We finished our discussion of the mechanics of response surface methods. We now need to spend some time talking about choosing an appropriate experimental design for fitting a response surface. This is a section 11.4 in the textbook. The first thing I want to talk about are criteria. What are some of the features of a response surface design that would be desirable? Well, here's a list of 11 things that I can think of. Some maybe a little bit more important than others. You want a reasonable distribution of points in your design space because that's giving you information that covers your region of interest. You want to be able to be able to check the adequacy of your model, and be able to say test lack of fit if necessary. You'd like to have designs that accommodate blocking, if necessary. You'd like to have designs that perhaps allow you to add some higher-order terms if necessary. You'd like to have an internal estimate of error. So that means you need some replication. You'd like to have good precise estimates of your model coefficients. You'd like to be able to get a reasonable distribution of prediction variance throughout your design space. You'd like to have reasonable robustness against outliers or missing values. You don't want designs that are too big. You don't want designs that have too many levels of the design factors because that makes them more difficult to run. In the old days, we were always concerned about simplicity of calculation and model-fitting. Well, with modern regression software, that's really not much of a factor these days. What about designs for fitting first-order models? Remember, we have to talk about both the first-order and second-order case. The first-order case is easy. The orthogonal first-order designs are really the best designs to use. So this ensures that the variance of your regression coefficients is minimized. These are be optimal designs. So 2_k designs and fractional 2_k designs that are at least of resolution three are all good choices. You can add center points to those designs. I think many times that is a very good idea so that you can test for curvature. The second-order case is much more interesting, and there are lots of different choices. The central composite design is very, very widely used. You see central composite designs in figure 11.20 in the slab. Basically, a central composite design is a 2_k with axial runs added along the x_1, x_2 on up to x_k coordinate axes and of course, there will be runs at the center. In general, there will be n_F factorial runs, and that will either be 2_k or sum 2_k minus p number of runs. There will be 2 times k axial runs and n_c center runs. I think that in most cases, you're going to be using either the full factorial or certainly a resolution five fraction because you have to be able to estimate all the main effects and all of the two-factor interactions. So if you do use a fractional design in the cube, it's got to be a design of resolution at least five or a full factorial. A very important property of second order models is rotatability. A design is said to be rotatable if the contours of constant standard deviation or variance of predicted response are concentric circles or concentric spheres. This is a rotatable central composite design for two variables. You notice that what the property of rotatability does is it means that the variance of the predicted response only increases as you change the distance away from the design center. It's constant on circles or constant on spheres. Now this is a really important property, I think, because it ensures that your prediction of, let's say the mean response or your your prediction of new observations only deteriorates as the point that you are making that estimate or prediction at gets further and further away from the center. So direction in which the point you want to make predictions or estimate mean response at, the precision of your estimate or prediction there does not depend on the direction in which you're moving away from the center. This is, I think, a really nice idea because while you've probably centered this design and what you think is a very good guess of the optimum, the optimum is likely to be someplace other than at the center. It could be over here, it could be over here, it could be over here, and you don't know which direction you're going to need to move in. So let's make sure that the response, your precision with which you either make predictions or make estimates that it doesn't deteriorate as a function of direction, it only deteriorates as a function of distance. How do you make a central composite design rotatable? That's the question. It's easy. You make that Alpha value in your coded units, you make Alpha equal to the fourth root of F, where F is the number of factorial runs. That guarantees a rotatable design. Rotatability is a spherical property. It's an excellent value of alpha to use. On the other hand, many people like spherical central composite designs. In a spherical central composite design, alpha is equal to the square root of k. That means that all of the design points are the same distance away from the center. There is very little difference in terms of prediction variance deterioration between the spherical CCD and the rotatable CCD. In fact, many people actually prefer the spherical CCD because you can show that, in many cases, the prediction variance at the boundary is slightly smaller with the spherical CCD than with the rotatable CCD. How many center runs should you put in a CCD? Well, if there isn't a sphere, you've got to include some reasonable number of center runs to get stable prediction variance near the center of the region. Three to five center runs is usually a very good choice. More than that really doesn't really benefit you, and fewer than that tends to make the prediction variance at the center bigger than the prediction variance in the middle of the region. So you want to have enough runs at the center to flatten that prediction variance surface. Three to five center runs usually works very, very well. Here's another widely used design for fitting second-order models, the Box-Behnken design. The nice thing about the Box-Behnken design is that all the factors only have three levels, minus one, zero, and plus one. Box and his PhD student, Behnken, produced this design. They created this design with the explicit purpose of having a design that had only three levels of the factors, but which was either rotatable or approximately rotatable. Box-Behnken designs all have this approximate property of rotatability. If you look at the geometry of the design, you can see that the design points are not at the corners of the cube, but rather they are at the centers of the edges of the cube. So these design points are not really on a cube, they're on a sphere. The radius of that sphere is square root two. You can see that by looking at any of the design points other than the center. You notice that the distance of any of those design points away from the center is the square root of two. So this is a design on a sphere. So it has to have at least 3-5 center points in order to give you good distribution of prediction variants throughout the design space. Now, another way to create a design on a cube is to use a central composite design, but make the alpha values equal to 1. That puts the axial values in the centers of the faces of the cube. This is also a very good choice of design for fitting a second-order model. But it's not rotatable. It's not a rotatable design because the alpha value is not equal to the fourth root of F. But nonetheless, this is still a very good choice of design. Here is a face-centered cube with three-center runs. I've shown you the plot of the standard deviation of the predicted response in the x_1, x_2 direction with x_3 equal to 0. You notice that even though we don't have contours of prediction variants that are concentric circles, it's still a very well-behaved function. It's very flat near the center and only starts to get larger as we go near the boundaries of the region. There are other designs that can be used, and there are references in the textbook to these. They are equiradial designs. These are designs in two-dimensions where all of the runs are the same distance away from the center. The rotatable central composite design in two-dimensions would be an example of an equiradial design. It would have eight points that are all the same distance away from the center. That distance would be the square root of two, 1.414. So it would be an octagon design. But there are other designs like a hexagon that could be used, that would be equiradial and they would allow you to fit the second-order model. Now, you would have to add center runs to those designs. They are designs on circles. Small composite designs are another choice. I don't really recommend small composite designs because I don't think they're prediction variance properties are very good. I would not be inclined to use those. Hybrid designs are another possibility. These are relatively small designs. They have excellent prediction variance properties. They are nearly rotatable in some cases, but they have in some cases unusual factor levels and more than just three factor levels sometimes. They're not as popular because many people consider them to be harder to run. So these are designs with some references for these in the book and then you can look at these. Some computer programs have these designs available, but not very many of them, and frankly, they're not used very much. Computer-generated designs, optimal designs are also very popular for fitting second-order models, and we're going to talk about that in some more detail later. Before we conclude our discussion on designs for fitted response surface models, we ought to talk a little bit about graphical evaluation of response surface designs. We've talked a little bit about this before. Fraction of design space plots, which are discussed back in chapter six in the book are useful. But there's also something called a variance dispersion graph which can be useful. These various dispersion graphs were developed back in the late 1980s. Basically, a VDG or a variance dispersion graph, is a graph that displays the minimum, maximum, and average prediction variance for your design and a particular model versus the distance of the design point from the center of the region. That distance usually varies between zero and the square root of k, which most people would take to be the most remote point in the region that you might be interested in. Typically, these plots graph the scaled prediction variance, which is the variance function multiplied by N, the number of runs, divided by Sigma square. Getting Sigma square out of the scale prediction variance eliminates an unknown constant. Multiplying by N, many people like to do that because that way you can compare designs with different numbers of runs. Weighting by N makes that comparison a little bit more straightforward for many people. Here are variants dispersion graphs for a central composite design. They're both central composite designs for K equal to 3. Up at the top is a central composite design with alpha equal to 1.68. That happens to be the rotatable value. You notice that you have only one land. That's because the minimum, and the maximum, and the average prediction variants are the same. You notice that we have a very stable prediction variants out to about, somewhere around about a distance of 1.5 units from the center of the design. Then, it begins to take off, it begins to accelerate. The bottom graph shows you the same design, K equal to three, four center runs, but now alpha equal to 1.732, the spherical value. You notice that there's really almost no difference between the minimum and the maximum and the average value, again, out to about 1.4 or 1.5. Then you start to see a little bit of difference between them, but they're really not a lot of difference between the min-max and average value. You also notice that the maximum value here is a little bit less than the maximum value here in the rotatable design. So that's why people sometimes prefer the spherical CCD in comparison to the rotatable CCD. Here is another CCD. This one has four variables, alpha is equal to 2, so it's the rotatable value. But the different lines here show you what happens to the prediction variance as you change the number of center runs. You notice that if you have only one center run, you have huge prediction variants at the center. Then it drops quickly as you go toward the boundary and then comes back up again. But this is a very unstable picture. A lot of prediction variance at the center, increasing the number of center runs. Look what it does, it reduces that prediction variance at the center until we get to about four or five center runs. Then we start to get a very stable distribution of prediction variance over a big portion of the design space out to about a radius of 1.5. So that's where that advice of 3-4-5 center runs for central composite designs come from. You can look at lots of different CCDs and construct pictures like this to give you information about how many center runs would be helpful. Okay. So that gives you a good idea about graphical methods for designs. More about design coming up.
