In our last class, we talked about resolution three designs. And this lecture continues that discussion. What we're going to be talking about here are the Plackett-Burman family of designs. The these are a different class of resolution 3 designs. They're often called nongeometric designs, because the number of runs in a Plackett-Burman design only has to be a multiple of 4. In the regular 2 to the K- P system, the number of runs is power of 2. So multiples of 4 are like 4, 8, 12, 16, 20 and so on and so forth. Obviously, there are multiples of 4 that are powers of 2, such as 8 and 16. But there are lots of multiples of 4 that are not powers of 2. And so these designs can fit sometimes rather nicely into the gaps between the regular 2 to the K- P fractions. And of course the reason they're called non geometric designs is because they don't fit necessarily nicely on a cube, the way the 2 to the K- P designs do. There's some information in the textbook about the construction of these designs, if you were going to manually build one. But generally, they are available in computer software, and so you don't need to worry about manually constructing the designs. Here is one of the most widely used Plackett-Burman designs. This is the case of a 12 run Plackett-Burman design that can accommodate up to 11 factors. So every one of these 11 columns represents one of your design factors, a through K, and there are 12 runs and you notice that everything is at 2 levels. Now, this is a resolution 3D design. And as the book tells you, these designs really have kind of messy Alias structures. For example, in this 12 run design that we just looked at, every main effect is partially aliased with every two-factor interaction that does not include that effect. So for instance, the AB interaction is aliased with main effects CD on out decay. In other words, it's aliased with with all nine main effects that don't include A and B. And the AC interaction is a list with nine main effects, B, D and so forth on out to K that don't include A and C. In other words, each main effect is aliased with a very large number of two-factor interactions. How many? Well, if you have 11 factors, 11 things taken 2 at a time is 55. And so if you figure out that the interactions are not aliased with the main effects involved in that interaction, that means that each one of your main effects has 45 two-factor interaction aliases. Wow, in the 12 run design, this is what the aliases look like in general. That's the Alias of A. Each one of the 45 two-factor interactions and that Alias chain is weighted by a constant, either plus or minus 1/3. And this waiting that you see of two-factor interactions occurs throughout the Plackett-Burman series. And now, in other Plackett-Burman designs, the constants may be different than a plus or minus 1/3, but this kind of relationships occur throughout the system. This is an example of what we call a non-regular design. In a regular fractional factorial design, the Alias Matrix is made up of either zeros or plus ones and minus ones. In other words, every effect is either independent uncorrelated with other effect, or it is completely aliased or completely confounded with it. That's the +1 of the -1. In non-regular designs, these entries that are between -1 and +1 that are not all zero, create this partial aliaship with aliasing which can sometimes be extremely useful. Let's look at the projection of the 12 run design. Up at the top is the projection of our 12 run Plackett-Burman design into three factors. And when you look at this design, you notice that it contains one, two, three, four, five, six, seven. It contains seven unique trials. Now, if we have only three factors, how many parameters do we need to estimate? We need to estimate the three main effects and we need to estimate the three two-factor interactions. This design has enough runs for us to be able to estimate the complete main effects, plus two-factor interaction model in three factors. So this design has projectivity 3. It's not a full factorial, you'll notice that there's one test combination that doesn't appear, and there are four tests combinations that have an extra run. It's not a full factorial, there is some correlative structure here between these effects. But one can still fit the main effects plus the two-factor interactions in any subset of three of the 11 factors, projectivity three. This is a really nice property of Plackett-Burman designs. Now, the projection in the four factors is really not quite as attractive. That's what you see down at the bottom of the page. And we don't have enough runs there to be able to to fit the complete four-factor model with main effects and two-factor interactions. We just don't have enough runs to be able to do that. But you still could get some improved estimates of main effects and you might be able to estimate some of the two-factor interactions that you be interested in. As I said at the start, one of the potential drawbacks of a Plackett-Burman design or any non regular design in general is the complexity of the Alias relationships. For example, many people are kind of frightened by that, with 12 runs and 11 factors. Every main effect is aliased with every two-factor interaction, not involving itself. And that leads to what we saw previously, every two-factor interaction Alias chain has 45 two-factor interactions included, kind of scary. Partial aliasing can possibly complicate interpretation. If there are a larger or moderately large number of big interaction terms. But on the other hand, the partial aliasing does in some cases give us the opportunity to be able to untangle some main effects in two-factor interactions. If there are not a large number of active effects. So, while I think we should use these designs carefully, they sometimes present some excellent opportunities to run a single experiment and be able to identify some main effects and some two-factor interactions that might be important. And here's an example that illustrates this. This is example 8.8 from the book. This is a Plackett-Burman design for an experiment that involves 12 factors. Now, the smallest regular fractional factorial that you could use for 12 factors is a 16 run regular fraction. That would be the 2 to the 12 minus 8. In that particular design, all 12 main effects are aliased with for two-factor interactions. And then there are three chains of two-factor interactions that each contains six interaction terms. And you can find that design in the textbook. My guess is that if there are any really significant two-factor interactions beyond just maybe one, it might be very difficult to untangle the relationships in this experiment using this design without additional runs. So instead of this 16 run regular fraction, suppose we decide to use a 20 run Plackett-Burman design. Now, this has more runs than the smallest regular fraction. Yes, but it contains a lot fewer runs then would be required by either a full fold-over or a partial fold-over. The full fold-over would require an additional 16 runs. The partial fold-over could require of an additional eight. So this is not a bad choice of design. We created this design in jump, and and I'll show you the test matrix for this along with the observed data in just a moment. We're going to look at the Alias Matrix for this design as well. And when you look at the Alias Matrix, you will notice that there are two-factor interactions that have constants in front of them, constants that multiply them that are not unity. And that's because it's a regular design. And this could give you some flexibility in potentially estimated interaction terms. We're also going to take a look at the jump analysis of this design. And the analysis procedure we're going to use is forward stepwise regression. And the way forward stepwise regression works is we enter the variables in the model one at a time, starting with those that appear to be the most important. And we're going to consider main effects in two-factor interactions as the possible candidate variables that would be of interest. And we're going to take a look at that, and then we'll see exactly how this unfolds. So here's the design, and here's the Alias Matrix for that design. Now, the way you interpret this Alias Matrix is here are the model terms that were interested in. And here are the interaction terms that they could be aliased with. And it requires two rows to be able to display this because of the very large number of two-factor interactions. Now, when you look at this Alias Matrix, the first thing you notice is there are no plus ones or minus ones. It's either 0 plus or minus 0.2, or in a few cases, some plus or minus point 6's. So the Alias relationships here are rather different than they would be in a regular design. So now, we do the stepwise regression analysis, and we're going to be doing forward selection. And this is the the initial solution where the only thing we have fit is the intercept. And then here's the final solution after we've gone through all of the stepwise operations. The column labeled estimate in this table is the one to take a look at. Those are the estimates of all of the factor effects. And you notice that the intercept is 200 x 1, the coefficient is 8. For x2, the coefficient is about 9.9. For x4, it's about 12 and 1/2. For x5, it's about 2 and 1/2. For the x1 x2 interaction, it's about -12 and 1/2. And for x1 x4 interaction, it's about 9 and 1/2. So those who appear to be pretty large affects, everything else is 0. Everything else is 0, those are the only terms that are entered in the equation. So here's a summary of the stepwise process. These are the terms that were included at each step, x2 was first, x4 was next, x1 x2 was next, x1 x4 was next, and then x5. So we would conclude at the end of this experiment, and by the way x1 was put in early too. We would conclude that x1, x2, x4 and x5 are important plus the two-factor interactions x1, x 2 and x1, x4. So that would be our interpretation of the results. Now, it turns out in this example, I made up the data. The data for this experiment were simulated from a model, and here's the model that was used to create the data. In this model, the random error term Epsilon was a normal random variable with mean 0 and standard deviation 5. So all of the effects here are about between one and a half and maybe a little more than two standard deviations in magnitude. And the important factors are rather x1, x2, x4 and then these two interactions x1, x2 and x1, x4. Well, what did our stepwise regression method find? It found all of these terms. It found x1, x2, x4, x1 x2 and x1 x 4, but it also included a term that's not important, x5. The partial Alias in here in the Plackett-Burman design is what enabled us to be able to estimate these interaction terms, even though this is a resolution three design. Now, you might look at this and you say, well, didn't you really get the wrong answer because you concluded x5 was important and it's not in your simulation model. Well, that's true, but what we've made here is we've made a type one error. And my view is that type 1 errors in screen and experiments are not nearly as critical as type 2 errors. You want to be sure that you get at least all of the right factors, and having an extraneous factor misidentified as being important isn't really the worst thing. Because eventually, you will find out that factor is not important and you will drop it from the results of excellent studies. But keeping a factor that's not important for additional experimentation is not nearly as disastrous or critical as failing to identify a factor that's important early on. Because when we fail to find an important factor, that factor gets ignored. And if it turns out to really be important later on, you may miss a lot of useful information. So I consider this to be a very successful example of a Plackett-Burman design. And there are other types of non-regular designs that can be used in situations like this with equally useful results.
