We've talked about the basic idea of a one-half fraction, now I need to show you how to construct a one-half fraction. This table shows you the general method. If you want to construct a half fraction of a two to the k, the first thing you do is you write down the columns associated with what I call the basic design. The basic design is always a full factorial that has the same number of runs as the desired fraction. In this case, we want a half fraction of a two to the three, the two to the three is an eight run design, the half fraction would be a four run design. So the basic design would be a two square. So the first panel of this table shows you a two-square design in factors A and B. Now, the design that you see there has the correct number of runs, but it's missing a column, you need a column for factor C. So we choose a generator for that column. The generator for any column, say the new column C must be one of the interaction columns from the basic design. Now, there's only one interaction column in the basic design and that's AB. So setting C equal to AB would be the way to construct this one-half fraction. So in the second panel of the table, you see C equal to AB, that generator being used to construct the column for factor C. So now we have a half fraction of a two to the three. Which one is it? Well, if you multiply both sides of that generator, C equal to AB by C, you get I equal to ABC, so this is the principle fraction. If you wanted the alternate fraction, you would have simply chosen the negative of that generator that is C equal to minus AB. Then when you multiply the column A times column B, you would change the sign of that product and that would give you the design in the panel on the far right of the table. That's the alternate fraction with I equal to minus ABC. One of the things that I mentioned that make fractional factorial designs work really well is this projection principle. Here's an illustration of the projection principle for our half-fraction of a two to the three. If you imagine a flashlight shining along that axis, notice that that would project that cube into the square that you see back here in the AB plane, and there would be one run at each corner of the cube. So you've projected out factor C. By the way, you could project out factor A and get a full factorial in B and C or you could project out factor B and get a full factorial of A and C. In general, a one-half fraction of a two to the k will project into a full factorial in any k minus one of the original factors. Let's look at an example. I'm going to show you an example of a two to the four minus one-half fraction of a two to the four. This is going to be a simulated example because I'm going to actually use the data from the resonant plant experiment back in Chapter 6. Now remember, that was a full two to the four. So the way I'm going to do my simulation is I'm going to set up my half fraction, and then I'm going to choose the runs from the half fraction out of the full factorial back in Chapter 6 and we'll see how that works out, we'll see what conclusions we get. So we need to construct a one-half fraction of a two to the four. So how would we do that? Well, the one-half fraction of a two to the four would have eight runs. So the basic design would be a full factorial that has eight runs and that's the two to the three, in factors AB and C. So here's my basic design. That design has eight runs, that's the right number of runs, it only has three columns, I need to generate column D. So what am I going to use as a design generator? Well, I could use any of the interaction columns from my two to the three, but it turns out that D equal to ABC is the best choice of generator because it gives me the highest possible resolution. So here are the treatment combinations that we would choose out of the full two to the four back in Chapter 6 and over here are the observed filtration rates. The actual design is shown on the cubes at the bottom of the page. Look at the pattern. The pattern of locations where we actually have those runs is very similar, it's really identical to the same pattern that we saw in the half fraction of the two to the three. So let's estimate the factor effects. We can do that with software. By the way, the alias structure is shown on the right. This is a resolution for design, notice that because I is equal to ABCD, the alias of A would be, BCD and the alias of B would be ACD and C would be aliased with ABD and so on and then the AB interaction would be aliased with CD. In fact, every main effect is aliased with another two-factor interaction. So what is the resolution of this design? It's resolution four, isn't it? Main effects are clear of two-factor interactions, they're aliased with the three-factor interactions, but the two-factor interactions are aliased with each other. Let's look at interpreting the results. Well, the main effect of A looks large, and so does the main effect of C and D. The main effect of B looks pretty small. So with A, it's either A or BCD or maybe both. Since three-factor interactions don't occur very often, I'm going to assume that that's the main effect of A and I'm going to make the same assumptions for the main effects of C and D. Now let's look at the AB interaction. Notice that the AB interaction is essentially zero, very small. So how do you interpret that? Well, simple interpretation would be the AB and it's alias CD are both negligible. The other interpretation is that AB and CD are both large, but they're about the same magnitude and they have opposite signs so they're canceling each other out. Which one of those two sets of assumptions do you think is most reasonable? Well, I think I would go with the simpler interpretation that both AB and CD are negligible. Now let's look at AC, that's a large effect. It's either AC or BD or both. I'm going to assume it's AC, because the chances of BD being significant if the main effect of B is not significant by itself are pretty small. Those kinds of two-factor interactions occur much less often than interactions which involve two significant main effects. Since A and C are both significant, I'm going to assume that that's the AC interaction. I'm going to make a similar interpretation when it comes to AD plus BC. So my conclusions here are that the active factors are A and C and D, three of the main effects and then AC and AD. By the way, that does turn out to be exactly the correct interpretation because it's the same results that we got when we ran the full factorial back in Chapter 6. What I've used here in interpreting the results is I've relied on a principle in science and is known as Ockham's razor. William of Ockham was a naturalist, a British scientist and his principle stated that when you're confronted with multiple interpretations of some physical or some observed phenomena, the simplest interpretation is usually the best. That can be very useful in interpreting the results of fractional factorials. Now one of the things that I think is important with a fractional factorial is to leave enough resources so that you can do some confirmation experiments after you've run the fraction, and there's several ways that you could do that. You could add the alternate fraction. If you had resources to do it, you could add the alternate fraction and now you'd have clean estimates of all of the factor effects, but that's an expensive proposition. Let's talk a little bit more about effects sparsity. I utilized my understanding of effect sparsity to interpret the results of this last experiment. This phenomena has been observed by experimenters in many fields for just decades, but there's a recent paper that provided a little bit more objective information about this. These authors examined a 133 response variables from published experiments that had up to seven factors. They actually got the data for these experiments and they reanalyzed all of these responses. They found that in the experiments they studied that a little over 40 percent of the main effects were active and that generally the size of a main effect was larger than that of an active two-factor interaction up to twice as large. The percentage of active two-factor interactions overall was about 11 percent, but remember two-factor interactions are much more common than main effects. For example, if you have four main effects, you have six two-factor interactions. So two-factor interactions occur pretty often. Interactions beyond order two were very,very rare. They also reported some conditional percentages about active two-factor interactions. A two-factor interaction was active and both main effects were active about a third of the time. Two-factor interaction was active but only one of the main effects involved in that interaction was active, was less than five percent of the time and the two-factor interactions were active and neither of the main effects involved who were active less than one percent of the time. Now, my experience has been a little bit different than these authors. I think a lot of the examples that they looked at were more mechanical systems or mechatronic type systems. I've looked at a lot of chemical and biological type experiments as well and I've found that the two-factor interactions were more common. I think that in those kinds of systems, two-factor interactions occur probably at least half the time, and the three-factor interactions occur up to perhaps as many as five percent of the time. So I still think that this paper gives you some good guidance, but I think that two-factor interactions are more active in general types of experiments than they found. You can also sometimes get a lot of information about the results of an experiment by looking at the cube plot. Here's a projection of this design into the three factors A, B and C. Now this is a full factorial in these factors, and I think you can get some very good information about performance from looking at this result. The best results appear to be along that upper edge where factors A and B are at the high-level and C is also at the low-level. So that gives you a good idea about the practical interpretation of a fractional factorial and the last thing that we probably should mention is confirmation experiments. We talked about that briefly. How do you do with confirmation experiment? Well, I think a very good way to do it is use the model from your experiment to predict the response at a test combination that you think you might be interested in, not one of the points in the current design. Pick a point that's not one of the current design, then run that test combination and compare predicted and observed. For example, 8.1. Suppose you wanted to run the point plus, plus, minus, plus. So that's A, B and D at the high-level and C at the low-level. Well, the predicted response at that point is shown in the regression equation that you see at the bottom of the page. Here's the intercept and these are the main effects of A, C, D and the AC and AD interaction. Now remember those regression coefficients are exactly half of the factorial effect. So when you plug in the coordinates of A, C and D in here, the predicted response turns out to be 100.25. If you go back to the full factorial where that point was run and remember that point was not run in our fraction, but if you go back to the full factorial, you will notice that the actual response is a 104 units. So in this case, the predicted and the actual agree pretty closely. If the predicted and actual had not agreed. If it was a big difference between them, that would point to the possibility that you had not interpreted the results correctly and that there maybe some other effect or interaction effect that's active that you need to reconsider and so maybe additional experimentation would be useful.