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# Two level factorial designs

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### Two level factorial designs

1. 1. Two-Level Factorial Designs Presented by: Juanito S. Chan, PIE
2. 2. Standard 23 Design • Where do the factors (independent variables) appear in this table? • Where do the responses (dependent variable) appear in this table? • What do the –1 and +1 mean? • Should these experimental runs be made in the order they are shown? Run A B C 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
3. 3. Standard 23 Design • Factors are A, B, C • Responses do not appear in this table? • Choose a high and a low value for each factor. o -1 means set factor to low level in this run o +1 means set factor to high level in this run • Run order should be randomized o Failure to randomize very risky for factor C, since it has runs 1-4 at low level and 5-8 at high level Run A B C 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
4. 4. Maximize Reaction Yield 23 Factorial Design • Objective: maximize reaction yield • Factors: o A = catalyst weight percent (1,2) o B = reaction time, hours (1,2) o C = temperature, °F (200,250) • Response: Reaction yield, %
5. 5. Maximize Reaction Yield Run Catalyst Weight % Reaction Time, hr Temperature, °F Yield, % 1 1 1 200 65.3 2 2 1 200 81.3 3 1 2 200 53.3 4 2 2 200 69.9 5 1 1 250 61.8 6 2 1 250 77.4 7 1 2 250 73.9 8 2 2 250 89.9
6. 6. Now What? • Calculate effects of each factor and interaction • Decide which effects are important • Plan another, multilevel experiment focusing on the important variables
7. 7. Interactions Run A B C AB AC BC ABC Y 1 -1 -1 -1 +1 +1 +1 -1 65.3 2 +1 -1 -1 -1 -1 +1 +1 81.3 3 -1 +1 -1 53.3 4 +1 +1 -1 69.9 5 -1 -1 +1 61.8 6 +1 -1 +1 77.4 7 -1 +1 +1 73.9 8 +1 +1 +1 89.9 Note that each factor is tested at each level 4 times. -1 × -1 = +1
8. 8. Investigating Interactions • You set the value for each factor in each experiment • The interactions happen naturally o You do not set some level of AB interaction; it happens automatically because of the levels you set for A and B individually • Interactions are a physical reality of the system, and will happen whether you calculate an effect for them or not
9. 9. How to Calculate Effects • High Total = sum of all response values when the factor is at the +1 level • Low Total = sum of all response values when the factor is at the –1 level • Difference = (High Total) – (Low Total) o Note that you can also calculate the difference by multiplying each +1 or –1 by the response for its row, then summing all the values in the column. That is what your book says. • Effect = Difference / (# runs at each level)
10. 10. Effects A Cat. Wgt. % B Rxn Time C Temp AB AC BC ABC High Total Low Total Diff Effect On Y
11. 11. Effects A Cat. Wgt. % B Rxn Time C Temp AB AC BC ABC High Total 318.5 287.0 303.0 286.9 285.9 310.4 286.3 Low Total 254.3 285.8 269.8 285.9 286.9 262.4 286.5 Diff 64.2 1.2 33.2 1.0 -1.0 48.0 -0.2 Effect On Y 16.05 .30 8.30 0.25 -0.25 12.00 -0.05
12. 12. “Scree Plot” to Identify Order of Importance -2 0 2 4 6 8 10 12 14 16 18 A BC C B AB AC Factor EffectonReactionYield
13. 13. Conclusions • Increasing catalyst weight % or increasing temperature will increase the yield o Increasing catalyst is most effective • Increasing reaction time itself has little effect on yield, but in combination with increased temperature multiplies the effect of temperature
14. 14. Comparison with OFAT • OFAT would reveal the effect of catalyst and temperature. • OFAT would not reveal the time-temperature interaction. • OFAT would not reveal the lack of time- catalyst and temperature-catalyst interaction.
15. 15. Adding a Factor • Adding a factor to a full factorial design doubles the number of experimental runs o 3 factors = 23 = 8 runs o 4 factors = 24 = 16 runs • If you are confident that an interaction is unimportant, you can substitute a new factor for that interaction term in the test matrix o 3-way interaction least likely to be important o Substitution of a factor for an interaction makes an unsaturated design
16. 16. Unsaturated Designs and Aliasing • If a factor replaces an interaction in the design, o You cannot tell the difference between the effect of the factor and the effect of the interaction o Interaction is an innate property of the system. You do not control whether or not it happens by deciding whether or not to study it. o Some or all of the effect you calculate for the new factor could be due to the interaction between other factors. o You cannot study how the new factor interacts with others