Design and Analysis of Screening Experiments Assuming Effect Sparsity

Many initial experiments for industrial and engineering applications employ screening designs to determine which of possibly many factors are significant. These screening designs are usually a highly fractionated factorial or a Plackett-Burman design that focus on main effects and provide limited information for interactions. To help simplify the analysis of these experiments, it is customary to assume that only a few of the effects are actually important; this assumption is known as ‘effect sparsity’. This dissertation will explore both design and analysis aspects of screening experiments assuming effect sparsity. In 1989, Russell Lenth proposed a method for analyzing unreplicated factorials that has become popular due to its simplicity and satisfactory power relative to alternative methods. We propose and illustrate the use of p-values, estimated by simulation, for Lenth t-statistics. This approach is recommended for its versatility. Whereas tabulated critical values are restricted to the case of uncorrelated estimates, we illustrate the use of p-values for both orthogonal and nonorthogonal designs. For cases where there is limited replication, we suggest computing t-statistics and p-values using an estimator that combines the pure error mean square with a modified Lenth’s pseudo standard error. Supersaturated designs (SSDs) are designs that examine more factors than runs available. SSDs were introduced to handle situations in which a large number of factors are of interest but runs are expensive or time-consuming. We begin by assessing the null model performance of SSDs when using all-subsets and forward selection regression. The propensity for model selection criteria to overfit is highlighted. We subsequently propose a strategy for analyzing SSDs that combines all-subsets regression and permutation tests. The methods are illustrated for several examples. In contrast to the usual sequential nature of response surface methods (RSM), recent literature has proposed both screening and response surface exploration using only one three-level design. This approach is named “one-step RSM”. We discuss and illustrate two shortcomings of the current one-step RSM designs and analysis. Subsequently, we propose a new class of three-level designs and an analysis strategy unique to these designs that will address these shortcomings and aid the user in being appropriately advised as to factor importance. We illustrate the designs and analysis with simulated and real data

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

Tailoring the Statistical Experimental Design Process for LVC Experiments

The use of Live, Virtual and Constructive (LVC) Simulation environments are increasingly being examined for potential analytical use particularly in test and evaluation. The LVC simulation environments provide a mechanism for conducting joint mission testing and system of systems testing when scale and resource limitations prevent the accumulation of the necessary density and diversity of assets required for these complex and comprehensive tests. The statistical experimental design process is re-examined for potential application to LVC experiments and several additional considerations are identified to augment the experimental design process for use with LVC. This augmented statistical experimental design process is demonstrated by a case study involving a series of tests on an experimental data link for strike aircraft using LVC simulation for the test environment. The goal of these tests is to assess the usefulness of information being presented to aircrew members via different datalink capabilities. The statistical experimental design process is used to structure the experiment leading to the discovery of faulty assumptions and planning mistakes that could potentially wreck the results of the experiment. Lastly, an aggressive sequential experimentation strategy is presented for LVC experiments when test resources are limited. This strategy depends on a foldover algorithm that we developed for nearly orthogonal arrays to rescue LVC experiments when important factor effects are confounded

Impurity rejection in the nickel laterite leach system

Atmospheric leaching (AL) of low-grade nickel laterite ores can produce a pregnant leach solution (PLS) containing significant amounts of impurities such as trivalent iron, aluminium and chromium ions. Purification of PLS by precipitation of the impurities with an alkaline reagent often causes an associated loss of nickel. This thesis documents an investigation of the physicochemical processes that occur during the precipitation of iron, aluminium and chromium from both synthetic and real nickel laterite AL leach liquors and associated nickel losses.A chemical equilibrium model in the Fe(III)–Ni(II)–H2SO4–H2O system was developed with the effects of ionic strength and temperature taken into account. This model was able to calculate the concentration distribution of iron and nickel species over the pH range from 0 to 4 and temperature from 25 to 100 °C, and predict the pH value of the solution. In addition, the model can calculate the saturation index of iron oxides such as goethite, ferrihydrite and schwertmannite to predict whether a specific iron oxide will precipitate or dissolve under particular conditions. The solubility of goethite, ferrihydrite and schwertmannite decreased substantially with increasing pH value. Goethite, ferrihydrite and schwertmannite were all undersaturated below pH 2. With increasing pH, ferric ions tended to precipitate first in the form of ferrihydrite followed by goethite and schwertmannite. A mixture was formed above pH 2.5, of which schwertmannite was the dominant phase.Considerable effort has been put into the experimental study on the relationships between impurities removal and nickel losses from nickel laterite AL liquors. The precipitation experiments were conducted in either single- or multi-stage simulation using synthetic and real PLS. For the single-stage precipitation experiments conducted using a synthetic PLS containing Fe(III)+Ni(II), the effects of the factors governing the iron precipitation process upon nickel losses were investigated by statistical analysis and modelling. Temperature, pH and the initial Fe/Ni ratio in PLS were found to be the important factors affecting iron removal efficiency and the level of nickel loss to solid. These factors were studied using a three-level Box-Behnken design combined with response surface methodology. Quadratic models were fitted to the experimental data, to enable construction of 3D response surfaces and corresponding contour plots. These graphs clearly demonstrated the links between responses and the interactions of factors.Further single-stage precipitation experiments performed using PLS containing Fe(III)+Ni(II)+Al(III), Fe(III)+Ni(II)+Cr(III), and Fe(III)+Ni(II)+Al(III)+Cr(III) showed that greater losses of nickel to solids occurred in the presence of aluminium and chromium. Increasing the pH value of solution and precipitation temperature favored the removal of iron, aluminium and chromium, but at a cost of greater nickel losses. By carefully controlling pH and temperature using a multi-stage precipitation process, however, the iron, aluminium and chromium can be effectively rejected with a minimal nickel loss and desirable sludge properties. The optimum conditions for a multi-stage precipitation process were found to be at pH 3 and 55 ºC in the first stage followed by a second stage operated at pH 3 and 85 ºC. Using this precipitation procedure, as much as 95% iron and chromium together with above 80% aluminium can be removed; the level of nickel loss to the solid can be reduced to below 1%. The sludge showed a fast settling rate of 5.05 m/h with the addition of a cationic flocculant. Similar satisfactory results were also obtained when performing this multi-stage precipitation procedure on real leach solutions.The effect of water salinity on impurities removal and nickel losses was also examined due to variable nature of process water available in Western Australia to process nickel laterite during atmospheric leaching. This was achieved by conducting single-stage precipitation experiments in Fe(III)+Ni(II)+Al(III)+Cr(III) systems with various amounts of sodium chloride added. The presence of high concentration of salts resulted in higher removal efficiencies for iron, aluminium and chromium, and less nickel losses to the solids, particularly when the precipitation reactions were carried out at 85 ºC. XRD analysis of the residues confirmed that the poorly structural-ordered schwertmannite and/or ferrihydrite were the dominant phases. Natrojarosite (NaFe3(SO4)2(OH)6) can be detected when the precipitation reaction was conducted at pH 2 and 85 ºC from synthetic solution with high salinity.The presence of large amounts of poorly structural-ordered schwertmannite and ferrihydrite in the iron-rich residues complicates mineralogical identification using routine XRD technique. A comprehensive characterization was performed using a combination of several techniques that include selective Acidified Ammonium Oxalate (AAO) dissolution, differential XRD, SEM and FTIR spectroscopy. These techniques in combination allowed reliable mineralogical identification for samples containing high proportions of schwertmannite and ferrihydrite. The effects of foreign metallic cations on the crystallization, dissolution behaviour and surface sulphate coordination were investigated. The results suggested that the presence of goethite in the precipitates can be identified after removing the schwertmannite and/or ferrihydrite. Nickel, aluminium and chromium retarded the transformations of schwertmannite and/or ferrihydrite to goethite, but aluminium and chromium supressed the formation of 6-line ferrihydrite. Also, aluminium and chromium influenced the symmetry of the sulphate absorbed onto the iron-rich precipitates. The structural order of the phases became less pronounced with the presence of foreign metallic cations, particularly aluminium and chromium. Aluminium and chromium can strongly stabilize iron-rich precipitates making these resistant to leaching by AAO solution. FTIR analysis confirmed the presence of goethite in the bi-metallic precipitates and suggested that the sulphate is present to a greater extent in lower symmetry environments

Attitudes towards old age and age of retirement across the world: findings from the future of retirement survey

The 21st century has been described as the first era in human history when the world will no longer be young and there will be drastic changes in many aspects of our lives including socio-demographics, financial and attitudes towards the old age and retirement. This talk will introduce briefly about the Global Ageing Survey (GLAS) 2004 and 2005 which is also popularly known as “The Future of Retirement”. These surveys provide us a unique data source collected in 21 countries and territories that allow researchers for better understanding the individual as well as societal changes as we age with regard to savings, retirement and healthcare. In 2004, approximately 10,000 people aged 18+ were surveyed in nine counties and one territory (Brazil, Canada, China, France, Hong Kong, India, Japan, Mexico, UK and USA). In 2005, the number was increased to twenty-one by adding Egypt, Germany, Indonesia, Malaysia, Poland, Russia, Saudi Arabia, Singapore, Sweden, Turkey and South Korea). Moreover, an additional 6320 private sector employers was surveyed in 2005, some 300 in each country with a view to elucidating the attitudes of employers to issues relating to older workers. The paper aims to examine the attitudes towards the old age and retirement across the world and will indicate some policy implications

Attitudes towards old age and age of retirement across the world: findings from the future of retirement survey

The 21st century has been described as the first era in human history when the world will no longer be young and there will be drastic changes in many aspects of our lives including socio-demographics, financial and attitudes towards the old age and retirement. This talk will introduce briefly about the Global Ageing Survey (GLAS) 2004 and 2005 which is also popularly known as “The Future of Retirement”. These surveys provide us a unique data source collected in 21 countries and territories that allow researchers for better understanding the individual as well as societal changes as we age with regard to savings, retirement and healthcare. In 2004, approximately 10,000 people aged 18+ were surveyed in nine counties and one territory (Brazil, Canada, China, France, Hong Kong, India, Japan, Mexico, UK and USA). In 2005, the number was increased to twenty-one by adding Egypt, Germany, Indonesia, Malaysia, Poland, Russia, Saudi Arabia, Singapore, Sweden, Turkey and South Korea). Moreover, an additional 6320 private sector employers was surveyed in 2005, some 300 in each country with a view to elucidating the attitudes of employers to issues relating to older workers. The paper aims to examine the attitudes towards the old age and retirement across the world and will indicate some policy implications