Robert E. Maurer

Two-Level Factorial Designs

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.

Introduction to Taguchi Methods

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.

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

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.

Some Further Issues in One-Factor Designs and ANOVA

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 e’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.

Two-Level Fractional-Factorial Designs

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.

One-Factor Designs and the Analysis of Variance

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.

Introduction to Response-Surface Methodology

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).

Confounding/Blocking in 2k Designs

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.

Designs with Factors at Three Levels

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.

Multiple-Comparison Testing

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 H0(μ1 = μ2 = μ3 = … = 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 Scheffe test as a post hoc study for multiple comparisons.