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2018 | Buch

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Experimental Design

With Application in Management, Engineering, and the Sciences.

verfasst von: Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This text introduces and provides instruction on the design and analysis of experiments for a broad audience. Formed by decades of teaching, consulting, and industrial experience in the Design of Experiments field, this new edition contains updated examples, exercises, and situations covering the science and engineering practice. This text minimizes the amount of mathematical detail, while still doing full justice to the mathematical rigor of the presentation and the precision of statements, making the text accessible for those who have little experience with design of experiments and who need some practical advice on using such designs to solve day-to-day problems. Additionally, an intuitive understanding of the principles is always emphasized, with helpful hints throughout.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction to Experimental Design

Abstract

Experimentation is part of everyday life. Will leaving 30 minutes earlier than usual in the morning make it easier to find a legal parking space at work? How about 20 minutes earlier? Or only 10 minutes earlier? Can I increase my gas mileage by using synthetic oil? Will my problem employees make more of an effort to be on time if I make it a practice to stop by their office to chat at the start of the day? Will a chemical reaction be faster if the amount of a specific reagent is increased threefold? How about if the temperature is increased by 10 °C? Will the yield increase if an extraction is carried for 40 minutes instead of 20 minutes?

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Statistical Principles for Design of Experiments

Frontmatter

Chapter 2. One-Factor Designs and the Analysis of Variance

Abstract

We begin this and subsequent chapters by presenting a real-world problem in the design and analysis of experiments on which at least one of the authors consulted. At the end of the chapter, we revisit the example and present analysis and results. The appendices will cover the analysis using statistical packages not covered in the main text, where appropriate. As you read the chapter, think about how the principles discussed here can be applied to this problem.

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Chapter 3. Some Further Issues in One-Factor Designs and ANOVA

Abstract

We need to consider several important collateral issues that complement our discussion in Chap. 2. We first examine the standard assumptions typically made about the probability distribution of the ε’s in our statistical model. Next, we discuss a nonparametric test that is appropriate if the assumption of normality, one of the standard assumptions, is seriously violated. We then review hypothesis testing, a technique that was briefly discussed in the previous chapter and is an essential part of the ANOVA and that we heavily rely on throughout the text. This leads us to a discussion of the notion of statistical power and its determination in an ANOVA. Finally, we find a confidence interval for the true mean of a column and for the difference between two true column means.

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Chapter 4. Multiple-Comparison Testing

Abstract

So far, we have seen a couple of statistical tests which can indicate if a factor has an impact on the response or not, and which would make us reject or accept H ₀(μ ₁ = μ ₂ = μ ₃ = … = μ _C); however, they do not show how the means differ, if, indeed, they do differ. In this chapter, we will discuss the logic and Type I errors in multiple-comparison testing. We then present several procedures which can be used for multiple comparison of means, such as Fisher’s Least Significant Difference (LSD) test, Tukey’s HSD test, the Newman-Keuls test, and Dunnett’s test. Finally, we discuss the Scheffé test as a post hoc study for multiple comparisons.

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Chapter 5. Orthogonality, Orthogonal Decomposition, and Their Role in Modern Experimental Design

Abstract

In Chap. 2, we saw how to investigate whether or not one factor influences some dependent variable. Our approach was to partition the total sum of squares (TSS), the variability in the original data, into two components – the sum of squares between columns (SSB_c), attributable to the factor under study, and the sum of squares within a column (SSW), the variability not explained by the factor under study, and instead explained by “everything else.” Finally, these quantities were combined with the appropriate degrees of freedom in order to assess statistical significance. We were able to accept or reject the null hypothesis that all column means are equal (or, correspondingly, reject or accept that the factor under study has an impact on the response). In Chap. 4, we discussed multiple-comparison techniques for asking more detailed questions about the factor under study; for example, if not all column means are equal, how do they differ? We now present a more sophisticated, flexible, and potent way to analyze (or “decompose”) the impact of a factor on the response, not limited to pairwise comparisons.

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Identifying Active Factors

Frontmatter

Chapter 6. Two-Factor Cross-Classification Designs

Abstract

Chapter 2 introduced one-factor designs – experiments designed to determine whether the level of a factor, the (one) independent variable, affects the value of some quantity of interest, the dependent variable. By way of example, we considered whether device/usage influences battery life. We expanded on this initial analysis by introducing multiple-comparison testing and orthogonal breakdowns of sums of squares.

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Chapter 7. Nested, or Hierarchical, Designs

Abstract

In previous chapters, we have seen one- and two-factor designs and how to analyze them. In a factorial design, such as the one discussed in Chap. 6, the levels of the two factors – for convenience, A and B – are crossed, that is, every level of B will occur with every level of A. In practice, this means that, if we have three levels per factor, we will have nine experimental runs (without considering replicates). In this chapter, we will see a different type of design, called nested designs, where the levels of factor B will occur only at certain levels of A. For instance, we can have three levels of A and nine levels of B, but levels 1–3, 4–6, and 7–9 of B will only occur when the levels of A are 1, 2, and 3, respectively.

Paul Berger, Robert Maurer, Giovana B. Celli

Chapter 8. Designs with Three or More Factors: Latin-Square and Related Designs

Abstract

When more than two factors are under study, the number of possible treatment combinations grows exponentially. For example, with only three factors, each at five levels, there are 5³ = 125 possible combinations. Although modeling such an experiment is straightforward, running it is another matter. It would be rare to actually carry out an experiment with 125 different treatment combinations, because the management needed and the money required would be great.

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Studying Factors’ Effects

Frontmatter

Chapter 9. Two-Level Factorial Designs

Abstract

We now change our focus from the number of factors in the experiment to the number of levels those factors have. Specifically, in this and the next several chapters, we consider designs in which all factors have two levels. Many experiments are of this type. This is because two is the minimum number of levels a factor can have and still be studied, and by having the minimum number of levels (2), an experiment of a certain size can include the maximum number of factors. After all, an experiment with five factors at two levels each contains 32 combinations of levels of factors (2⁵), whereas an experiment with these same five factors at just one more level, three levels, contains 243 combinations of levels of factors (3⁵) – about eight times as many combinations! Indeed, studying five factors at three levels each (3⁵ = 243 combinations) requires about the same number of combinations as are needed to study eight factors at two levels each (2⁸ = 256). As we shall see in subsequent chapters, however, one does not always carry out (that is, “run”) each possible combination; nevertheless, the principle that fewer levels per factor allows a larger number of factors to be studied still holds.

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Chapter 10. Confounding/Blocking in 2 k Designs

Abstract

The topic of this chapter is useful in its own right, and absolutely essential to understanding the subject of fractional-factorial designs discussed in Chap.11. Imagine coming to a point in designing our experiment where we have settled on the factors and levels of each factor to be studied. Usually this will not be an exhaustive list of all the factors that might possibly influence the experimental response, but a bigger list would likely be prohibitive and, even then, not truly exhaustive. There are always factors that affect the response but that cannot be fully identified. Of course, if we are fortunate, these unidentified factors are not among the most influential (often the intuition of good process experts contributes to such “luck”). Ideally, we would like to have all of these other factors held constant during the performance of our experiment; unfortunately, this is not always possible. In this chapter, we discuss a potentially-powerful way to mitigate the consequences if we can’t. We focus on 2^k factorial designs; however, the concepts and reasoning involved apply to all experimental designs.

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Chapter 11. Two-Level Fractional-Factorial Designs

Abstract

We continue our examination of two-level factorial designs with discussion of a design technique that is very popular because it allows the study of a relatively large number of factors without running all combinations of the levels of the factors, as done in our earlier 2^k designs. In Chap. 10, we introduced confounding schemes, where we ran all 2^k treatment combinations, although in two or more blocks. Here, we introduce the technique of running a fractional design, that is, running only a portion, or fraction, of all the treatment combinations. Of course, whatever fraction of the total number of combinations is going to be run, the specific treatment combinations chosen must be carefully determined. These designs are called fractional-factorial designs and are widely used for many types of practical problems.

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Chapter 12. Designs with Factors at Three Levels

Abstract

Sometimes, we wish to examine the impact of a factor at three levels rather than at two levels as discussed in previous chapters. For example, to determine the differences in quality among three suppliers, one would consider the factor “supplier” at three levels. However, for factors whose levels are measured on a numerical scale, there is a major and conceptually-different reason to use three levels: to be able to study not only the linear impact of the factor on the response (which is all that can be done when studying a factor that has only two levels), but also the nonlinear impact. The basic analysis-of-variance technique treats the levels of a factor as categorical, whether they actually are or not. One (although not the only) logical and useful way to orthogonally break down the sum of squares associated with a numerical factor is to decompose it into a linear effect and a quadratic effect (for a factor with three numerical levels), a linear effect, a quadratic effect, and a cubic effect (for a factor with four numerical levels), and so forth.

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Chapter 13. Introduction to Taguchi Methods

Abstract

We have seen how, using fractional-factorial designs, we can obtain a substantial amount of information efficiently. Although these techniques are powerful, they are not necessarily intuitive. For years, they were available only to those who were willing to devote the effort required for their mastery, and to their clients. That changed, to a large extent, when Dr. Genichi Taguchi, a Japanese engineer, presented techniques for designing certain types of experiments using a “cookbook” approach, easily understood and usable by a wide variety of people. Most notable among the types of experiments discussed by Dr. Taguchi are two- and three-level fractional-factorial designs. Dr. Taguchi’s original target population was manufacturing engineers, but his techniques are readily applied to many management problems. Using Taguchi methods, we can dramatically reduce the time required to design fractional-factorial experiments.

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Regression Analysis, Response Surface Designs, and Other Topics

Frontmatter

Chapter 14. Introduction to Simple Regression

Abstract

In previous chapters, we have had data for which there has been a dependent variable (Y ) and an independent variable (X – even though, to be consistent with the notation that is close to universal in the field of experimental design, we have been using factor names, A, B, etc., or “column factor” and “row factor,” instead of, literally, the letter X ). The latter has been treated mostly as a categorical variable, whether actually numerical/metric or not. Often, we have had more than one independent variable. Assuming only one independent variable, if we want to say it this way (and we do!), we can say that we have had n (X, Y ) pairs of data, where n is the total number of data points. With more than one independent variable, we can say that we have n (X ₁, X ₂, …, Y ) data points.

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Chapter 15. Multiple Linear Regression

Abstract

In the previous chapter, we discussed situations where we had only one independent variable (X ) and evaluated its relationship with a dependent variable (Y ). This chapter goes beyond that and deals with the analysis of situations where we have more than one X (predictor) variable, using a technique called multiple regression. Similarly to simple regression, the objective here is to specify mathematical models that can describe the relationship between Y and more than one X and that can be used to predict the outcome at given values of the predictors. As we did in Chap. 14, we focus on linear models.

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Chapter 16. Introduction to Response-Surface Methodology

Abstract

Until now, we have considered how a dependent variable, yield, or response depends on specific levels of independent variables or factors. The factors could be categorical or numerical; however, we did note that they often differ in how the sum of squares for the factor is more usefully partitioned into orthogonal components. For example, a numerical factor might be broken down into orthogonal polynomials (introduced in Chap. 12). For categorical factors, methods introduced in Chap. 5 are typically employed. In the past two chapters, we have considered linear relationships and fitting optimal straight lines to the data, usually for situations in which the data values are not derived from designed experiments. Now, we consider experimental design techniques that find the optimal combination of factor levels for situations in which the feasible levels of each factor are continuous. (Throughout the text, the dependent variable, Y, has been assumed to be continuous.) The techniques are called response-surface methods or response-surface methodology (RSM).

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Chapter 17. Introduction to Mixture Designs

Abstract

In previous chapters, we discussed situations where our factors, or independent variables (X’s), were categorical or continuous, and there were no constraints which limited our choice of combinations of levels which these variables can assume. In this chapter, we introduce a different type of design called a mixture design, where factors (X’s) are components of a blend or mixture. For instance, if we want to optimize a recipe for a given food product (say bread), our X’s might be flour, baking powder, salt, and eggs. However, the proportions of these ingredients must add up to 100% (or 1, if written as fractions), which complicates our design and analysis if we were to use only the techniques covered up to now.

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Chapter 18. Literature on Experimental Design and Discussion of Some Topics Not Covered in the Text

Abstract

In this last chapter, we have two goals. The first is to acquaint the reader with several references in the field of experimental design and the second is to introduce some additional topics in experimental design and provide references for them.

Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Backmatter

Titel: Experimental Design
verfasst von: Paul D. Berger
Robert E. Maurer
Giovana B. Celli
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-64583-4
Print ISBN: 978-3-319-64582-7
DOI: https://doi.org/10.1007/978-3-319-64583-4