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Fractional Factorials

2 _k p Designs

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2k-p Designsk factorstwo levels for each factorwill only run 2-p of the possible factor

combinationswill only run 2k-p observations total

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Basic facts about fractional factorialswill only have 2k-p-1 degrees of freedom totalcan only estimate that many terms in the

model (actually less if we are to have degrees of freedom for error)

many terms are confounded with each other

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What do we need to do?determine which terms we are most

interested in (Main Effects, two-way Interactions, etc.)

figure out which treatment combinations to run

then figure out which terms in the Model are confounded with each other

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Let’s startThe minimum we want out of any fractional

factorial is to retrieve Main EffectsPick an Identity to generate incomplete

Blocks for the fractional factorialUse Orthogonal Contrasts to get the

Identity!!!(knew Dr. Tom was going over those for some

reason :p)

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Easiest case, 2k-1, a half fractionRecall the three factor design with A, B and CThere were eight treatment combinationsLets look at the contrast for ABC (1) a b ab c ac bc abc ABC - + + - + - - +

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This Identity gives us two (incomplete) Blocks

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How do we know which treatment combinations go in the minus (-) block and which in the plus (+) block?See what factors are in the experimentFor the 23-1 it is Factors A, B and CList the treatment combinations, i.e. (1), a, b,

ab, c, ac, bc, abcIf I block on ABC, then count the number of

letters in a treatment combination whose upper cases are in the blocking term, here it is ABC

If that number is odd, it goes into the one block, otherwise it goes into the other block (useful for the next HW problem)

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The IdentityI=ABC is the “Identity”The Identity gives us the incomplete blocksThe Identity tells us the pattern of

confoundingA=A2BC=BC, for exampleand how do we know that, pray tell?

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It’s all Group theory, of course! A semigroup has an operator * such that:

*:( * )* *( * )G G Ga b c a b c

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For fractional factorials, the Contrasts for the Effects (plus a vector of 1’s) form a semigroup If G is our semigroup, then the members of G

include the contrasts:(-1,+1,-1,+1,-1,+1,-1,+1) for A(-1,-1,+1,+1,-1,-1,+1,+1) for B(-1,-1,-1,-1,+1,+1,+1,+1) for Cand A*B (which we will denote as AB from now on)

gives(+1,-1,-1,+1,+1,-1,-1,+1)as long as * means the pair-wise product of elements

for the vectors. We could have done this in terms of Matrices but we’ll stick to the shorthand notation.

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Is our semigroup, G a Group? Yes, as long as we include: AA=A2=I and I=(1,1,1,1,1,1,1,1). In Group theory, I is referred to as the identity, although for Fractional Factorials this term has a double meaning for determining “aliases”. Aliases tell us which terms are confounded with which other terms.

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Back to the 23-1

For the 23-1 design, we run one incomplete blockThe Identity, I=ABC, gave us our incomplete

blocks, and we run one of them (doesn’t matter which for the analysis)

The Identity also tells us which terms are confounded with each other

So A=BC, B=AC and C=BCSo at least the Main Effects are not confounded

with each otherThis is the minimum we want out of any fractional

factorial

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What about more factors?A 23-1 design is not too useful since we have

no degrees of freedom for error (though you can at least estimate the Main Effects)

For more factors you just need the Identity to specify the design (很容易！！ )

For half fractions we always use the highest order interaction

So for a 25-1 design we use I=ABCDE for the Identity to generate the design

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What if….My client comes in with six factorsI recommend a 26-1 fractional factorialMy client asks how many observationsYou say 32Client says that will use up the rest of the

research moneySo how about a 26-2 fractional factorial?Your client can afford 16 observations and

have some money left over for further experiments!

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How to set up a 26-2 fractional factorial?

We need to divide up our treatment combinations in half, then divide those in half again

This gives us a ¼ fractionWe need to choose two terms in the Model to

set up two incomplete blocksHow about ABCD and CDEF?

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What are the incomplete blocks that my treatment combinations go to?

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What is the pattern of confounding?Well, if we ran only ¼ of the possible combinations, you

can show that terms must be confounded in groups of four

I=ABCD=CDEF doesn’t give all of the confounded terms, but if we include one more term…

I=ABCD=CDEF=ABEFWhere did ABEF come from?ABCDCDEF=ABC2D2EF=ABEFABEF is called the Generalized Interaction of ABCD

and CDEFThat part was not intuitive (unless you are a Math

head), but everything is accounted for now

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Review of the basics:We generate the design by choosing one or more

terms in the model to set up our incomplete blocksThe number of terms we chose determines the

degree of the fractional factorialOne term gives a 2k-1, two terms gives a 2k-2, three

terms gives a 2k-3 and so forthThe terms, along with the generalized

interactions, give the IdentityThe Identity tells us what is confounded with whatWhew!!

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Back to the 2k-2

The Identity was I=ABCD=CDEF=ABEFThis tells us that:A=BCD=ACDEF=BEFB=ACD=BCDEF=AEFAB=CD=ABCDEF=EFetc.So main effects are not confounded with each

other, but two ways are confounded with each other

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How many fractional factorials are there and how do I choose which one to use?Lots of possibilities existYou decide based upon which terms you are

interested in estimating/testingGenerally, this means you want Main effects

at the minimumYou don’t want Main effects confounded with

things that are likely to be significantLow order terms are likely to be significant,

high order terms are less likely to be insignificant

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Is there an easy way to tell?For once, the answer is yes (WooHoo!, 太好了！ , Super!)Look at the Identity (example:

I=ABCD=CDEF=ABEF)Count the number of letters in the shortest

string of letters (other than I), that is the Resolution of the design (for the example above it is Resolution IV)

The higher the Resolution the betterMost Statistical Software packages will

actually generate the designs for you if you ask for designs of a certain Resolution for a given number of factors (they give you the treatment combinations to run also)

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For example:For a 23-1 with I=ABC, the Resolution is IIIFor a 24-1 with I=ABCD, the Resolution is IVFor a 26-2 with I=ABCD=CDEF=ABEF, the Resolution

is IVFor a 26-2 with I=ABC=DEF=ABCDEF, the Resolution

is IIIResolution IV is better then Resolution III since Main

effects are confounded with three way and higher terms, while the Resolution III design confounds Main effects with two-way interactions which are more likely to be significant than higher order terms

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Does this get more complicated?Of courseYou may need to run the experiment in BlocksYou may have more than two levels in some of your

factorsYou may want to fit a curvilinear response if most

variables are quantitative (Chemical Engineers do this all the time, it’s called Response Surface Methodology)

Go see a Statistical consultant who understands this stuff, you are in over your head here

The semester is now over! Except for presentations, that is.