Video 1. What is Fractional Factorial Design of Experiments and How to Use it Like an Expert.

Fractional Factorial Design runs only a fraction of the full factorial design to screen the most important variables/factors those affect the response the most. For an example, a 2⁷ design of experiment with seven variables of two levels for each factor will require 128 unique experiments to complete one full replication of the design. Running 128 experiments will be unnecessary and wasteful. Therefore, a fraction of the 128 experiments are run first to understand whether all the seven variables are affecting the response or not. For an example, a 1/16^th of the 128 experiments results in only eight experiments, which has seven degrees of freedom, can be utilized to get enough information for the seven variables to screen out the most important ones. A stronger justification for the fractional factorial design of experiments has been observed through the Principal of Factor Sparsity.

1.1. The Principal of Factor sparsity, Pareto Principal or the 80/20 Rule

Italian economist and sociologist Vilfredo Pareto (1848-1923) discovered that 80% wealth in Italy belonged to about 20% of the population. There are many situations follow the same principal. Such as roughly 80% effects come from 20% causes. Joseph Juran (1904-2008) suggested this principal as the Pareto Principal after the name of Vilfredo Pareto. In reality, only a few factors produce the significant effect in an experiment that involves many factors. In the design and analysis of experiments, the Pareto Principal is known as the Principal of Factor Sparsity ((Box, Hunter et al. 2005; Montgomery 2013)).

2. Primary Basics of the 2^k Fractional Factorial Design of Experiments: The One-Half Fraction, 2^k-1 Design

An introduction to the one-half fraction and its justification is provided in Video 2.

Video 2. Introduction to 2K Fractional Factorial Design of Experiments DOE and the One-Half Fraction Explained

Video 3 demonstrates the details on the primary basics, including the definitions and explanations for associated terms and the systematic process of developing the fractional factorial design.

Video 3. Introduction to the Primary Basics of the Fractional Factorial Design of Experiments DOE Explained

Table 1 shows the one full replication of the 2³ design. Assume that we just want to screen the factors or to see the importance of the three variables first before we invest more time into them. Three degrees of freedom are required to get information on three variables/factors. Three degrees of freedom can be obtained from four experiments. Therefore, half of the experiments, four out of the eight can be systematically conducted to determine the effects of the three factors, A, B, and C.

Table 1. The 2³ Full Factorial Design of Experiments

As the higher-order interactions are considered the least important, the ABC effect can be sacrificed. To sacrifice ABC, either all the positive (or highs) treatment combinations (a, b, c, and abc) or all the negative (or lows) treatment combinations ((1), ab, ac, bc) of the ABC effect can only be run (Table 2). By dividing the experiments into two, the one fraction is known as the One-Half Fraction of the 2^k design (2³ design in this example).

Table 2. The 2³ Factorial Design of Experiments with two blocks

Video 4 shows the details on the two one-half fractions of the 2³ design.

Video 4. Introduction to Basic One-Half Fractional Factorial 2k Design of Experiments DOE Details Explained

Assume that the positive fraction is used for the fractional design shown in Table 3. This one-half fraction with all positive treatment combinations is known as the principal fraction (Table 3), while the other half with the negative treatment combinations is known as the alternative or complementary fraction. The ABC interaction for the 2³ design is known as the design generator, simply because the word ABC is generating this design. The Generator is also known as word, probably because of it is used very often in fractional designs and easier to say. Moreover, the higher order interaction terms also look like a word.

Table 3. The Principal One-Half Fraction of the 2³ Factorial Design of Experiments

Table 4. The Effects of the One-Half Fraction of the 2³ Factorial Design

Table 4 shows the signs for the treatment combination of a factor stays the exact same for the multiplications between the generator ABC and the factor A, B, and C. Therefore, the word ABC is called the identity. Therefore, a relationship for the word ABC or the generator of the design can be developed as in Equation 1, which is known as the defining relation.

Equation 1

The rest of the relationships can be developed using the defining relation of the design such as in Equation 2. Any relationship can be developed by multiplying both sides of the defining relation. For an example, if the relationship for the factor A is desired, A is multiplied with both sides of the defining relation, results in A = BC (Equation 2).

Equation 2

Similarly, the relationships for all other effects (e.g. B =AC, and C=AB in this 2³ design) can be developed. As A and BC are related and indistinguishable from each other for the one-half of the 2³ design, A and BC is aliased with each other. Similarly, B is aliased with AC, and C is aliased with AB. The word “alias” is used to call the relationship between effects or when they are indistinguishable from each other.

Table 4 shows the main, the interaction effects and their treatment combinations. An effect can be easily estimated by simply multiplying the factor/effect column with the treatment combination column. The effect is estimated by the half of the contrast or the linear combinations (Equation 3).

Equation 3

Similarly, the interaction effects can also be estimated by multiplying the interaction column with the treatment combination column, results in Equation 4.

Equation 4

Again, the Equation 3 & Equation 4 show that [A] =[BC], [B] = [AC], and [C]=[AB]. Therefore, the A effect is indistinguishable from the BC effect and so on.

The other one-fraction (the alternative or complementary fraction) of the design is shown in Table 5. While, the defining relation I = ABC used for the principle fraction, I = -ABC defining relation for the complementary fraction produces the exact same results (Table 6).

Table 5. The Complementary One-Half Fraction of the 2³ Factorial Design

Table 6. The Effects of the Complementary One-Half Fraction of the 2³ Factorial Design

And the relationships for each effect can be developed from this defining relation as in Equation 5.

Equation 5

Similarly, the relationships for all other effects (e.g. B =-AC, and C=-AB in this 2³ design) can be developed.

Table 6 shows the main and the interaction effects and their treatment combinations for the complementary fraction of the one-half fraction. An effect can be easily estimated by simply multiplying the effect column with the treatment combination column. The effect is estimated by the half of the contrast or the linear combinations (Equation 6).

Equation 6

Similarly, the interaction effects can also be estimated by multiplying the interaction column with the treatment combination column, results in Equation 7.

Equation 7

As the principal and complementary one-half fractions complete the full replication, the full estimate of the main effect can be calculated using the Equation 3 & Equation 6. For an example,

Equation 8

Which can be verified from the 2³ full factorial design in Table 1.

Similarly, the interaction effects can be estimated from Equation 4 & Equation 7.

Equation 9

Which can be verified from the 2³ full factorial design in Table 1.

2.1 Graphical Representation of the One-Half Fraction

Figure 1 shows the graphical representation of the two one-half fractions of the 2³ design.

Figure 1. The Two One-Half Fractions of the 2³ Design

Video 5 demonstrates a simple process of designing a one-half fraction of the 2³ design.

Video 5. How to Design a One-Half Fractional Factorial 2k Design of Experiments Easiest Method in MS Excel

3. Design Resolution

Video 6 explains the design resolution in fractional factorial design of experiments.

Video 6. What is Design Resolution in 2k Fractional Factorial Design of Experiments DOE Explained Example

For the one-half fraction design in Table 7, the number of letters in the generator (or the word or the defining relation) of the design determine the resolution number of the design. For an example, for the one-half fraction design, the design is called a “resolution of three” if a three-letter generator, ABC is used in developing the design. For the one-half fraction design, if a five-letter generator word is used, the design is called a “resolution of five.”

Table 7. One-Half Fraction Design Generator, Resolution, Alias Structures, the resolution

The alias structure for the word ABC is A=BC, B = AC, and C = AB. Therefore, the main effect is aliased with the two-factor interaction in a resolution III design, and no main effects are aliased with any other main effect. The aliased structure for the word ABCD is A = BCD, B = ACD, C = ABD, D = ABD, and AB = CD, AC = BD, and AD = BC. Therefore, the main effects are aliased with the three-factors interactions and the two-factors interactions are aliased with the other two-factors interactions in a resolution IV design. And, no main effects are aliased with any other main effect. Therefore, the general definition for the resolution for any fractional design (e.g. half, quarter, 1/8^th, 1/16^th, etc.) can be developed as in Table 8. To determine the correct resolution of a design, the “must not aliased with” column shall be primarily used. Resolution can be thought of as the screen resolution. Higher resolution will produce better output but cost more. Although the higher resolution of the design has been suggested (Box, Hunter et al. 2005; Montgomery 2013), lower resolution would work just fine for screening variables. As the fractional factorial design is primarily utilized for screening factors/variables, resolution of III will make more sense than any higher resolution design in screening purpose to reduce the total number of initial experiments. Nevertheless, when possible, higher should be preferred over lower resolution without increasing the number of experiments in the screening process.

Table 8. Determine the Design Resolution Rules

4. One-Quarter Fraction 2^k-2 Design of Experiment

2^k design with four factors/variables requires 16 experiments for the full replication. Therefore, quarter of the 16 experiments will results in only four experiments producing only three degrees of freedoms. Therefore, quarter fraction design is applicable for five or more factors/variables. In the one-half fraction 2^k-1 design, only one generator word or the defining relation was required to develop the design. Therefore, in one-quarter fraction 2^k-2 design, two generator word (or the defining relation) is required. Therefore, in a 2^k-p fractional design, p number of defining relation is required. A one-quarter fraction of five factors, design is provided in Table 9 using the defining relation I = ABD and I = BCE.

Table 9. One-Quarter Fraction Design of Resolution 3

The Detail Explanations for the one-quarter design is provided in Video 7.

Video 7. Introduction to Basic One Quarter Fractional Factorial 2k Design of Experiments Details Explained

Video 8 shows the process of design a one-quarter fraction using MS Excel.

5. Alias structure

To correctly develop the alias structure of any design, follow the steps below. Video 10 demonstrates the following steps to develop the alias structure of a design systematically.

1. Calculate the total number of effects of the design (Video 9). Total number of effects in a factorial design is 2^k -1.

2. Write all factorial effects (Video 9).

3. Multiply the main effects with all the defining relations, including their generalized interactions and place them side by side with an equal sign in-between (Table 10).

4. Multiply the remaining two-factor interactions with all the defining relations, including their generalized in….follow step 3 (Table 10).

Keep doing it until all the effects are in the table (e.g. Table 10). In the 2^5-2 quarter design, four columns multiplied by the eight rows of the table minus the Identity, I in the first cell of the Table 10 equal to all 31 effects.

Video 9. How to Calculate and Write All Effects in 2k Factorial Design of Experiments DOE Systematic Flawless

Video 10. How to Write Alias Structure in 2K Fractional Factorial Design of Experiments DOE Systematic

The alias structure for this one-quarter design, can be found in Table 10.

Table 10. Alias structure of one-quarter design,

5.1 Why the full alias structure?

Image in a musical band, the vocal and drummer are needed to coordinate/interaction to produce a better-quality music. While the drummer can produce some appealing music alone, the vocal can only sing if the drummer is in there. Therefore, the data analysis results may show that the main effect of the drummer factor is significant while the main of the vocal factor is insignificant with respect to the audience rating. However, their interactions are observed to be significant too. In this scenario if the two-factor interaction is aliased with the main effects, it will be impossible to independently know whether the interaction is significant or not. Without knowing all the alias structures of the design, the conclusions maybe misleading. In this hypothetical music band, the “alias” can be thought of as if the drummer and the vocal are the same person. When the drummer (factor A) is aliased with the drummer and the vocal, they are the same person.

6. One-Eighth Fraction 2^k-3 Design of Experiment

One-eighth fraction will only be applicable for the 6 or more factors screening experiments. Video 11 shows the design process for the one-eighth fraction 2^6-3 design.

Video 11. How to Design a One-Eighth Fractional Factorial 2k Design of of Experiments DOE Easiest Method in MS Excel

Simply, generate D from the interaction of AB, E from the interaction of AC and F from the interaction of BC. Therefore, the defining relation here is I = ABD = ACE = BCF. The alias structure for the design can be found Table 12.

Table 11. One-Eight Fraction design

Table 12. Alias structure of One-Eight Fraction design

7. Fractional Factorial Design for the Lowest Numbers of Runs Possible Upto 15 Variables

The fractional factorial designs with the lowest number of possible runs from 3 to 15 factors/variables are provided, including their alias structure in the MS Excel File 1. Readers are welcome to download the designs and use it.

MS Excel File 1. Fractional Factorial Design for the Lowest Numbers of Runs Possible from 3 to 15 Factors/Variables with The Alias Structure

8. Analysis Example

The performance of a student depends on so many factors, including study (A), exercise (B), nutrition (C), party (you know!) (D), instructor (E), program (F), University (G), family life (H) and work life (J). An education researcher needs a total of 512 distinctly different students to complete a full factorial design of experiments with 9 factors/variables. Because it will be very difficult to get experimental units with the specific characteristics, including all 9 combinations of the factors, he/she wants to reduce the number of factors as much as possible. The researcher has decided to run 2^9-5; 1/32 fraction of 9 factors in 16 runs. After a couple of months of search, the researcher has found 16 students with the treatment combination matches as described in the data table below.

She has utilized the following design generator to develop the experiment

E=ABC F=BCD G=ACD H=ABD J=ABCD

The alias structure of the design can be found in Table 13.

Table 13. Alias structure of 1/32 Fraction design

Table 14. Data of the 1/32 Fraction design

Video 12 demonstrations the process of fractional factorial data analysis using Minitab.

Video 12. Fractional Factorial DOE Data Analysis Example Minitab

8.1 Fractional Factorial Analysis Step #1. Run the Analysis using only the main effects in the model to screen out all the insignificant variables.

The analysis output is provided in Table 15. Factor A and B are observed to significant with respect to the GPA.

Table 15. Fractional Factorial Analysis Step #1. Factor Screening Step

Table 15. Fractional Factorial Analysis Step #1. Factor Screening Step

*The analysis was performed using the Minitab version 19

8.2 Step #2. Rerun the analysis using the full model for the significant variables only, after screening out the insignificant variables/factors.

The analysis output is provided in Table 16. The interaction AB is observed to be significant.

Table 16. Fractional Factorial Analysis Step #2. Rerun the Analysis with Important Factors

8.3 Step #3. Run the post-hoc analysis

As the interaction is observed to be significant, the main effects are not omitted from the post-hoc analysis. The post-hoc analysis for the interaction is provided in Table 17.

Table 17. Fractional Factorial Analysis Step #3. Tukey Method Post-hoc Analysis

8.4 Step #4. Draw Conclusions.

While the combination of both the high-level of factor A (study) and the high-level factor B (exercise) are observed to be the best with respect to the response (GPA), only exercise does not produce a better response (GPA) if the study level is low (Table 17).

9. Plackett-Burman Design

Video 13 and Video 14 demonstrate the design of the Plackett-Burman fractional factorial design using MS Excel and Minitab, respectively.

Video 13. Plackett-Burman Fractional Factorial Design of Experiments DOE Using MS Excel Easiest Way Explained

Video 14. Plackett Burman Fractional Factorial Design of Experiments DOE Using Minitab Easiest Method Explained

The Plackett-Burman Fractional Factorial Design was developed in 1946 for screening a long list of variables/factors (Plackett and Burman 1946). The design is only of resolution of three. Therefore, it uses the lowest number of experiments. In factor, for many designs, it uses lower number of experiments than the regular fractional factorial design of experiments without losing much of the advantages of the regular fractional factorial design, such as the projectivity property of the regular fractional factorial design. The Plackett-Burman design uses a simple algorithm to generate the design for any number of factors with the lowest possible runs Table 18.

Table 18. Plackett-Burman Design Generator (Plackett and Burman 1946)

The following steps can be taken to generate the design using the Plackett-Burman Generator. However, the Video 13 will be easier to follow to generate the design.

1. Copy the relevant design generator below the first factor 1.
2. The last generator number (zero (0) for this example) is the first number for the second factor
3. The first number of the first factor is the second number of the second factor
4. The second number of the first factor is the third number of the second factor
5. The last standard order # 8 is the low-level of all factors used in the study
6. Rest of the factors can be design following the same algorithm.

MS Excel File 2 provides the Placket-Burman Design up to 47 variables. Readers are welcome to download and use them.

MS Excel File 2. Table 18. Plackett-Burman Fractional Factorial Design to 47 Variables