Welcome back to our experimental design course. Here's another topic that is, I think, part of the response surface world. It's not as new as the computer experiments area, but it is nonetheless an application of growing importance and this is the area of experiments with mixtures. What is a mixture experiment? Well, in a mixture experiment, the factors in the experiment, design variables, are the components or the ingredients of a mixture, and as a result of that, their levels are not independent. They can't be independently adjusted. For example, if x1 and x2 and on up to xp, denote the proportions of the components of a mixture, then each of these x's are some proportion between zero and one, and the sum of those component proportions, x1 plus x2 plus and up to xp adds to one. That's a 100% of the mixture. Well, what does this look like? Well, here's an example. Here is a two component mixture, and x1 varies between zero and one and x2 varies between zero and one. Now, the only place in this design space where we can have a feasible mixture is when x1 plus x2 is equal to one. Where does that occur? That only occurs along this diagonal. So all of our design points have to occur along that line and for every design point, x1 plus x2 must add to one. Here's a three-component mixture. So I start with a cube which is what we would have, if we had independently adjustable parameters. Where in this cube is x1 plus x2 plus x3 equal to one? Well, these vertices all have one of the x is equal to one and the other two equal to zero. All along any of these edges, two of the components are varying between zero and one and the third component is zero. These are representing binary blends of these three ingredients. Then in the center of this oddly looking shaped region, x1, x2, and x3 are all between zero and one. So your mixture experiments have to lie either at the corners or along these edges, or in the center of this region. That region on that three-dimensional picture that we saw a moment ago is called a simplex, and this is a simplex region for four-three components. Notice, it's an equilateral triangle. There is a coordinate system called trilinear coordinates that we impose on the simplex. The way this coordinate system works is all along any one edge or flat of the system, two of the components are varying between zero and one and the third one is zero. This is the x2, x3 flat and all along that flat x1 is equal to zero. Here we are at the mid point. X2 and x3 are both equal to 0.5. As we start to move along the x3 axis, what we're doing is we're increasing the proportion of x1, but the relative proportion of x2 and x3 is decreasing proportionately. For example, when we're six tenths of the way along that axis, x1 is equal to 0.6, but x2 and x3 are only equal to 0.2, and we have an x2 axis and we have an x3 axis. So any point in this simplex region can be described or defined in terms of a set of trilinear coordinates with x1, x2, and x3 summing to one. Simplex regions and simplex designs are frequently used for mixture experiments. The study of these experimental designs and mixtures, in general, started with work by Henry Scheffe, a very famous statistician. Scheffe proposed using what he called lattice designs. These are simplex lattice designs at the top for three components and at the bottom for four components. This is a three-two lattice. In other words, it has three components and you can fit a quadratic model to this data. This is a three-three lattice, it's a three-component mixture, but you can fit a cubic model to the data. This is a four-two lattice and this is a four-three lattice. Notice that the four dimensional simplex is really a pyramid. It has four vertices and each face of the pyramid is an equilateral triangle. He also proposed a different design called a simplex centroid design and you are looking at a three-component and a four-component simplex centroid design on this slide. Basically, the simplex centroid design has the vertices along with the centers of edges and then it has the centroid that is in the face of each of the equilateral triangles. What kind of models do you fit to this data? Well, you can't fit standard polynomial models and it's really easy to illustrate. Let's consider a linear model, let's consider the x matrix for a linear model. So if we want to fit a first order model, then we'll have x1, x2 on out to xp and we'll have rows that represent the design, and those are coordinates of x1 to xp. Then we'll have ones in each row for the intercept. Well, you're going to have a problem trying to fit that model because if you sum all of these coordinates together, they add to one. So there's an exact linear dependency in the columns of x and you can't fit a standard linear model. What we typically do is, we fit the Scheffe Linear Model that you see here.We essentially drop the intercept and that gets rid of the ill-conditioning or singularity in the x matrix. This is sometimes called the shi-fei canonical model. Here's the quadratic model. Notice that it doesn't have any squared terms because if you try to put in squared terms, you'd get the same kind of ill-conditioning that we had with the intercept. So it only has the linear blending terms and the quadratic blending terms. Here is the full cubic model and it's a fairly large model. Most people don't use that model. I would say they use the special cubic model more than the full cubic model. So this has the linear blending terms, the binary blending terms, and then these ternary blending terms that you see involving three of the components. Here's an example of a three-component mixture experiment, and this comes from the definitive reference on Mixture Experiments, a textbook by John Cornell. I knew John Cornell from early days in graduate school. We were both students at Virginia Tech and we lived in the same apartment complex and it was even worse than that. We lived in adjacent apartments and so we became very good friends. Over the years I learned a lot about mixture experiments by osmosis just by being around John. He did his dissertation on mixtures, but before he came back to school, he worked in a company that made textile fibers. He used to talk about this experiment all the time, and then he ended up writing about it in his book. It's an experiment in which three components, polyethylene, polystyrene, and polypropylene, are blended together to create a material fiber that's going to be spun into yarn for making draperies. The response variable that we're interested here is something called thread elongation. The way this is computed is to suspend a weight from a sample of the fiber, and you use up a known weight and you see how much elongation is produced. You don't want very much elongation because if you have high thread elongation, you will have draperies that tend to sag when they're manufactured. So here is the design that John used. If you look carefully at this design, you notice that it has the vertices and it has the centers of edges, and the vertices were replicated twice, and the centers of the edges were replicated three times. Then the data here, these are the average elongation values. Notice that there are no ternary blends. John said the reason there are no ternary blends is because the experimenters here didn't want the complexity in manufacturing of having to deal with all three of these materials. They were willing to use combinations of two of them, but not all three. So he avoided like an overall centroid, for example, in the middle of this simplex. So here are the results. By the way, the choice to replicate the pure blends only once or twice and three replicates to the binary blends, is somewhat arbitrary. In fact, it's quite reasonable to adopt a different replication strategies depending on where you think the error might be larger or smaller over the design space. John was able to compute an estimate of the standard error because of these replicates, and the standard error estimate and the standard deviation for the noise here was 0.85. Then he fit the second-order mixture polynomial, that has the three linear terms and the three binary blending terms. Notice that beta_3 is the largest of the coefficients, and so component three, which is polypropylene, probably produces yarn with the highest elongation, and then high elongation is good. However, beta_12 and beta_13 are positive, so blending components 1 and 2 or 1 and 3 produces higher elongation values and you get by just averaging the elongation of the pure blends. So there's what we call synergistic blending going on here. Components 2 and 3 have antagonistic blending because there's a negative sign in front of beta_23. The figure on the right is a contour plot of constant estimated yarn elongation. You notice that the highest values of elongation seem to occur along this x1, x3 flat. So components 1 and 3 should be chosen with probably about 80 percent component 3 and 20 percent component 1. That would probably give you the best results in terms of achieving high thread elongation. Here's another design that is very often used in addition to the standard simplex designs that you saw a few minutes ago. Remember Cornell used a 3, 2 lattice design. Here is the 3, 2 lattice design that has been augmented. Notice we've augmented it with these three axial runs. Some people call those axial check blends, but this is a very nice way to put additional runs into the center of a simplex design. It turns out that these runs are located along the x1, x2, and x3 axis, so that there is far away from the other design points as you can get. So notice that each one of these points is about two-thirds of the way along that axis in comparison to the centroid. The centroid is at one-third and this is in two-thirds. So it's really halfway between the centroid and the opposed vertex. This is a very good design, which I strongly recommend that you think about this design as a way to add interior runs to say a 3, 2 lattice design.