Consequences of using estimated response values from negligible interactions in factorial designs

This article analyzes the increase in the probability of committing type I and type II errors in assessing the significance of the effects when some properly selected runs have not been carried out and their responses have been estimated from the interactions considered null from scratch. This is done by simulating the responses from known models that represent a wide variety of practical situations that the experimenter will encounter; the responses considered to be missing are then estimated and the significance of the effects is assessed. Through comparison with the parameters of the model, the errors are then identified. To assess the significance of the effects when there are missing values, the Box-Meyer method has been used. The conclusions are that 1 missing value in 8 run designs and up to 3 missing values in 16 run designs experiments can be estimated without hardly any notable increase in the probability of error when assessing the significance of the effects.Peer ReviewedPostprint (author's final draft

Contributions to the planning and analysis of factorial designs

The thesis is structured in five articles. In the first article, two methods are compared to analyze the significance of the effects: the Lenth method and the one based on the estimation of the variance of the effects from interactions that can be considered negligible from scratch. For the most common factorial designs and in a set of scenarios that seek to reflect the situations that the experimenter can find in practice, simulation techniques are used to identify the errors that are committed with each method. Based on the analysis of the results obtained, we recommend in which situations it is more appropriate to use one method or the other. The second article analyzes the problem of estimating the results of experiments that could not be performed based on the expression of the interactions that can be considered negligible. The variance of the estimated values depends on what these values are and also on the interactions considered negligible. All possible encountered situations are analyzed and tables are presented with the values that can be estimated with minimum variance depending on the type of design and the contrasts available to perform the estimates. The third article deals with the same problem as the second but analyzing the impact of the estimate not on the variance of the estimated response values but on the variance of the effects and also on the correlations among them. The analysis of all the situations that can be given in the most common designs, allows us to make recommendations about what experiments should be skipped in the case that, due to time or budgetary constraints, all runs indicated by the factorial design cannot be executed.La tesi està estructurada en cinc articles. En el primer es comparen dos mètodes per analitzar la significació dels efectes: el mètode de Lenth i el basat en l’estimació de la variància dels efectes a partir de les interacciones que a priori es poden considerar nul·les. Pels dissenys factorials més habituals i en un conjunt d’escenaris que pretenen reflectir les situacions que l’experimentador es pots trobar a la pràctica, es fan servir tècniques de simulació per identificar els errors que es cometen amb cada mètode. A partir de l’anàlisi dels resultats obtinguts es recomana en quina situació és més adequat fer servir un mètode o l’altre. El segon article analitza el problema d’estimar els resultats d’experiments que no s’han pogut realitzar a partir de l’expressió de les interaccions que es poden considerar negligibles. La variància dels valors estimats depèn de quins siguin aquests valors i també de les interaccions considerades negligibles. S’analitzen totes les situacions que es poden donar i es presenten unes taules amb els valors que es poden estimar amb variància mínima segons el tipus de disseny i de quins siguin els contrastos disponibles per fer les estimacions. El tercer article aborda el mateix problema que el segon però analitzant l’impacte de l’estimació no en la variància dels valors estimats sinó en la variància dels efectes i també en les correlacions entre ells. L’anàlisi de totes les situacions que es poden donar en els dissenys més habituals permet fer recomanacions sobre quins experiments convé deixar de fer en el cas que per restriccions de temps o pressupostaries no es puguin fer tots el que indica el disseny factorial. Fer servir resultats estimats en comptes de fer els experiments, té conseqüències negatives encara que es facin servir les millors opcions possibles. Aquestes conseqüències negatives s’estudien en el quart article i fan referencia a l’augment de la probabilitat d’error, tant de tipus I com de tipus II, en l’anàlisi de la significació dels efectes. Aquest article aporta un conjunt de gràfics que representen les probabilitats d’error amb valors estimats en front el cas que s’hagin fet tots els experiments. Finalment, els cinquè article té una estructura similar al primer. Es compara el mètode de Lenth, el més popular i el que apareix més sovint en els paquets de software estadístic, amb el mètode de Box-Meyer. Aquest últim és un mètode bayesià que pràcticament no es fa servir, segurament per la seva dificultat tant a nivell conceptual com de càlcul. Però aquestes dificultats perden rellevància amb l’ús generalitzat dels ordinadors. L’article posa de manifest que en molts casos el mètode de Box-Meyer dona millor resultats que el de Lenth

Design of Experiments for Screening

The aim of this paper is to review methods of designing screening experiments, ranging from designs originally developed for physical experiments to those especially tailored to experiments on numerical models. The strengths and weaknesses of the various designs for screening variables in numerical models are discussed. First, classes of factorial designs for experiments to estimate main effects and interactions through a linear statistical model are described, specifically regular and nonregular fractional factorial designs, supersaturated designs and systematic fractional replicate designs. Generic issues of aliasing, bias and cancellation of factorial effects are discussed. Second, group screening experiments are considered including factorial group screening and sequential bifurcation. Third, random sampling plans are discussed including Latin hypercube sampling and sampling plans to estimate elementary effects. Fourth, a variety of modelling methods commonly employed with screening designs are briefly described. Finally, a novel study demonstrates six screening methods on two frequently-used exemplars, and their performances are compared

Minimum aberration designs for discrete choice experiments

A discrete choice experiment (DCE) is a survey method that givesinsight into individual preferences for particular attributes.Traditionally, methods for constructing DCEs focus on identifyingthe individual effect of each attribute (a main effect). However, aninteraction effect between two attributes (a two-factor interaction)better represents real-life trade-offs, and provides us a better understandingof subjects’ competing preferences. In practice it is oftenunknown which two-factor interactions are significant. To address theuncertainty, we propose the use of minimum aberration blockeddesigns to construct DCEs. Such designs maximize the number ofmodels with estimable two-factor interactions in a DCE with two-levelattributes. We further extend the minimum aberration criteria toDCEs with mixed-level attributes and develop some general theoreticalresults

Techniques for sensitivity analysis of simulation models: A case study of the CO2 greenhouse effect

Environment;Simulation;environmental economics

Optimization of a Centrifugal Compressor Using the Design of Experiment Technique

Centrifugal compressor performance is affected by many parameters, optimization of which can lead to superior designs. Recognizing the most important parameters affecting performance helps to reduce the optimization process cost. Of the compressor components, the impeller plays the most important role in compressor performance, hence the design parameters affecting this component were considered. A turbocharger centrifugal compressor with vaneless diffuser was studied and the parameters investigated included meridional geometry, rotor blade angle distribution and start location of the main blades and splitters. The diffuser shape was captured as part of the meridional geometry. Applying a novel approach to the problem, full factorial analysis was used to investigate the most effective parameters. The Response Surface Method was then implemented to construct the surrogate models and to recognize the best points over a design space created as based on the Box-Behnken methodology. The results highlighted the factors that affected impeller performance the most. Using the Design of Experiment technique, the model which optimized both efficiency and pressure ratio simultaneously delivered a design with 3% and 11% improvement in each respectively in comparison to the initial impeller at the design point. Importantly, this was not at the expense of sacrificing range, of critical concern in compressor design

Regression Models and Experimental Designs: A Tutorial for Simulation Analaysts

This tutorial explains the basics of linear regression models. especially low-order polynomials. and the corresponding statistical designs. namely, designs of resolution III, IV, V, and Central Composite Designs (CCDs).This tutorial assumes 'white noise', which means that the residuals of the fitted linear regression model are normally, independently, and identically distributed with zero mean.The tutorial gathers statistical results that are scattered throughout the literature on mathematical statistics, and presents these results in a form that is understandable to simulation analysts.metamodels;fractional factorial designs;Plackett-Burman designs;factor interactions;validation;cross-validation

Selection of best conditions of inoculum preparation for optimum performance of the pigment production process by Talaromyces spp. using the Taguchi method

Process optimisation techniques increasingly need to be used early on in research and development of processes for new ingredients. There are different approaches and this article illustrates the main issues at stake with a method that is an industry best practice, the Taguchi method, suggesting a procedure to assess the potential impact of its drawbacks. The Taguchi method has been widely used in various industrial sectors because it minimises the experimental requirements to define an optimum region of operation, which is particularly relevant when minimising variability is a target. However, it also has drawbacks, especially the intricate confoundings generated by the experimental designs used. This work reports a process optimisation of the synthesis of red pigments by a fungal strain, Talaromyces spp. using the Taguchi methodology and proposes an approach to assess from validation trials whether the conclusions can be accepted with confidence. The work focused on optimising the inoculum characteristics, and the studied factors were spore age and concentration, agitation speed and incubation time. It was concluded that spore age was the most important factor for both responses, with optimum results at 5 days old, with the best other conditions being spores concentration, 100,000 (spores/mL); agitation, 200 rpm; and incubation time, 84 h. The interactive effects can be considered negligible and therefore this is an example where a simple experimental design approach was successful in speedily indicating conditions able to increase pigment production by 63% compared to an average choice of settings

Contributions to the planning and analysis of factorial designs

Tesi per compendi de publicacions.The thesis is structured in five articles. In the first article, two methods are compared to analyze the significance of the effects: the Lenth method and the one based on the estimation of the variance of the effects from interactions that can be considered negligible from scratch. For the most common factorial designs and in a set of scenarios that seek to reflect the situations that the experimenter can find in practice, simulation techniques are used to identify the errors that are committed with each method. Based on the analysis of the results obtained, we recommend in which situations it is more appropriate to use one method or the other. The second article analyzes the problem of estimating the results of experiments that could not be performed based on the expression of the interactions that can be considered negligible. The variance of the estimated values depends on what these values are and also on the interactions considered negligible. All possible encountered situations are analyzed and tables are presented with the values that can be estimated with minimum variance depending on the type of design and the contrasts available to perform the estimates. The third article deals with the same problem as the second but analyzing the impact of the estimate not on the variance of the estimated response values but on the variance of the effects and also on the correlations among them. The analysis of all the situations that can be given in the most common designs, allows us to make recommendations about what experiments should be skipped in the case that, due to time or budgetary constraints, all runs indicated by the factorial design cannot be executed.La tesi està estructurada en cinc articles. En el primer es comparen dos mètodes per analitzar la significació dels efectes: el mètode de Lenth i el basat en l’estimació de la variància dels efectes a partir de les interacciones que a priori es poden considerar nul·les. Pels dissenys factorials més habituals i en un conjunt d’escenaris que pretenen reflectir les situacions que l’experimentador es pots trobar a la pràctica, es fan servir tècniques de simulació per identificar els errors que es cometen amb cada mètode. A partir de l’anàlisi dels resultats obtinguts es recomana en quina situació és més adequat fer servir un mètode o l’altre. El segon article analitza el problema d’estimar els resultats d’experiments que no s’han pogut realitzar a partir de l’expressió de les interaccions que es poden considerar negligibles. La variància dels valors estimats depèn de quins siguin aquests valors i també de les interaccions considerades negligibles. S’analitzen totes les situacions que es poden donar i es presenten unes taules amb els valors que es poden estimar amb variància mínima segons el tipus de disseny i de quins siguin els contrastos disponibles per fer les estimacions. El tercer article aborda el mateix problema que el segon però analitzant l’impacte de l’estimació no en la variància dels valors estimats sinó en la variància dels efectes i també en les correlacions entre ells. L’anàlisi de totes les situacions que es poden donar en els dissenys més habituals permet fer recomanacions sobre quins experiments convé deixar de fer en el cas que per restriccions de temps o pressupostaries no es puguin fer tots el que indica el disseny factorial. Fer servir resultats estimats en comptes de fer els experiments, té conseqüències negatives encara que es facin servir les millors opcions possibles. Aquestes conseqüències negatives s’estudien en el quart article i fan referencia a l’augment de la probabilitat d’error, tant de tipus I com de tipus II, en l’anàlisi de la significació dels efectes. Aquest article aporta un conjunt de gràfics que representen les probabilitats d’error amb valors estimats en front el cas que s’hagin fet tots els experiments. Finalment, els cinquè article té una estructura similar al primer. Es compara el mètode de Lenth, el més popular i el que apareix més sovint en els paquets de software estadístic, amb el mètode de Box-Meyer. Aquest últim és un mètode bayesià que pràcticament no es fa servir, segurament per la seva dificultat tant a nivell conceptual com de càlcul. Però aquestes dificultats perden rellevància amb l’ús generalitzat dels ordinadors. L’article posa de manifest que en molts casos el mètode de Box-Meyer dona millor resultats que el de Lenth.Postprint (published version

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