Understanding Fractional Factorials in Experimental Designs
Explore the concept of fractional factorials in experimental designs, including the basics, factors, terms estimation, confounding, and practical considerations for running treatment combinations. Learn how to generate incomplete blocks, use orthogonal contrasts, identify confounded terms, and allocate treatment combinations efficiently.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
E N D
Presentation Transcript
k p 2 _ Designs 1
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 2
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 3
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 4
Lets 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) 5
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 bc abc ABC - + + - + - - + 6
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) 8
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? 9
Its all Group theory, of course! A semigroup has an operator * such that: G G = *: ( * )* a b G a *( * ) b c c 10
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 done this in terms of Matrices but we ll stick to the shorthand notation. 11
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. 12
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 13
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 ( ) 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 14
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! 15
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 fraction We need to choose two terms in the Model to set up two incomplete blocks How about ABCD and CDEF? 16
What are the incomplete blocks that my treatment combinations go to? 17
What is the pattern of confounding? Well, if we ran only 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 18
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!! 19
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 20
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 21
Is there an easy way to tell? For once, the answer is yes (WooHoo!, , 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) 22
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 23
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. 24
