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2K Factorial Design of Experiments

Learning Outcomes

After successfully completing the Module 5 2K Factorial Design of Experiments, students will be able to

  1. Explain the 2K design and analysis of experiments
  2. Develop the data layout, structure, and the coding system of the factor levels for a 22 design
  3. Graphically represent the 22 design
  4. Develop formulas for the contrast, effect, estimate, sum of square, and the ANOVA table for the 22 design
  5. Design a 2K experiment, showing up to 8 variables, using MS Excel
  6. Show the flawless system of writing the treatment combinations for the 2K design
  7. Develop the formula for the contrast, effect, estimate, sum of square, and ANOVA table for the 2K design
  8. Using MS Excel, show the manual calculation procedures for the contrast, effect, estimate, and sum of square
  9. Perform the 2K design and analysis using statistical software, including Minitab, SPSS, and SAS
  10. Interpret the analysis results in the context of the problem

1. What is a 2K Factorial Design of Experiment?

Video 1. Introduction to 2K Factorial Design of Experiments DOE Formula Equation Explained with Examples.

As the factorial design of experiments are primarily used for screening variables, using only two levels is enough to determine whether a variable is significant to affect a process or not. If k number of variables/factors are studied to determine/screen the important ones, the total number of treatment combinations for a k number of factors can be calculated as in Equation 1. Therefore, this screening technique is known as the 2K design of experiments. More specifically, this experiment should be named as the completely randomized 2K factorial design of experiments. Recent popular textbooks on the design of experiment refer this design as the 2K design (Box, Hunter et al. 2005; Montgomery 2019), while the earlier texts refer the design as 2n design (Kempthorne 1952; Yates 1978; Hinkelmann and Kempthorne 2008), 2f (Hicks 1964), 2p, and so on. However, recently, 2K name has been popular for the factorial design of experiments with multiple factors with two levels for each factor.

Equation 1

The design is conducted very systematically, will be explained here, so that no data is wasted even if the insignificant variables are deleted from the study. Table 1 shows the layout, data structure and the coding systems for the levels of the factors of a 22 design, which is the basic to all 2K factorial design of experiments. The formulas or equations for the 22 will be shown first and then it will be generalized for the 2K factorial design of experiments. The process of formula development will provide us with behind the seen picture of a 2K design. Understanding this step will significantly impact learning the 2K design and analysis of experiment and interpreting the results in the context of the problems. Moreover, the formula development process will be extremely useful in understanding the Module 7 fractional factorial design, which is the ultimate goal of learning the factorial design of experiments.

Statistical software such as MS Excel, Minitab, SAS, and SPSS will be demonstrated for both design and analysis of the 2K factorial design of experiments. Moreover, for detail understanding of the concepts of the 2K factorial design of experiments, the manual calculations will be shown using MS Excel without requiring plugging the numbers in the formula. Rather than plugging the numbers (or responses) in the formulas, a method in MS Excel is shown in section 2.4 to produce all necessary results.

1.1. Layout/Data Structure and the Level Coding System of the Basic 22 Factorial Design of Experiment

The low level of a variable (or factor) is generally the current level, exiting level, or the control level, while the high level is used for the treatment level. For an example, testing whether a medicine works or not, the medicine is generally the high level while the low level is the control level, which is the placebo in general. Testing a new fertilizer against nothing or the current fertilizer, hoping that the new fertilizer is better or higher in value with respect to the yield (response or the dependent variable). The word ‘treatment combination” is inherited from the agriculture and medical fields where plants and animals are treated with multiple fertilizers, medicines, and so on. For an example, when the low levels of both factors are used, meaning that no treatment is applied, the treatment combination is defined as the “control,” which is denoted by “(1).” Simply “1” without the parenthesis is also used to represent “control” in some equations. Similarly, when only the high level of factor A is applied, the treatment combination is denoted as “a.” When only the high level of factor B is applied, the treatment combination is denoted as “b.” “ab” is used for the treatment combinations when both high level of both factors are applied. The treatment combination notations, (1), a, b, and ab are used for the total responses at that particular treatment combination.

Table 1. Layout, Data Structure, & Level Coding System of a Factorial Design of Experiment

1.2. Graphical Representation of the Basic 22 Factorial Design of Experiment

An example graphical representation of the 22 factorial design of experiment is provided in Figure 1.

Figure 1. 22 Factorial Design of Experiments with two levels for each factor (independent variable, x). The response (dependent variable, y) is shown using the solid black circle with the associated response values.

1.3. How to Develop the Formulas for the Main and the Interaction Effects of the Basic 22 Factorial Design?

The calculation of the effects from a 22 factorial design of experiment is described with examples in Video 2 (both Video 1 and are the same Video 2).

Video 2. Introduction to 2K Factorial Design of Experiments DOE and Formula Equation Explained with Examples.

1.3.1. Method 1. The Basic

The calculation of the main effect A was described in the Module 4 Factorial Design of Experiments. The main effect of A is calculated by subtracting the average responses at the low-level from the average responses at the high-level.

Therefore, the main effect of the temperature factor can be calculated as in Equation 2.

Equation 2

Using the treatment combination notations, the Equation 2 can be written as in Equation 3, and therefore, the Equation 4.

Equation 3

Equation 4

Similarly, the equations for the main effect of B and the interaction effect of AB can be developed for the 22 design of experiments.

Equation 5

Equation 6

1.3.2. Method 2. The -1/+1 Coding System

Using the -1/+1 coding system, the total effect can also be calculated by simply multiplying the factor A column with the treatment combination column (Table 1), which results in the Equation 7. Total effect of

Equation 7

As there are two replications for A, the average effect is calculated as in Equation 8.

Equation 8

Similarly, the equations for the main effect of B and the interaction effect of AB can be developed as in Equation 5 and Equation 6.

1.3.3. Method 3. The Commonsense System

Another way of developing the equation for the main effect of factor A is subtracting all treatment combinations without “a” in them from all the treatment combinations that contain “a” in them. Think about all the experimental units that received a fertilizer as compared to those did not receive the fertilizer. Therefore, the main effect of

Similarly, the equations for the main effect of B and the interaction effect of AB can be developed as in Equation 5 and Equation 6.

1.4. Contrast, Effect, Estimate, Sum of Square, and ANOVA Table for the Basic 22 Factorial Design?

Video 3 explains the process of developing the ANOVA table, including the contrast, effect, estimate, and sum of square for the basic 22 design.

Video 3. Contrast, Effect, Sum of Square, Estimate Formula, ANOVA table for 2K factorial design of experiment.

1.4.1. The Contrast

The contrast is defined by the total responses as described in Equation 9.

Therefore, the contrast of

Equation 9

1.4.2. The Effect

The equations for effects are described in Equation 6.

1.4.3. The Estimate

The estimates for each of the effects are simply the one-half of their respective effect. Therefore, the equations for the effects are written as.

Equation 10

1.4.4. The Sum of Square, SS

Sum of square is simply the average of the square of the contrast. For an example, the sum of square for A, B and the interaction effect can be calculated using the following equations.

Equation 11

The total sum of square, SST can be calculated as in Equation 12.

Equation 12

The sum of square for the experimental error can be calculated as in Equation 13.

Equation 13

1.4.5. ANOVA Table for a 22 Factorial Design of Experiment

The ANOVA table for a 22 factorial design of experiment can be developed as Table 2. One exception is the levels of A and B are denoted by small a and b respectively.

Table 2. ANOVA table for a 22 factorial design of experiment

1.5. Practice Problem on 22 Factorial Design of Experiments

Assume that four replications were made for the human comfort study provided in Table 3.

Table 3. Practice Problem for a 22 Factorial Design of Experiment

1.5.1. Questions

  1. Calculate the main and the interaction effects
  2. Calculate the estimates for A, B, and AB
  3. Write down the estimated regression equation
  4. Calculate the Sum of Squares
  5. Develop the ANOVA Table
  6. Interpret the effects in the context of the problem
  7. Which effects are likely to be significant based on the effect calculation? Why?

1.5.2. Solution

  1. Effect of A = 7.5, B = 1.75, & AB = 1
  2. Estimate of A = 3.75, B = 0.875, & AB = 0.5
  3. Estimated regression Equation:
  4. Sum of Square of A = 225, B = 12.25, & AB = 4
  5. ANOVA table

6. The comfort level increases by 7.5 if the temperature increases from 0 to 75-degree Fahrenheit. The comfort level increases by 1.75 if the humidity increases from 0 to 35 percent. As compared to the low level, at the high level of the temperature, the comfort increases by 1 if the humidity increases from 0 to 35 %.

7. The effect is very large for the temperature factor as compare to the main effect of humidity and the interaction effect. Therefore, the temperature factor is likely to be significant. As the experimental error is small as compare to the main and the interaction effects, results in high f-value. Therefore, both the main effects of A and B and the interaction could be significant too.

2. The 2K Factorial Design & Analysis of Experiments

2.1 How to Systematically Design the 2K Factorial Experiments Using MS Excel.

The step by step development and design process of the 2K factorial design of experiment is described using MS Excel with examples in Video 4. Originally, the method of development of the 2K design was described in many classic (Kempthorne 1952; Yates 1978) and recent texts (Box, Hunter et al. 2005; Montgomery 2019). However, in manual calculations for understanding the detail design and analysis process, utilizing MS excel reduces the work about 99.99% [a bit exaggerated].

Video 4. Design Layout and Construction of 2K Factorial Design of Experiments DOE Using MS Excel Easiest Way.

The development starts with the basic 22 design. The step by step development process of the 2K is provided below. However, watching the video will be easier to follow than this text, probably.

1. Create the basic 22 design as in Table 4.

Table 4. The basic 22 design, including the standard order (SO) and the treatment combinations (TRT).

2. To add the third variable (factor), C to this 22 basic design, all treatment combinations of the basic design are added with both the low and the high level of the third variable, C. Add a column after the variable B and name it as “C.”

3. Fill the empty cells with all low level of C.

4. Add another four rows

5. Copy the basic design in the empty cells below the A & B columns.

6. Now add the high level to the empty cells below the C column.

7. Fill the empty cells of the TRT column with the appropriate treatment combinations

8. Similarly, fourth, fifth and more variables can be added to the design as needed. MS Excel File 1 provides an example containing 2K design up to eight variables, 28 is provided here.

2k design with all treatment combinations upto 8 variables.xlsx

MS Excel File 1. 2K design development process in MS Excel

2.2. How to Systematically Flawlessly Develop/Write the Treatment Combinations for Exhaustively Long List of Variables.

Originally, this method of writing the treatment combination was shown in the one of the most popular classic texts in DOE (Kempthorne 1952) (Table 5). However, with the invention of the great MS Excel, writing the treatment combination can be performed even more easily and flawlessly. The step by step process of writing the treatment combinations is described in the Video 5.

Video 5. Write Treatment Combinations Systematically and Flawlessly in Design and Analysis of Experiments DOE

Table 5 shows the step by step process of writing the treatment combinations.

  1. Write the treatment combinations for the basic 22 design.
  2. Add a column next to it and type c to add the factor C with it.
  3. Now use the treatment combinations for the three variables A, B, and C to develop the treatment combinations for four variables A, B, C, and D.
  4. And so on.

Table 5. Flowless system for writing the treatment combinations

2.3 Development of the Generic Formulas for the 2K Factorial Design of Experiments

To develop the generic formulas, let’s start the design with three variables can be found in Table 6. The use of the -1/+1 coding system in MS Excel provides an advantage to multiply columns easily to find the interaction effects. For an example, the AB interaction column is simply the multiplication of the main effect columns A and B.

Table 6. 23 factorial design

Now, the total effect of A (known as the contrast of A) can be calculated from the Table 6 by simply multiply the column A and the treatment combination column, TRT. Another way of find the total effect of A is subtracting all treatment combinations without “a” from all the treatment combinations with “a” in them. Think about all the experimental units that received a fertilizer (e.g. factor A) and all the experimental units that did not receive the fertilizer. Therefore, the total effect of A (contrast of A) can be written as in Equation 14.

Equation 14

Table 6 also shows that there are four replications for each level of the factor A. Therefore, the average effect can be calculated as.

Equation 15

As the replication for the factor increases by the power of 2 for each additional factor in a 2K design (Equation 1), to generalize for the higher number of factors, the Equation 15 can be modified as Equation 16.

Equation 16

Therefore, the main effect for A and the other effects can be generalized for k number of variables (factors) with n replications as in Equation 17.

Equation 17

Similarly, the Equation 18 represents the estimates (one-half of the effect) for the 2K design, which can be developed following the basic 22 design in Equation 10.

Equation 18

Similarly, the Equation 19 represents the sum of square for the 2K design, which can be developed following the basic 22 design described in Equation 11.

Equation 19

2.4. The use of MS Excel in the Manual Analysis of a 2k Factorial Design of Experiment

While the formulas provide many insights into the 2K factorial design of experiments, the use of MS Excel provides much easier alternative way to calculate the contrasts, effects, estimates, sum of squares and the development of the ANOVA table is shown in Video 6.

Video 6. Contrast, Effect, Estimate, and Sum of Square Calculation in 2K Design of Experiments Using MS Excel.

MS Excel File 2 shows the manual calculation procedure to find the contrasts, estimates, effects, sum of squares, and ANOVA table for the problem # 6.7 in page 293 from a popular Design and Analysis of Experiments Textbook (Montgomery 2013). The problem or the design consists of four variables affecting a chemical process. Two replications were collected. The MS Excel File 2 shows an easier alternative way to find the contrasts, effects, estimates, sum of squares, and the development of the ANOVA table without plugging the numbers in the formulas, rather a much easier process is applied using the basic MS excel functions, which has been proven to be very useful when a dedicated statistical software is unavailable. Even when a dedicated statistical software is available, over a decade of experience in DOE, the author finds the use of MS excel procedure is very useful in many design and analysis of experiments, including the 2K factorial design of experiments. As the data structure and layout is already there in the MS Excel, as soon as the data collection is completed, the analysis will be generated automatically with a little tweak in the formula. It may even take less time than running analysis in a statistical software.

MS Excel in the manual analysis of a 2k factorial design of experiments Montgomery 8th Ed Solution to 6.7.xlsx

MS Excel File 2. MS Excel in the manual analysis of a 2k factorial design of experiments Montgomery 8th Ed Solution to Problem # 6.7

2.5 2K Factorial Design of Experiments Analysis Using MS Excel, Minitab, SPSS, and SAS

The Video 7 demonstrates the analysis of 2K factorial design of experiments using four population software, including MS Excel, Minitab, SPSS, and SAS.

Video 7. Comparisons between MS Excel, Minitab, SPSS, and SAS in Design and Analysis of Experiments DOE

2.6 Practice Problem on 2K Factorial Design of Experiment

A factorial design of five variables with three replications is provided in MS Excel File 3.

Practice Problem 2 on 2K Factorial Design of Experiment TOE.xlsx

MS Excel File 3. Practice Problem on 2K Factorial Design of Experiment

2.6.2. Questions

1. Using MS Excel, calculate contrast, effect, estimate, sum of squares for all main and all interaction effects

a. Interpret one significant main effect and one significant two-way interaction in the context of the problem.

2. Write down the estimate regression equation

a. Explain one linear coefficient (effect estimate) in the context of the problem

3. Develop the ANOVA Table

4. How the contrast, effect, estimate, and the sum of square are related? How, they are connected to the statistical significance? Which effects are larger? How likely are they contributing to the sum of square?

2.6.2. Solution

The complete solution is provided in MS Excel File 4.

Practice Problem 2 on 2K Factorial Design of Experiment TOE Solution.xlsx

MS Excel File 4. Solution to the Practice Problem on 2K Factorial Design of Experiment

1. Check the MS Excel File 4 for the detail Solution. Table 7 shows the solution summary. The process of finding the numbers in Table 7 is shown in MS Excel File 4.

Table 7. Solution Summary

2. Regression Equation

Y = 14.917 + 1.000 A + 1.271 B + 1.521 C + 1.042 D + 1.333 E - 0.271 A*B - 0.063 A*C - 0.583 A*D - 0.625 A*E + 0.375 B*C + 0.021 B*D - 0.438 B*E - 0.313 C*D + 0.188 C*E + 0.792 D*E + 0.250 A*B*C - 0.396 A*B*D - 0.604 A*B*E - 0.271 A*C*D - 0.438 A*C*E + 0.375 A*D*E - 0.167 B*C*D + 0.917 B*C*E - 0.104 B*D*E + 0.188 C*D*E + 0.667 A*B*C*D + 0.667 A*B*C*E - 0.062 A*B*D*E + 0.104 A*C*D*E - 0.375 B*C*D*E - 0.250 A*B*C*D*E

3. ANOVA table

4. All main effects and a few interactions are significant shown in the ANOVA table with bold texts.