In this lesson, we go through Central Composite Designed Experiments, and what to do if your designed experiment exhibits curvature. If you have any questions, or would like help on your own central composite designed experiments, leave a comment below!
When we design a 2^K factorial experiment, we have the option of adding center points. Now typically and for reasons that are beyond what we can go into here, you’ll add about three to six center points and the purpose of that is to test the assumption of linearity.
When we’ve got these three variables we make the assumption that between the endpoints they’re linear. Well, that may or may not be true and it’s nice to find out. You can begin to visualize what’s going on by looking at these effects plots. Here we have the effect of variable A and here’s the center point. It’s right at the midpoint and the question is, is that far enough away from the line to indicate curvature? If we look in the session window, we’ll find that we have a new line here, it’s called center point. If I read across, that does have the magic p-value less than point .05. There’s probably, no, there is enough curvature there to be statistically detected. We don’t have a linear model. Now what do we do?
Here you can see a three-dimensional model of what we’ve done. We can represent one variable as left to right, another one as front-to-back, and a third one as up and down. In the middle of that cube we put our center points. There’s good news and there’s bad news. The good news is, that if you get a significant center point, it means you do have curvature and there’s a way to handle that. The bad news is that it is theoretically possible for you to not get a significant center point and still have curvature, but that’s way less likely. The other bad news is that we are not able to estimate which variable contributes how much curvature given a simple model like this.
What do we do then? We go back and we don’t throw our data away, we keep it. We just add some additional points and what we do then is this. We add what are called star points and each of these imaginary rods that supporting a star point goes right through the center of each of the squares that are the faces of the cube. These points that we will add in after the fact, fall on the surface of the same sphere as the corner points.
Once we have those, we can very handily estimate curvature and tell which variable it’s coming from. When you go into the designer here, if you choose central composite, and it says here star for curvature, meaning we’re going to add the star points. QuikSigma will automatically generate the coordinates for those star points. So all you have to do then is copy and paste your old data into this and then take the star point data and you’ll have a design.
One thing you do have to remember to do is to turn on squared terms. Curvature comes from terms with exponents, and this model will handle exponents up to second power. So turn on the squared terms, and then over under the optimizer, once you’ve analyzed, you may find that your variables produce a nonlinear output. Just like before, you can move those around and decide where you want them for the output that you want.
Also note, that in the session window you’ll have some new terms. Here we’ve got term A and term B. We’re going to get an A squared turn and a B squared turn in addition to our AB interaction term. So, as they say, the mathematics take on a new richness. You can build some extremely sophisticated models with this. You can design robust processes that are stable, and that don’t vary when there’s a little perturbation, and it’s just a absolutely outstanding optimization tool.