12 Fractional factorial designs

A \(2^k\) full factorial requires \(2^k\) runs. Full factorials are seldom used in practice for large k (k>=7). For economic reasons fractional factorial designs, which consist of a fraction of full factorial designs are used. There are criteria to choose “optimal” fractions.

12.1 Example - Effect of five factors on six properties of film in eight runs

The following example is taken from Box, Hunter, and Hunter (2005).

Five factors were studied in 8 runs. The factors were:

  • Catalyst concentration (A)
  • Amount of additive (B)
  • Amounts of three emulsifiers (C, D, E)

Polymer solutions were prepared and spread as a film on a microscope slide. Six different responses were recorded.

run A B C D E y1 y2 y3 y4 y5 y6
1 -1 -1 -1 1 -1 no no yes no slightly yes
2 1 -1 -1 1 1 no yes yes yes slightly yes
3 -1 1 -1 -1 1 no no no yes no no
4 1 1 -1 -1 -1 no yes no no no no
5 -1 -1 1 -1 1 yes no no yes no slightly
6 1 -1 1 -1 -1 yes yes no no no no
7 -1 1 1 1 -1 yes no yes no slightly yes
8 1 1 1 1 1 yes yes yes yes slightly yes
  • The eight run design was constructed beginning with a standard table of signs for a \(2^3\) design in the factors A, B, C.

  • The column of signs associated with the BC interaction was used to accommodate factor D, the ABC interaction column was used for factor E.

  • A full factorial for the five factors A, B, C, D, E would have needed \(2^5 = 32\) runs.

  • Only 1/4 were run. This design is called a quarter fraction of the full \(2^5\) or a \(2^{5-2}\) design (a two to the five minus two design). In general a \(2^{k-p}\) design is a \(\frac{1}{2^p}\) fraction of a \(2^k\) design using \(2^{k-p}\) runs.

12.2 Effect Aliasing and Design Resolution

A chemist in an industrial development lab was trying to formulate a household liquid product using a new process. The liquid had good properties but was unstable. The chemist wanted to synthesize the product in hope of hitting conditions that would give stability, but without success. The chemist identified four important influences: A (acid concentration), B (catalyst concentration), C (temperature), D (monomer concentration). His 8 run fractional factorial design is shown below.

test A B C D y
1 -1 -1 -1 -1 20
2 1 -1 -1 1 14
3 -1 1 -1 1 17
4 1 1 -1 -1 10
5 -1 -1 1 1 19
6 1 -1 1 -1 13
7 -1 1 1 -1 14
8 1 1 1 1 10

The signs of the ABC interaction is used to accommodate factor D. The tests were run in random order. He wanted to achieve a stability value of at least 25.

The factorial effects and Normal, half-Normal, and Lenth plots are below.

(Intercept)           A           B           C           D         A:B 
      29.25       -5.75       -3.75       -1.25        0.75        0.25 
        A:C         B:C         A:D         B:D         C:D       A:B:C 
       0.75       -0.25          NA          NA          NA          NA 
      A:B:D       A:C:D       B:C:D     A:B:C:D 
         NA          NA          NA          NA 

    alpha       PSE        ME       SME 
 0.050000  1.125000  4.234638 10.134346 

Even though the stability never reached the desired level of 25, two important factors, A and B, were identified. This Normal and half-Normal plots indicate the importance of these factors, although factor B is not significant according to the Lenth plot.

What information could have been obtained if a full \(2^5\) design had been used?

Factors Number of effects
Main 5
2-factor 10
3-factor 10
4-factor 5
5-factor 1

There are 31 degrees of freedom in a 32 run design. But, are 16 used for estimating three factor interactions or higher. Is it practical to commit half the degrees of freedom to estimate such effects? According to effect hierarchy principle three-factor and higher not usually important. Thus, using full factorial wasteful. It’s more economical to use a fraction of full factorial design that allows lower order effects to be estimated.

Consider a design that studies five factors in 16 run. A half fraction of a \(2^5\) or \(2^{5-1}\).

Run B C D E Q
1 -1 1 1 -1 -1
2 1 1 1 1 -1
3 -1 -1 1 1 -1
4 1 -1 1 -1 -1
5 -1 1 -1 1 -1
6 1 1 -1 -1 -1
7 -1 -1 -1 -1 -1
8 1 -1 -1 1 -1
9 -1 1 1 -1 1
10 1 1 1 1 1
11 -1 -1 1 1 1
12 1 -1 1 -1 1
13 -1 1 -1 1 1
14 1 1 -1 -1 1
15 -1 -1 -1 -1 1
16 1 -1 -1 1 1

The factor E is assigned to the column BCD. But, the column for E is used to estimate the main effect of E and also for BCD. So, this design cannot distinguish between E and BCD. The main factor E is said to be aliased with the BCD interaction.

This aliasing relation is denoted by \[E = BCD \thinspace or \thinspace I = BCDE,\] where \(I\) denotes the column of all +’s.

This uses same mathematical definition as the confounding of a block effect with a factorial effect. Aliasing of the effects is a price one must pay for choosing a smaller design.

The \(2^{5-1}\) design has only 15 degrees of freedom for estimating factorial effects, it cannot estimate all 31 factorial effects among the factors B, C, D, E, Q.

The equation \(I = BCDE\) is called the defining relation of the \(2^{5-1}\) design. The design is said to have resolution IV because the defining relation consists of the “word” BCDE, which has “length” 4.

Multiplying both sides of \(I = BCDE\) by column B \[B = B \times I = B \times BCDE = CDE\], the relation \(B = CDE\) is obtained. B is aliased with the CDE interaction. Following the same method all 15 aliasing relations can be obtained.

\[B = CDE, C = BDE, D = BCE, E = BCD, \\ BC = DE, BD = CE, BE = CD, \\ Q = BCDEQ, BQ = CDEQ, CQ = BDEQ, DQ = BCEQ, \\ EQ = BCDQ, BCQ = DEQ, BDQ = CEQ, BEQ = CDQ\]

Each of the four main effects \(B, C, D, E\) is respectively aliased with \(CDE, BDE, BCE,BCD\). Therefore, the main effects of \(B,C,D,E\) are estimable only if the aforementioned three-factor interactions are negligible. The other factorial effects have analogous aliasing properties.

12.3 Example - Leaf Spring Experiment

The following example is from Wu and Hamada (2009). An experiment to improve a heat treatment process on truck leaf springs. The heat treatment that forms the camber in leaf springs consists of heating in a high temperature furnace, processing by forming a machine , and quenching in an oil bath. The free height of an unloaded spring has a target value around 8in. The goal of the experiment is to make the variation about the target as small as possible.

Five factors were studied in this \(2^{5-1}\) design.

Factor Level
B. Temperature 1840 (-), 1880 (+)
C. Heating time 23 (-), 25 (+)
D. Transfer time 10 (-), 12 (+)
E. Hold down time 2 (-), 3 (+)
Q. Quench oil temperature 130-150 (-), 150-170 (+)
B C D E Q y
-1 1 1 -1 -1 7.7900
1 1 1 1 -1 8.0700
-1 -1 1 1 -1 7.5200
1 -1 1 -1 -1 7.6333
-1 1 -1 1 -1 7.9400
1 1 -1 -1 -1 7.9467
-1 -1 -1 -1 -1 7.5400
1 -1 -1 1 -1 7.6867
-1 1 1 -1 1 7.2900
1 1 1 1 1 7.7333
-1 -1 1 1 1 7.5200
1 -1 1 -1 1 7.6467
-1 1 -1 1 1 7.4000
1 1 -1 -1 1 7.6233
-1 -1 -1 -1 1 7.2033
1 -1 -1 1 1 7.6333

The factorial effects are estimated as before.

(Intercept)           B           C           D           E           Q 
      15.27        0.22        0.18        0.03        0.10       -0.26 
        B:C         B:D         C:D         B:E         C:E         D:E 
       0.02        0.02       -0.04          NA          NA          NA 
        B:Q         C:Q         D:Q         E:Q       B:C:D       B:C:E 
       0.08       -0.17        0.05        0.03          NA          NA 
      B:D:E       C:D:E       B:C:Q       B:D:Q       C:D:Q       B:E:Q 
         NA          NA        0.01       -0.04       -0.05          NA 
      C:E:Q       D:E:Q     B:C:D:E     B:C:D:Q     B:C:E:Q     B:D:E:Q 
         NA          NA          NA          NA          NA          NA 
    C:D:E:Q   B:C:D:E:Q 
         NA          NA 

Notice that the factorial effects are missing for effects that are aliased. The Normal, half-Normal, and Lenth plots are below.

    alpha       PSE        ME       SME 
0.0500000 0.0606000 0.1557773 0.3162503 

12.4 Example - Baking Cookies

Consider a factorial design to study the effects of the amounts of three factors on the taste of chocolate chip cookies.

Factor Amount
Butter 10g (-1), 15g (+1)
Sugar 1/2 cup (-1), 3/4 cup (+1)
Baking powder 1/2 teaspoon (-), 1 teaspoon (+)

Taste will be measured on a scale of 1 (poor) to 10 (excellent). A full factorial will require \(2^3 = 8\) runs.

The 8 runs of the full factorial design tell the experimenter how to set the levels of the different ingredients (factors).

Run butter sugar powder
1 -1 -1 -1
2 1 -1 -1
3 -1 1 -1
4 1 1 -1
5 -1 -1 1
6 1 -1 1
7 -1 1 1
8 1 1 1

In the first run the experimenter will bake chocolate chip cookies with 10g butter, 1/2 cup sugar, and 1/2 teaspoon of baking powder; the second run will use 15g butter, 1/2 cup sugar, and 1/2 teaspoon of baking powder; etc.

But, the experimenter decides to also study baking time on taste, but can’t afford to do more than 8 runs since each run requires a different batch of cookie dough. He wants to test if a baking time of 12 minutes versus 16 minutes has an impact on taste

So, he decides to use the three factor interaction between butter, sugar, and powder to assign if baking time will be 12 minutes (-1) or 16 minutes (+1) in each of the 8 runs.

Run butter sugar powder baking time
1 -1 -1 -1 -1
2 1 -1 -1 1
3 -1 1 -1 1
4 1 1 -1 -1
5 -1 -1 1 1
6 1 -1 1 -1
7 -1 1 1 -1
8 1 1 1 1

In this factorial design with four factors in 8 runs the experimenter will bake the cookies with 10g butter, 1/2 cup sugar, 1/2 teaspoon of baking powder, and baking time 12 minutes in the first run; in the second run use 15g butter, 1/2 cup sugar, and 1/2 teaspoon of baking powder, and 16 minutes baking time; etc.

Let \(A\)=butter, \(B\)=sugar, \(C\)=powder, \(D\)=baking time. But, \[D = ABC\] since he used the three factor interaction to assign baking times to each run. In other words \(D\) is aliased with \(ABC\).

This type of design is called a \(2^{4-1}\) fractional factorial design. Instead of using a full factorial or \(2^4 = 16\) runs to study 4 factors we are using \(\frac{1}{2}2^4 = 8\) runs.

The aliasing relation \[D = ABC \Rightarrow I = ABCD,\] where \(I\) is the column of \(+1\)s.

The aliasing relation also means that other factors in the design have aliases. We can find these aliases by multiplying \(I = ABCD\) by all the possible main and interaction effects.

\[A = BCD, B = ACD, C = ABD, D = ABC, AB = CD, AC = BD, BC = AD\]

All the main effects are aliased with three factor interactions, and two factor interactions with two factor interactions.

Suppose that the experimental runs were conducted and the following results were obtained.

Run butter sugar powder baking time taste (y)
1 -1 -1 -1 -1 9
2 1 -1 -1 1 4
3 -1 1 -1 1 7
4 1 1 -1 -1 1
5 -1 -1 1 1 2
6 1 -1 1 -1 5
7 -1 1 1 -1 3
8 1 1 1 1 10

The factorial effect of baking time (\(D\)) is really the effect of \(D+ABC\). In other words the effects of \(D\) and \(ABC\) are confounded. They cannot be separately estimated which is why \(ABC\) is called an alias of \(D\).

This means that effect of \(D\)

\[\frac{1}{4}\left(-y_1+y_2+y_3-y_4+y_5-y_6-y_7+y_8\right)=-2.5\]

is really the effect of \(D+ABC\) or baking time plus the interaction of butter, sugar, and baking powder. In order for this to be equal to the effect of baking time (\(D\)) we must assume the three-way interaction (\(ABC\)) is small enough to be ignored (i.e., the factorial estimate of \(ABC\) is close to 0).

If the experimenter were to use a full factorial then he would require \(2^4 = 16\) different batches of cookies. In a full \(2^4\) design he would be estimating 4 main effects, 6 two-way interactions, 4 three-way interactions, and 1 four-way interaction. If we assume that we can ignore three-factor and higher order interactions then a 16 run design is being used to estimate then a 16 run design is being used to estimate 10 effects. Fractional factorials use these redundancies by arranging that lower order effects are confounded with higher order interactions that are assumed negligible.

12.5 Questions

  1. Consider the table of contrasts from a \(2^{k-p}\) design. The four factors investigated are A,B,C,D.
##    C  B  A  D
## 1 -1 -1 -1 -1
## 2  1 -1 -1  1
## 3 -1  1 -1  1
## 4  1  1 -1 -1
## 5 -1 -1  1  1
## 6  1 -1  1 -1
## 7 -1  1  1 -1
## 8  1  1  1  1
  1. What are the values of \(k\) and \(p\)?
  2. What is the defining relation? What is the resolution of this design?
  3. What are the aliasing relations?

12.6 Answers to Questions

    1. There are 4 factors so \(k=4\) in 8 runs so, \(p=1\).

    2. By inspection of the table we can see that \(D=ABC\). The defining relation is \(I=ABCD\). The defining relation has 4 letters so the resolution is IV.

    3. The aliasing relations are: \(D=ABC, A=BCD,B=ACD,C=ABD, AB=CD,AC=BD, AD=BC\).

References

Box, George EP, J Stuart Hunter, and William Gordon Hunter. 2005. Statistics for Experimenters: Design, Innovation, and Discovery. Vol. 2. Wiley-Interscience New York.