In our last lecture, we did a pretty thorough job of analyzing a one-half fraction, a half fraction of a 2 to the 4. Here's another example of a half fraction. This is a half fraction of a 2 to the 5. So this would be five factors in 16 runs, and the Table 8.5 shows you the setup for this design. Notice that the basic design would have 16 runs, so it's a full 2 to the 4, and then E was chosen to be equal to ABCD. That's the highest-order interaction among the columns in the basic design. That ensures that I've got a Resolution V fraction because the defining relation is now I equal to ABCDE. So A, for example, would be aliased with B, C, D, E. In fact, every main effect, would be aliased with a four-factor interaction and every two-factor interaction would be aliased with a three-factor interaction. What would have happened if I had chosen a different column generator? Suppose I had chosen I equal to ABC. Well, what would have happened? Well, A would've been aliased with BC. Uh-oh. That's not very good. That's a Resolution III design. So choosing anything other than E equal to ABCD would be a bad choice. If you chose D equal to ABC, that would be nearly as bad choice because now main effects would be aliased with two-factor interactions. But your two-factor interaction AB would be aliased with CD. So that's a Resolution IV designs. So anything other than E equal to ABCD leads to a design of lower resolution. Generally, you always want to get the highest resolution possible. So this design is in the book. Here's the data over in the column labeled yield, and here's the resulting analysis of this design using JMP, and I looked at all of the main effects and all of the two-factor interactions, and use the JMP screening platform, and so the analysis that's being done here is with Lynth's method. If we look at the magnitudes of the, now these are regression coefficients, and if we look at the pseudo t-values and the pseudo p-values, we see right away, that main effects of X2, X1, and X3, and the X1, X2 interaction emerge as being significant. So the interpretation here is pretty straightforward. What about follow-up strategies from a fractional factorial? Well, here's an illustration that gives you a good understanding of that, I think. After you run a fractional factorial, some of the things that you can do are to, for example, perform one or more confirmation runs, and we saw an example of that. You can add more runs to resolve ambiguity in interpretation due to aliasing, and we'll see a couple of examples of that. You might decide to change the scale on one or more factors and rerun the experiment. Because perhaps you think that upon reflection, you didn't vary the factors over a big enough range, and that's why the factor may be turned out to be non-significant. So that's a useful thing to consider. You might replicate one or more of the runs in the experiment to perhaps verify that you got the correct results. Maybe there's some concern about having the correct results. You might also replicate some runs to get an estimate of error. You might drop factors from the experiment and add others and then rerun a new experiment. For example, you might have started with five-factors, but you might have had a candidate list of maybe 8-10 factors. So you run the first experiment and perhaps you drop two or three of them and you go back to the original list and add two or three more, run a new experiment. You may decide to move to a new experimental region because you don't think the results that you're getting in the region which you've run the current experiment are as good as you could possibly achieve. So you might think about moving the region of experimentation a bit. Then you might consider adding additional runs that would allow you to estimate higher-order terms, such as quadratic terms. If you've added center points, for example, and seen an indication of curvature, then you might add axial runs so that you could fit a second-order model. So these are all strategies that you need to think about following running a fractional factorial. Here's another example of a fractional factorial. This is a one-quarter fraction. This is a one-quarter fraction of a 2 to the 6. The 2 to the 6 would be a 64 run design, the one-quarter fraction would have 16 runs. So the basic design would be a 2 to the 4, and so if you look at the first four columns, those are the columns of a 2 to the 4 in standard order. The column generators I chose where E equal to ABC and F equal to BCD. Now, you're wondering why I did that, and the reason I did that is because that gives me a design of highest possible resolution. In fact, this is a Resolution IV design, as we will see shortly. The complete defining relation for this design is I equal to ABCE and BCDF, and then the product of those two columns, ADEF, also must be included in the defining relation. Now why is that true? Well, there's a column here, essentially, AB. This column produces an identity column I equal to ABCE, and this column produces an identity column, I equal to BCDF. So those two columns, if multiplied together, must produce another Identity column in the table. So that's where ADEF comes from, and you can get ADEF simply by multiplying ABCE times BCDF. When we multiply those together, the Bs cancel out and the Cs cancel out, and you get ADEF as the remaining factor. The remaining word in the defining relation. These three interaction terms are called words. Something interesting. As soon as I write out the defining relation, I immediately know the resolution. It turns out that the resolution of a fractional factorial design is the number of letters in the shortest word in the defining relation. In this case, our defined in relation all three words have four letters. So this is a Resolution IV design. The Table 8.8, shows you the alias structure for this design. This is the complete alias set. You notice that main effects are aliased with three-factor and higher-order interactions. Two-factor interactions aliased with each other. In fact, the two factors are aliased in pairs, except for this one group. That's where you have three two-factor interactions aliased together, and then you have two three-factor interaction alias chains that you see down at the bottom of the table. So the interpretation of this design would probably not be a whole lot more difficult than the interpretation of the half fraction of the 2 to the 4. Because the main effects are clear of two-factor interactions, and except for one chain, the two-factor interactions are aliased in pairs. This is the principle fraction. Notice that I was equal to plus ABC and F was equal to minus BCD. But there are three other alternate fractions. You could choose E equal to plus ABC and F equal to minus BCD. Or you could choose E equal to minus ABC and F equal to plus BCD, or E and F are both equal to minus ABC. Now, what would that do? Well, that would change the signs in either this column, or this column, or both columns. These alternate fractions are unique, that is, they don't overlap at all with the principal fraction. What are some of the possible uses of that? Well, one possible use is perhaps there's something you don't like about the principle fraction. One of the things that experimenter should always do before you run a fractional factorial design is the team should sit down and carefully examine all of the runs and make sure that those runs are indeed feasible, that there're runs that you can and do want to do that nothing unusual, nothing strange is going to happen. I even like to suggest that people guess at what they think the response is going to be. That could really get you thinking about the runs that you've got to make. Now, suppose you do this and you find that one of these runs, say this run down here at the bottom, you have everything at the high level. Suppose that melts the reactor and you just can't do that. So how can you get around that? Simple, switch to an alternate fraction because this run only occurs in the principle fraction, it only occurs there. So if you switch to an alternate fraction, you get rid of that run. Now, there's no guarantee that the alternate fractions still doesn't contain infeasible runs, so you need to go back and check for that. But as a general rule of thumb you can usually avoid runs that you don't want to make by simply using one of the alternate fractions in the design. The projection property is one of the things that make these designs useful. We talked briefly about that before. What about projection of this design that we have here, the quarter-fraction of a two to six into subsets of our original six factors. Well, it turns out that any subset of the original six factors that is not a word in the complete defining relation gives you a full factorial. So look at your subsets of four. A, B, C, D would be a full factorial, A, B, C, F would be a full factorial. But A, B, C, E would produce a replicated half-fraction, and that replicated half-fraction would still be a Resolution IV design. So the projection capabilities, the projection features of these designs, can be extremely useful. The textbook has a pretty complete example of a one-quarter fraction of the two to the six. It's Example 8.4, it's an injection molding process with six factors. The textbook shows you the design matrix, it takes you through all the computations so that you can calculate the effects. We use normal probability plotting there to estimate the effects. Two of the factors, A and B and the AB interaction emerges being important. I'd urge you to take a look at that example. We do some residual analysis in that example, that's fairly interesting. The residual analysis indicates that there may be some settings of the factors that produce more variability in response than others. That's sometimes called a dispersion effect. I think that's an interesting part of this example that you probably should take a look at. The general case of a two to the k minus p fractional factorial design is also discussed in this chapter. It's Section 8.4. In general, a two to the k minus one is one-half fraction, a two to the k minus two is a quarter-fraction, a two to the k minus three is a one-eighth fraction, and in the general case, a two to the k minus p is a one over two to the p fraction. These designs are always constructed by starting with the basic design and then adding p columns to that basic design. That means that you have to specify p independent generators to construct those columns. It's really important to choose the generators to maximize resolution, and Table 8.4 can help you do that. I'll show you that table in just a moment. There's a projection rule that is also useful. A design of resolution R contains full factorials in any subset of r minus one of the original factors. So a Resolution III design, for example, contains full factorials and any subset of two factors. A Resolution IV design contains four factorials in any subset of three factors and so on. This is very useful. We'll talk about blocking in just a moment. Here's the table that I wanted to show you. This is Table 8.14, and this table contains four designs, a selection of designs with up to 15 factors. The design generators that you would use for various fractions to create a design that has the maximum resolution. Generally, you're not going to create these designs by hand, you're going to do them using computer software. The JMP computer software builds designs using generators that are chosen from this table. So you always maximize the resolution of the design. Now, in some fractional factorials, we may actually be able to block the design. Let's go back and look at this one-quarter fraction for just a moment. This is the quarter-fraction of the two to the six. You notice that there are two alias chains down at the bottom of this table. These alias chains contain only three-factor interactions. You could choose one of these alias chains. Let's say this one, and then any one of the three-factor interactions in that alias chains such as ABD could be used to break this design up into two blocks and ABD would be confounded with blocks, you would not loose information on anything else. One last point for the general case of the two to the k minus p, resolution may not be everything that you want to consider in choosing a design. You might also be interested in designs that are of minimum aberration. Now, what do we mean by minimum aberration? Well, let me show you an example that illustrates this. Let's talk about the two to the seven minus two. Let's consider the Resolution IV case. This is a 32 run design. There are three sets of generators that you can use that lead to a Resolution IV design. Design A, has generators F equal to ABC and G equal to BCD. The basic design would have columns A, B, C, D, E. But if you choose those generators, then notice that it is Resolution IV because all the words have four letters, then here are the complete set of two-factor interaction aliases, shown below that. Now Design B, we could choose F equal to ABC and G equal to ABE. The complete defining relation for that design is shown in Column B. Well, that is a Resolution IV design because the number of letters in the shortest word in that defining relation is four. Now, look at the aliases. They're far fewer two-factor interaction aliases in Design B. But in Design C, suppose we choose F equal to ABCD and G equal to ABDE. Now, look at the defining relation. Again, it's still Resolution IV because the number of letters in the shortest word, CEFG is four, but look at the two-factor interaction aliases. A minimum aberration design minimizes the number of interactions of that particular resolution. In this case, we only have three two-factor interaction alias chains. Otherwise, we have six or seven two-factor interaction alias chains with the other choices. Minimum aberration designs are highly desirable. All the designs in this Table 8.14 are not only minimum aberration, but maximum resolution. Most computer packages use designs from this table or ones that are isomorphic to them to create their fractional factorials. So that's an overview of the general case of the two to the k minus p and some guidance about how you might go about constructing those designs. See you next time.
