# How to Optimize Using RSM and Central Composite Design

I am learning about Central Composite Designs (CCD) and I am wondering how one chooses the center and the range that the factorial corners span.

For example, if I am optimizing the yield of a chemical reaction with pH (0-14) and temperature (273-373 K), does choosing to take data at the extremes really give the RSM model enough data to find the maximum in "1 shot"? Or would I initially follow a CCD, use RSM to predict maximum, do another CCD with a new center, use new RSM, repeat etc until the solution converges?

If the repetition until convergence is the case, is it advisable to start with the maximum ranges for the variables? Do I ever sample more closely together?

I guess I am confused as to the overall steps.

• There is a tutorial vignette in the rsm package in R that goes through the steps. Also good textbooks by Myers and Montgomery, Box, Hunter, and Hunter, and Hamada and Wu. Oct 10, 2020 at 2:58

Here are two important things to consider:

1. As in all statistics contexts, we assume that data are measured with error. The larger the experimental region, the more accurately we are able to estimate the equation of the response surface, assuming that our model is correct.

2. As has been stated by GEP Box and others, all models are wrong, but some are useful. CCDs are great for estimating second-order (quadratic) surfaces. That model is useful if the experimental region is small enough that the second-order equation is a reasonable approximation to the true response surface. But if the experimental region is too large, then the second-order model may no longer be very useful.

Notice that (1) says you want a far-ranging experiment, and (2) says you want a smaller scope. All this said, you could counteract (1) by replicating -- i.e., run two or more CCDs on the selected region; so that is an option as well.

Note that (2) answers "no" to your first question -- one CCD over the whole region is not enough to find the maximum with any assurance. The surface may not be second-order and may not be well approximated by a second-order model over that wide a range. There is also not all that much data in one CCD, but (1) says that if indeed the actual surface is second-order, then a CCD over the whole region of operability makes the best use of that small amount of data.

So, what to do? This really depends on how much error there is; and you may not know much about that until you have some data. I'd suggest running a first-order experiment (just the four corners plus a few center points) over a rectangle comprising, say, the middle third or middle half of the range of each variable. You need the experimental region to be big enough to observe the trends in the response surface. You get two things out of fitting a first-order model to these results: a measure of error SD from the center points, and a lack-of-fit test for the model.

If the LOF is significant, that suggests the first-order model is not enough, and you should do a second experiment on the axis ("star") points plus perhaps a few more center points. Between those two experiments, you now have a CCD (which you should model with an additional factor for the block effect --- which half of the experiment the data come from). That will give you an idea of where the optimum lies.

If the LOF is non-significant, then it suggests we are sitting on some kind of planar area --- a hillside on the response surface. You may use the steepest ascent method to lead you in the direction to center the next experiment.

In either case, you need to do another experiment with a new center (at the estimated optimum or some place along the steepest-ascent path), and possibly a new range of predictors (smaller if you were able to observe very clear trends, larger if there is a lot of noise in the data).

You keep doing this until you find an optimum and are able to more-or-less confirm it. I have left out the details of how to fit the models and do the tests, how to choose axis points, and the fact that you need to use coded data when estimating a steepest-ascent path. So you need to read those details carefully in any of a number of texts. I think Statistics for Experimenters (2nd ed) by Box, Hunter, and Hunter (aka $$BH^2$$) is a good choice.