A Unified Approach to Factorial Designs with Randomization Restrictions

Minimum-aberration two-level fractional factorial split-plot designs

Abstract: It is often impractical to perform the experimental runs of a fractional factorial in a completely random order, In these cases, restrictions on the randomization of the experimental trials are imposed and the design is said to have a split-plot structure. We rank these fractional factorial split-plot designs similarly to fractional factorials using the aberration criterion to find the minimum-aberration design. We introduce an algorithm that constructs the set of all nonisomorphic two-level fractional factorial split-plot designs more efficiently than existing methods. The algorithm can be easily modified to efficiently produce sets of all nonisomorphic fractional factorial designs, fractional factorial designs in which the number of levels is a power of a prime, and fractional factorial split-plot designs in which the number of levels is a power of a prime.

Designing fractional factorial split-plot experiments with few whole-plot factors

Abstract: When it is impractical to perform the experimental runs of a fractional factorial design in a completely random order, restrictions on the randomization can be imposed. The resulting design is said to have a split-plot, or nested, error structure. Similarly to fractional factorials, fractional factorial split-plot designs can be ranked by using the aberration criterion. Techniques that generate the required designs systematically presuppose unreplicated settings of the whole-plot factors. We use a cheese-making experiment to demonstrate the practical relevance of designs with replicated settings of these factors. We create such designs by splitting the whole plots according to one or more subplot effects. We develop a systematic method to generate the required designs and we use the method to create a table of designs that is likely to be useful in practice.

Strip-plot configurations of fractional factorials

Abstract: Running industrial experiments in strip-plot configurations can be a useful method of reducing cost. This article presents an effective procedure for constructing strip-plot arrangements of fractional factorial designs that can be used for m-level or mixed-level designs. The procedure consists of three steps: (1) Identify a suitable row design, (2) identify a suitable column design, and (3) select a latin-square fraction of the product of the designs in (1) and (2). Several examples are used to demonstrate the procedure.

A Note on the Definition of Resolution for Blocked 2 k–p Designs

Abstract: When two-level fractional factorial designs are blocked, the application of the standard definition of resolution requires careful consideration. Sometimes linear contrasts that superficially appear to be estimates of higher-order interaction effects are in reality estimates of first-order effects. Experimenters may therefore inadvertently choose designs that are of lower resolution than intended or unknowingly confound important effects. In this note, I discuss this subtle problem and propose an additional rule to the usual definition of resolution that provides a conservative but more realistic assessment of the resolution. With this more realistic characterization, experimenters are provided with a warning about possible confounding. I also show that my amendment to the definition of resolution may be useful when characterizing designs in which several two-level contrasts are combined to accommodate factors with four or more levels

Combinatorics of Finite Geometries

Abstract: In this lecture we intend to present a brief survey of very recent results. We shall be interested in the development of some topics considered in section 3.2 of Dembowski’s book [21] (Combinatorics of finite planes) and in problems connected with the existence of finite geometrical structures.