Listy Biometryczne-Biometrical Letters vol.36(1), 1999, pp.1-14

In a more geometrical fashion than by the customary matrix approach the analysis of lattice and lattice square designs is revisited. Pointing out canonical subspaces in treatment effects subspace contained in observation space of which any canonical vector c by orthogonal projection on block effects subspace followed by orthogonal projection on treatment space will be turned into lc, where 0 < l < 1 is the associated canonical value, forms a central theme. Best estimator of the treatment effect vector in intra-block analysis follows immediately, as well as the residual variance factors for estimated treatment pair differences after an advantageous reparameterization. Block effects within superblocks supposed to be random lead to a reformulation of using interblock information, and the value of all l will be reduced by a positive factor w smaller than 1 (or w_r and w_c for lattice squares), canonical spaces remaining unaltered. For exploring the required ratio of block variance(s) to residual variance under the normality assumption, application of REML, modified by Kitanidis to an iterative Gauss-Newton procedure extended with line search, is recommended in preference to relying on expected mean squares which may only provide initial values.

lattice design; canonical values and spaces, treatment reparameterization for pairwise comparison variances; recovery of interblock information; REML estimation of variance components; Gauss-Newton iteration; lattice square design