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This assignment focuses on the application of fractional factorial designs at 2-levels using a 2k-p structure to explore multiple factors efficiently. It emphasizes the construction of designs from full factorials and the significance of interaction columns. We’ll also analyze data using normal QQ-plots and compare findings with Lenth’s method. Practical exercises include designing experiments with limited trials, understanding minimum aberration designs, and evaluating factor impacts on outcomes. This will enhance your ability to conduct effective experiments with reduced resource usage.
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Stat 470-15 • Today: Start Chapter 4 • Assignment 3: 3.14 a, b (do normal qq-plots only),c, 3.16, 3.17 • Additional questions: 3.14 b (also use the IER version of Lenth’s method and compare to the qq-plot conclusions), 3.19
Fractional Factorial Designs at 2-Levels • Use a 2k-p fractional factorial design to explore k factors in 2k-p trials • In general, can construct a 2k-p fractional factorial design from the full factorial design with 2k-p trials • Set the levels of the first (k-p) factors similar to the full factorial design with 2k-p trials • Next, use the interaction columns between the first (k-p) factors to set levels of the remaining factors
Example • Suppose have 7 factors, each at 2-levels, but only enough resources to run 16 trials • Can use a 16-run full factorial to design the experiment • Use the 16 unique treatments for 4 factors to set the levels of the first 4 factors (A-D) • Use interaction columns from the first 4 factors to set the levels of the remaining 3 factors
Example • Would like to have as few short words as possible • Why?
How can we compare designs? • Resolution
Example • Suppose have 7 factors, each at 2-levels, but only enough resources to run 32 trials • Can use a 27-2 fractional factorial design • Which one is better? • D1: I=ABCDF=ABCEG=DEFG • D2: I=ABCF=ADEG=BCDEG
Example • Suppose have 8 factors (A-H), each at 2-levels, but only enough resources to run 32 trials • Can use a 28-3 fractional factorial design • Table 4A gives the minimum aberration (MA) designs for 8, 16, 32 and 64 runs • From Table 4A.3, MA design gives: • 6=123 • 7=124 • 8=1345
Example • Table 4A.3, MA design is: • 6=123 • 7=124 • 8=1345 • Design for our factors: • Word length pattern:
Example • Speedometer cables can be noisy because of shrinkage in the plastic casing material • An experiment was conducted to find out what caused shrinkage • Engineers started with 6 different factors: • A braiding tension • B wire diameter • C liner tension • D liner temperature • E coating material • F melt temperature
Example • Response is percentage shrinkage per specimen • There were two levels of each factor • A 26-2 fractional factorial • The purpose of such an experiment is to determine which factors impact the response
Example • Constructing the design • Write down the 16 run full factorial • Use interaction columns to set levels of the other 2 factors • Which interaction columns do we use? • Table 4A.2 gives 16 run MA designs • E=ABC; F=ABD
Example • Results
Example • Which effects can we estimate? • Defining Contrast Sub-Group: I=ABCE=ABDF=CDEF • Word-Length Pattern: • Resolution:
Example • Effect Estimates and QQ-Plot:
Comments • Use defining contrast subgroup to determine which effects to estimate • Can use qq-plot or Lenth’s method to evaluate the significance of the effects • Fractional factorial designs allow you to explore many factors in relatively few trials • Trade-off run-size for information about interactions
