] e

6 [8,] e [9,] e [10,] e [11,] e [12,] e [13,] e [14,] e [15,] e [16,] e [17,] e [18,] e [19,] e [20,] e [21,] e [22,] e [23,] e [24,] e [25,] e [,7] [,8] [,9] [,10] [,11] [,12] [1,] [2,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [10,] [11,] [12,] [13,] [14,] [15,] [16,] [17,] [18,] [19,] [20,] [21,] [22,] [23,] [24,] [25,] [,13] [,14] [,15] [,16] [,17] [,18] [,19] 6

7 [1,] [2,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [10,] [11,] [12,] [13,] [14,] [15,] [16,] [17,] [18,] [19,] [20,] [21,] [22,] [23,] [24,] [25,] [,20] [,21] [,22] [,23] [,24] [,25] [1,] 3.796e [2,] e [3,] 5.821e [4,] e [5,] 1.257e [6,] 2.672e [7,] e [8,] 2.085e [9,] 2.049e [10,] 2.947e [11,] 9.233e [12,] 3.713e [13,] 7.011e [14,] 2.782e [15,] 4.318e [16,] 6.456e [17,] 1.501e [18,] 1.704e [19,] e

8 [20,] 1.017e [21,] 3.346e [22,] 1.518e [23,] 4.327e [24,] 4.697e [25,] 1.878e Compute internally studentized residuals h<-diag(h) h ## [1] ## [9] ## [17] ## [25] r<-e/sqrt(mse*(1-h)) r ## ## ## ## ## ## ## 25 ## Compute externally studentized residuals dmse<-((n-3)*mse-e^2/(1-h))/(n-3-1) t<-e/sqrt(dmse*(1-h)) t

9 t1<-rstudent(dfit) t Hypothesis testing for outliers qt(1-0.05/(2*25),n-3-1) [1] t[9] Compute PRESS and SSE ei<-e/(1-h) ei ei^ e e e e e e e e e e e e e e+00 9

10 e e e e e e e e e e e-02 PRESS<-sum(ei^2) PRESS [1] 459 SSE<-sum(e^2) SSE [1] Table 4.1 cbind(e,d,r,h,ei,t,ei^2) e d r h ei t e e e e e e e e e e e e e e e e e e e e e e e e e-02 10

11 2 Residual plots 2.1 Normal Q-Q plot of residuals dfit <- lm(time ~ cases + distance, data=delivery) qqnorm(residuals(dfit)) qqline(residuals(dfit)) Normal Q Q Plot Sample Quantiles Theoretical Quantiles 2.2 Normal Q-Q plot of residuals qqnorm(residuals(dfit)) qqline(residuals(dfit)) 11

12 Normal Q Q Plot Sample Quantiles Theoretical Quantiles # residuals are from a heavy-tailed distribution 2.3 Normal Q-Q plot of studentized residuals qqnorm(r, xlab="normal Quantiles", ylab="studentized residual Quantiles") qqline(r) 12

13 Normal Q Q Plot Studentized residual Quantiles Normal Quantiles 2.4 Residual plots plot(dfit$fit,dfit$res,xlab="fitted Value",ylab="Residual", pch=16) title(main="residual plot") abline(h=0) 13

14 Residual plot Residual Fitted Value plot(dfit$fit,r,xlab="fitted Value",ylab="Studentized residual", pch=16) title(main="residual plot") abline(h=0) 14

15 Residual plot Studentized residual Fitted Value plot(dfit$fit,r,xlab="fitted Value",ylab="Studentized residual", pch=16) title(main="residual plot") abline(h=0) identify(dfit$fit,r) 15

16 Residual plot Studentized residual Fitted Value integer(0) plot(dfit$fit,t,xlab="fitted Value", ylab="externally Studentized residual", pch=16) title(main="residual plot") mtext(side=3, line=0, text="residual vs. fitted") abline(h=0) 16

17 Residual plot residual vs. fitted Externally Studentized residual Fitted Value attach(delivery) The following objects are masked from delivery (position 3): cases, distance, observation, time plot(cases, dfit$res,xlab="cases",ylab="residual", pch=16) title(main="residual plot") mtext(side=3, line=0, text="residual vs. cases") abline(h=0) 17

18 Residual plot residual vs. cases Residual Cases plot(distance, dfit$res,xlab="distance",ylab="residual", pch=16) title(main="residual plot") mtext(side=3, line=0, text="residual vs. distance") abline(h=0) 18

19 Residual plot residual vs. distance Residual distance 2.5 Series plot ts.plot(dfit$res,xlab="observation number",ylab="residual") title(main="series plot") abline(h=0) 19

20 Series plot Residual observation number 2.6 Scatter plot lagres<-0 for(i in 1:24){lagres[i+1]=dfit$res[i]} lagres[1]<-na lagres [1] NA [8] [15] [22] cbind(dfit$res, lagres) lagres NA 20

21 plot(lagres, dfit$res,xlab="lagged residual",ylab="residual", pch=16) title(main="scatter plot") mtext(side=3, line=0, text="residual vs. lagged residual") 21

22 Scatter plot residual vs. lagged residual Residual lagged residual 2.7 Partial residual plot dfit <- lm(time ~ cases + distance, data=delivery) dfit$coef (Intercept) cases distance pres<-dfit$res+dfit$coef['cases']*cases plot(cases, pres, xlab="cases",ylab="time(adjusted)", pch=16) title(main="partial residual plot") 22

23 Partial residual plot Time(adjusted) Cases 2.8 Partial regression plot dfit1 <- lm(time ~ distance, data=delivery)$res dfit2 <- lm(cases ~ distance, data=delivery)$res plot(dfit2, dfit1, xlab="residual(cases distance)", ylab="residual(time distance)", pch=16) title(main="partial regression plot") 23

24 Partial regression plot residual(time distance) residual(cases distance) dfit3 <- lm(time ~ cases, data=delivery)$res dfit4 <- lm(distance ~ cases, data=delivery)$res plot(dfit4, dfit3, xlab="residual(distance cases)", ylab="residual(time cases", pch=16) title(main="partial regression plot") 24

25 Partial regression plot residual(time cases residual(distance cases) 2.9 Both Partial regression and Partial residual plots dfit <- lm(time ~ cases + distance, data=delivery) dfit$coef (Intercept) cases distance par(mfrow=c(1,2)) dfit1 <- lm(time ~ distance, data=delivery)$res dfit2 <- lm(cases ~ distance, data=delivery)$res plot(dfit2, dfit1, xlab="residual(cases distance)", ylab="residual(time distance)", pch=16) title(main="partial regression plot") abline(0,dfit$coef[2]) 25

26 dfit <- lm(time ~ cases + distance, data=delivery) dfit$coef (Intercept) cases distance pres<-dfit$res+dfit$coef[2]*cases plot(cases, pres, xlab="cases",ylab="time(adjusted)", pch=16) title(main="partial residual plot") abline(0,dfit$coef[2]) Partial regression plot Partial residual plot residual(time distance) Time(adjusted) residual(cases distance) Cases par(mfrow=c(1,1)) 3 Lack of fit test 26

27 xx<-c(1.0, 1.0, 2.0, 3.3, 3.3, 4.0, 4.0, 4.0, 4.7, 5.0, 5.6, 5.6, 5.6, 6.0, 6.0, 6.5, 6.9) yy<-c(10.84, 9.30, 16.35, 22.88, 24.35, 24.56, 25.86, 29.16, 24.59, 22.25, 25.90, 27.20, 25.61, 25.45, 26.56, 21.03, 21.46) g<-lm(yy ~ xx) summary(g) Call: lm(formula = yy ~ xx) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) *** xx ** --- Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 4.08 on 15 degrees of freedom Multiple R-squared: 0.487,Adjusted R-squared: F-statistic: 14.2 on 1 and 15 DF, p-value: anova(g) Analysis of Variance Table Response: yy Df Sum Sq Mean Sq F value Pr(>F) xx ** Residuals Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 plot(xx,yy, main="scatter plot") abline(g$coef) ga <- lm(yy ~ factor(xx)) points(xx, ga$fit, pch=18) 27

28 Scatter plot yy xx anova(g,ga) Analysis of Variance Table Model 1: yy ~ xx Model 2: yy ~ factor(xx) Res.Df RSS Df Sum of Sq F Pr(>F) ** --- Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 28