This is the last lecture from Chapter 7, and it extends this idea of confounding to a little more complicated situation. We looked at an example that had two blocks. It's always easy to confound a two to the k single replicate design in two blocks. We simply take the highest order interaction and we use that to construct the blocks confounding that factorial effect with blocks. But what if we need more than two blocks? Well, the two-level factorial can be confounded in either two blocks, four blocks, eight blocks. In general, two to the p blocks, where p is, of course, greater than one. How do we do this? Well, for four blocks, it's really pretty easy. Pick two factorial effects. Now, in the two columns for those factorial effects, you will have four different sign combinations: minus-minus, plus-plus, minus-plus, and plus-minus. Those sign combinations determine the assignment of the factorial test combinations to the four blocks. Now, when you pick those two effects to confound, you automatically confound a third effect. Why? Because remember in the table plus and minus signs, any two columns define a third column, the product of any two columns is a third column in the table. You would expect that to be the case because you have to have three degrees of freedom for the four blocks. We're going to look at an example of this in just a moment. Here is a two to the five design. This is a 32 run design in four blocks, and I chose ADE and BCE as the factorial effects to confound with blocks. It turns out that the product column of that, which is ABCD would also be confounded with blocks. In other words, in Block 1, both ADE and BCE are negative, but ABCD would be positive. In Block 2, one is plus and minus, and ABCD is minus. In Block 3, another one is minus, the other one is plus giving you a minus sign and ABCD, and then the other one, Block 4, both are positive. Both chosen effects are positive, and so the ABCD effect would also be positive. So it's easy to do this, it's really quite simple to do this. Here is a Table 7.9 from the textbook that suggests what I think are optimal blocking arrangements for the two to the k. So for up to seven factors, this shows you how you can create these designs in block sizes, in some cases that are as small as two. It shows you the effects that would be chosen to generate the blocks, and then it shows you the complete set of factorial effects would be confounded with blocks if you make such a choice. The case of four blocks is pretty easy, we just talked about that. The case of eight blocks is a little bit more complicated. With eight blocks, for example, the smallest reasonable design is a two to the four and eight blocks, the block sizes would be two. You would have to confound three of your two-factor interactions to choose the blocks, and then many other two-factor interactions along with a four-factor interaction end up being confounded. So as the block sizes gets smaller, you lose information on more potentially interesting effects. So you do have to be a little careful about how you deploy this. By the way, computer software will do this for you automatically. Packages like JMP, or Design-Expert, or Minitab typically have choices for all of their two-level factorial designs that incorporate blocking. They may not have the number of blocks or the block sizes that you see in this table, but they will have many others that are consistent with that. There is one other topic in this chapter that you probably could be interested in, we're not going to talk about it in detail, and this is called partial confounding. This is a technique that could be used with replicated to the k designs where the block sizes are so small that you cannot run a complete replicate in an individual block. For example, a two to the three, we're going to do two replicates of a two to the three. But the block sizes cannot be larger than four. So how could you do this? Well, you could confound ABC and replicate one, and ABC and replicate two. So ABC will lose information on ABC throughout the entire design. Well, another strategy would be to confound ABC in the first replicate and then a different interaction maybe AB in the second replicate. So now, you would get some information on AB from replicate one and some information on ABC from replicate two. So this technique called partial confounding can be useful in some cases. There is a discussion of it in the book and there is a complete example. So if you think that is something you might one day find useful, that would be a good reference for you to take a look at. This ends Module 7. This went by very quickly. The concepts here are not very difficult. They're embedded in computer software, and you'll see more about that when we do some software demo here to support this module. We're ready to move on to Module 8, which is an extremely important set of material on fractional factorial two-level designs. That's it. Thanks for watching, see you next time.
