Stat-Ease Blog

Most people who have been exposed to design of experiment (DOE) concepts have probably heard of factorial designs—designs that target the discovery of factor and interaction effects on their process. But factorial designs are hardly the only tool in the shed. And oftentimes to properly optimize our system a more advanced response surface design (RSM) will prove to be beneficial, or even essential.

Let’s start by introducing the Stat-Ease® software menu options for RSM designs. Once we understand the alternatives we can better understand when which might be most useful for any given situation and why optimal designs are great—when needed.

• First on the list is the central composite design (our software default)
• Next is the Box-Behnken design
• And third is something called optimal design

The natural question that often pops up is this. Since optimal designs are third on our list, are we defaulting to suboptimal designs? Let’s dig in a bit deeper.

The central composite design (“CCD”) has traditionally been the workhorse of response surface methods. It has a predictable structure (5 levels for each factor). It is robust to some variations in the actual factor settings, meaning that you will still get decent quadratic model fits even if the axial runs have to be tweaked to achieve some practical values, including the extreme case when the axial points are placed at the face of the factorial “cube” making the design a 3-level study. A CCD is the design of choice when it fits the problem and generally creates predictive models that are effective throughout the design space--the factorial region of the design. Note that the quadratic predictive models generally improve when the axial points reside outside the face of the factorial cube.

When a 5-level study is not practical, for example, if we are looking at catalyst levels and the lower axial point would be zero or a negative number, we may be forced to bring the axial points to the face of the factorial cube. When this happens, Box-Behnken designs would be another standard design to consider. It is a 3-level design that is laid out slightly differently than a CCD. In general, the Box-Behnken results in a design with marginally fewer runs and is generally capable of creating very useful quadratic predictive models.

These standard designs are very effective when our experiments can be performed precisely as scripted by the design template. But this is not always the case, and when it is not we will need to apply a more novel approach to create a customized DOE.

Optimal designs are “custom” creations that come in a variety of alphabet-soup flavors—I, D, A, G, etc. The idea with optimal designs is that given your design needs and run-budget, the optimization algorithm will seek out the best choice of runs to provide you with a useful predictive model that is as effective as possible. Use of the system defaults when creating optimal designs is highly advised. Custom optimal designs often have fewer runs than the central composite option. Because they are generated by a computer algorithm, the number of levels per factor and the positioning of the points in the design space may be unique each time the design is built. This may make newcomers to optimal designs a bit uneasy. But, optimal designs fill the gap when:

• The design space is not “cuboidal”— there are constraints on the operating region that make the design space lopsided or truncated.
• There are categoric or discrete numeric factors to deal with.
• The expected polynomial model is something other than a full quadratic.
• You are trying to augment an existing design to expand the design space or to upgrade to a higher order model.

The classic designs provide simple and robust solutions and should always be considered first when planning an experiment. However, when these designs don’t work well because of budget or practical design space constraints, don’t be afraid to go “outside the box” and explore your other options. The goal is to choose a design that fits the problem!