DOE with curvature - Minitab Tutorial

- In 2K factorial designs, the assumption is that the response Y maps to a straight line equation between the low and high settings of the axis. We see that Y=f(x) is a first order equation. Here is the form of a first order equation, showing the many facts and interaction. What if it's not a straight line and there is curvature? In that case, a first order model is inadequate and will not be valid. A second order or quadratic equation is required, where the Y=f(x) is not a straight line or a flat surface. You can see it here. To evaluate whether the model has any significant quadratic components, we need to test for curvature. This is done by including center points when creating a 2K factorial design. Let's illustrate with this example. What if the CTQ requirement for paper is whiteness? In order to determine optimal settings, a pulp and paper manufacturer wants to study the effects of two variables. The center of bleach concentration and temperature on the whiteness of the paper, whiteness being the response. So a two to the power of two factorial design is selected. For this example, the selected low and high levels of the factors are in this table. If you suspect that there is curvature in the response, design the experiment with center points. To do this, create an experimental design with one replicate and five center points. The center points are the middle values of both variables, in this case, concentration at 4.75% and temperature at 157.5 degrees. Now, let's run this factorial design with centerpoints using minitab. To get started, open up minitab with a clean worksheet. We need to create the worksheet for the DOE with the two factors, one replicate and five center points. To do that, go up to the stat menu, DOE, factorial, create factorial design. We have our two factors and the design we want is one with five center points and one replicate which means two to the power of two is four, four plus five is nine, so we should expect nine runs. The factors are, type in concentration, and we have that at 3.75% and 5.75% for the high. Temperature is our second factor, it's numeric also, 152.5 as the low setting and 162.5 for the high setting. There we have it. Go to options, typically we would randomize runs in the real experiment, but to have consistent easy follow along for this course, let's un-randomize it and click okay. Here we have the nine runs in standard order, no results yet. But wait, we have the results of the experiment already created, so let's go up to the exercise files and download DOECurvature.mtw and open that file, you have the whiteness results in C7. To analyze this DOE, we have to go up to stat, DOE, factorial, analyze factorial design. The response is whiteness, our terms are already there and one quick note, include center points in the model. That being said, let me just point out something real quick. In the worksheet is that center points are indicated by zero and if you notice the values 4.75 is in the middle of 3.75 and 5.75 and 157.5 degrees is between 152.5 and 162.5. Literally the center of the low and high values of all factors and these are the results. Anyway, go back to okay. Graphs, ask for the four in one residuals, for the effects pareto is good enough, okay and okay. We have the residual plots, look pretty decent, not the world's greatest but normal enough. Then we have the session window, let's make this bigger, with the details of the results. Here we see that nova table with p values that are very significant in the terms as well as the interactions between concentration and temperature. But more importantly, the last line here, we have curvature. Since we had center points, the DOE model tested for curvature and the p value is significant, less than 0.05. That means that there is curvature in the response. It's no longer a straight line between a low and high level settings. So let's look at the factorial plots. Go up to stat menu, DOE, factorial, factorial plots. We have the terms there, okay. Here we have the interaction plot where the center point is the dot that we see here, based on the legend it's brown or red and we see that it does not rest on the line, so there is curvature in the response. The real response is between the low, the center to the high setting. Similarly for the other line, it's a curve. In the main effects, we see much more clearly the true response is from this dot, connect the dots. That's a true response curve, as you can see it's not a straight line, it's a curve in both cases, so there is curvature. It is significant from that nova table, therefore the factorial design will fail us. We have to use a different model to model curvature and that model is called response surface methods.