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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 29-5; 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 demonstrates the process of fractional factorial data analysis using Minitab.
Step #1
Step #1
Run the Analysis using only the main effects in the model to screen out all the insignificant variables
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 be 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
Step #2
Step #2
Rerun the analysis using the full model for the significant variables only, after screening out the insignificant variables/factors
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
Step #3
Step #3
Run the post-hoc analysis
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
Step #4
Step #4
Draw Conclusions
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).
Second Example
Second Example
Fractional Factorial Design
Fractional Factorial Design
In a screening multiple factors study with seven (7) variables, the lowest number of runs possible is eight (8) using the one-eight fractional factorial design of experiments. The data is provided in the table below.
In a screening multiple factors study with seven (7) variables, the lowest number of runs possible is eight (8) using the one-eight fractional factorial design of experiments. The data is provided in the table below.
Questions
Questions
- Perform the analysis and draw conclusion. You must use the step by step procedure in analyzing fractional factorial design of experiments.
- Additional Questions
- Calculate the contrasts for the effects
- Calculate the estimates for the effects
- Find the Generator/Identifier/Word
- Write down the full alias structure of the design
- What is the design resolution of this design? Is there a higher resolution possible? Why? Why not? Is higher resolution(s) necessary?
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