2^{K - p} Fractional Factorial Designs

Even smaller fractions for more sample efficient experiments.

Kris Sankaran true
11-23-2021

Readings 8.3, 8.4, Rmarkdown

  1. \(2^{K - 1}\) 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.

  2. To construct a \(2^{K - p}\) design, we can follow a standard recipe,

  1. Here’s the recipe in action. Suppose we want a \(2^{6 - 2}\) design. This studies 6 factors using 16 (not 64) runs.
A B C D
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A B C D E = ABC F = BCD
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  1. Let’s analyze this design,
  1. 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

  1. Let’s use the \(2^{6 - 2}\) design that we just constructed on a dataset about injection molding1 The 6 factors are,
# 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.

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

  1. 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.

  1. 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

  1. 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.

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


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