11.2.2 - Box-Behnken Designs

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Box-Behnken Designs

Another class of response surface designs are called Box-Behnken designs. They are very useful in the same setting as the central composite designs. Their primary advantage is in addressing the issue of where the experimental boundaries should be, and in particular to avoid treatment combinations that are extreme. By extreme, we are thinking of the corner points and the star points, which are extreme points in terms of region in which we are doing our experiment. The Box-Behnken design avoids all the corner points, and the star points.

One way to think about this is that in the central composite design we have a ball where all of the corner points lie on the surface of the ball. In the Box-Behnken design the ball is now located inside the box defined by a 'wire frame' that is composed of the edges of the box. If you blew up a balloon inside this wire frame box so that it just barely extends beyond the sides of the box, it might look like this, in three dimensions. Notice where the balloon first touches the wire frame; this is where the points are selected to create the design.

Minitab Output

Therefore the points are still on the surface of a ball, but the points are never further out than the low and high in any direction. In addition, there would be multiple center points as before. In this type of design you do not need as many center points because points on the outside are closer to the middle. The number of center points are again chosen so that the variance of  is about the same in the middle of the design as it is on the outside of the design.

In Minitab we can see the different designs that are available. Listed at the bottom are the Box-Behnken Designs.

Box-Behnken Designs

A Box-Behnken (BB) design with two factors does not exist. With three factors the BB design by default will have three center points and is given in the Minitab output shown above. The last three observations are the center points. The other points, you will notice, all include one 0 for one of the factors and then a plus or minus combination for the other two factors.

If you consider the BB design with four factors, you get the same pattern where we have two of the factors at + or - 1 and the other two factors are 0. Again, this design has three center points, and a total of 27 observations.

Comparing the central composite design with 4 factors, which has 31 observations, a Box-Behnken design only includes 27 observations. For 5 factors, the Box-Behnken would have 46 observations, and a central composite would have 52 observations if you used a complete factorial, but this is where the central composite also allows you to use a fractional factorial as a means of making this experiment more efficient. Likewise for six factors, the Box-Behnken requires 54 observations, and this is the minimum of the central composite design.

Both the CCD and the BB design can work, but they have different structures, so if your experimental region is such that extreme points are a problem then there are some advantages to the Box-Behnken. Otherwise, they both work well.

The central composite design is one that I favor because even though you are interested in the middle of a region, if you put all your points in the middle you do not have as much leverage about where the model fits. So when you can move your points out you get better information about the function within your region of experimentation.

However, by moving your points too far out, you get into boundaries or could get into extreme conditions, and then enter the practical issues which might outweigh the statistical issues. The central composite design is used more often but the Box-Behnken is a good design in the sense that you can fit the quadratic model. It would be interesting to look at the variance of the predicted values for both of these designs. (This would be an interesting research question for somebody!) The question would be which of the two designs gives you smaller average variance over the region of experimentation.

The usual justification for going to the Box-Behnken is to avoid the situation where the corner points in the central composite design are very extreme, i.e. they are at the highest level of several factors. So, because they are very extreme, the researchers may say these points are not very typical. In this case the Box Behnken may look a lot more desirable since there are more points in the middle of the range and they are not as extreme. The Box-Behnken might feel a little 'safer' since the points are not as extreme as all of the factors.

The Variance of the Predicted Values

Let's look at this a little bit. We can write out the model:

\(\hat{y}_{x}\) = b0 + b1x1 + b2x2 + b11x12 + b22x22 + b12x1x2

Where the b0, b1, etc are the estimated parameters. This is a quadratic model with two x's. The question we want to answer is how many center points should there be so that the variance of the predicted value, var(\(\hat{y}_{x}\)) when x is at the center is the same as when x is at the outside of the region?

See handout Chapter 11: Supplemental Text Material. This shows the impact on the variance of a predicted value in the situation with k = 2, full factorial and the design has only 2 centerpoints rather than the 5 or 6 that the central composite design would recommend.

What you see (S11-3) is that in the middle of the region the variance is much higher than further out. So, by putting more points in the center of the design, collecting more information there, (replicating a design in the middle), you see that the standard error is lower in the middle and roughly the same as farther out. It gets larger again in the corners and continues growing as you go out from the center. By putting in enough center points you balance the variance in the middle of the region relative to further out.

Another example (S11-4) is a central composite design where the star points are on the face. It is not rotatable design and the variance changes depending on which direction you're moving out from center of the design.

It also shows another example (S11-4), also a face-centered design with zero center points, which shows a slight hump in the middle on the variance function.

Notice that we only need two center points for the face centered design. Rather than having our star points farther out, if we move them closer into the face we do not need as many center points because we already have points closer to the center. A lot of factors affect the efficiencies of these designs.

Rotatability

Rotatability is determined by our choice of alpha. A design is rotatable if the prediction variance depends only on the distance of the design point from the center of the design. This is what we were observing previously. Here in the supplemental material (S11-5) is an example with a rotatable design, but the variance contours are based on a reduced model. It only has one quadratic term rather than two. As a result we get a slightly different shape, the point being that rotatability and equal variance contours depend both on the design and on the model that we are fitting. We are usually thinking about the full quadratic model when we make that claim.