THE ORTHOGONAL PROJECTION MATRIX FOR THE UNBALANCED FACTORIAL DESIGN

The design matrix X for an unbalanced factorial design can be expressed as a product of two matrices T and X_o, where T is replication matrix and X_o is the corresponding balanced design matrix. We suggest an efficient method for computing the orthogonal projection matrix Px=X(X'X)^-X' for an unbalanced model y=Xβ+e using the nonzero eigenvalues and eigenvectors of X'_oX_o. Also we can easily find a regular rule for the nonzero eigenvalues and the corresponding eigenvectors.