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2^{K - p} Fractional Factorial Designs

Even smaller fractions for more sample efficient experiments.

Author

Affiliation

Kris Sankaran

Published

Nov. 23, 2021

DOI

Readings 8.3, 8.4, Rmarkdown

$2^{K - 1}$ designs allow us to draw conclusions about $K$ factors using only half of the runs that would be required for a full $2^{K}$ factorial. It’s possible to generalize these ideas so that smaller fractions ( $\frac{1}{4}, \frac{1}{8}, \dots$ of samples are required. When only a fraction $\frac{1}{2^{p}}$ is required, we call the resulting design a $2^{K - p}$ fractional factorial design.
To construct a $2^{K - p}$ design, we can follow a standard recipe,

First, build a full factorial design from $K - p$ of the factors. The associated full factorial is called the basic design.
We then need $p$ generating relations, which confound the remaining $p$ factors with terms from the full factorial design.
The complete defining relation for a design is the set of columns that equal $I$ , the identity column. These can be found by looking at generating relations and their products
The extended table specifies which factors should be active in each experimental run.

Here’s the recipe in action. Suppose we want a $2^{6 - 2}$ design. This studies 6 factors using 16 (not 64) runs.

First, create a full factorial on the first four factors. Note that I’m only writing contrasts for main effects.

A	B	C	D
-	-	-	-
+	-	-	-
-	+	-	-
-	-	+	-
-	-	-	+
+	+	-	-
+	-	+	-
+	-	-	+
-	+	+	-
-	+	-	+
-	-	+	+
+	+	+	-
+	+	-	+
+	-	+	+
-	+	+	+
+	+	+	+

Next, study the last two factors through the defining relations $E = ABC$ and $F = BCD$ . Our hope is that by defining them as high-order interactions, we have a chance at higher resolution.
The resulting design is found by multiplying signs across the defining columns. For example, the first entry for E is -, because $ABC = - \times - \times - = -$ .

A	B	C	D	E = ABC	F = BCD
-	-	-	-	-	-
+	-	-	-	+	-
-	+	-	-	+	+
-	-	+	-	+	+
-	-	-	+	-	+
+	+	-	-	-	+
+	-	+	-	-	+
+	-	-	+	+	+
-	+	+	-	-	-
-	+	-	+	+	-
-	-	+	+	+	-
+	+	+	-	+	-
+	+	-	+	-	-
+	-	+	+	-	-
-	+	+	+	-	+
+	+	+	+	+	+

Let’s analyze this design,

The associated complete defining relations are $I = ABCE = BCDF = ADEF$ . To see this, notice $E^2 = ABCE$ , but any term squared is just $I$ . The last relation comes from multiplying the two previous ones together.
The alias groups are complicated looking, but they can be found by simply multiplying the defining relations by each of the factors and combinations of factors. In practice, these would be found using code (see below). $\begin{align*} I&=A B C E=B C D F=A D E F \\ A&=B C E=D E F=A B C D F \\ B&=A C E=C D F=A B D E F \\ C&=A B E=B D F=A C D E F \\ D&=B C F=A E F=A B C D E \\ E&=A B C=A D F=B C D E F \\ F&=B C D=A D E=A B C E F \\ A B&=C E=A C D F=B D E F \\ A C&=B E=A B D F=C D E F \\ A D&=E F=B C D E=A B C F \\ A E&=B C=D F=A B C D E F \\ A F&=D E=B C E F=A B C D \\ B D&=C F=A C D E=A B E F \\ B F&=C D=A C E F=A B D E \\ A C D&=B D E=A B F=C E F \\ A B D&=C D E=A C F=B E F \end{align*}$
From the alias groups, we can tell that the resolution is 4. Two-way interactions are confounded with one another, but not with any main effects.

For an exercise, you can try going through this process using an alternative confounding structure: Set $E = ABCD$ , and $F = ABC$ . It’s somewhat tedious, but will build your confidence with this type of design.

Code Example

Let’s use the $2^{6 - 2}$ design that we just constructed on a dataset about injection moldingAccording to Example 8.4 in the book, this is how paper clips are made. The 6 factors are,

A: mold temperature
B: screw speed
C: holding time
D: cycle time
E: gate size
F: holding pressure

# A tibble: 6 × 7
      A     B     C     D     E     F Shrinkage
  <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>     <dbl>
1    -1    -1    -1    -1    -1    -1         6
2     1    -1    -1    -1     1    -1        10
3    -1     1    -1    -1     1     1        32
4     1     1    -1    -1    -1     1        60
5    -1    -1     1    -1     1     1         4
6     1    -1     1    -1    -1     1        15

$Visualizing a $2 ^ {6 - 2}$ design for the injection experiment. Each column is a factor, each row is a run.$

Figure 1: Visualizing a $2 ^ {6 - 2}$ design for the injection experiment. Each column is a factor, each row is a run.

We can look at the aliasing structure by inspecting the design matrix.

Let’s make a Daniel plot and look at effects.

Based on the Daniel plot, it seems that a submodel based on just A, B, and AB should be sufficient (perhaps an argument could be made for ACD and AD, though).

            Df Sum Sq Mean Sq F value   Pr(>F)    
A            1    770     770   37.15 5.38e-05 ***
B            1   5077    5077  244.90 2.39e-09 ***
A:B          1    564     564   27.21 0.000216 ***
Residuals   12    249      21                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Both mold temperature and screw speed affect shrinkage. Moreover, there is an interaction; i.e., screw speed has a different effect on shrinkage depending on the underlying mold temperature.

To evaluate the model, let’s visualize the residuals.

Notice that the residuals have lower variance at the setting $C = -1$ . Even though $C$ doesn’t have a strong effect, that might be a useful fact, in case we want to select a configuration with the optimal response and low variability around that response.

Evaluating Designs

In general, there will be several ways of constructing any $2^{p}$ design. How do we know which one to use? Here are some criteria.

Resolution: We’ve discussed this before. Remember, a design has resolution $R$ if no $p$ -factor effect is aliased with an effect with fewer than $R - p$ factors.
Aberration: This is the number of defining words of length equal to the resolution. Lower is better.

For example the example above had words $ABCE, BCDF$ and $ADEF$ and resolution 4, so aberration is 3.

Footnotes

According to Example 8.4 in the book, this is how paper clips are made.[↩]

A	B	C	D
-	-	-	-
+	-	-	-
-	+	-	-
-	-	+	-
-	-	-	+
+	+	-	-
+	-	+	-
+	-	-	+
-	+	+	-
-	+	-	+
-	-	+	+
+	+	+	-
+	+	-	+
+	-	+	+
-	+	+	+
+	+	+	+

A	B	C	D	E = ABC	F = BCD
-	-	-	-	-	-
+	-	-	-	+	-
-	+	-	-	+	+
-	-	+	-	+	+
-	-	-	+	-	+
+	+	-	-	-	+
+	-	+	-	-	+
+	-	-	+	+	+
-	+	+	-	-	-
-	+	-	+	+	-
-	-	+	+	+	-
+	+	+	-	+	-
+	+	-	+	-	-
+	-	+	+	-	-
-	+	+	+	-	+
+	+	+	+	+	+

A	B	C	D
-	-	-	-
+	-	-	-
-	+	-	-
-	-	+	-
-	-	-	+
+	+	-	-
+	-	+	-
+	-	-	+
-	+	+	-
-	+	-	+
-	-	+	+
+	+	+	-
+	+	-	+
+	-	+	+
-	+	+	+
+	+	+	+

A	B	C	D	E = ABC	F = BCD
-	-	-	-	-	-
+	-	-	-	+	-
-	+	-	-	+	+
-	-	+	-	+	+
-	-	-	+	-	+
+	+	-	-	-	+
+	-	+	-	-	+
+	-	-	+	+	+
-	+	+	-	-	-
-	+	-	+	+	-
-	-	+	+	+	-
+	+	+	-	+	-
+	+	-	+	-	-
+	-	+	+	-	-
-	+	+	+	-	+
+	+	+	+	+	+

A	B	C	D
-	-	-	-
+	-	-	-
-	+	-	-
-	-	+	-
-	-	-	+
+	+	-	-
+	-	+	-
+	-	-	+
-	+	+	-
-	+	-	+
-	-	+	+
+	+	+	-
+	+	-	+
+	-	+	+
-	+	+	+
+	+	+	+

A	B	C	D	E = ABC	F = BCD
-	-	-	-	-	-
+	-	-	-	+	-
-	+	-	-	+	+
-	-	+	-	+	+
-	-	-	+	-	+
+	+	-	-	-	+
+	-	+	-	-	+
+	-	-	+	+	+
-	+	+	-	-	-
-	+	-	+	+	-
-	-	+	+	+	-
+	+	+	-	+	-
+	+	-	+	-	-
+	-	+	+	-	-
-	+	+	+	-	+
+	+	+	+	+	+