/pdf/optimal-balanced-fractional-3-spm-factorial-designs-of-4cw1dip2fp.pdf

Optimal balanced fractional $3\spm$ factorial designs of resolution ${\rm V}$ and balanced third-order designs

Combinatorialists often consider a balanced incomplete-block design to consist of a set of points, a set of blocks, and an incidence relation between them which satisfies certain conditions. To a statistician, such a design is a set of experimental units with two partitions, one into blocks and the other into treatments; it is the relation between these two partitions which gives the design its properties. The most common binary relations between partitions that occur in statistics are refinement, orthogonality and balance. When there are more than two partitions, the binary relations may not suffice to give all the properties of the system. I shall survey work in this area, including designs such as double Youden rectangles.

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Relations among partitions

Suppose that t experiments are conducted simultaneously on the same set of experimental units. For example, suppose that t mutually orthogonal latin square experiment designs are used for the t experiments on n2 experimental units. Statistical literature is voluminous on construction of such designs, but contains relatively little and incomplete results on statistical analyses for such designs. Six statistical analyses are presented for a pair of orthogonal latin square experiment designs. Then, the methods are generalized for t mutually orthogonal experiment designs. The results are also extended to a set of t mutually balanced Youden experiment designs.

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Statistical Analyses for Multistage Experiment Designs

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Statistical Analyses for Multistage Designs

Some results are given on the form of the information matrices, canonical efficiency factors and average efficiencies for various classes of row-column designs. All the designs considered are structurally incomplete, that is not all cells of the design are assigned treatments (or treatment combinations), and the designs have column balance, that is every pair of columns in the design intersects rows in exactly f non-empty cells, where f is independent of the choice of columns. An upper bound is given for the average efficiency factor of a row-column design which has adjusted orthogonality and is arranged as a square layout with one empty cell in each row and column.

Structurally incomplete row-column designs with adjusted orthogonality

A general construction is given for m-way completely variance balanced designs where each factor has v levels, m is any integer less than or equal to k, and N = vk, where k = 2λ+1 and v = 4λ3 is a prime power. The construction gives rise to a variety of designs, easily enumerated, with the same parameters pairwise but with differing variance properties. For m = 3 there are only two distinct designs possible, and their relative efficiency is shown to be .

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W.T. Federer

Papers

Multidimensional Balanced Designs