Even smaller fractions for more sample efficient experiments.
\(2^{K - 1}\) 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,
A | B | C | D |
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- | - | - | - |
+ | - | - | - |
- | + | - | - |
- | - | + | - |
- | - | - | + |
+ | + | - | - |
+ | - | + | - |
+ | - | - | + |
- | + | + | - |
- | + | - | + |
- | - | + | + |
+ | + | + | - |
+ | + | - | + |
+ | - | + | + |
- | + | + | + |
+ | + | + | + |
-
, because \(ABC = - \times - \times - = -\).A | B | C | D | E = ABC | F = BCD |
---|---|---|---|---|---|
- | - | - | - | - | - |
+ | - | - | - | + | - |
- | + | - | - | + | + |
- | - | + | - | + | + |
- | - | - | + | - | + |
+ | + | - | - | - | + |
+ | - | + | - | - | + |
+ | - | - | + | + | + |
- | + | + | - | - | - |
- | + | - | + | + | - |
- | - | + | + | + | - |
+ | + | + | - | + | - |
+ | + | - | + | - | - |
+ | - | + | + | - | - |
- | + | + | + | - | + |
+ | + | + | + | + | + |
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.
# 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
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
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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.
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.
For example the example above had words \(ABCE, BCDF\) and \(ADEF\) and resolution 4, so aberration is 3.
According to Example 8.4 in the book, this is how paper clips are made.↩︎