10001036

-- Diagram from Wiki

It is easier to imagine relation in three spatial vectors by their angles, than by their correlations. For standardized and s , , cosines of three angles of the triangular pyramid determinate the correlation matrix, thus, all statistics of the regressions and . Unexpected but imaginative results on the impact of introducing are --

1. Both s are nearly independent of . Togethor they predict almost perfectly ( and ).

2. Both s are almost perfectly correlated with . Togethor, one of the regressive coefficient is significantly negative (, and respectively).

3. Redundancy (Cohen, Cohen, West, & Aiken, 2003) increases to full and then decreases to zero and even negative (, and closes from to then to ).

cy1 <- 89; ## \angle YX_1 cy2 <- 89; ## \angle YX_2 c12 <- 177.9; ## \angle X_1X_2 N <- 100; rawdata=TRUE; ## S <- matrix(rep(1,9),3); S[1,2]<-S[2,1]<-cos(cy1/180*pi); S[1,3]<-S[3,1]<-cos(cy2/180*pi); S[2,3]<-S[3,2]<-cos(c12/180*pi); ## require(MASS);## install.packages('MASS'); ## x<-mvrnorm(n=N,mu=c(0,0,0),Sigma=S,empirical= TRUE); Y<-x[,1];X_1<-x[,2];X_2<-x[,3]; ## colnames(x)<-colnames(S)<-rownames(S)<-c('Y','X_1','X_2'); R2<-matrix(rep(NA,3),nrow=3); colnames(R2)<-c('R^2'); rownames(R2)<-c('Y = b_1*X_1 + e','Y =b_2*X_2 + e','Y =b_1*X_1 + b_2*X_2 + e'); lm1 <- lm(Y~0+X_1); lm2 <- lm(Y~0+X_2); lm12 <- lm(Y~0+X_1+X_2); R2[,1] <- c( summary(lm1)$r.squared, summary(lm2)$r.squared, summary(lm12)$r.squared); ## require(Rpad); require(R2HTML); HTMLon(); HTML(R2); R2[1,1]+R2[2,1]-R2[3,1] summary(lm1); summary(lm2); summary(lm12); cat('\ncorr') HTML(S); if (rawdata) { cat('\nraw data'); HTML(x); } HTMLoff();

--
Cohen, J., Cohen, P., West, S. G., & Aiken, L. S.  (2003). Applied multiple regression/correlation analysis for the behavioral sciences(3rd ed.) Mahwah, NJ: Lawrence Erlbaum Associates.