Interpreting model matrix columns when using contr.sum
With the following example using contr.sum for both factors,
dd <- data.frame(a = gl(3,4), b = gl(4,1,12)) # balanced 2-way model.matrix(~ a * b, dd, contrasts = list(a="contr.sum", b="contr.sum"))
(Intercept) a1 a2 b1 b2 b3 a1:b1 a2:b1 a1:b2 a2:b2 a1:b3 a2:b3 1 1 1 0 1 0 0 1 0 0 0 0 0 2 1 1 0 0 1 0 0 0 1 0 0 0 3 1 1 0 0 0 1 0 0 0 0 1 0 4 1 1 0 -1 -1 -1 -1 0 -1 0 -1 0 5 1 0 1 1 0 0 0 1 0 0 0 0 6 1 0 1 0 1 0 0 0 0 1 0 0 7 1 0 1 0 0 1 0 0 0 0 0 1 8 1 0 1 -1 -1 -1 0 -1 0 -1 0 -1 9 1 -1 -1 1 0 0 -1 -1 0 0 0 0 10 1 -1 -1 0 1 0 0 0 -1 -1 0 0 11 1 -1 -1 0 0 1 0 0 0 0 -1 -1 12 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 ... I have two questions: (1) I assume the 1st column (under intercept) is the overall mean, the 2rd column (under a1) is the difference between the 1st level of factor a and the overall mean, the 4th column (under b1) is the difference between the 1st level of factor b and the overall mean. Is this interpretation correct? (2) I'm not so sure about those interaction columns. For example, what is a1:b1? Is it the 1st level of factor a at the 1st level of factor b versus the overall mean, or something more complicated? Thanks in advance for your help, Gang