THE ORTHOGONAL PROJECTION MATRIX FOR THE UNBALANCED FACTORIAL DESIGN
Byung Chun Kim, Jong‐Tae Park · Journal of the Japanese Society of Computational Statistics · 1992
The design matrix X for an unbalanced factorial design can be expressed as a product of two matrices T and Xo, where T is replication matrix and Xo 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 Xo'Xo. Also we can easily find a regular rule for the nonzero eigenvalues and the corresponding eigenvectors.