STAT-4204 - Notes - 14 - Fractional Factorial Designs II PDF

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Prof: AR Driscoll...

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Daniel T. Eisert

STAT-4204

14 – Fractional Factorial Designs II (Ch. VIII) STAT-4204: Experimental Designs October 22, 2019 (Week 9)

Class Business Quarter Fraction Designs

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Class assignment next class. Project proposal due this Sunday, October 27.  Example: 25-2 = 8 runs (1/4 fraction design: 2-2 = 1/4) Generate runs for three factors; let software choose D, E A B C D=AB E=AC + + + + + + + + + + + + + + Intercept = ABD = ACE (Intercept confounded with ABC and ACE) So, D = AB = ACDE (multiply by D) E = ABDE = AC Defining Relation: I =ABD = ACE = BCDE ABD, ACE are interaction generators Since BCDE is the product of both interaction generators, it is called the generalized interaction. Resolution: III (length of the smallest word in the defining relation). This means that the main effects are confounded with two-factor interactions (one minus the resolution). Alias Structure: I =ABD = ACE =BCD E A=BC =CE = ABCDE (multiplied by main effect A) B= AD = ABCD =CDE (multiplied by main effect B) C=ABCD = AE = BDE (multiplied by main effect C) D= AB = ACDE= BCE (multiplied by main effect D) E= ABDE = AC = BCD (multiplied by main effect E) … BC = ACD =ABE = DE (multiplied by interaction BC) … Can keep going for all terms. ***Bad design if you are less than Resolution III design; main effects would be confounded with other main effects! ***

Example (Test Question)

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Example: 26-2 Design; need two generators for a quarter fraction design I =ABCE = ACE (generating interactions) 1. Complete the defining interaction

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Daniel T. Eisert

STAT-4204 I =ABCE = BCE = A (Resolution I design); this is bad because we’ve lost out ability to estimate A since A is confounded with the intercept  Therefore, this is a bad design. 

Example: 26-2 Design I =ABCE = ABDE =CD (Resolution II design); main effects are confounded with main effects (bad design) C=ABE =ABCDE = D (multiplied by C)

For half fractions, you should be able to specify the defining relation I =(highest order interaction) . For quarter fractions or lower, you do not have to specify the design generators, but you should be able to find the generalized inverse and determine the resolution to decide if it is a good design. Test Question Choose the higher resolution if you have the same number of runs.

Fractional Factorial Design

Motivation for fractional factorials is to minimize the size of the design with an emphasis is on factor screening; efficiently identify the factors with large effects (then follow up). - There may be many factors of interest - Always run unreplicated but with center points Sparsity Effects Principle - There may be many factors, but few are important - System is likely dominated by main effect and low-order interactions Projection Property - [Test Question] Every fractional factorial design contains full factorial designs in fewer factors - 28 design is fully replicated in three factors. - Can “step back down” to a lower level. - If you have a design in four factors and one is insignificant, can jump down to three factors. Sequential Experimentation - Can add runs to a fractional factorial design to resolve difficulties (or ambiguities) in interpretation. Ockham’s Razor: pick the simplest model or interpretation (same in regression).

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