Joint Identification of Location and Dispersion Effects in Unreplicated Two-Level Factorials

Most procedures that have been proposed to identify dispersion effects in unreplicated factorial designs assume that location effects have been identified correctly. Incorrect identi- fication of location effects may impair subsequent identification of dispersion effects. We develop a model for joint identification of location and dispersion effects that can reliably identify active effects of both types. The joint model is estimated using maximum likelihood, and hence effect selection is done using a specially derived information criterion. An exhaustive search through a limited version of the space of possible models is conducted. Both a single-model output and model averaging are considered. The method is shown to be capable of identifying sensible location-dispersion models that are missed by methods that rely on sequential estimation of location and dispersion effects

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

The design and analysis of computer experiments

Computer simulation modeling is an important part of engineering design. The process of creating quality products under variable conditions is called robust engineering design. Frequently these simulation models are expensive to run and it can take many runs to find the appropriate parameter settings of the engineering design. To reduce the cost of robust engineering design, it has been proposed that statistical models be used to predict the results of the simulator at unobserved values of the inputs. This involves running experiments on the computer simulation models, or computer experiments. Computer experiments have no random error and frequently involve a large number of experimental factors; because of this, there are many reasons why standard prediction and design methods may not work well. Methods from spatial statistics, frequently refered to as kriging, are used to predict new observations of the simulation model. A generalized linear model with unknown covariance parameters is used on several examples of high dimensions. The model requires estimation of the covariance function parameters and methods are described for parameter estimation and model building. Latin hypercube sampling is used for the experimental design. Latin hypercube sampling is as easy to use as Monte Carlo sampling and has been shown to have better estimation properties. The space filling properties of Latin hypercube sampling are investigated here and shown to fill the design space more uniformly than Monte Carlo sampling. These statistical methods are applied to two circuit simulation models. The results show that these methods work well on computer experiments and can form the basis for a methodology in robust engineering design. Although these methods have been applied to circuit design problems, the methods are applicable to a wide variety of computer simulation models. Robust engineering design received a great deal of attention when Taguchi’s ideas were introduced. An investigation into the methods that Taguchi introduced revealed some shortcomings from a statistical view. General conclusions are drawn about the efficacy of traditional Taguchi methods compared with the prefered model-based approach of the thesis

Estimation and Inference for 2k-p Experiments with Beta Response

Fractional factorial experiments are widely used in industry and engineering. The most common interest in these experiments is to identify a subset of the factors with the greatest effect on the response. With respect to data analysis for these experiments, the most used methods include linear regression, transformations, and the Generalized Linear Model (GLM). This thesis focuses on experiments whose response is measured continuously in the (0,1) interval (if y ∈(a,b), then (y-a)/(b-a) ∈ (0,1)). Analyses for factorial experiments in (0,1) are rarely found in the literature. In this work, advantages and drawbacks of the three mentioned methods for analyzing data from experiments in (0,1) are described. Here, as the beta distribution assumes values in (0,1), the beta regression model (BRM) is proposed for analyzing these kinds of experiments. More specifically, the necessity of considering variable dispersion (VD) and using linear restrictions on parameters are justified in data from 2k and 2k and 2k-p experiments. Thus, the first result in this thesis is to propose, develop, and apply a restricted VDBRM. The restricted VDBRM is developed from frequentist perspective: a penalized likelihood (by means of Lagrange multipliers), restricted maximum likelihood estimators with their respective Fisher Information Matrix, hypothesis tests, and a diagnostic measure. Upon applying the restricted VDBRM, good results were obtained for simulated data, and it is shown that the hypothesis related to 2k and 2k-p experiments are a special case of the restricted model. The second result of this thesis is to explore an integrated Bayesian/likelihood proposal for analyzing data from factorial experiments using the (Bayesian and frequentist) simple BRM's. This was done upon employing at prior distributions in the Bayesian BRM. Thus, comparisons between confidence intervals (frequentist case) and credibility intervals (Bayesian case) on the mean response are done with good and promisory results in real experiments. This work also explores a technique for choosing the best model among several candidates which combine the Half-normal plots (given by the BRM) and the inferential results. Starting from the active factors chosen from each plot, subsequently the respective regression models are fitted and, finally, by means of information criteria, the best model is chosen. This technique was explored with the following models: normal, transformation, generalized linear, and simple beta regression for real 2k and 2k- p experiments: into the greater part of the examples considered for the Bayesian and frequentist BRM's, results were very similar (using at prior distributions). Moreover, four link functions for the mean response in the BRM are compared: results highlight the importance to study each problem at hand.Resumen. Los experimentos factoriales fraccionados se usan ampliamente en la industria y en la Ingeniería. El interés más común en estos experimentos es identificar el subconjunto de factores que tiene mayor efecto sobre la respuesta. Con respecto al análisis de datos de dichos experimentos, los métodos más usados incluyen regresión lineal, transformaciones y Modelo Lineal Generalizado (MLG). Esta Tesis se enfoca en experimentos cuya respuesta está medida continuamente en el intervalo (0,1), (si y ∈ (a,b), entonces y (y-a)/(b-a) ∈ (0,1)). En la literatura se encuentran pocos análisis de experimentos con esta respuesta. En este trabajo, se describen ventajas y desventajas de las tres metodologías mencionadas en experimentos con esta respuesta. Acá, como la distribución beta asume valores en (0,1), se propone el modelo de regresión beta (MRB) para analizar estos datos. Más específicamente, se justifica la necesidad de modelar la dispersión variable y usar restricciones sobre los parámetros se justifican en datos de experimentos 2k y 2k-p. De este modo, el primer resultado de esta Tesis es proponer, desarrollar y aplicar un modelo de regresión beta con dispersión variable y restricciones en los parámetros (MRBDV restringido). El modelo es desarrollado desde la perspectiva clásica: una función de verosimilitud penalizada (con multiplicadores de Lagrange), estimadores de máxima verosimilitud restringidos con su respectiva matriz de Información de Fisher, tests de hipótesis y una medidad de bondad de ajuste. Al aplicar el MRBDV restringido, se obtuvieron buenso resultados para datos simulados y se mostró que las hipótesis asociadas con experimentos 2k y 2k-p son un caso especial del modelo restringido. El segundo resultado de esta Tesis es explorar una propuesta integrada bayesiana/verosimil para analizar datos de experimentos factoriales usando los dos MRB (bayesiano y clásico). Esto se hizo al emplear distribuciones a priori planas (poco informativas) en el modelo bayesiano. Así, las comparaciones entre intervalos de confianza y de credibilidad presentaron buenos resultados y promisorios en experimentos factoriales reales. Esta Tesis tambien explora una técnica para elegir el mejor modelo entre varios candidatos, el cual combina los Half-normal plots (dados por el BRM) y resultados inferenciales. Partiendo de los efectos activos según cada gráfico, posteriormente se ajustan los modelos de regresión respectivos y, finalmente, por medio de criterios de información, se escoge el mejor modelo. Esta técnica fue explorada con los siguientes modelos: normal, transformaciones, MLG y MRB simple para datos reales de experimentos 2k y 2kDoctorad

Bayesian Design and Analysis of Small Multifactor Industrial Experiments

PhDUnreplicated two level fractional factorial designs are a common type of experimental design used in the early stages of industrial experimentation. They allow considerable information about the e ects of several factors on the response to be obtained with a relatively small number of runs. The aim of this thesis is to improve the guidance available to experimenters in choosing a good design and analysing data. This is particularly important when there is commercial pressure to minimise the size of the experiment. A design is usually chosen based on optimality, either in terms of a variance criterion or estimability criteria such as resolution. This is given the number of factors, number of levels of each factor and number of runs available. A decision theory approach is explored, which allows a more informed choice of design to be made. Prior distributions on the sizes of e ects are taken into consideration, and then a design chosen from a candidate set of designs using a utility function relevant to the objectives of the experiment. Comparisons of the decision theoretic methods with simple rules of thumb are made to determine when the more complex approach is necessary. Fully Bayesian methods are rarely used in multifactor experiments. However there is virtually always some prior knowledge about the sizes of e ects and so using this in a Bayesian data analysis seems natural. Vague and more informative priors are 6 explored. The analysis of this type of experiment can be impacted in a disastrous way in the presence of outliers. An analysis that is robust to outliers is sought by applying di erent model distributions of the data and prior assumptions on the parameters. Results obtained are compared with those from standard analyses to assess the bene ts of the Bayesian analysis