So let's look at an example to end this module. We said in the prior video that you should always include as many factors as you possibly can in a set of experiments. Do you remember why we recommend that? If not, please review the prior video again. In this example we are going to use 7 factors, and the fewest possible experiments; that's 8 experiments. We are going to screen out which of those 7 factors really affect our outcome. So it is a screening design with 8 experiments and a resolution of III. I could choose more experiments, and then go to higher and higher resolutions. But let's see what happens when we start with just eight experiments and seven factors. With eight experiments, we have factors A, B and C to form a full factorial in eight rows. The tradeoff table tells us to generate factors D, E, F and G. Now notice that this is a 2^{7 - 4} design. So this design has p=4. These 4 generators, can be used to create the columns for the remaining factors in my system. And here's the completed table. I can go ahead and run the experiments and start my analysis. But the whole purpose of the tools introduced in this module is all about checking your aliasing before you start the analysis. Let's go do that. Our 4 generators are rearranged over here. I equals ABD, I equals ACE, and so on. How many words in our defining relationship? Two to the power of p and with p=4 in this case, that equals 16 words. That's a lot of words to figure out, but let's give it a try. The first few words are easy. Take the rearranged generators individually: I = ABD = ACE = BCF = ABCG That's 5 of them. Now we can add to that the combinations two at a time: (ABD)(ACE) = BCDE. The next combination two at a time is: (ABD)(BCF) = ACDF. You can prove to yourself that those are the remaining four (CDG, ABEF, BEG, AFG). Now we've got 11 words so far in our defining relationship. The next step is to take our generators three at a time: (ABD)(ACE)(BCF) = DEF Try the next three (ADEG, CEFG, BDFG). So, there we have a total of 15. And the final combination is to use all four generators multiplied together. And that simplifies to ABCDEFG. So, here's our complete defining relationship. Now, let's go try and calculate the aliasing for factor A. If we go and do that, we get this very long expression over here. I've highlighted only the two-factor interactions that are confounded with the main effect of A. I can create this list of aliases for the seven main effects in my design. This illustrates the tremendous confounding that takes place in the very dense designs at the far right-hand side of the trade-off table. Remember, instead of doing two to the seven, which equals 128 experiments, we've done 8. There's going to be a steep price to pay for this reduction in work. Now let's go and look at the numbers from the outcome variable, and how to continue on with the analysis. And as you'll see, and this is very typical, the analysis goes much quicker than the planning. Here's the code that you can use to analyze this design. Please copy and paste it from the website. We recommend that you always clear your environment from prior work. This is because you might have a variable with the same name from a different analysis; this will avoid any confusion. Build the linear model in exactly the same way as you created the design on paper. First, define the three variables that you start with: A, B, and C. Next, generate the remaining four factors using the definitions from the tradeoff table. When you inspect these variables in the console, you should get exactly what you had on paper. Now, add the outcome values recorded for the eight experiments. I'm going to take them from the standard order table. When you are ready to visualize your linear model, load the PID package, using the "library" command. You would have installed this package if you had been following prior videos. I will quickly note that R packages are frequently updated. You should check for updates regularly, as demonstrated here. So use the "paretoPlot(...)" command and let's examine the output. We can see here that the factors C, A and G are significant and have a negative, reducing effect, on the outcome variable. Factor E is a little smaller. And factors B, D and F have small to negligible coefficients. Note however, when we say factor A up here is important, it is really A that is aliased with a variety of two factor and higher interactions. As long as the assumption is true that those two factor and higher order interactions are small, or zero, then that bar in the Pareto plot essentially represents the effect of A. What about the unimportance of small effects down here? They can be removed -- judiciously. As long as you are confident that when you varied factor B, you did so over a large enough range to affect the outcome variable meaningfully, then you can be sure that this Pareto plot shows that factor B really has no significant effect on the outcome. It is safe to remove it. In other words, we have screened factor B out of consideration. So let's go remove factors B, F, and D for those reasons. By removing these three factors, we've reduced ourselves from 7 to 4 factors, but we've still have done eight experiments. We might as well have done the experiments with only factors A, C, G and E present. Note however that we do not have to redo the experiments. If you refit the model in R with only these four factors you get exactly the same coefficients as before. This is due to the independence property that's built into the model's design. Those of you with a least-squares background, will recognize that the columns in this matrix are independent, so when you rebuild the model you will get the same results. So, there's that; we've essentially found ourselves a system with four factors in eight experiments. We've eliminated three unimportant variables, as we've learned that they have little effect on our outcome. We have retained four important factors that we know affect our outcome. We will see in the following module that we can focus our future attention on these important factors now, to optimize the system. So that's the end of this module. For advanced students I do want to point out two other reduced designs. The first, is a Plackett-Burman design, the regular tradeoff table shows that you can do 4, 8, 16, 32, 64, and so on runs. But what if you had a budget, for example for 24 runs. That's more than 16 but not quite enough for 32. Well Placket-Burman design works well for these cases where you have a budget that is a multiple of four but not one of the existing powers in the table. So a budget of 20, 24, 28, and so on. I'm not going to go into the details of the Placket-Burman design, but now that you know the terminology, you can go search for more information. The final type of design to be aware of is a class of designs called the "Definitive Screening Design", and here's a link that you can read up some more information. These designs are a type of optimal design. Let's quickly define the term "optimal", here. It means, that the experiments selected, obey some sort of criterion, and they're optimized to meet that criterion. The great thing about an optimal design is that they can be very flexible. For example, if you had a limited budget you can create an optimal design for a given number of factors you are investigating to maximize one of these optimality criteria to fit your budget. A computer algorithm is used to find the settings for each one of the budgeted number of runs, so that the optimization criterion is maximized. In other words the computer is designing the experiments for you. And there's several of those criteria available. This is where the topic of experimental design quickly becomes more mathematical than this course is intended for. So I'm going to leave you at reading this link for more information, and you can quickly see that these modern, computer-created designs, have some very distinct advantages. So on reflection this has been a long module of the course. It is imperative that you work on case studies, and preferably with your own data to solidify your knowledge. This can be a tough topic to grasp, so don't be afraid to watch these videos several times, and to ask questions. Working with fractional factorials is a bit like playing with fire, the only way to learn is to burn your fingers. So go ahead, play with the fire, but preferably on a system that has no painful penalty. Like making biscuits or trying out recipes for good coffee, or preferably both.