# How do Fractional Factorial Designs relate to 'Response Surface Methodology'?

Fractional factorial designs (FFDs) and Response Surface Methodology (RSM) are both approaches to extracting some information about how multiple interacting factors affect a response variable. Their value lies in the fact that they involve much less time and investment to perform than the classic (and easier-to-understand) full-factorial experiment. FFDs and RSM are uncommon in my field and I'm trying to understand some basics about them.

Are FFDs mainly thought of as a component of RSM? If so, what does RSM look like without FFDs? And how are FFDs used if not as part of RSM?

• Fractional Factorial designs can be used as a part of response surface designs, but they can also be used otherwise. The best, first, reference is Box, Hunter & Hunter, see stats.stackexchange.com/a/30123/11887 Commented Mar 29, 2023 at 22:57
• @kjetilbhalvorsen Thanks! Looking into it now.
– mkt
Commented Mar 30, 2023 at 10:23

I would argue that the analysis FFDs are a subset of the analysis techniques for RSM. Furthermore, the analogy I would use is the comparison of a multiple regression model where you are deciding between including an ordinal variable as a purely categorical variable (estimating each levels response separately) vs as a scalar variable (where you can assume some element of continuity across the spacing of the ordinal levels).

What is true in both FFD and RSM is that you are not examining every possible combination of experimental cells possible. With FFD this is often because if the number of cells can become quite large (10 dichotomous factors would required $$2^{10}=1024$$ experimental cells), and with RSM you have continuous variables (so a selection of all possibilities is impossible). However, with careful selection of a subset of the possible combinations, you obtain enough information to tease out most 1st and 2nd order effects.

I am currently teaching a DoE course, and I would recommend the textbook we are using: D.C. Montgomery's Design and Analysis of Experiments. (We are using the 10th ed.) This book provides a very detailed breakdown of the variations of FFDs you might encounter (chapters 8 & 9). The coverage of RSM is very thorough (chapter 11), and the information is presented in a manner that shows how multiple regression analyses of the models from FFDs and RSMs are similar in nature.

I hope this response is useful.

• Thank you, Gregg. My impression is that FFD is often used for continuous variables as well, by picking 2 levels, and kjetil b halvorsen's comment above suggests that it is also used outside of the RSM framework. Both seem to contradict your answer a bit - could you comment on that?
– mkt
Commented Apr 5, 2023 at 8:42
• @mkt Regarding continuous variables variables, FFD can be used...but as you suggest, it would be with variables to a restricted specified level (say room temp set by the researcher to 65 and 85 degrees). As for the second question, my answer is from the view point that both models are essential multiple regression models. My read of the "outside of the framework" idea is about possible research questions that you can examine with FFD designs...but I'm just guessing. Commented Apr 5, 2023 at 11:30
• @mkt: RSM is about optimization, so part is how you get from a first design, analyse, and use it to define new experimental runs to hunt the optimum. That is mostly done with continuous variables, if you have only two-level factors it is not necessary. Commented Apr 6, 2023 at 2:28
• FFD is also a protocol for optimization. Even if you have only two-levels, if you have multiple dichotomous factors, you may still wish to know which combination gives the largest effect. From Montgomery (cited above) p.274 "A major use of fractional factorials is...in experiments in which many factors are considered and the objective is to identify those factors (if any) that have large effects." Commented Apr 6, 2023 at 13:19

A brief summary of what I've learned:

Response Surface Methodology (RSM) is a type of adaptive or sequential design. Sequential designs involve multiple rounds of experimentation, with the choice of treatments in each later round being dependent on the data accumulated from completed rounds. RSM is usually used to identify some sort of optimum, often for industrial production. Sequential designs also exist outside of the RSM framework, however, such as Bayesian Adaptive Experimental Design.

Fractional Factorial Designs (FFDs) are a specific group of experimental designs. They allow useful information to be extracted from relatively tiny experiments, especially in situations with a large number of predictors. The central (quite reasonable) assumption is that higher-order interactions are less important than lower-order ones and main effects, which is called the "sparsity-of-effects principle". Experimental designs can therefore be scaled down by neglecting the higher-order interactions. This scaling down involves intentionally confounding (‘aliasing’) combinations of factors relative to the full factorial experiment, with the consequence that one cannot estimate each separate main effect and interaction term. We can still learn a great deal of useful information from the data despite this limitation, though.

How RSM and FFDs interact:

FFDs are most often used as a component of RSM. But they can in principle be used independently, either as part of a one-shot experiment, or as part of a sequential design approach that does not involve RSM. They seem unlikely to be very useful when used in a one-shot experiment, which is probably why they are so tightly connected to RSM.

Additionally, RSM can take as its input experimental designs that are not FFDs. Central Composite Designs and Box-Behnken designs are two other commonly-used designs.

My thanks to kjetil b halvorsen and Gregg H for their input and the suggested resources. I was not able to lay my hands on the Montgomery book but found the Box, Hunter & Hunter book very useful.