Commutative orthogonal block structure and error orthogonal models

A model has orthogonal block structure, OBS, if it has variance-covariance matrix that is a linear combination of known pairwise orthogonal orthogonal projection matrices that sum to the identity matrix. These models were introduced by Nelder is 1965, and continue to play an important part in randomized block designs. Two important types of OBS are related, and necessary and sufficient conditions for model of one type belonging to the other are determined. The first type, is that of models with commutative orthogonal block structure in which T, the orthogonal projection matrix on the space spanned by the mean vector, commutes with the orthogonal projection matrices in the expression of the variance-covariance matrix. The second type, is that of error orthogonal models. These results open the possibility of deepening the study of the important class of models with OBS.