Hello. We're back again continuing to talk about fractional factorial designs and in this class, we're going to talk about the special case of Resolution III designs in a bit more detail. Now, remember Resolution III designs are fractional factorial designs where the main effects are aliased with two-factor interactions. You may also have some two-factor interactions aliased with each other, but the real defining feature of these designs is that the main effects are aliased with two-factor interactions. These designs are frequently used for screening experiments. For example, you can accommodate between five and seven factors in eight runs with Resolution III, and between nine and 15 factors in 16 runs for Resolution III. So these designs can be fairly small yet they can accommodate a reasonably large number of factors. Saturated versions of these designs are sometimes used. A saturated design has K equal to N minus one factor, where N is the number of runs. So seven factors of eight runs would be a saturated design, fifteen factors in 16 runs would be saturated and so on. We're going to take a look at an example in Table 8.19, and this example is a two to the seven minus four Resolution III design. So it is seven factors in eight runs. Here's the design. If you read about the experiment in the book, it tells you that these are factors associated with our focus time experiment. I won't go into all the details of this because you can read about the experimental details in the book. It is seven factors in eight runs, its Resolution III and the alias relationships are shown at the bottom of the design matrix. You notice that every main effect is aliased with three two-factor interactions. For example, A is aliased with BD, and with CE, and with FG. So what this example is going to allow us to illustrate is sequential assembly of fractions to separate alias defects, and it's clear we need to do that in this example, and then it's likely that we're going to need to do that in this example, because if we have more than two factors that appear to be important, the aliasing is going to take over. If we only have two factors that appear to be important, then probably it's just the main effect of those two factors. But this is an example that we're going to see shortly where we do have to do some additional runs to dealias, main effects in two-factor interactions. There are two types of schemes that are used to do this. The common approach, the standard approach is something called fold-over. In fold-over, what we end up doing is running another experiment that is exactly the same size as the first one. So if we needed to fold-over this example that we're just looking at, it would be another eight run design. Now, there's a couple of ways to do fold-over. In one approach, the fold-over consists of the same design that we ran previously, but we switched the signs in one column. In other words, the pluses become minuses, the minuses become plus. Switching the signs in one column when we combined the two designs, it enables us to estimate the factor and all of its two-factor interactions. So if you switch the sign in column A, for example, you don't only get to estimate A free of other two-factor interactions, but you can estimate all the two-factor interactions that involve A. Now, the other approach for using a fold-over is to switch all the signs that has changed the signs in all columns and when you combine that design with the first one, that enables you to dealias all main effects from the two-factor interaction alias chains. Now, the two-factor interactions still remain alias, but at least you can dealias all the main effects from all of the two-factor interactions. The book talks a bit about how to mathematically find the defining relation for a fold-over. Not terribly important because most of this is done by computer. One thing I do caution you about, be careful because the rules that have just given you only work for Resolution III designs. That is this single factor fold-over in this full fall-over, only works for Resolution III. There are other rules that we can use for, say, Resolution IV designs. As I hinted at the start, fold-over is only one way to dealias interactions. There are other methods for adding runs to fractions to dealias effects that you might be interested in, but we're not going to talk about those in this section. Let's take a look at this example, this autofocus tab experiment that we introduced briefly a few minutes ago. Here's the design matrix again, and the alias chains are also shown down at the bottom of the page and now I've also included in those alias chains the estimates of the effects. You notice that the main effect of A is large, the main effect of B is large, the main effect of D is large, and then everything else is apparently pretty small, their estimates are fairly close to zero. These three seem to dominate. So the simplest interpretation of this experiment would be that it's the main effect of A, and the main effect of B, and the main effect of D. But when you examine the alias chains a little bit more closely, you can see that there are other possible interpretations. For example, it could be the main effects of A and B, but it could also be the AB interaction instead of the main effect of B. Or it could be D plus the two-factor AB interaction. It could be B and D as the main effects and the BD interaction, or the main effect of A plus the BD interaction. Or it could be A and D main effects and the AD interaction. Or the the main effect of AD together. It's not clear, the results are ambiguous here as to how we would go about interpreting the results. Now, if you have some prior knowledge or some underlying science that might suggest a particular interpretation, you could possibly rely on that. But I remember George Box saying one time that in screening experiments where you have lots of factors, the experimenters really often don't know which main effects might be important and to expect them to have insight about which interactions are important is probably stressing things a little bit too far. You can see that when you project this design into the three factors, A, B, and D. When you get to look at that picture on the right, Figure 8.23, you notice that what you have is essentially a half fraction of the 2 to the 3. So that's why we have these main effects and two-factor interactions aliased with each other in the projection. Now, if there only been two factors that were large, this thing would have projected into a two squared design, and it would have made the interpretation much simpler. So let's run the fold-over. So Table 8.22 is the fold-over experiment. So to get this design matrix, all we did was reverse the signs in the original test matrix. It turns out that what that is really equivalent to doing is changing the signs on the generators that involve these two-factor interactions, that is, instead of D equal to plus AB, in this design, D is equal to minus AB, and E is equal to minus AC instead of plus AC, and F is equal to minus BC. So this design is really the same as the first one, with just the signs in those columns reversed. But it turns out you can create the test matrix by just switching all the signs. You don't have to really worry too much about changing the signs on a specific pair or a set of columns. However, in changing that design with the fold-over, what you do is you change the signs on the two-factor interactions in the alias relationships. Those aliased relationships are shown down at the bottom of the table on the left. You notice that now A is aliased with minus BD, and minus CE, and minus FG, for example. In fact, all of those aliased relationships are exactly the same as they were before, except the signs of the two-factor interactions are switched. I've also shown you the estimates of the parameters from this group of eight runs. Again, the A effect is large and the B effect is large, and the D effect is large. So now how do we actually analyze the results? Well, you could of course just combine the two designs into 1160 in one experiment and analyze it again. You could do that, but you don't really have to do that. If you take the estimate of A from the first fraction and add to that, the estimate of a from the second fraction, all of those two-factor interactions will cancel out. It turns out that when you do that, what's left is the A main effect, and the difference in the estimates of the effects turns out to be 1.48. You can do that for every single pair of these effect estimates. The first column of this table shows you what happens when you take the effect estimates from the first group of eight runs and add the effect estimates from the second set of eight runs from the fold-over to it and then average. You get all of the main effects isolated, free and clear of two-factor interactions. When you look at those effects from the combined set of runs, you noticed that only two of them are really large now, B and D. Now, if you take the effect estimates from the first group of runs and then subtract the effect estimates from the fold-over, the second group of runs from that, you will actually remove all of the main effects. The main effects will cancel out, and what you'll be left with are the two factor interaction alias chains. So we average the numerical results from that process and that's what's shown in the second column of this table at the lower right. Now, we have all of the two factor interaction alias chains isolated. Well, one of those two-factor interaction alias chains is pretty large. BE plus CE plus FG. So at least one of those two factor interactions is very likely to be significant. Which one? Well, since the B and D main effects are important and BD is in this alias chain, the simplest interpretation is that this is the BD interaction. So I really relying on our old principle of Occam's razor again, and to conclude that what we have here as two significant main effects and a significant two factor interaction. And the technique of fold-over is what enabled us to untangle these relationships with a couple of relatively small experiments. Remember, this fold-over technique that we illustrated here, either running a mirror image design with all the signs reversed or switching the sides on a single column. These tricks that I've shown you only work in Resolution III. There are other fold-over rules that we can use for, say, a Resolution IV design. If you want to look at the textbook, you'll find some reading there about two things that would possibly be of interest. Mathematically, finding the defining relation for a fold-over. That's something that might be of interest. You don't really need to worry about that in practice much because the computer programs that we use will take care of this thing. Sometimes I think blocking is an important consideration in the fold-over design because the second group of runs may be run after the first group sometime in the future, maybe a day or several days later, or it may use different material. So blocking could be an important consideration. The nice thing is that when you fold-over a design, there is some blocking going on automatically. For example, in this experiment, in the first group of runs, D was equal to AB. So there's a part of the defining relation that involves the positive ABD interaction, and then there's a part that involves the positive ACE interaction, and a part that involves the positive BCF interaction. While in the second group of runs, those interaction terms are all negative. So in the first group of runs, you've got an interaction term or a set of interaction terms that are positive, and in the second group of runs, they're negative. So those sets of interaction effects collectively are confounded with blocks. Okay, so this has been a quick introduction to Resolution III designs of a particular type. And we'll see that there is another type of Resolution III designs that we'll talk about a little later.