Abstract
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.
Original language | English |
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Qualification | Doctor of Philosophy |
Awarding Institution |
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Supervisors/Advisors |
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Award date | 21 May 2007 |
Place of Publication | Eindhoven |
Publisher | |
Print ISBNs | 978-90-386-3860-7 |
DOIs | |
Publication status | Published - 2007 |