I want to conclude this section of material on designs for response surface methods, by discussing for a few minutes a new and I think powerful and useful class of response surface design called Definitive Screening Designs. These designs were introduced by Bradley Jones and Chris Nachtsheim in 2011. They, they found these designs, or they found that these designs can be constructed by use of an optimal design algorithm. They referred to these as definitive screening designs, because they're small enough to allow screening of potentially a lot of factors, yet they can often accommodate fitting complete second-order models without additional runs. So we can, in a sense, think of these as one step response surface designs, combining screening and optimization in the same experiment. The table that you see, Table 11.15, shows you the general structure of these designs. Notice that every design, every defensive screening design is made up of m fold over pairs. Now what do we mean by fold over pair? Well, by fold over pair, we mean that, the non-zero entries in the first row, have different signs than the non-zero entries in the second row. The entries are either plus one or minus one. So if you have a plus one in the first row, the corresponding entry in the second row will be minus one. Then every pair of rows has to zeros. You notice that those pairs of zeros march down the diagonal, until we get to the bottom. So there are two m runs that are made up of m fold over pairs. Then there's a final center run that's added. So these designs have a total of 2m plus 1 runs, where m is the number of variables in the experiment. There is of course, one factor level at its center for all variables in addition to the overall center run. These designs have a lot of really nice properties. First of all, they're relatively small. The number of runs is only one more than twice the number of factors. So this is really quite small. Unlike resolution III designs, the main effects are completely independent of two- factor interactions. So your estimates of main effects are not biased, by the presence of active two-factor interactions. Whether or not the interactions are included in the model or not. Unlike resolution IV designs, the two-factor interactions are not completely aliased or not completely confounded with each other. Although they may be correlated, there may be some non-zero correlation. Unlike the standard two-level resolution III, IV, and V designs with added center runs, you can actually fit all of the quadratic effects in models composed of any number of linear and quadratic main effects. The quadratic effects are orthogonal to the main effects, but they're not completely confounded, although somewhat correlated with the interaction effects. If you have a design with six or more factors, these designs are capable of fitting all possible full quadratic models involving three or fewer factors. So if you have six or seven or eight factors and you can reduce it to three active factors. You can fit a complete quadratic model, in any subset of three of those factors and with very good statistical efficiency. These designs can be constructed using an optimal design tool. But, a year or so after they were introduced, a new construction method was proposed that uses conference matrices. Now, a conference matrix C, is an n-by-n matrix that has diagonal elements equal to zero. All the off-diagonal elements are either plus one or minus one. They all have the property that c prime c is a multiple of the identity matrix. For an n by n conference matrix C, C transpose to C is equal to n-minus 1 times I. Conference matrices originally arose in telephone engineering types of problems. They were used in constructing ideal telephone networks, conferencing networks from transformers. These conference networks, these telephone conference networks were represented mathematically by conference matrices. There are other applications, but just for example, here is a six by six conference matrix. So the 13 run definitive screening design would be found by taking every row of this conference matrix, folding it over, and then adding a row of zeros at the bottom. So if C is the conference matrix, then the definitive screening design, design matrix would be made up of C and then a matrix that is the negative of C, and then a row of zeros. So this gives us a total of m equal to 2n plus 1 runs Here are a whole set of definitive screening designs for you to look at, for factors from 4-12. Software has these designs imbedded in it. You can construct definitive screening designs for lots and lots of situations. Originally, DSDs were set up for use with continuous factors, but you can modify them to include, two level categorical factors. Here's how you do that. You set up the usual DSD for continuous factors, except you add two rows of zeros instead of one. Then you change the zeros in the column for your categorical variables to either plus one or minus one. If the zeros are the added rows of zeros make all the categorical variables, minus one for the first row, and plus one for the second row. If the zeros are from the conference matrix and its fold over, make the factor minus one for the first row and plus one for the second row. Here's an example for four continuous factors and two categorical factors. The DSD for two factors or for six factors with two added zeros is shown in the display, that you see here at the bottom. So here are the two edit zeros, and here is the standard DSD for six factors. Well, we follow the rules given above. Now you notice that, rows E or columns E and F, all the zeros have been changed to either plus one or minus ones, according to the rules that we just discussed. Now we have a definitive screening design that will accommodate four continuous factors and two categorical, two level factors. When we do this, the main effects of your categorical variables have some correlation with other factors, but generally the correlations are pretty small. You can also construct DSDs and orthogonal blocks. The procedure is pretty easy. All you do is you create the standard DSD for continuous factors, except you add as many rows of zeros as there are blocks. Then you arrange the design in standard order, so that each pair of runs is the initial run and it's fold over. Then you assign the first fold-over pair to the first block, the second fold over pair to the second block, and so on until you get the last block. Then continue again assigning the next fold over pair to the first block. Then assign each center run to a separate block. Here's a six-factor example in two blocks. So here's my first fold over pair. It goes into block 1. Here's my second fold over pair, block 2. Here's my third fold over pair, now we're back to block 1 again and so on and so forth and here are my two added rows of zeros. The first row goes in block 1 and the second row goes in block 2. It is very easy to do. Computer software will construct these designs for you.
