A generating algorithm for linear trend-free and nearly linear trend-free block designs

The theory and properties of trend-free (TF) and nearly trend-free (NTF) block designs are wel1 developed. Applications have been hampered because a methodology for design construction has not been available.

This article begins with a short review of concepts and properties of TF and NTF block designs. The major contribution is provision of an algorithm for the construction of linear and nearly linear TF block designs. The algorithm is incorporated in a computer program in FORTRAN 77 provided in an appendix for the IBM PC or compatible microcomputer, a program adaptable also to other computers. Three sets of block designs generated by the program are given as examples.

A numerical example of analysis of a linear trend-free balanced incomplete block design is provided.     

Some adjusted orthogonal,variance balanced,multidimensional block designs

A multidimensional block design (MBD) is an experimental design with d > 1 blocking criteria geometrically represented as a d-dimensional lattice with treatment varieties assigned to some or all nodes of the lattice. Intrablock analysis of variance tables for some special classes of two- and three-dimensional block designs with some empty nodes are given. Design plans and efficiencies for 31 two-dimensional designs, each universally optimal in defined classes of designs, and 7 three-dimensional designs, each nearly optimal in defined classes of designs, are listed in the appendices. A need for such designs is apparent when the blocking criteria are implemented successively and empty nodes do not represent wasted experimental units.     

Interblock information in multidimensional block designs

In many experimental situations, d-way heterogeneity among experimental units may be controlled through use of multiple blocking criteria. In some cases it is reasonable to regard some or all of the block effects as random. Then the model is mixed and observations within blocks are correlated. Very general estimators of treatment effects and their dispersion matrix with recovery of interblock information are provided. They apply to designs with d > 1 blocking criteria that may be crossed, nested, or a combination thereof. These general results may be specialized to provide analyses of new classes of MBD's or used directly for numerical analyses of designs in the general class, perhaps through use as the basis for very general computer programs. Estimation of variance components is discussed, and an example is provided to illustrate adaptation of the general results.