Regression Analysis lab 3. 1 Multiple linear regression. 1.1 Import data. 1.2 Scatterplot matrix
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1 Regression Analysis lab 3 1 Multiple linear regression 1.1 Import data delivery<-read.csv(file="d:/chilo/regression 3/delivery.csv", header=t) delivery observation time cases distance Scatterplot matrix plot(delivery[,-1], main=("scatter plot matrix")) 1
2 scatter plot matrix time cases distance D Scatter plot library(lattice) cloud(time ~ distance * cases, data = delivery, pch=1, cex=2, screen = list(z = 120, x = -90, y = -60)) 2
3 time cases distance 1.4 Compute estimated coefficients, fitted values and residuals by formula attach(delivery) n<-length(delivery$time) n [1] 25 delivery[,-c(1,2)] cases distance
4 X<-cbind(1,delivery[,-c(1,2)]) X 1 cases distance
5 y<-delivery$time y [1] [12] [23] 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 Xty<-t(X) %*% y Xty cases distance beta<-xtxi %*% t(x) %*% y beta cases distance
6 yhat<-x %*% beta yhat [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,] e<-y-yhat e [1,] [2,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [10,] [11,]
7 [12,] [13,] [14,] [15,] [16,] [17,] [18,] [19,] [20,] [21,] [22,] [23,] [24,] [25,] cbind(y,yhat,e) y [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,]
8 1.5 Compute the estimate of variance yty<-t(y) %*% y yty [1,] t(beta) %*% t(x) %*% y [1,] SSE<-yty-t(beta) %*% t(x) %*% y SSE [1,] sigmahat2<-sse/(n-3) sigmahat2 [1,] sigmahat2<-sum(e^2)/(n-3) sigmahat2 [1] Fit a multiple linear regression attach(delivery) The following objects are masked from delivery (position 3): cases, distance, observation, time dfit <- lm(time ~ cases + distance, data=delivery) summary(dfit) Call: lm(formula = time ~ cases + distance, data = delivery) 8
9 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 dfit$fit dfit$res sigmahat2<-sum(dfit$res^2)/(n-3) sigmahat2 [1] Compute confidence intervals by formula 9
10 qt(.975, n-3) [1] # 95% C.I. for beta2 # beta2<-beta[3,1] beta2 distance SE.beta2<-sqrt(sigmahat2*XtXi[3,3]) SE.beta2 [1] qt(.975, n-3) [1] c(beta2-qt(.975, n-3)*se.beta2, beta2+qt(.975, n-3)*se.beta2) distance distance # 95% C.I. for beta1 # beta1<-beta[2,1] beta1 cases SE.beta1<-sqrt(sigmahat2*XtXi[2,2]) SE.beta1 [1] qt(.975, n-3) [1] c(beta1-qt(.975, n-3)*se.beta1, beta1+qt(.975, n-3)*se.beta1) cases cases
11 # 95% C.I. for beta0 # beta0<-beta[1,1] beta SE.beta0<-sqrt(sigmahat2*XtXi[1,1]) SE.beta0 [1] qt(.975, n-3) [1] c(beta0-qt(.975, n-3)*se.beta0, beta0+qt(.975, n-3)*se.beta0) Confidence intervals for model parameters dfit <- lm(time ~ cases + distance, data=delivery) confint(dfit, level=0.95) 2.5 % 97.5 % (Intercept) cases distance Compute confidence interval for variance MSE<-sum(dfit$res^2)/(n-3) qchisq(.975,n-3) [1] qchisq(.025,n-3) [1] c((n-3)*mse/qchisq(.975,n-3),(n-3)*mse/qchisq(.025,n-3)) [1]
12 1.10 Compute confidence interval for the mean value of y when x=x0 x0<-c(1,8,275) muhat<-t(x0) %*% beta muhat [1,] qt(.975, n-3) [1] bm<-qt(.975, n-3)*sqrt(mse*(t(x0) %*% XtXi %*%x0)) bm [1,] 1.57 c(muhat-bm, muhat+bm) [1] Compute prediction interval for individual value of y when x=x0 x0<-c(1,8,275) y0<-t(x0) %*% beta y0 [1,] qt(.975, n-3) [1] bm<-qt(.975, n-3)*sqrt(mse*(1+t(x0) %*% XtXi %*%x0)) bm [1,] 6.94 c(y0-bm, y0+bm) [1]
13 1.12 Compute ANOVA by formula sumy<-sum(delivery$time) SST<-yty-(sumy)^2/n SST [1,] 5785 SSR<-t(beta) %*% t(x) %*% y-(sumy)^2/n SSR [1,] 5551 SSE<-SST-SSR SSE [1,] MSR<-SSR/2 MSR [1,] 2775 MSE<-SSE/(n-3) MSE [1,] Fratio<-MSR/MSE Fratio [1,] qf(0.95,2,n-3) [1] pvalue<-1-pf(fratio,2,n-3) pvalue [1,] 4.441e-16 13
14 1.13 Compute ANOVA table dfit <- lm(time ~ cases + distance, data=delivery) anova(dfit) Analysis of Variance Table Response: time Df Sum Sq Mean Sq F value Pr(>F) cases < 2e-16 *** distance *** Residuals Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' Compute coefficient of determination R2 and adjusted R2 dfit <- lm(time ~ cases + distance, data=delivery) summary(dfit) 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 R2<-SSR/SST R2 14
15 [1,] adjr2<-1-(n-1)/(n-3)*(1-r2) adjr2 [1,] t test for individual beta # Hypothesis testing for beta2 # beta2<-beta[3,1] beta2 distance SE.beta2<-sqrt(MSE*XtXi[3,3]) SE.beta2 [1,] t2<-beta2/se.beta2 t2 [1,] pvalue2<-2*(1-pt(abs(t2),n-3)) pvalue2 [1,] # Hypothesis testing for beta1 # beta1<-beta[2,1] beta1 cases
16 SE.beta1<-sqrt(MSE*XtXi[2,2]) SE.beta1 [1,] t1<-beta1/se.beta1 t1 [1,] pvalue1<-2*(1-pt(abs(t1),n-3)) pvalue1 [1,] 3.255e-09 # Hypothesis testing for beta0 # beta0<-beta[1,1] beta SE.beta0<-sqrt(MSE*XtXi[1,1]) SE.beta0 [1,] t0<-beta0/se.beta0 t0 [1,] pvalue0<-2*(1-pt(abs(t0),n-3)) pvalue0 [1,] Partial F test for individual beta 16
17 dfit1 <- lm(time ~ cases + distance, data=delivery) # Full model dfit2 <- lm(time ~ cases, data=delivery) # Reduced model anova(dfit2,dfit1) Analysis of Variance Table Model 1: time ~ cases Model 2: time ~ cases + distance Res.Df RSS Df Sum of Sq F Pr(>F) *** --- Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' Simultaneous confidence intervals for model parameters rocket<-read.csv(file="d:/chilo/regression 2/rocket.csv", header=t) rocket observation strength age attach(rocket) 17
18 The following object is masked from delivery (position 3): observation The following object is masked from delivery (position 4): observation n<-length(age) n [1] 20 rocket.lm<-lm(strength~age, data=rocket) summary(rocket.lm) Call: lm(formula = strength ~ age, data = rocket) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** age e-10 *** --- Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 96.1 on 18 degrees of freedom Multiple R-squared: 0.902,Adjusted R-squared: F-statistic: 165 on 1 and 18 DF, p-value: 1.64e-10 xbar<-mean(age) xbar [1] sumx<-sum(age) sumx [1] sumx2<-sum(age^2) sumx2 [1]
19 Sxx<-sumx2-(sumx)^2/n Sxx [1] 1107 beta1<-coef(rocket.lm)[2] beta1 age beta0<-coef(rocket.lm)[1] beta0 (Intercept) 2628 MSE<-sum((resid(rocket.lm))^2)/(n-2) MSE [1] 9236 SE.beta1<-sqrt(MSE/Sxx) SE.beta1 [1] SE.beta0<-sqrt(MSE*(1/n+xbar^2/Sxx)) SE.beta0 [1] #Bonferroni's method qt(1-.025/2, n-2) [1] # 95% joint C.I. for beta0 and beta1 # c(beta0-qt(1-.025/2, n-2)*se.beta0, beta0+qt(1-.025/2, n-2)*se.beta0) (Intercept) (Intercept) c(beta1-qt(1-.025/2, n-2)*se.beta1, beta1+qt(1-.025/2, n-2)*se.beta1) age age
20 #Scheffe S-method dd<-(2*qf(1-.025, 2, n-2))^0.5 dd [1] 3.02 # 95% joint C.I. for beta0 and beta1 # c(beta0-dd*se.beta0, beta0+dd*se.beta0) (Intercept) (Intercept) c(beta1-dd*se.beta1, beta1+dd*se.beta1) age age #Confidence region for beta0 and beta1 install.packages("ellipse") Error: trying to use CRAN without setting a mirror library(ellipse) Warning: package ellipse was built under R version ellipse(rocket.lm, which=c(1,2), level=.95) (Intercept) age [1,] [2,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [10,] [11,] [12,] [13,] [14,] [15,] [16,] [17,]
21 [18,] [19,] [20,] [21,] [22,] [23,] [24,] [25,] [26,] [27,] [28,] [29,] [30,] [31,] [32,] [33,] [34,] [35,] [36,] [37,] [38,] [39,] [40,] [41,] [42,] [43,] [44,] [45,] [46,] [47,] [48,] [49,] [50,] [51,] [52,] [53,] [54,] [55,] [56,] [57,] [58,] [59,] [60,] [61,] [62,]
22 [63,] [64,] [65,] [66,] [67,] [68,] [69,] [70,] [71,] [72,] [73,] [74,] [75,] [76,] [77,] [78,] [79,] [80,] [81,] [82,] [83,] [84,] [85,] [86,] [87,] [88,] [89,] [90,] [91,] [92,] [93,] [94,] [95,] [96,] [97,] [98,] [99,] [100,] plot(ellipse(rocket.lm), type = 'l') points(beta0,beta1,pch=18) 22
23 age (Intercept) 23
Figure 1: The fitted line using the shipment route-number of ampules data. STAT5044: Regression and ANOVA The Solution of Homework #2 Inyoung Kim
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