In our last lecture we talked about the combined array approach as an alternative to Gucci's crossed array. And I want to illustrate the combined array for you and show you a modeling approach that is much superior to what he suggested doing because it takes full advantage of the interactions and both main effects and interactions of all your control and all your noise factors. It makes sense to fit a model that includes controllable and noise factors and all of their interactions and there's a way to do that by using what we call a combined array design and a response model. So to illustrate, suppose that we have two controllable variable variables, X1 and X2, and a single noise factor and that both of these factors can be expressed in the usual sort of coded units centered at zero, lower and upper limits that some plus or minus a probably plus or minus 1 and we want to first order model. Well a first order model containing these three factors would have the main effects of X1 and X2 and X1 X2 interaction. And then the main effect of your noise variable, let's call it Z1 and then the two factor interactions between X1 and Z1 and X2 + Z1, and so there is a model that has the main effects of both the controllable factors, the noise factor and their interactions. So this model has all the interaction effects that you need and at least one of these regression coefficients Delta 1 1, Delta 2 1 that multiplied the interactions between the control of the noise factors. At least one of those has to be significant, it's got to be non zero, otherwise, we don't have a robust design problem. The advantage of this is that we can run a single experimental design called a combined array that allows us to fit all of these model parameters very efficiently. For example, we might use a 2 to the 3 factorial in the previous example. So we're going to have everything expressed in coded units. We're going to assume that the noise variables have expected value 0 and some variance sigma square Z. And once we fit this response model that you see here, it would be easy to find a model for the mean, just simply take the expected value with respect to Z and Epsilon of that model. And so equation 12.2 would be the the model for the mean and now all you would do is replace the betas by the estimates beta hat. To find the model for the variance of the response, you could use the so-called transmission of era approach or propagation of era approach, and this is really nothing more than doing a Taylor series of first order expansion of your model around say Z1 equal to 0. When you do that first order Taylor series expansion, this is what you get. And this is the remainder in the Taylor series, ignore the remainder. Now the variance of Y can be found by simply applying the variance operator across this last expression ignoring r, so when we do that, this is the equation that we get for the transmitted variance, and you notice that that it's entirely in terms of are controllable variables x1 and x2. So the Z subscript on that variance operator is just reminding me that both Z1 and Epsilon are random variables in that expression. So these are just simple models, and the mean and variance models involve only the controllable variables, which is really nice. So you could theoretically set those control variables to get a target value for the mean and minimize the variability easily. Even though the variance model involves only the controllable variables, it also has the interaction regression coefficients between the controllable and the noise variables and this is how the noise variable influences the response. The noise variable is a quadratic function of the controllable variables and the variance model apart from sigma square turns out to be just the square of the slope of the fitted response surface model in the direction of the noise variable. So you could think of the standard deviation of the transmitted variability is just the slope of your fitted response model in the direction of the noise variables. So how do we operationalize this? Perform an experiment that is appropriate to fit your response model. Then replace the unknown regression coefficients in the mean and variance models with the least squares estimates from the response model. Replace sigma square in the variance model by the residual mean square error from your Anova for the response model. And then go through the optimization for the mean and variance using multiple response surface methods like we talked about back in Chapter 11. And we can actually generalize these results In terms of Matrix math. K Controllable variables are a noise variables, then this is the general form of the Response model. F of x, this portion contains only the controllable variables and H of x and z are the terms that involve the main effects of the noise factors and the interactions between the controllable and noise factors. And typically that's going to look something like this, main effects of the noise variables and then the interactions between control and noise. And the structure that you choose for F of x depends on what type of model you think is appropriate, it could be first order model [COUGH] could be first order model with interaction, could be a second order model. We're going to assume that the noise variables have mean 0, they're uncorrelated, variance sigma square Z and that the noise variables and the random errors are also zero, could have zero correlation. So your mean model is just f of x, and the variance model is a little bit more complicated looking, it looks like equation 12.6, but that actually simplifies quite easily because this derivative is actually very easy to compute. So to illustrate this, let's go back to a previous example from chapter 6 or module six, four factors in a two to the four factorial and we are studying the filtration rate of a chemical product. And let's say factor A is difficult to control, A was temperature, and we ran a full 2 to the 4 factorial and we fit the model that you see here. So this is the final model. That's the response model and Z1 is my noise variable. Okay. So using our equations for the noise variable and the mean model, noise model and the mean model, there is the mean model and here is the transmitted variance model. This noise factor temperature, we need to study how to find the levels of X2 X3 and X4 that minimize the variability that's transmitted from that temperature variable, and you notice that that X4 is not important. So there are only two process variables here, that controllable variables that are really important X2 and X3. And we do have interactions involving X2 and X3 that are important. So the variance model, if we substitute in sigma square Z of 1 and sigma square hat of 19.51, which is the residual mean square from your variance model. This is the final form of the transmitted variance model, and let's take a look at the form of these models, here is the mean model with temperature zero, that's the mid-level of the temperature variable which is not controllable, are easily controllable. And here is the standard deviation of the variance of the transmit, this is the transmitted standard deviation the so-called propagation of error and you notice that there is a quadratic bend here, there's a bend in the surface. It's because this is a second-order model. So let's suppose we want a mean filtration rate of about 0.75, or rather of about 75 with minimum variability. Well here is the contour of 75, if you go in this direction the filtration rate increases, and here is the minimum value of this propagation of error, this transmitted noise, which keeps it at or below 5.5, and so here is a narrow little operating window up here which will allow us to achieve the desired results. Concentration at the high-level stirring rate near the middle of the level, that would give us the best results regardless of the variability and temperature. Here's a somewhat more complicated example. This is an experiment running a semiconductor manufacturing plant. We have two controllable variables and three noise variables, [SOUND] here's the combined array, a fairly big experiment combined array design. Its a 20-3 run variation of a central composite design. Now, the way this design was created was by starting with a standard central composite design for five factors. The cube part of that design is a two to the five minus one and then the axial runs associated with the three noise variables were deleted. Now, why did I drop those? Well, because you're not going to fit quadratic terms in the noise variable so you don't need those runs. However, we can estimate all of the main effects. We can estimate the interactions between the controllable factors the interactions between the noise variables and the interactions between the controllable and the noise variable. And so here is your fitted response model, to all of these factors and here is the mean model, and there is the formula for the transmitted variance. So the figure on the right is the Contour plot of the standard deviation of transmitted variability, the so-called transmitted variance or propagation of error. And here is the Contour plot of the mean model. Well, we want the mean to be greater than 30 and we want the standard deviation of transmitted variance to be less than 5. And so here is an overlay plot that shows you a feasible region, which you can operate this process. So this response surface model and getting explicit mathematical models for the mean response and for the transmitted variability is really the right way to solve this robust design problem.