where represents a diagonal matrix with (matrix) elements on the leading diagonal, each b equal to , a matrix of all 1s (which is also ). t×t To see whether the assumption random blocks affects the estimation of the fixed effects - recall that - we need to look at the matrix for the RCB design: H The inverse of exists and will clearly be a block diagonal matrix with diagonal matrices each being the inverse of . There is a standard formula for such a matrix. Let be a nonsingular matrix and let both and be column vectors. So, here we have , , . is a diagonal matrix composed of such matrices. b Next we look at the individual matrices and for random blocks. Now with our parameterization of the design matrix has columns each of which contains X b cells containing 1 and the remaining cells contain 0. Also, an entry of 1 is unique in any row, so must equal . Next, for each block in the diagonal matrix above, simply adds the numbers in the columns of the design matrix in the block under consideration. However, in that (and every) block, X
where represents a diagonal matrix with b (matrix) elements on the leading diagonal, each equal to , a t×t matrix of all 1s (which is also ). To see whether the assumption random blocks affects the estimation of the fixed effects - recall that - we need to look at the matrix H for the RCB design: The inverse of exists and will clearly be a block diagonal matrix with diagonal matrices each being the inverse of . There is a standard formula for such a matrix. Let be a nonsingular matrix and let both and be column vectors. So, here we have , , . is a diagonal matrix composed of b such matrices. Next we look at the individual matrices and for random blocks. Now with our parameterization of the design matrix X has columns each of which contains b cells containing 1 and the remaining cells contain 0. Also, an entry of 1 is unique in any row, so must equal . Next, for each block in the diagonal matrix above, simply adds the numbers in the columns of the design matrix X in the block under consideration. However, in that (and every) block,
An%20introduction%20to%20REML.docx