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

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

Review of Sensitivity Analysis Methods and Experience for Geological Disposal of Radioactive waste and Spent Nuclear Fuel

This reports gives an overview of sensitivity methods (screening, global and response surface based) that are suitable for safety analysis of a repository for radioactive waste or spent nuclear fuel. The theorerical background of the methods, their limitations and suitability for different analyses are discussed and illustrated by examples.JRC.F.7-Energy systems evaluatio

An Integrated Probability-Based Approach for Multiple Response Surface Optimization

Nearly all real life systems have multiple quality characteristics where individual modeling and optimization approaches can not provide a balanced compromising solution. Since performance, cost, schedule, and consistency remain the basics of any design process, design configurations are expected to meet several conflicting requirements at the same time. Correlation between responses and model parameter uncertainty demands extra scrutiny and prevents practitioners from studying responses in isolation. Like any other multi-objective problem, multi-response optimization problem requires trade-offs and compromises, which in turn makes the available algorithms difficult to generalize for all design problems. Although multiple modeling and optimization approaches have been highly utilized in different industries, and several software applications are available, there is no perfect solution to date and this is likely to remain so in the future. Therefore, problem specific structure, diversity, and the complexity of the available approaches require careful consideration by the quality engineers in their applications

Biometrical approaches for analysing gene bank evaluation data on barley (Hordeum spec.)

This thesis explored methods to statistically analyse phenotypic data of gene banks. Traits of the barley data (Hordeum spp.) of the gene bank of the IPK-Gatersleben were evaluated. The data of years 1948-2002 were available. Within this period the ordinal scale changed from a 0-5 to a 1-9 scale after 1993. At most gene banks reproduction of accessions is currently done without any experimental design. With data of a single year only rarely do accessions have replications and there are only few replications of a single check for winter and summer barley. The data of 2002 were analysed separately for winter and summer barley using geostatistical methods. For the traits analysed four types of variogram model (linear, spherical, exponential and Gaussian) were fitted to the empirical variogram using non-linear regression. The spatial parameters obtained by non-linear regression for every variogram model then were implemented in a mixed model analysis and the four model fits compared using Akaike's Information Criterion (AIC). The approach to estimate the genetical parameter by Kriging can not be recommended. The first points of the empirical variogram should be explained well by the fitted theoretical variogram, as these represent most of the pairwise distances between plots and are most crucial for neighbour adjustments. The most common well-fitting geostatistical models were the spherical and the exponential model. A nugget effect was needed for nearly all traits. The small number of check plots for the available data made it difficult to accurately dissect the genetical effect from environmental effects. The threshold model allows for joint analysis of multi-year data from different rating scales, assuming a common latent scale for the different rating systems. The analysis suggests that a mixed model analysis which treats ordinal scores as metric data will yield meaningful results, but that the gain in efficiency is higher when using a threshold model. The threshold model may also be used when there is a metric scale underlying the observed ratings. The Laplace approximation as a numerical method to integrate the log-likelihood for random effects worked well, but it is recommended to increase the number of quadrature points until the change in parameter estimates becomes negligible. Three rating methods (1%, 5%, 9-point rating) were assessed by persons untrained (A) and experienced (B) in rating. Every person had to rate several pictograms of diseased leaves. The highest accuracy was found with Group B using the 1%-scale and with Group A using the 5%-scale. With a percentage scale Group A tended to use values that are multiples of 5%. For the time needed per leaf assessment the Group B was fastest when using the 5% rating scale. From a statistical point of view both percent ratings performed better than the ordinal rating scale and the possible error made by the rater is calculable and usually smaller than with ratings by rougher methods. So directly rating percentages whenever possible leads to smaller overall estimation errors, and with proper training accuracy and precision can be further improved. For gene banks augmented designs as proposed by Federer and by Lin et al. offer themselves, so an overview is given. The augmented designs proposed by Federer have the advantage of an unbiased error estimate. But the random allocation of checks is a problem. The augmented design by Lin et al. always places checks in the centre plot of every whole plot. But none of the methods is based on an explicit statistical model, so there is no well-founded decision criterion to select between them. Spatial analysis can be used to find an optimal field layout for an augmented design, i.e. a layout that yields small least significant differences. The average variance of a difference and the average squared LSD were used to compare competing designs, using a theoretical approach based on variations of two anisotropic models and different rotations of anisotropy axes towards field reference axes. Based on theoretical calculations, up to five checks per block are recommended. The nearly isotropic combinations led to designs with large quadratic blocks. With strongly anisotropic combinations the optimal design depends on degree of anisotropy and rotation of anisotropy axes: without rotation small elongated blocks are preferred; the closer the rotation is to 45° the more squarish blocks and the more checks are appropriate. The results presented in this thesis may be summarised as follows: Cultivation for regeneration of accessions should be based on a meaningful and statistically analysable experimental field design. The design needs to include checks and a random sample of accessions from the gene pool held at the gene bank. It is advisable to utilise metric or percentage rating scales. It can be expected that using a threshold model increases the quality of multivariate analysis and association mapping studies based on phenotypic gene bank data.Die statistische Auswertbarkeit von phänotypischen Genbankdaten war Aufgabe dieser Arbeit. Boniturdaten von Gerste (Hordeum spec.) des IPK, Gatersleben, der Jahre 1948-2002 Standen zu Verfügung. Die Skalierung der Ordinal-Bonituren war ab 1993 von 0-5 auf 1-9 Intervalle umgestellt worden. Dem Erhaltungsanbau lag kein Versuchsdesign zu Grunde. Die Daten je eines Jahres hatten nur wenige Wiederholungen je eines einzigen Standards innerhalb der Winter- bzw. Sommergerste, von anderen Akzessionen gab es nur vereinzelt Wiederholungen. Der Datensatz 2002 wurde getrennt für Sommer- und Wintergersten mit geostatistischen Verfahren ausgewertet. An das jeweilige empirische Variogramm diverser Merkmal wurden vier Variogramm-Modelle (linear, sphärisch, exponentiell und Gauß) mittels nichtlinearer Regression angepasst. Deren geostatistischen Parameter wurden in ein Gemischtes Modell integriert und danach anhand des Akaikeschen Informationskriterium (AIC) verglichen. Der vordere Bereich des Variogramms, der von besonderem Interesse ist, sollte dabei gut angepasst werden. Als günstig erwiesen sich das sphärische und das exponentielle geostaistische Modell. Ein Nugget-Effekt wurde häufig benötigt. Die geringe Zahl an Standards und Wiederholungen erschwert es, den Nugget und damit den genetischen Effekt gut zu schätzen. Kriging zum Schätzen des genetischen Effekts kann nicht empfohlen werden. Das Schwellenwertmodell ermöglicht, mehrjährige ordinale Daten verschiedener Boniturskalen gemeinsam auszuwerten. Das Schwellenwertmodell lieferte bessere Ergebnisse als eine Analyse der Daten mit einem gemischten Modell. Das Schwellenwertmodell kann auch bei Boniturnoten mit zugrunde liegender metrischer Skala verwendet werden. Die Laplace-Approximation zur numerischen Integration der log-Likelihood über die zufälligen Effekte erwies sich als geeignete. Die Anzahl der Quadraturpunkte sollte jedoch erhöht werden, bis die Änderung der Parameter vernachlässigbar ist. Drei Boniturskalen (1%-, 5%, 9er Bonitur) wurden von geübte und ungeübte Boniteure auf Piktogramme von Getreideblätter mit Mehltaubefall angewandt. Die genauesten Schätzungen gelangen den Geübten mit der 1% Skala und den Ungeübten mit der 5% Skala. Bei der 1% Skalierung neigten die Ungeübten dazu, Vielfache von 5 häufiger zu vergeben. Die Geübten war eindeutig mit der 5% Bonitur am schnellsten. Die meisten Boniteure, besonders die Ungeübten, bevorzugten die 9er Bonitur. Aus statistischer Sicht sind beide Prozentbonituren angemessener. Der Fehler des Boniteurs ist dabei berechenbar und in der Regel kleiner als der bei groberer Skalenunterteilung. Bonitieren mit Prozentskalen führt zu geringeren Schätzfehlern und Schätzübungen erhöhen die Genauigkeit und Präzision. Für Genbankdaten bieten sich Augmented Designs (AD) an, wie sie von Federer und Lin et al. Vorliegen, daher wurde ein Übersichtsartikel verfasst. Beide ADs haben Vor- und Nachteile. Federers Designs schätzen den Fehler unverzerrt, aber die zufällige Verteilung der Standards kann zu Problemen führen. Das AD von Lin et al. platziert Standards in die Mitte der Großparzelle, beruht jedoch nicht auf einer expliziten statistischen Methode, daher gibt es kein offensichtlich bestes Modell zur Schätzung. Räumliche Statistik kann genutzt werden, um Augmented Designs zu optimieren, d.h. eine möglichst kleine Grenzdifferenz (LSD) zu erhalten. Die durchschnittliche Varianz einer Differenz (a.v.d.) und die durchschnittliche quadrierte LSD wurden zum Designvergleich genutzt. Ein theoretischer Ansatz, der auf zwei anisotropen Modellen und verschiedenen Rotationen der Anisotropie-Achse zur Hauptachse des Feldversuchs beruhte zeigte, dass bis zu fünf Standards je Block empfehlenswert sind. Bei nahezu Isotropie sind große quadratische Blöcke empfehlenswert. Bei Anisotropie ist die Blockform von der Intensität der Anisotropie und der Rotation der Achsen zueinander abhängig: ohne Rotation sind schmale lange Blocks günstig, je näher die Rotation bei 45° liegt, um so quadratischer sollte der Block sein und umso mehr Standards sollten Verwendung finden. Zusammenfassend kann gesagt werden: Um phänotypische Merkmalsdaten von Akzessionen zu erhalten, die für statistische Auswertung geeignet sind, ist es nötig, dass der Erhaltungsanbau auf einem sinnvollen und statistisch auswertbaren Versuchsdesign beruht, dass wiederholte Standards und dass eine zufällige Auswahl der angebauten Akzessionen aus der Gesamtheit garantiert ist. Des Weiteren ist es sinnvoll metrische oder Prozentboniturskalen zu verwenden. Es ist davon auszugehen, dass die Anwendung des Schwellenwertmodells bei Boniturnoten sowie die Verwendung von metrischen oder Prozentskalen die Qualität multivariater Auswertungen sowie Assoziationsstudien mit phänotypischen Genbankdaten verbessern