# Fractional factorial design

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• #30920

JM
Participant

I am studying DOE in BB training and need some help.  What does (k-p) signify in a fractional factorial design.  For example what is a 2 7-3 with resolution IV? Also, how many different +/- treatment combinations does this design have? Thanks for your help.

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#81157

MMBB
Participant

I suppose you understand that a 2^k design provides all combinations of k variables. A 2^2 design has four treatment combinations (tc’s) involving two variables. A 2^3 design has eight tc’s involving three variables.
In a 2^(k-p) design, you have 2^(k-p) tc’s involving k variables, instead of the usual 2^k tc’s. When p=1, we’d call this a half fraction. When p=2, this is a quarter fraction. You can think of the fraction as 1/(2*p). What you are doing is running just a fraction of the tc’s that might have been run in the 2^k design.
To create one of these fractional factorials, you actually start out with the 2^(k-p) full factorial — for a 2^(5-2), which looks at five variables, this would be starting out with a 2^3 full factorial, with 8 runs.
This 2^3 design already has 3 variables. To run the other 2 variables you would assign each to an interaction’s terms. This gets kind of complicated, but in Excel you can create these by assigning -1’s to low levels and +1’s to high levels. The interations columns are then calculated by multiplying the respective -1’s & +1’s for the involved terms (such as AB or ABC).
But which interaction should you use? This can be tricky, since you want to pick interactions that allow you to understand interactions cleanly – without nasty confounding. This is where good statistical software or DOE design tables are very usefull.
To answer your specific question, a 2^(7-3) design has 2^4 tc’s, or 16.
Again, this all becomes MUCH easier with the help of good DOE software such as MINITAB, JMP, StatGraphics, Statistica, and others.
Hope that helps.

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#81158

Charles H
Participant

The source for the following is: Understanding Industrial Designed Experiments, 4th Edition, Schmidt/Launsby, Air Academy Press
k = number of factors
q = size of fraction         q = 1 (1/2 fraction)  q = 2 (1/4 fraction)

Resolutions:
Res V: main effects OK (aliased with 4-way interactions)   2-way interactions OK (aliased with 3 way interactions)
Res IV: main effects OK (aliased with 3-way interactions)  some (or all) 2-way interactions aliased with other 2-ways
Res III: some (or all) main effects alieased with 2-way interactions.
A 2^7-3 will have 16 runs, n reps = 3

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#81173

Manee
Participant

If you have 7 variables and 2 levels, you can use orthogonal array L8 and make 8 tests to study all variables.
Manee

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#81174

RBC
Participant

Answers from others inform you what you need to know but I have a curiousity question – Did the training company / instructors internal not cover this in you training and if so – what did they give you??

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#81184

chhabra
Participant

JM,
Resolution IV is the length of the shortest word to the right of the defining relation.
2(7-3) design will have 16 runs
where7 is the nos of factors

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#81229

sari
Member

2^(k-p)
2 = signifies two levels
k = # of factors
p = # of generators
example : 2^(4-1) design is a half fraction, 8 runs.
the generator is D=ABC because you aliased factor D to highest order interaction. From the generator, you can can get the aliasing pattern:
D=ABC
DB=AC, etc….
notice the aliasing pattern, main effect equals 3-way int; 2 way equals 2 way. This is a resolution IV design.
Hope this will confuse you further :-).

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#81285

MMBB
Participant

…and provide absolutely no ability to estimate interaction effects.

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#89155

zuheyr alsalihi
Member

Hello
I am a newcomer to this topic. Do youknow if there is a free source code available for the Fractional Factorial design? Or an algorithm that I can program. It seems to me that a lot of data has to be input to the program.
If anybody can help please, I will greatly appreciate.
Best regards,
Zuheyr

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#89166

asa
Participant

Dear Sari :
You consider only control factors and what about noise factors ?
Where we place noise factor in DOE ?
If we avoid noise factors then we will neglect products characteristics hence product will not as per customers requirement.
Cheers !!!

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