5.3.3.9. Three-level full factorial designs

5. Process Improvement
5.3. Choosing an experimental design
5.3.3. How do you select an experimental design?

5.3.3.9. Three-level full factorial designs

Three level designs are useful for investigating quadratic effects - but require many runs

The simplest 3 level design - with only 2 factors

The model and treatment runs for a 3 factor 3 level design

The three-level design is written as a 3^k factorial design. It means that k factors are considered, each at 3 levels. These are (usually) referred as low, intermediate and high levels. These levels are numerically expressed as 0, 1, and 2. One could have considered the digits -1, 0, and +1 but this may be confusing with respect to the 2 level designs, since 0 is reserved for center points., so we stick with the 0, 1, 2 scheme... The reason that the three-level designs were proposed is to model possible curvature in the response function or because one is experimenting with nominal factors, each having 3 levels. A third level facilitates investigation of a quadratic relationship between the response and each of the factors.

Unfortunately, the three-level design is prohibitive in terms of the number of runs, and thus in terms of cost and effort. For example a two-level design with center points is much less expensive while it still is a very good (and simple) way to establish the presence or absence of curvature.

The 3² design

This is the simplest three level design. It has two factors, each at three level. The 9 treatment combinations for this type of design
can be pictorially shown as follows:

FIGURE 3.23 A 3² Design Schematic

A notation such as 20 means that factor A is at its high level (2) and factor B is at its low level (0).

The 3³ design.

This is a design that consists of three factors, each at three levels. It can be expressed as a 3 x 3 x 3 = 3³ design. The model for such an experiment is

where each factor is included as a nominal factor rather than a continuous variable. In such cases, main effects have 2 degrees of freedom, two-factor interactions have 2² = 4 degrees of freedom and k-factor interactions have 2^k degrees of freedom. The model contains 2 + 2 + 2 + 4 + 4 + 4 + +8 = 26 degrees of freedom and this uses up all the information in the experiment, leaving no degrees of freedom for error, unless replication exists. Replication is generally unnecessary, however, as one typically assumes there are no three-factor interactions, and uses these 8 degrees of freedom for error estimation.

In this model we see that i = j = k = 1, 2, 3 making 27 treatments.
These treatments may be displayed as follows:

TABLE 3.37 The 3³ Design

The design can be pictorially represented by

FIGURE 3.24 A 3³ Design Schematic

Two types of fractions of 3^k designs are employed:

Box-Behken designs whose purpose is to estimate a second order model for quantitative factors (discussed earlier in section 5.3.3.6.2)

3^k-p orthogonal arrays.