Many Factors at Many Levels, but One Factor at a Time

Let us first illustrate a case in which two or more factors have a combined influence but are experimented with one factor at a time. Example 7.1 We call our experimenter here “coffee lover.” He guessed that “great” coffee is the result of adding to the fresh brew of a given strength of a

Factorial Design, the Right Way

When two or more factors, each in two or more levels, are to be tested for their combined effect on the quality characteristic—the dependent variable—-factorial design is appropriate. The follow­ing is an example of three factors at two levels. Example 7.2 In a sheet metal press, it is the intention to measure the quantity

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Too Many Factors on Hand?

Efficient as it is, even factorial design can get choked with too many treatments in industrial experiments, where it is not uncommon to face as many as ten to fifteen factors threatening to act simultaneously on the outcome. Consider an experiment in which there are six factors—this is not too many in several industrial

“Subjects-and-Controls” Experiments

Situations wherein several causes acting together result in one or more noticeable effects are not rare. Our coffee lover’s experiment was but one example. If the intention of the experiment is to study the effect of a new cause in addition to the existing ones, then a comparison between two cases, one with the

Combined Effect of Many Causes

In the preceding example of an experiment on the benefit of a new plant food, it was agreed by implication that the addition of the plant food to the nourishment protocol of a subject plant did not, by its presence, in any way influence the other items of care. Suppose that a particular plant’s

Unavoidable (“Nuisance”) Factors

In the context of many factors acting together, some uninten­tional, often unperceived or unavoidable, factors that cannot be accounted for may influence, to an unknown extent, the quality characteristic. For example, vibrations in the building, noise, illu­mination, room temperature, humidity, and so forth, can have an effect when the experiment involves only inanimate objects.

Designing Factors

The most striking feature that distinguishes experiments with designed factors from those without them is that, in the former, all the factors are simultaneously designed into the experiment, whereas in the latter, experiments are done one factor at a time, like experiments with only one independent variable. That designing factors is more efficient than

Experiments with Designed Factors

Consider the following eight experiments with combinations: Out of these, each of the following four pairs provides results for comparing the effect of ^2 with that of ap The other four pairs serve to compare the effect of b>2 with that of bp Finally, the following four pairs provide the bases to compare the

Matrix of Factors

The situation of having three variables, each at two levels, can be represented pictorially, as shown in Figure 8.1. Each factor combination of the eight combinations we have dealt with as “designed factors” is represented by a vertex of the orthorhom­bic volume; the space represented by this shape is often referred to as the

Remarks on Experiments with Two-Level Factors

In multifactor experiments, particularly when the number of fac­tors is high, testing at two levels of factors is quite adequate to decide (1) if one or more of the factors is ineffective, (2) the rela­tive extent of effectiveness of each factor, and (3) if two or more of the factors interact, meaning that their

Response of Multifactor Experiments

Whereas results of single-factor or one-factor-at-a time experi­ments can usually be compiled as y = fx), the response to multi­factor experiments needs a different treatment for analysis. We will start in this chapter with the simplest situation: two factors, at two levels each. We further assume that the factors are quanti­tative. But to keep

Experiments with More Factors, Each at Two Levels

As a way of preparing for factorial experiments with more than two factors, let us slightly modify the data in Table 8.5, using the “-1” and “+1” symbolism in place of “low” and “high” for the factor levels. The modification results in Table 8.6. An explanation of the last two rows of this table

Fractional Factorials

Having arrived at the logical end point for discussing full-facto­rial experiments with four factors, we need to reflect on what lies ahead. We mentioned early in this chapter that in many contexts, for example, manufacturing, confronting a large number of fac­tors is not uncommon. If the number of factors is ten, for instance, even

Varieties of Factors

1. Quantitative versus Qualitative Factors For convenience, we have so far considered a factor as an inde­pendent variable in a function of the kind y = fx). In this equa­tion, for every different numerical value of x, there is a corresponding numerical value of y. Therefore, x, then, is obvi­ously a quantitative factor. The

Levels of Quantitative and Qualitative Factors

Considerations, of quantity or quality levels, is an inseparable part of the selection of factors. First of all, the difference relative to a given variable may be taken either as a distinction between (or among) factors or as different levels of one factor. Men and women, in an experiment dealing with physical strength, may

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Limitations of Experiments with Factors at Two Levels

Chapter 8 dealt with designing many factors together, all at only two levels. When we are beset with a large number of factors, it is possible that one or more among those are spurious. It is desir­able to eliminate those early in planning the experiment. If the main effect of a particular factor is

Four-Level Factorial Experiments

We take the two factors as a and b, and designate the levels of The sixteen combinations can be lined up as follows: We notice in the above matrix that In the first column, the factor is common, and the only variables are the levels of b: bi, b>2, b3, and b4. In the

Interactions in Experimental Research

In most cases of research, particularly with quantitative parame­ters, data as depicted in Figure 9.1, with absolutely parallel lines, are rather rare. A set of data as shown in Figure 9.3, instead, is one of many possibilities. We will examine Figure 9.3 for interac­tion and for main effects. We notice in this figure some

Main Effects in Experimental Research

For observation of the main effects, we import the data in Table 9.1, augmented with some derived, additional information, now shown in Table 9.2. We observed in Figure 9.1 that there is practically no interac­tion between the factors a and b, but that has no bearing what­ever on the presence or absence of the

More on Interactions in Experimental Research

Interactions between factors, when there are only two, are easy to identify in two-way tables, as in Table 9.1, or even make visible in figures such as Figures 9.1 to 9.3. The presence of interactions complicates the inter­pretation of the experimental results relative to the benefits or harmful effects of individual factors. If there