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The successful use of fractional factorial designs is based on three key ideas:

Fractional Factorial. The successful use of fractional factorial designs is based on three key ideas: The sparsity of effects principle. When there are several variables, the system or process is likely to be driven primarily by some of the main effects an low order interactions.

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The successful use of fractional factorial designs is based on three key ideas:

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  1. Fractional Factorial • The successful use of fractional factorial designs is based on three key ideas: • The sparsity of effects principle. When there are several variables, the system or process is likely to be driven primarily by some of the main effects an low order interactions. • The projection property. Fractional factorial designs can be projected into stronger designs in the subset of significant factors. • Sequential experimentation.

  2. Fractional Factorial For a 24 design (factors A, B, C and D) a one-half fraction, 24-1, can be constructed as follows: Choose an interaction term to completely confound, say ABCD. Using the defining contrast L = x1 + x2 + x3 + x4 like we did before we get:

  3. Fractional Factorial

  4. Fractional Factorial Hence, our design with ABCD completely confounded is as follows: The fractional factorial design

  5. Fractional Factorial Each calculated sum of squares will be associated with two sources of variation.

  6. Fractional Factorial Lets clean a bit:

  7. Fractional Factorial Lets reorganize: Complete 23 Design

  8. Fractional Factorial So to analyze a 24-1 fractional factorial design we need to run a complete 23 factorial design (ignoring one of the factors) and analyze the data based on that design and re-interpret it in terms of the 24-1 design.

  9. Fractional Factorial Resolution: • Many resolutions the three listed in the book are: • Resolution III designs: No main effect is aliased with any other main effect, they are aliased with two factor interactions and two factor interactions are aliased with each other. Example 2III3-1 with ABC as the principle fractions. • Resolution IV designs: No main effect is aliased with any other main effect or any two factor interaction, but two factor interactions are aliased with each other. Example, 2IV4-1 with ABCD as the principle fraction. • Resolution V designs. No main effect or two-factor interactions is aliased with any other main effect or two-factor interaction, but two-factor interactions are aliased with three factor interactions. Example, 2V5-1 with ABCDE as the principle fraction.

  10. Fractional Factorial Example:

  11. Fractional Factorial Assuming all factors are fixed, the linear model is as follows:

  12. Fractional Factorial If we still cant run this design all at once, we can block; that is we can implement a group-interaction confounding step. We can confound the highest level interaction of the 23 design, as we did before.

  13. Group-Interaction Confounded designs Partial confounding: Confounding ABC replicate 1:

  14. Group-Interaction Confounded designs Partial confounding: Confounding ABC replicate 1:

  15. Fractional Factorial For a 24 design (factors A, B, C and D) a one-quarter fraction, 24-2, can be constructed as follows: Choose two interaction terms to confound, say ABD and ACD, these will serve as our principle fractions. The third interaction, called the generalized interaction, that we confounded in the way is: A2BCD2 = BC. Need two defining contrasts L1 = x1 + x2 + 0+ x4 and L2 = x1 + 0 + x3 + x4

  16. Fractional Factorial

  17. Fractional Factorial

  18. Fractional Factorial

  19. Fractional Factorial Hence, our design with ABCD completely confounded is as follows:

  20. Fractional Factorial One of the possible one-quarter designs is:

  21. Fractional Factorial Each calculated sum of squares will be associated with four sources of variation.

  22. Fractional Factorial Each calculated sum of squares will be associated with four sources of variation.

  23. Fractional Factorial The above is not quite satisfactory because we are aliasing some of the main effects with other main effects; i.e. the resolution is not good enough!!!

  24. Fractional Factorial What happens after analyzing the data: Can do a confirmatory experiment, complete the block!!

  25. Fractional Factorial

  26. Fractional Factorial

  27. Fractional Factorial

  28. Fractional Factorial Hence, our design with ABCD completely confounded is as follows:

  29. Fractional Factorial Each calculated sum of squares will be associated with four sources of variation.

  30. Fractional Factorial Each calculated sum of squares will be associated with four sources of variation.

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