Fractional Factorials

2k  p _ Designs
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
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
 many terms are confounded with each other
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
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
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
- + + - + +
This Identity gives us two (incomplete) Blocks
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)
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?
It’s all Group theory, of course! A semigroup has
an operator * such that:
*: G  G  G
(a * b) * c  a *(b * c)
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
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.
Is our semigroup, G a Group?
Yes, as long as we include:
AA=A2=I and
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.
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
This is the minimum we want out of any fractional factorial
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
 So for a 25-1 design we use I=ABCDE for the Identity to
generate the design
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
 So how about a 26-2 fractional factorial?
 Your client can afford 16 observations and have some
money left over for further experiments!
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?
What are the incomplete blocks that my
treatment combinations go to?
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…
Where did ABEF come from?
ABEF is called the Generalized Interaction of ABCD and
That part was not intuitive (unless you are a Math head),
but everything is accounted for now
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
Back to the 2k-2
The Identity was I=ABCD=CDEF=ABEF
This tells us that:
 etc.
 So main effects are not confounded with each other, but
two ways are confounded with each other
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
 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
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)
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
 For a 26-2 with I=ABC=DEF=ABCDEF, the Resolution is
 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
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.