Learning Outcomes

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

1. Explain the 2^K 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 2^K experiment, showing up to 8 variables, using MS Excel
6. Show the flawless system of writing the treatment combinations for the 2^K design
7. Develop the formula for the contrast, effect, estimate, sum of square, and ANOVA table for the 2^K design
8. Using MS Excel, show the manual calculation procedures for the contrast, effect, estimate, and sum of square
9. Perform the 2^K design and analysis using statistical software, including Minitab, SPSS, and SAS
10. Interpret the analysis results in the context of the problem

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 2^K factorial design of experiments. The formulas or equations for the 2² will be shown first and then it will be generalized for the 2^K factorial design of experiments. The process of formula development will provide us with behind the seen picture of a 2^K design. Understanding this step will significantly impact learning the 2^K 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 2^K factorial design of experiments. Moreover, for detail understanding of the concepts of the 2^K 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.

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.

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

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

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

1.4.2. 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.

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.

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

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

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.

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.

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 2^K design up to eight variables, 2⁸ is provided here.

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.

2.3 Development of the Generic Formulas for the 2^K 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.

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.

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

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

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

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

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

While the formulas provide many insights into the 2^K 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.

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

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

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

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

Next Topic

Blocking and Confounding in 2k Design