Confidence interval example Stat 542 Peter Hoff University of - PowerPoint PPT Presentation

Copula modeling Confidence interval example Stat 542 Peter Hoff University of Washington

Copula modeling Expression data from leukemia patients 10 ● ● ● 8 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● X[,2] ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● 2 4 6 8 10 12 X[,1]

Copula modeling Summary statistics n<-dim(X)[1] ; p<-dim(X)[2] xbar<-apply(X,2,mean) Sig<-cov(X) > n [1] 143 > xbar [1] 6.598292 4.043136 > Sig [,1] [,2] [1,] 4.653583 3.076522 [2,] 3.076522 4.353581

Copula modeling Univariate tests and CIs xint<-t.test(X[,1])$conf.int yint<-t.test(X[,2])$conf.int > xint [1] 6.241685 6.954900 > yint [1] 3.698214 4.388057

Copula modeling Univariate confidence region - not simultaneous 10 ● ● ● 5.5 ● ● ● ● ● ● ● ● ● ● 8 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 4.5 ● ● ● 6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● X[,2] ● ● ● ● X[,2] ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 3.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2 4 6 8 10 12 5 6 7 8 X[,1] X[,1]

Copula modeling Simultaneous confidence ellipse fq<-qf(c(.25,.5,.75,.9,.95),p,n-p) sc<- (n-1)*p/(n-p) # about p tq<- sc*fq Smh<-eigen(cov(X)/n) Smh<-Smh$vec%*%diag(sqrt(Smh$val))%*%t(Smh$vec) for(r in k) { theta<-seq(0,2*pi,length=100) z<-sqrt(r)*cbind(sin(theta),cos(theta)) x<- t( t(z%*%Smh)+xbar ) lines(x,col="green") }

Copula modeling Simultaneous confidence ellipse 10 ● ● ● 5.5 ● ● ● ● ● ● ● ● ● ● 8 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 4.5 ● ● ● 6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● X[,2] ● ● ● ● X[,2] ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 3.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2 4 6 8 10 12 5 6 7 8 X[,1] X[,1]

Copula modeling Simultaneous prediction ellipse 10 ● ● ● 8 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● X[,2] ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● 2 4 6 8 10 12 X[,1]