## missing values in RBD ## table 5.8, page 136 y1<-c(8.4,12.8,9.6,9.8,8.4,8.6,8.9,7.9) y2<-c(9.4,15.2,9.1,8.8,8.2,9.9,NA,8.1) y3<-c(9.8, 12.9, 11.2, 9.9, 8.5, 9.8, 9.2, 8.2) y4<-c(12.2, 14.4, 9.8, 12.0,8.5,10.9, 10.4, 10) ################################### # data augmentation before modeling ################################### y2[7]<-9.14 cbind(y1,y2,y3,y4) ## table 5.9 y<-c(y1,y2,y3,y4) trt<-rep(1:4,rep(8,4)) subj<-rep(1:8,4) > fit0<-lm(y~factor(subj)+factor(trt)) > anova(fit0) Df Sum Sq Mean Sq F value Pr(>F) factor(subj) 7 78.820 11.260 17.1841 2.219e-07 *** factor(trt) 3 12.939 4.313 6.5823 0.002612 ** Residuals 21 13.760 0.655 ## ANOVA without adjusting for missing value fit1<-lm(y~factor(trt)+factor(subj)) anova(fit1) Df Sum Sq Mean Sq F value Pr(>F) factor(trt) 3 12.939 4.313 6.5823 0.002612 ** factor(subj) 7 78.820 11.260 17.1841 2.219e-07 *** Residuals 21 13.760 0.655 ## ANOVA without adjusting for missing value ############################# # leave missing values as NAs ############################# y2[7]<-NA y<-c(y1,y2,y3,y4) fit2<-lm(y~factor(trt)+factor(subj),na.action=na.omit) Df Sum Sq Mean Sq F value Pr(>F) factor(trt) 3 12.577 4.192 6.0934 0.00406 ** factor(subj) 7 78.422 11.203 16.2832 5.413e-07 *** Residuals 20 13.760 0.688 ## SSBl here is the same as SSBl(adj) with data augmentation fit3<-lm(y~factor(subj)+factor(trt),na.action=na.omit) Df Sum Sq Mean Sq F value Pr(>F) factor(subj) 7 78.157 11.165 16.2282 5.565e-07 *** factor(trt) 3 12.842 4.281 6.2218 0.00369 ** Residuals 20 13.760 0.688 ## SSTr here is the same as SSTr(adj) with data augmentation ## The Sum. Sq.'s here are sequential > contrasts(factor(trt)) 2 3 4 1 0 0 0 2 1 0 0 3 0 1 0 4 0 0 1 #treatment contrasts, trt=1 is the comparison group ### to match the top portion of table 5.12 (p.142) newtrt<-ifelse(trt==4,0,trt) fit4<-lm(y~factor(subj)+factor(newtrt),na.action=na.omit) summary(fit4,correlation=T) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 10.9768 0.4874 22.522 1.11e-15 *** factor(subj)2 3.8750 0.5865 6.607 1.96e-06 *** factor(subj)3 -0.0250 0.5865 -0.043 0.966424 factor(subj)4 0.1750 0.5865 0.298 0.768500 factor(subj)5 -1.5500 0.5865 -2.643 0.015612 * factor(subj)6 -0.1500 0.5865 -0.256 0.800760 factor(subj)7 -0.5393 0.6400 -0.843 0.409361 factor(subj)8 -1.4000 0.5865 -2.387 0.026979 * factor(newtrt)1 -1.7250 0.4147 -4.159 0.000485 *** factor(newtrt)2 -1.2946 0.4340 -2.983 0.007355 ** factor(newtrt)3 -1.0875 0.4147 -2.622 0.016324 * Correlation of Coefficients: (Intercept) factor(subj)2 factor(subj)3 factor(subj)4 factor(subj)2 -0.60 factor(subj)3 -0.60 0.50 factor(subj)4 -0.60 0.50 0.50 factor(subj)5 -0.60 0.50 0.50 0.50 factor(subj)6 -0.60 0.50 0.50 0.50 factor(subj)7 -0.58 0.46 0.46 0.46 factor(subj)8 -0.60 0.50 0.50 0.50 factor(newtrt)1 -0.43 0.00 0.00 0.00 factor(newtrt)2 -0.43 0.00 0.00 0.00 factor(newtrt)3 -0.43 0.00 0.00 0.00 factor(subj)5 factor(subj)6 factor(subj)7 factor(subj)8 factor(subj)2 factor(subj)3 factor(subj)4 factor(subj)5 factor(subj)6 0.50 factor(subj)7 0.46 0.46 factor(subj)8 0.50 0.50 0.46 factor(newtrt)1 0.00 0.00 0.00 0.00 factor(newtrt)2 0.00 0.00 0.12 0.00 factor(newtrt)3 0.00 0.00 0.00 0.00 factor(newtrt)1 factor(newtrt)2 factor(subj)2 factor(subj)3 factor(subj)4 factor(subj)5 factor(subj)6 factor(subj)7 factor(subj)8 factor(newtrt)1 factor(newtrt)2 0.48 factor(newtrt)3 0.50 0.48