Fractional Factorials

1. Fractional Factorials

2. 2k-p Designs • k factors • two levels for each factor • will only run 2-p of the possible factor combinations • will only run 2k-p observations total

3. Basic facts about fractional factorials • will only have 2k-p-1 degrees of freedom total • can 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

4. 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

5. Let’s start • The minimum we want out of any fractional factorial is to retrieve Main Effects • Pick an Identity to generate incomplete Blocks for the fractional factorial • Use Orthogonal Contrasts to get the Identity!!! • (knew Dr. Tom was going over those for some reason :p)

6. Easiest case, 2k-1, a half fraction • Recall the three factor design with A, B and C • There were eight treatment combinations • Lets look at the contrast for ABC (1) a b ab c ac bcabc ABC - + + - + - - +

7. This Identity gives us two (incomplete) Blocks

8. How do we know which treatment combinations go in the minus (-) block and which in the plus (+) block? • See what factors are in the experiment • For the 23-1 it is Factors A, B and C • List the treatment combinations, i.e. (1), a, b, ab, c, ac, bc, abc • If 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)

9. The Identity • I=ABC is the “Identity” • The Identity gives us the incomplete blocks • The Identity tells us the pattern of confounding • A=A2BC=BC, for example • and how do we know that, pray tell?

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

11. 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 C and 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 don’t this in terms of Matrices but we’ll stick to the shorthand notation.

12. 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.

13. Back to the 23-1 • For the 23-1 design, we run one incomplete block • The 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=BC • So at least the Main Effects are not confounded with each other • This is the minimum we want out of any fractional factorial

14. 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

15. What if…. • My client comes in with six factors • I recommend a 26-1 fractional factorial • My client asks how many observations • You say 32 • Client says that will use up the rest of the research money • So how about a 26-2 fractional factorial? • Your client can afford 16 observations and have some money left over for further experiments!

16. 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 fraction • We need to choose two terms in the Model to set up two incomplete blocks • How about ABCD and CDEF?

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

18. 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=ABEF • Where did ABEF come from? • ABCDCDEF=ABC2D2EF=ABEF • ABEF is called the Generalized Interaction of ABCD and CDEF • That part was not intuitive (unless you are a Math head), but everything is accounted for now

19. Review of the basics: • We generate the design by choosing one or more terms in the model to set up our incomplete blocks • The number of terms we chose determines the degree of the fractional factorial • One term gives a 2k-1, two terms gives a 2k-2, three terms gives a 2k-3 and so forth • The terms, along with the generalized interactions, give the Identity • The Identity tells us what is confounded with what • Whew!!

20. Back to the 2k-2 The Identity was I=ABCD=CDEF=ABEF This tells us that: • A=BCD=ACDEF=BEF • B=ACD=BCDEF=AEF • AB=CD=ABCDEF=EF • etc. • So main effects are not confounded with each other, but two ways are confounded with each other

21. How many fractional factorials are there and how do I choose which one to use? • Lots of possibilities exist • You decide based upon which terms you are interested in estimating/testing • Generally, this means you want Main effects at the minimum • You don’t want Main effects confounded with things that are likely to be significant • Low order terms are likely to be significant, high order terms are less likely to be insignificant

22. 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 better • Most 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)

23. For example: • For a 23-1 with I=ABC, the Resolution is III • For a 24-1 with I=ABCD, the Resolution is IV • For a 26-2 with I=ABCD=CDEF=ABEF, the Resolution is IV • For a 26-2 with I=ABC=DEF=ABCDEF, the Resolution is III • Resolution 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

24. Does this get more complicated? • Of course • You may need to run the experiment in Blocks • You may have more than two levels in some of your factors • You 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.