Equivalences in design of experiments

The statistical theory of experimental designs was initiated by Fisher in the 1920s in the context of agricultural experiments performed at the Rothamsted Experimental Station. Applications of experimental designs in industry started in the 1930s, but really took off after World War II. The second half of the 20th century witnessed both a widespread application of experimental designs in industrial settings and tremendous advances in the mathematical and statistical theory. Recent technological developments in biology (DNA microarrays) and chemical engineering (high-throughput reactors) generated new challenges in experimental design. So experimental designs is a lively subject with a rich history from both an applied and theoretical point of view. This thesis is mainly an exploration of the mathematical framework underlying factorial designs, an important subclass of experimental designs. Factorial designs are probably the most widely used type of experimental designs in industry. The literature on experimental designs is either example-based with lack of general statements and clear definitions or so abstract that the link to real applications is lost. With this thesis we hope to contribute to closing this gap. By restricting ourselves to factorial designs it is possible to provide a framework which is mathematically rigorous yet applicable in practice. A mathematical framework for factorial designs is given in Chapter 2. Each of the subsequent chapters is devoted to a specific topic related to factorial designs. In Chapter 3 we study coding full factorial designs by finite Abelian groups. This idea was introduced by Fisher in the 1940s to study confounding. Confounding arises when one performs only a fraction of a full factorial design. Using the character theory of finite Abelian groups we show that definitions of so-called regular fractions given by Collombier (1996), Wu and Hamada (2000) and Pistone and Rogantin (2005) are equivalent. An important ingredient in our approach is the special role played by the cosets of the finite Abelian group. We moreover use character theory to prove that any regular fraction when interpreted as a coset is an orthogonal array of a certain strength related to the resolution of that fraction. This is a generalization of results by Rao and Bose for regular fractions of symmetric factorial designs with a prime power as the number of levels. The standard way to analyze factorial designs is analysis of variance. Diaconis and Viana have shown that the well-known sums of squares decomposition in analysis of variance for full factorial designs naturally arises from harmonic analysis on a finite Abelian group. We give a slight extension of their setup by developing the theoretical aspects of harmonic analysis of data structured on cosets of finite Abelian groups. In Chapter 4 we study the estimation of dispersion parameters in a mixed linear model. This is the common model behind modern engineering approaches to experimental design like the Taguchi approach. We give necessary and sufficient conditions for the existence of translation invariant unbiased estimators for the dispersion parameters in the mixed linear model. We show that the estimators for the dispersion parameters in Malley (1986) and Liao and Iyer (2000) are equivalent. In the 1980s Box and Meyer initiated the identification of dispersion effects from unreplicated factorial experiments. They did not give an explicit estimation procedure for the dispersion parameters. We show that the well-known estimators for dispersion effects proposed by Wiklander (1998), Liao and Iyer (2000) and Brenneman and Nair (2001) coincide for two-level full factorial designs and their regular fractions. Moreover, we give a definition for MINQUE estimator for the dispersion effects in two-level full factorial designs and show that the above estimators are MINQUE in this sense. Finally, in Chapter 5 we study a real-life industrial problem from a two-step production process. In this problem an intermediate product from step 1 is split into several parts in order to allow further processing in step 2. This type of situation is typically handled by using a split-plot design. However, in this specific example running a full factorial split-plot design was not feasible for economic reasons. We show how to apply recently developed analysis methods for fractional factorial split-plot designs developed by Bisgaard, Bingham and Sitter. Finally, we modified the algorithm in Franklin and Bailey (1977) to generate fractional factorial split-plot designs that identify a given set of effects while minimizing the number of required intermediate products.