The specification of rank reducing observation sets in experimental design

Improved conditions for the robustness of binary block designs against the loss of whole blocks

“…Godolphin (2006) pointed out that block designs that are equally efficient are not necessarily equally vulnerable to observation loss, and a similar conclusion can be anticipated if there is a tendency for whole blocks to be invalidated. It is sensible to concentrate on the concurrence or weighted concurrence between any pair of treatments and compare κ * or λ * with the bounds given in Theorems 5 or 6.…”

“…Researchers have investigated methods for guarding against a disconnected design because of observation loss during the experiment; see the survey of methods given by Godolphin (2004Godolphin ( , 2006 which considers cases where it is not possible to anticipate beforehand which observations may be missing. However, there are several experiments where the loss of a whole block is more likely than the loss of an individual observation; for example if the experimental units are the leaves on a plant then any damage to the plant will invalidate the block.…”

New formulations for recursive residuals as a diagnostic tool in the fixed-effects linear model with design matrices of arbitrary rank

“…It is suggested that the Z matrix be computed as part of the standard output since it may have additional value for examining the suitability of the model, in view of the definitive properties obtained for Z, both in the present paper and in the work of Godolphin (2006) and elsewhere, and it is reasonable to suppose that other useful properties of the Theil factorization may emerge from further consideration.…”

“…It also seems likely The derivation of the n − r columns of Z is an essential preliminary step for the formulation of recursive residuals and diagnostic procedures based upon them. This is, however, achieved in a straightforward way; see Godolphin (2006). Indeed, it is interesting to note that the Theil factorization (2.4) of the design matrix, on which these results are based, does not depend on the computation of eigenvalues or eigenvectors of X X as is required, for example, by the singular value decomposition; see Belsley et al (1980, §3.2).…”

“…Godolphin (2006) showed that all rank reducing observation sets of Types I-III can be determined from an examination of the matrix (2.3). Therefore it is interesting to ask if a useful representation for the recursive residuals can be found from Z.…”