In the previous video, I showed you a process where we selected half the number of experiments, and I showed you that I used a rule where C = A*B. Now we shouldn't automatically use these rules. Let me show you where I obtained it from. What I'm showing you here is a table that describes how you can do less work in a way that you can still recover most of the information. This table doesn't really have a name that I'm aware of, so I call it a 'trade-off table' because it allows me to plan and budget for a set of experiments where I can trade off the costs with the information I hope to obtain. You're going to see this table several times again throughout the rest of the course but for now: remember how in the previous class we did four experiments and we had three factors in our system? On this table, we can locate ourselves over here in the top left corner, with four runs and three factors. In this entry in the table, we are told to generate C as the product of AB. Now remember in the wastewater example, where we had factors C, T and S? That would correspond to setting S as the product of C*T if we were using those factor names. When you're dealing with your case and your factor names are different, simply replace them with A, B, Cs temporarily to use this trade-off table. Now I guess you're curious about that plus and minus sign here. If I use the minus, it says to generate -C as the product of AB. In our wastewater example, that would translate to -S is the product of CT. If you created your design following the rule with the minus sign, you'd notice that you would end up with the closed circles on the original cube, rather than the open circles. But notice the symmetry. The closed circles still have that same collapsibility we noticed earlier with the open circles. Now with so many learners from a wide range of backgrounds in this course, I know that some of you might have difficulty with the math that follows. Be patient. Watch it all unfold here on the screen. And I'll explain it in the end in plain language. We're hoping to appeal to those of you with a technical background, and also those of you with a non-technical background. So let's get started with that math. Start by writing out the standard order table. Three factors means that there are eight experiments. But in the last video, I showed you that the best four experiments to pick are either the open circles or the closed circles. Let's work with the open circles, they corresponded to rows two, three, five, and eight from the original standard order table. So I'm going to remove those other four rows, the experiments we didn't run, and leave only the four behind that we actually ran. Now I'm going to slightly rearrange them for you. Put row five first, and then original row two and three, and then finally the original eighth row becomes our fourth row. Notice that columns A and B are in standard order and that column C is the product of A and B. In a prior video I showed you the matrix form for a three factor system, in the context of the wastewater treatment example. The matrix form adds extra columns for the two factor interactions and an extra column for the three factor interaction. There were eight unknowns in that example, in that vector b. So I'm going to remove those other four rows, the experiments we didn't run, and leave only the four behind that we actually ran. Now I'm going to slightly rearrange them for you. Put row five first, and then the original row two and three, and then finally the original row eight becomes our fourth row. Examine any patterns you notice in the eight columns. Did you pick up on the similar columns in the matrix? For example, notice that the column AB matches the column for C. And that the B column has the same recurring pattern as the AC column, minus minus plus plus. The BC column matches A. And finally, the ABC column matches the intercept column. What you need to take away from this explanation is that you notice the same pattern reoccurring in certain columns. What this means is we won't be able to tell those columns apart from each other. Telling columns apart is critical. In the prior four factorials, perhaps you noticed that each column was unique and independent. This helps us know the unique contribution of that factor to the outcome, the y variable. What does it mean if we cannot tell the columns apart? Let's use A and the BC columns as an example. By doing this fractional factorial, from a mathematical perspective, those two columns appear to be the same. We use the word alias to describe the situation. Now, you may be familiar with the word alias as used in the English language. For example, if I asked you, who is Jorge Mario Bergoglio? You may not necessarily know. But you do know his alias, Pope Francis. We all have aliases. In the real world, my family and friends know me by my full name, Kevin George Dunn. But, online, my alias is my email address or my username for a website. So an alias simply means we have another name for something else. Back to these experiments, where the alias for A is B times C. There are three other aliases, B is the alias for A times C, C is the alias for A times B, and that one was intentional, remember. We chose to set C as the product of A and B in our design. Finally, the last alias is the three factor interaction between A, B, and C, is aliased with the intercept. Now when I talked about not being able to distinguish between A and the BC interaction, or not being able to tell the columns apart, that is a consequence of aliasing. The effects of A and the effects of the BC interaction are aliased or mixed up with each other. So if there's a large effect size, it might be due to A, or it might be due to the BC two factor interaction. We won't know until we run more experiments. The term confounding is used to describe what is happening here. That means, after doing the reduced set of four experiments, you will never really be sure whether it was A that caused the change in the outcome, or whether it was the two factor interaction BC. Confounded is a word that means confused, it means the effect of A is confused or mixed up with or confounded with the BC interaction. We cannot tell them apart, A is an alias for BC and BC is an alias for A. That's the price we pay for doing half the experiments, and in some cases, it is a price that is worth paying. Let's review the math. Those with a background in this sort of thing will recognize that one way to solve a set of underdetermined equations where there are more unknowns than equations, is to collapse these columns together to achieve a square system that can be solved. Now you can see algebraically how this confounding develops. The coefficient for A in the model is a combination of the original A, plus BC. Similarly, the coefficient for B is the sum of the original B plus AC, and so on. These four entries here are our aliases. It also explains why you see only four coefficients in the R software outputs and NA values for the remainder of them. R notices that there are aliased terms in the full system that we requested a model for, but it will only report the coefficient for one of the aliases. To end off the class, you might be concerned that you've lost quite a bit of information by using a fractional factorial. In the next video we see the situation isn't quite so bad. Using prior knowledge about our process we can actually benefit from knowing about the aliases. Now hopefully this video has not left you too confused or confounded. Just to summarize, there are three new terms that you must be comfortable with from today's class, half fractions, aliasing, and confounding. We're going to use them a lot more in the next class.