Stat-Ease Blog

Response surface methods (RSM) pave the way to the pinnacle of process improvement. However, the central composite design (CCD)—the most common layout for RSM (pictured in Figure 1 for three factors)—traditionally limits the region of prediction to the cubical core. This conservative view avoids dangerous extrapolation out to the far reaches of the space defined by the axial ranges of the star points. This article lays out a less-limiting (but still reasonably safe) approach to optimization based on using a specified standard error (SE) of prediction as the boundary for searching out the optimal process setup.

Three different methods for defining the search area will be detailed for a four-factor CCD. The goal is to avoid extrapolating beyond where the data provides adequate knowledge about the response while maximizing the volume that will be explored.

Let’s compare three boundaries for defining the search area in the factor space, the first two of which do not make use of the SE:

1. Factorial bounded—the hypercube* with vertices at coded values ±1, thus each edge spans 2 coded units. The volume of this four-dimensional hypercube is 16 (=2x2x2x2). The maximum SE is 0.764, which occurs at the vertices (i.e., corners). See figure 2. For comparison’s sake, we will use this SE (0.764) as our benchmark—anything more than this will be deemed unacceptable.

2. Axial (star-point) bounded—a cube with vertices at ±2 to include the star runs.

The volume of this four-dimensional hypercube is huge: 256 coded units (=4x4x4x4), which offers big advantages for optimization. However, most of the volume (69%) exhibits an SE ≥ 0.764 (maximum is 2.963!). Therefore, this method must be rejected. See figure 3.

3. Standard error bounded—the area within SE ≤0.764.*

Once again looking at figure 3, the SE at the axial (star) points equals that of the ±1 factorial points. Limiting the standard error ≤0.764 produces a hypersphere with a radius of 2. The volume of this hypersphere is 78.96, almost five times larger than the ±1 factorial hypercube.

Summarizing the three methods of defining the search area in the factor space:

1. The factorial cube with vertices at ±1 may be too restrictive and may not include all the volume where acceptable predictions could be made.
2. A cube with vertices at ±2 that includes the axial runs is too liberal; most of the volume has poor predictions.
3. Defining the search area by standard error may prove insightful—it includes all the areas where acceptable predictions may reside.

Using standard error to constrain the optimization defines a search area that matches its properties:

• Spheres for rotatable CCDs. (Note: The above graphics and discussion assumed the choice of alpha values produced a spherical standard error plot).
• Cubes for face-centered CCDs.
• Irregular shapes for central composite designs with alpha between 1 and that recommended for rotatable designs, optimal designs, models for which model reduction was applied, and historical data.

An added bonus to using SE is that it adjusts the search area for reduced models and/or missing data.

It should be noted that it is assumed the design was sized for precision and contains enough data to make sound predictions within the cube (or hypercube). If the FDS is low (for example, below 80%), then making good predictions within the cube is already challenged. Extending the search zone outside the cube would exacerbate things further.

Another caveat is the assumption that a quadratic model pertains outside the design cube. The primary purpose of axial points in a central composite design is to fortify the estimates of quadratic terms to be applied within the cube. Sometimes the specified quadratic model performs well inside the cube, but extrapolation becomes dangerous due to higher-order behavior beyond the faces of the cube. Checking the diagnostic plots for anomalous behavior of the axial points can provide some assurance that the quadratic model is useful beyond the cube.

So, the key takeaway is this. Adding standard error to the search criteria and expanding the factor ranges beyond the edges of the factorial cube can be helpful for making judicious extrapolations beyond the edges of the cube. Simply applying the highest standard error found within the cube to regions outside the cube is a reasonable place to start, especially when the FDS performance of the design is over 80%. It is advisable to treat any interesting discoveries as tentative until verified by confirmation runs, augmented designs, or an entirely new design focused on the projected area of interest.

For more information on how to include standard error in the optimization module, see: Extrapolating a Response Surface Design in the Stat-Ease software Help menu.