# lect20 - IE 410 Design of Experiments Lecture 20 Chapter 8...

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Unformatted text preview: IE 410 Design of Experiments Lecture 20 Chapter 8: Fractional Factorial Designs When k gets large, 2 k designs require many runs. Example : the 2 6 design requires 64 runs even without replicates. We are often interested in factor screening experiments with 10 or even 15 factors. Let’s examine the 2 6 a bit more: A 2 6 produces 63 df allocated as follows: 6 df for main effects 15 df for 2 way interaction effects 42 df for higher order interactions We might guess that we could get information on main effects and lower order interactions without the need for 64 runs. Fractional factorial designs are the answer. 1/2 fraction of a 2 3 or: the 2 3-1 design. We will investigate k=3 factors with a total of 4 runs. Suppose our design consisted of the following 4 runs: a, ab, ac, abc Note that all four runs have factor A at the high level. Thus we have NO INFORMATION about the effect factor A has on the response Y. Let's look at the "+ and -" table for the 2 3 We chose the 4 runs indicated by the "*"'s cel l ( I ) A B C AB AC BC ABC - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (1 ) + - - - + + + - a + + - - - - + + * b + - + - - + - + ab + + + - + - - - * c + - - + + - - + ac + + - + - + - - * bc + - + + - - + - abc + + + + + + + + * In the design above, we lost info on A. Better to loose info on the ABC interaction instead, right? Thus we could take the 4 runs in which ABC is "high". Or alternatively, the four runs with ABC "low". cel l ( I ) A B C AB AC BC ABC - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (1 ) + - - - + + + - a + + - - - - + + * b + - + - - + - + * ab + + + - + - - - c + - - + + - - + * ac + + - + - + - - bc + - + + - - + - abc + + + + + + + + * Removing rows not included in the experiment yields: cel l ( I ) A B C AB AC BC ABC - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - a + + - - - - + + * b + - + - - + - + * c + - - + + - - + * abc + + + + + + + + * We could have chosen the runs where ABC was "low" cel l ( I ) A B C AB AC BC ABC - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (1 ) + - - - + + + - ab + + + - + - - - ac + + - + - + - - bc + - + + - - + - These two designs are referred to as "the two 1/2 fractions of the 2^3 factorial design generated from ABC". ABC is called the "fraction generator" Let's examine either of these two designs more closely. We observe that: The A and BC effects are completely confounded. The B and AC effects are completely confounded. The C and AB effects are completely confounded. Also observe that: The (I) and ABC columns are all "+" 's Thus if the contrast (a+abc) - (b+c) is significant, we can't tell if it is due to A or to BC....
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lect20 - IE 410 Design of Experiments Lecture 20 Chapter 8...

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