Well, welcome back to our experimental design class. This module deals with fractional factorial designs. We are continuing to work with the 2_k series of designs. Remember, Module 6 was all about an introduction to the 2_k. Module 7 was about blocking and confounding in those designs, and now we're ready to talk about fractional factorials. The text reference for this material is Chapter 8, and I think the motivation for doing fractional factorial should be pretty obvious. As the number of factors grows, the size of the basic 2_k gets out of hand pretty quickly, 2_4 is 16, that's not too bad, 2_5 is 32, maybe that's okay, 2_6 is 64, 2_7, 128, 2_10 is the 1012. So those designs are just getting way too big to be practical, and so we have to do something. In Chapter 6, one of the things we talked about doing was to use only a single replicant, and that can be okay when you have four or maybe five factors. But I think anytime you have five or more factors, you need to think about doing a fractional factorial. The primary emphasis on application of these designs is in factor screening. That is, we have a relatively large number of variables because we don't know much about the system or about the process. What we want to do is we want to quickly and efficiently with these few resources as possible, identify the factors that have large effects. These designs are often run as unreplicated factorials, like we talked about in Chapter 6. Sometimes we add center points to these designs, but they're almost always run as unreplicated factorials. This is the outline for Chapter 8 in the book, and we're going to cover essentially everything in this chapter. Although our discussion of some of the topics is not going to be as detailed as others, and I'm going to suggest that you really read and study this chapter in detail because it's one of the key chapters, I think, in the book. In fact, I think Chapters 5,6,7, and 8 are really key chapters for this material, and detailed reading and studying of those chapters will really pay a lot of dividends. So why do fractional factorial designs work? What is it that would make us think that running only a subset of the runs from a full factorial would work? Well, the general principles that drive this thing are shown here. First of all, sparsity of effects. Now, what is sparsity of effects mean? Well, it means that you may have lots of factors, but relatively few of these factors are important, and the system is likely to be dominated by main effects and low order interactions. The low-order interactions here, principally, are two-factor interactions. The second thing is the projection property. Every fractional factorial contains full factorials or stronger fractions, typically, in fewer factors. So that means that if you can drop or eliminate some of the factors from your fractional factorial, you could very well have a design left, an experiment left, that is the very design that you would probably wanted to run anyway if you knew what the important factors were at the start. Then finally, sequential experimentation. You can always add runs to a fractional factorial design to resolve any difficulties or ambiguities that you have in interpretation of the results. Here's the simplest case of a fractional factorial. It's the one-half fraction of the 2_k. This is section 8.2 in the book. Now, this design does not have 2_k runs. It has 2_k over 2 runs, and so it's usually called a 2_k minus 1. The simplest case of the half fraction would be three variables, the 2_3 minus 1. The table you see down at the bottom of the slide, Table 8.1 is the table of plus and minus signs that form the contrasts for the 2_3 factorial. Columns A, B, and C are the design matrix columns, and then the other columns are generated by multiplying the columns from A, B or C together. Now, we're only going to look at the top four runs in this table. That's going to be our one-half fraction, the runs A, B, C and ABC. Now, notice when you look at this table that all of the runs in this table are positive in both the identity column and the ABC column, and so that column, I equal to ABC, we often call that the defining relation for this particular fraction. Here's a picture that illustrates the design, figure 8.1. The left-hand side of this figure shows you the four runs that we just identified as our half fraction; A, B, C and ABC, and notice the geometry, see the geometry of the design, opposite points, opposite faces. Now, we could've used the bottom four runs in this table as a fractional factorial as well. Those runs are shown in the right-hand cube in Figure 8.1. So either one of these designs could be used as a half fraction. For the runs in the bottom half of the table, notice that I is equal to minus ABC. So we have two different fractions. When I is equal to ABC, we refer to that as the principal fraction and when I is equal to minus ABC, we refer to that as the alternate fraction. Now, I want you to take a look at this table of minus and plus signs one more time. Look at the contrast for estimating the effect of factor A, there it is, plus A minus B minus C plus ABC, that's the contrast and then we would divide that by two. Now, look at the contrast for computing BC. It's also plus A minus B minus C plus ABC. So in other words, the same contrast that we use for estimating A is the same contrast that would be used to estimate BC. In other words, A and BC cannot be distinguished from each other. This phenomenon is called aliasing, and it occurs in all fractional factorial designs. Now, if you look at this table again, you'll notice that B would be aliased with AC because the columns of minus and plus signs are the same, and that C is aliased with AB because the pattern of signs in those two columns are the same. So this phenomena called aliasing is something that occurs in all fractional factorial designs, and we can always determine the aliases directly from looking at the columns in the table of minus and plus signs. In our design, we saw that A is equal to BC, B is equal to AC, and C is equal to AB or in other words, the main effects here are aliased with two factor interactions. We don't have to use the table of minus and plus signs to find the aliases. We can find them directly from the defining relation, I equal to ABC. To do that, all you do is you multiply both sides of that defining relation by one of the effects. Suppose we're interested in the alias of A, so we multiply A times I equal to ABC. So on the left-hand side, that would be just A because A times I is A and A times ABC would be A square BC. But the square of column A is an identity column. So A would be equal to BC. Similarly, B would be equal to AC and C would be equal to AB. The textbook notation that we use is shown at the very bottom of the slide. A in brackets indicates that that's the estimate of A, and the arrow says A estimates A plus BC, and this says that B estimates B plus AC, and that C estimate C plus AB. So in other words, you're estimating the sum of those aliased effects. Now, what about the alternate fraction? Well, in the alternate fraction, things are slightly different. I equal to minus ABC is the defining relation. So that means that we have slightly different aliases. A would be equal to minus ABC, B would be equal to minus AC, and C would be equal to minus AB. We say that both of these designs belong to the same family, I equal to plus or minus ABC. So you could imagine running, say, the principal fraction first, and then if you decided later on, say, the next day or that afternoon to also run the alternate fraction, you can combine those two groups of runs to actually form a full factorial. That's a really simple example of sequential experimentation. But since ABC would be positive in the first group of four runs and negative in the second group of four runs, that strategy, that design would actually confound ABC with blocks. The blocks being the two time periods in which you actually run those runs. Let's talk for a minute about design resolution. Design resolution is a way to classify or categorize fractional factorial designs into important categories. A resolution III design is one in which main effects are aliased with two factor interactions. We just saw an example of that. That's the 2_3 minus 1, and lots of times, we will put a roman numeral subscript on the two to indicate the resolution. Resolution IV designs do not alias main effects with two-factor interactions, but two-factor interactions are aliased with each other. A good example of this is the half fraction of the 2_4, the 2_4 minus 1. Then resolution V designs, main effects are not aliased with two-factor interactions. Two-factor interactions are not alias with each other, but two-factor and three-factor interactions are aliased with each other, and a good example of that is a 2_5 minus 1, that's a one-half fraction of a 2_5. Resolution III and resolution IV designs are very extensively used as designs for screening. Resolution V designs would also be good screening designs. But for any more than about five factors, they turn out to be pretty large, and so resolution III and resolution IV designs are the ones that we typically use.