So welcome back to our experimental design class. This is the second lecture on nested designs. In the previous lecture, we talked about the simple case of two-stage nesting. But there are many, many variations of the nested design, and we're going to just briefly talk about some of them in this lecture. Staggered nested designs are a possibility that you sometimes consider. You find out in using nested designs when you have several levels of nesting, that you can get lots of degrees of freedom building up at lower levels. You may not really want that to happen. So you can stagger the designs in an approach that I'm going to show you in just a minute. Several levels of nesting occurs fairly often. I'm going to show you an alloy chemistry or alloy formulation example that has three levels of nesting so you can see how that strategy works. Then finally, there are experiments that contain both nested factors and factors that have a factorial or a crossed structure as well. I'll show you a fairly complete example of one of those. So here's an example of a staggered nested design. So we have lots of material and that is the stage one level. At stage two, we take two samples out of each lot. But then at stage three, we may not make the same number of measurements on each sample. For example, in this arrangement, we're only going to take one measurement from the second sample, but we'll take two measurements from sample one. So you could repeat this strategy across several lots, and that would give you a three stage staggered nested design. These designs can be analyzed in JMP and they can also be analyzed in Minitab. If you look at the supplemental text material for this chapter, you'll see an example. Here's an example with several levels of nesting, and this is an alloy formulation experiment. So we want to change the chemistry of, let's say, an aluminum alloy, and so we have two different formulations that we want to study. That's the stage one that you see up here. Now, when we produce this particular alloy chemistry, we do it in a furnace and we prepare heats. So we have a different heat, three of them, for alloy formulation one, and three heats for formulation two. Now clearly, this is nesting because heat one from alloy one cannot be the same as heat one from alloy chemistry two. So there's a level of nesting there. Now, when we actually want to test the material for hardness or strength, or what other factor we're interested in, we have to cast the material from that heat into ingots. So let's say we select two ingots at random from heat one, two ingots at random from heat two, and two ingots at random from heat three. Well, clearly, ingots have to be nested within heats because ingot one from heat one cannot be the same as ingot one from heat two and so on and so forth. We can arbitrarily renumber the heats 1, 2, 3, 4, 5, and 6 and the ingots 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. If you can arbitrarily renumber the levels of a factor, it's a nested factor. That's the key in recognizing factors that are nested as opposed to factors that are arranged in a factorial. So now here's a really interesting example. This example involves factors that are factorially arranged. That is that they're crossed with each other. But there's also nesting going on. So let's see how this experiment evolves or how it unfolds. Let's suppose that we're looking at an assembly task and we want to find out ways to improve the efficiency, the time, of the assembly task. So we have three different fixtures that we would like to experiment with. The fixture is something that really is utilized in making the assembly task easier for the operator. Then we have a couple of different workplace layouts that we want to use, and every fixture can be used with every layout. So fixtures are one factor, layouts are another factor, and they are crossed with each other so they have a factorial structure. But to actually execute the experiment, we need operators, and so we choose four operators at random. But unfortunately, we can't use the same four operators with layout one that we can with layout two because maybe they're located in different parts of the facility. So we can't really move those operators around. So operator one is nested within layout one, and operator one over here is nested within layout two, because they can't be the same. So the operator factor is a nested factor inside this factorial structure. So what does the statistical model look like for this experiment? Well, Mu is the overall mean, Tau is the, let's say the effect of the ith fixture. Beta sub j is the effect of the jth layout. Now, Gamma sub k within j is the effect of the jth operator within the kth operator within the jth layout. There can be an interaction between fixtures and layouts. That's the term you see here. Now, let's take a look at this term, Tau Gamma. That is an interaction between fixtures and operators. Why can we have a fixture operator interaction? Because every operator uses every fixture. But we can't have an operator times layout interaction because operators are nested within layouts and we can't have a three-factor interaction, why? Because one of your factors operators is nested. Then Epsilon ijk is the usual random error term. So this is what the statistical model looks like. In these types of experiments, figuring out what the statistical model is, is the secret to getting the analysis done correctly. Let's assume that fixtures and layouts are fixed factors, but the operators are random. So we will use a mixed model here. This would be a mixed model. If we use the restricted form of the mixed model, this is what the expected mean squares look like in this experiment. This shows you how the statistical testing would have to be done. This is the error term, this is the fixture operator interaction, this is the fixture layout interaction, the operator within layouts main effect, the layout main effect, and the operator main effect. So these expected mean squares show you how the statistical testing would have to be done. This is the Minitab analysis, and it does the statistical testing in the manner that we've suggested from looking at the expected mean squares, and what do we find? Well, we find out that just from looking at the P values, that there really isn't much difference in the layouts. The P value is quite large, but there is a very large operator effect and there is a very large fixture effect. There is no layout by fixture interaction, but there is a fixture by operator within layout interaction. The P value there is about 0.04. So that says that we can get estimates of these variance components. The fixture times operator within layout variance component is about 1.58 and the operator within layout variance component is about 1.61. Here's the analysis done with the unrestricted model, and here are the expected mean squares for the unrestricted model. Minitab just displays those for you. Again, conclusions are fairly similar. There is a big fixture effect, there is a fixture times operator within layout interaction, and this P value is not really all that small. But given that there is a fixture times operator within layout interaction, I would suspect that there is some operator within layout effect. Here are the variance component estimates, assuming the unrestricted model. JMP will also do this analysis. Here's the JMP analysis and of course, it gives you the REML estimates of the variance components. The advantage of REML is it does give you confidence intervals. So here is the confidence interval on operators within layouts variance component, and here's the confidence interval on operators times fixture within layout variants components. You notice that both of those intervals are pretty wide and that they do include zero, and the width of that interval is strictly a function of the fact that we don't have very many operators, we don't have very many layouts. It's a sample size issue. Now, here are your variance component estimates. Take a look at that. Now, those are from REML and you'll notice that they match the Minitab variance component estimates for the unrestricted model. That's because JMP assumes the unrestricted model in its analysis of nested designs.