Welcome back to our class on experimental design and we're still talking about fractional factorials. But in this class, we're going to talk about the special case of Resolution 4 and Resolution 5 designs. Now remember, in a Resolution 4 design, main effects are clear, free of two-factor interactions. But the two-factor interactions are aliased with each other. Resolution 4 designs, like their resolution three kindred spirits, are very often used in screening designs. The Resolution 5 designs are larger experiments. They often have more runs than we really want to use in an initial screen and experiment. Resolution 4 designs have to have at least twice as many runs as the number of factors. So for example, if you had six factors, the Resolution 4 design would have to have at least 12 runs. If you had eight factors, it would have to have at least 16 runs. The table that you see on this slide shows you the number of factors that can be used in a regular two to the k minus p design with 4, 8, 16, and 32 runs. For example, with 16 runs, that's a full factorial in four factors. It's a half fraction in five factors and it's a resolution for fraction with 6-8 factors and a resolution three fraction with between 9 and 15 factors. So that gives you an idea of what you can do with a fixed number of runs and again, these are regular fractions. So if effects are aliased, they are completely aliased or completely confounded. The non-zero entries in that alias matrix are either plus one or minus one. Resolution 4 designs can be employed for screening and when there is difficulty in interpretation, it is possible to do sequential experimentation with some form of fold-over. There's a paper referred to here that gives you a lot of details of that, there's also discussion of this in the book. When you fold over a Resolution 4 design, your objectives might include things like breaking as many of the two factor interaction alias chains as possible or breaking the two-factor interactions on a specific alias chain or breaking the two factor interaction aliases involving the specific factor, all of those are possibilities. Now, we can't use the full fold- over procedure that was given previously for Resolution 3 designs. It turns out that if you take a Resolution 4 design, and you switch all the signs, you get exactly the same design back but in a different run order. So you don't gain any additional information that can help you break alias chains with that technique, you need to do something different. Something that is very useful in Resolution 4 designs is to switch the signs in a single column. That allows all of the two-factor interactions involving that column to be separated so that you can estimate them. This is a very nice trick. Here's an example that we're going to use to illustrate this. This is an experiment from the semiconductor industry. It's a spin coating tool. This is used to apply photo-resist wafers. The wafers sits on something that looks like a phonograph turntable, it spins around, and then this resist fluid is sprayed onto the surface of the wafer and the idea is to run this tool at a set of conditions that give us a particular target thickness for the photo-resist layer. So there are six factors here: speed, that's RPM, acceleration, volume of photo-resist material that is being ejected, how long we run the process, the resist viscosity, and then the exhaust rate, and then what's measured is the thickness of the photo-resist layer in mils. So here's the results of analyzing this experiment. This is a half normal probability plot and it's clear that we have four very strong factors A, B, C, and E, and we're pretty confident that those are the main effects because in this design, the main effects are not aliased with any two-factor interactions, but the AB two factor interaction is aliased with CE. So how do we interpret this? Is this AB or is it CE or is it both? You could argue that it really could be both because main effects A, B, C, and E are all important and so it could be either AB or CE or both. We need to de-alias interactions and a logical way to do this would be to switch the signs in column A. Because when we switch the signs in column A, we de-alias all of the two-factor interactions involving A from anything else. So that's going to give us AB, AC, AD and by the way it's going to give us CE as well. So switching the signs in A is really a very good idea. So we could do another 16 runs. Finding the aliases here would involve using the alias matrix, if you wanted to manually do this. You can also find them from computer software. You can create the original 16 run design folded over and you can look at the alias relationships on computer software. So now we run the complete fold-over and here is the complete fold-over experiment for this example. So the bottom 16 runs are in a separate block, we're going to consider that to be a second block. On the right is the half normal probability plot for this modify, this expanded spin coater experiment with 32 runs. Look what we've got A, B, C, and E merge as important main effects and so does the CE interaction. So it's the CE interaction here, not AB, and not both of them, it's just the CE interaction that's important. Full fold-overs of Resolution 4 designs may not be necessary. In fact, a lot of people are reluctant to do them because potentially it's inefficient and it takes a lot of additional runs for. For example, in the spin coater experiment, there was seven degrees of freedom available to estimate two-factor interaction alias chains. So you add the fold-over 16 more runs, there are only 12 degrees of freedom now for estimating two-factor interaction aliased chains. So you did 16 additional runs but you only gained five more degrees of freedom so far as parameter estimation is concerned. A technique called partial fold-over or semi folding might be useful. Basically a partial fold-over or semi-fold only uses half the number of runs in a full factorial. Here's how you do this. Step 1, construct a single factor fold-over from the original design by changing the signs on a factor that's involved in one of your two factor interactions that you're interested in. Then only choose half of the full overruns. Choose those runs where the chosen factor is either all of its high level or all of its low level. Many people think that if you can select the level that you believe generates the most desirable response values, that that's a good idea. Sometimes that's hard to do, but as long as you pick the half of the runs that are either positive or the half of the runs that are either negative and do that half of the fold over that will enable you to de-alias interaction effects just like you could with a full fold-over. Here is the semi-fold or partial fold-over design for the spin coater experiment. When I did this, of course we changed the signs in column A, but I chose only the eight runs where A was negative to form the fold-over. Those are the runs that you see here, runs 17-24. Now the partial fold over design is not orthogonal. So it has some correlated parameter estimates, that's something that you will live with in these designs. The standard errors of the regression coefficients will be a little larger than they would be in an orthogonal design. But again not such a big problem that it gives us any difficulty in interpreting results. This is what the alias chains look like now and notice that AB, and CE are still now dealiased, still 12 degrees of freedom to estimate two-factor interaction alias chains. Instead of requiring 16 runs to do that, it only takes an extra eight. Here is the half normal probability plot of the effects from this partial fold over version of the original spin coater experiment. Just as we saw before, AB and CE are important main effects, and the CE interaction emerges as the important two-factor interaction in this experiment. So we get exactly the same results as we would've gotten with a full fold over with half the number of runs. So when you're thinking about folding over a resolution for design, think about the possibility of using a partial fold over. Resolution V designs, they can be large as we've talked about. There's an example of a two to the five minus two in the book. Generally, the two to the five minus one, is the smallest resolution five design, that's five factors and 16 runs. If you have six or more factors, these are designs with at least 32 runs. These are large experiments. In some cases, we can find non-regular designs that are effectively resolution five. We use optimal design tools to do that. Jump has excellent capability to do that and there are some examples for six and eight factors that are illustrated in the textbook. If you're interested in those that it'd be instructed to look at that. There's one other type of design that I would like to mention in concluding our coverage of this chapter and this is something called a supersaturated design. Now, we know a saturated design is a design where k, the number of factors is exactly equal to N minus one, where N is the number of runs. But in recent years, there's been a lot of interest and actually a lot of work done in developing and using something called super saturated designs. In a supersaturated design, the number of factors k exceeds N minus one. In fact, sometimes these designs contain quite a few more variables than they do runs. The idea of using supersaturated designs goes back into the late 1950s. There was a paper published by Frank [inaudible] in which he talked about generating what he called a random balanced designs. These turn out to be supersaturated designs. There was a lot of discussion of his paper and really a lot of criticism of these ideas because of the complexity of aliasing and difficulty in interpretation. There really wasn't much work done on supersaturated designs, the idea was neglected until the early '90s when this topic was resurrected, revisited. Since then, a number of authors have actually proposed methods for constructing supersaturated designs that are quite useful. Both heuristic methods and computer search methods can be used to create these designs and some people have even discussed using optimal design construction techniques to create these designs. Here is a design that can be used to create a super saturated design. Now, if you look carefully at this, you would recognize this as the 11 factor, 12-Run Plackett-Burman design that we talked about earlier in this chapter. This design is an example of a Hadamard Matrix design. A Hadamard Matrix is a square matrix, it's orthogonal. It has columns that are made up of either plus signs or minus signs and Hadamard Matrices are that can be used as the basis for constructing experimental designs. For example, the 12 Run Plackett-Burman design is a Hadamard Matrix design. So how could we use this design to create a supersaturated design? Well, it's really easy. In this case, all you do is you take half of the runs from that design, you take the six runs at the top. That is a supersaturated design for handling in this case up to 10 factors in six runs. You cannot handle this last column, but you can handle up to 10 factors in six runs with this design. Now, analysis of these designs gets tricky and there are special methods that can be used for doing this, some of which are things that you can do with computer software. There are references to analysis methods for some of these designs in the textbook. Okay, that is the end of Module 8, all about fractional factorial designs, and with module eight, we really concluding what I think of as the basic concepts of experimental design. We've talked about the basic tools of analysis, t-tests, analysis of variance, we've talked about blocking, and we've talked about factorial designs and variations of the standard factorial designs such as the two to the k, blocking in the two to the k, and then fractional factorial designs with factors at two levels. These are very basic fundamental design tools that will carry you through the successful use of experimental design in many practical situations in science, engineering, business, marketing, all the areas where you might want to use these methods.
