Hey,
I try to understand a basic statistic question. I am modelling a linear
regression Y=X*B+E. To compute the effect of ?group? the B-values of the
regressors/columns that code the interaction effects (col. 5-8 and col. 11-14,
see below) have to be weighted with non-zero elements within the contrast
"Group 1" minus "Group 2" (see below). My first
understanding was that the interaction effects add up to zero in each group. Why
do they get non-zero contrast weights? Do you recommend any literature
discussing this topic?
Many Thanks,
Stefan
The columns of matrix X dummy-codes
1. Data of Group1 (G1)
2. Data of Group2 (G2)
3. Data condition A1 (factor A)
4. Data condition A2 (factor A)
5. Data in cell A1G1
6. Data in cell A2G1
7. Data in cell A1G2
8. Data in cell A2G2
9. Data condition B1 (factor B)
10. Data condition B2 (factor B)
11. Data in cell B1G1
12. Data in cell B2G1
13. Data in cell B1G2
14. Data in cell B2G2
15. Mean
Example:
#create X
dimnames = list( paste("sub",c(1:8)),
c("Group1","Group2","A1","A2","A1G1","A2G1",
"A1G2","A2G2","B1","B2",
"B1G1","B2G1",
"B1G2","B2G2","mean" ))
X=matrix(nrow=8,ncol=15,dimnames=dimnames)
X[,1]= c(rep(1,4), rep(0,4))
X[,2]=c(rep(0,4),rep(1,4))
X[,3]=c(1,1,0,0,1,1,0,0);
X[,4]=c(0,0,1,1,0,0,1,1);
X[,5]=c(1,1,0,0,0,0,0,0);
X[,6]=c(0,0,1,1,0,0,0,0);
X[,7]=c(0,0,0,0,1,1,0,0);
X[,8]=c(0,0,0,0,0,0,1,1);
X[,9]=c(1, 0, 1, 0, 1, 0,1, 0);
X[,10]=c(0, 1, 0, 1, 0, 1, 0, 1);
X[,11]=c(1, 0, 1, 0,0, 0, 0, 0);
X[,12]=c( 0, 1, 0, 1,0, 0, 0, 0);
X[,13]=c( 0, 0, 0, 0, 1, 0, 1, 0)
X[,14]=c(0, 0, 0, 0, 0, 1, 0, 1)
X[,15]=c(1,1,1,1,1,1,1,1)
#y data
Y=c(1.2,5.1,1.7,7,6.1,8,1.1,2.5)
library(MASS)
ginv()
B=(ginv(t(X)%*%X) %*% t(X)) %*% Y
#contrast
c1=rep(0,15)
c1[1]=1
c1[c(5,6,11,12)]=1/2
c2=rep(0,15)
c2[2]=-1
c2[c(7,8,13,14)]=-1/2
#Group effect
mean(Y[1:4])-mean(Y[5:8])
c1%*%B+c2%*%B
--