In the Screening Design platform, the Design List provides the following types of designs:
• Two-Level Regular Fractional Factorial
A full factorial design has runs for all combinations of the levels of the factors. The sample size is the product of the levels of the factors. For two-level designs, this is 2k where k is the number of factors.
Full factorial designs are orthogonal for all effects. It follows that estimates of the effects are uncorrelated. Also, if you remove an effect from the analysis, the values of the other estimates do not change. Their p-values change slightly, because the estimate of the error variance and the degrees of freedom are different.
Full factorial designs allow the estimation of interactions of all orders up to the number of factors. However, most empirical modeling involves only first- or second-order approximations to the true functional relationship between the factors and the responses. From this perspective, full factorial designs are an inefficient use of experimental runs.
A regular fractional factorial design also has a sample size that is a power of two. For two-level designs, if k is the number of factors, the number of runs in a regular fractional factorial design is 2k – p where p < k. A 2k – p fractional factorial design is a 2–p fraction of the k-factor full factorial design. Like full factorial designs, regular fractional factorial designs are orthogonal.
A full factorial design for k factors provides estimates of all interaction effects up to degree k. But because experimental runs are typically expensive, smaller designs are preferred. In a smaller design, some of the higher-order effects are confounded with other effects, meaning that the effects cannot be distinguished from each other. Although a linear combination of the confounded effects is estimable, it is not possible to attribute the variation to a specific effect or effects.
In fact, fractional factorials are designed by deciding in advance which interaction effects are confounded with other interaction effects. Experimenters are usually not concerned with interactions involving more than two factors. Three-way and higher-order interaction effects are often assumed to be negligible.
Plackett-Burman designs are an alternative to regular fractional factorials for screening. The number of runs in a Plackett-Burman design is a multiple of four rather than a power of two. There are no two-level fractional factorial designs with run sizes between 16 and 32. However, there are 20-run, 24-run, and 28-run Plackett-Burman designs.
In a Plackett-Burman design, main effects are orthogonal and two-factor interactions are only partially confounded with main effects. By contrast, in a regular Resolution 3 fractional factorial design, some two-factor interactions are indistinguishable from main effects. Plackett-Burman designs are useful when you are interested in detecting large main effects among many factors and where interactions are considered negligible.
For most designs that involve categorical or discrete numeric factors at three or more levels, standard designs do not exist. In such cases, the screening platform generates main effects screening designs. These designs are orthogonal or near orthogonal for main effects.
For cases where standard mixed-level designs exist, the possible designs are given in the Design List. The Design List provides fractional factorial designs for pure three-level factorials with up to 13 factors. For mixed two-level and three-level designs, the Design list includes the complete factorials and the orthogonal-array designs listed in Table 10.1.
If your number of factors does not exceed the number for a design listed in the table, you can adapt that design by using an appropriate subset of its columns.
|
|
Number of Factors |
|
|---|---|---|
|
Design |
Two–Level |
Three–Level |
|
L18 John and L18 Taguchi |
1 |
7 |
|
L18 Chakravarty |
3 |
6 |
|
L18 Hunter |
8 |
4 |
|
L36 Taguchi |
11 |
12 |
Note: By default, Cotter designs are not included in the Design List. To include Cotter designs, deselect Suppress Cotter Designs in the Screening Design red triangle menu. To always show Cotter designs, select File > Preferences > Platforms > DOE and deselect Suppress Cotter Designs.
Cotter designs are useful when you must test many factors, some of which might interact, in a very small number of runs. Cotter designs rely on the principle of effect sparsity. They assume that the sum of effects shows an effect if one of the components of the sum has an active effect. The drawback is that several active effects with mixed signs might sum to near zero, thereby failing to signal an effect. Because of this false-negative risk, many statisticians discourage their use.
For k factors, a Cotter design has 2k + 2 runs. The design structure is similar to the “vary one factor at a time” approach.
The Cotter design is constructed as follows:
• A run is defined with all factors set to their high level.
• For each of the next k runs, one factor in turn is set at its low level and the others high.
• The next run sets all factors at their low level.
• For each of the next k runs, one factor in turn is set at its high level and the others low.
• The runs are randomized.
When you construct a Cotter design, the design data table includes a set of columns to use as regressors. The column names are of the form <factor name> Odd and <factor name> Even. They are constructed by summing the odd-order and even-order interaction terms, respectively, that contain the given factor.
For example, suppose that there are three factors, A, B, and C. Table 10.2 shows how the values in the regressor columns are calculated.
|
Effects Summed for Odd and Even Regressor Columns |
|
|
AOdd = A + ABC |
AEven = AB + AC |
|
BOdd = B + ABC |
BEven = AB + BC |
|
COdd = C + ABC |
CEven = BC + AC |
The Odd and Even columns define an orthogonal transformation. For this reason, tests for the parameters of the odd and even columns are equivalent to testing the combinations on the original effects.