- [Charlie] One of the last things
you're gonna learn in this course
is how to set up and analyze fractional factorial designs.
And so that's what I'm gonna be talking about
in this screencast.
Fractional factorial experiments are important
when full factorial designs are too large
or are not necessary.
So if this is the case
a fraction of the full design may be employed.
This is particularly useful
when we have higher order interactions
that are likely to be negligible
when we are screening for factors
that have large effect on the measured variable.
In particular, this is used early on
when you're trying to screen through the important factors.
We've got an example here.
We have a chemical reaction, A plus B going to C.
You hypothesize that there might be five things
five factors that have an effect on the reaction yield,
temperature, time, pH, pressure,
and concentration of substrate.
Because it is likely that only one
or two factors actually impact the reaction yield.
We may reduce the number of essential experiments
by first screening for the key factors
and so we've got a lot of factors here.
We have five, we're early on in our characterization,
higher order interactions are likely to be negligible
or we are not very interested
in those higher order interactions at this point
and so we're just kind of trying to screen through
to eliminate the insignificant factors.
This is an example of where we might wanna use
a fractional factorial design.
Let's talk about some terminology, a half fraction design.
That's when you're only carrying out half of the number
of experimental treatment combinations
as you would in a full factorial design.
We have k factors, but only two to the k minus one runs
and that's how we refer to these as
we refer to this as two to the k minus one.
For example, if we had five factors, there would be two
to the fifth or 32 runs in a full factorial design
but in a half fraction design, we only need two
to the five minus one or 16 runs.
A quarter fraction design,
we have k factors, but only two to the k minus two runs.
For example, if we had eight factors,
there'd be 256 runs in a full factorial design.
But in a quarter fraction design,
we would have two to the eight minus two equals 64 runs.
In general, we define a fractional design
as two to the k minus p.
So let's revisit a full factorial design.
We have two to the fifth.
We have 32 different possible experiments
or treatment combinations as shown here on the right.
The sparsity of effects principle basically tells us
that a system is generally dominated
by the main effects and low order interactions.
Three factor and higher order interactions
are generally negligible.
For a two to the fifth design,
we have five degrees of freedom
that correspond to main effects,
10 that correspond to two factor interactions
and 16 degrees of freedom that correspond to three,
four, and five factor interactions.
So based on the sparsity of effects principle
if we neglect the 16 higher order interactions
and use these degrees of freedom
these 16 degrees of freedom for error
it's quite easy to perform adequate statistical analysis
on the main and binary effects.
And this is what I've shown you
in some previous screencasts.
But now if we look at a two to the five minus one
half fractional factorial design
we would have only 16 experiments or observations.
We have 32 possible treatment combinations
shown over here on the right.
And so the question becomes how do we decide
which of these treatment combinations
to actually put together and perform?
For a two to the five minus one design
with only 16 observations,
that means we only have 15 degrees of freedom.
We can only assign estimates therefore to 15
of our total 31 total effects,
main binary, ternary, quaternary
and fifth order interactions.
So we basically only have half
the required degrees of freedom.
So how do we proceed?
Somehow we have to pair down these 32
possible treatment combinations down to 16.
Using the sparsity of effects principle.
We can make some assumptions
with respect to higher order terms.
We can assume that the third, fourth and fifth order
terms are negligible, but still even if we do that
if we neglect all the third order and higher interactions
we still don't have enough degrees of freedom for error
because we have 15 main effects and two factor interactions
and we only have 15 degrees of freedom.
And so when we're statistically analyzing
these fractional factorial designs
it becomes a little more complicated.
So let's start out with the simplest half fraction design
two to the three minus one.
Up above here this is a full two cube design,
which has eight possible experiment
or treatment combinations.
So how do we pair down
and make this a half fractional design?
Which of these treatment combinations
do we choose to eliminate in which do we choose
to actually collect data for?
If unreplicated, if any equals one,
then we'll have four observations
and the question becomes which four of the eight to use?
One option for this two to the three minus one design
is if we sort by column AB and we only retain those rows
that have AB equal to one.
If we did this, you notice that if we computed the contrast
the contrasts are computed by taking the dot product
of these columns with the actual treatment responses
we would not be able to tell the difference between A and B.
You notice that they're exactly the same nor AC and BC
those are exactly the same.
And C and ABC, those are exactly the same.
So we have a problem here.
This is a very poor choice since A and B
has an example are aliased.
That's a term that we're gonna use quite a bit
when we talk about fractional factorial designs.
If the AB effect, remember they're combined
because the computation of them them is exactly the same.
So if that ab effect is significant,
we don't know if it's due to effect A
or if it's due to effect B.
Another option for the two to the three minus one design
is if we sorted our eight original rows.
Remember, we're starting with this full
two to the three design
and we're trying to decide which of these four to keep.
So if we sorted by the first column when A equals one
we would retain these four treatment combinations.
Here we would not be able to tell the difference
between B and AB, nor C and AC, nor BC and ABC.
This is a little bit different
than the previous one that I showed
since the main effects B and C are aliased
with binary effects which are generally
not as significant as the main effects.
So this is a little bit better,
but we still have some improvements
that we can make for this two to the three minus one design
and that is when we sort by the last column, ABC equals one.
Again, we're starting with those eight full rows
of the two to the three design
and we're trying to decide which ones to keep.
If we sorted by the last column
ABC and only kept those where ABC equals plus one,
what happens here is we will not be able
to tell the difference between A and BC,
nor B and AC, nor C and AB.
But this is a lot better
because all of the main effects are aliased
with binaries of the other factors.
For example, A is aliased with the interaction BC.
And so this is actually the best possibility
for a two to the three minus one design.
Now, keep in mind this aliasing.
Again if we calculated the contrast for A and BC,
they're exactly the same.
Generally because of what we know about
higher order interactions, main effects generally
have more significance than binary interactions.
So if we determined that the A/BC effect was significant,
we could probably be confident
that most of that significance
was due to the main effect of A and not the binary BC.
Here ABC has been chosen as the design generator
so that's something that we'll talk about quite a bit
when we're talking about fractional factorial designs.
I equals ABC is the design generator.
If you multiply both sides of the design generator by C
you would get C equals ABC squared.
Anything squared is just equal to I,
which is equivalent to one.
And so really what we're saying is the C column up here
is equal to AB.
So we're only performing those experiments
where C equals the product of AB.
Table 14-30 in your book
outlines a bunch of different fractional factorial designs.
We have the number of factors, k, the fraction.
We've already talked about the two to the three minus one.
There's all sorts of different designs.
This table shows you the number of runs
and also the design generators.
So that C equals AB is what I just referred to.
You notice that there's plus and minus for all of these.
It's typical to choose the plus option
and that's known as the principle fraction.
The negative value is known as the alternate fraction.
So here we chose C equals AB
but we could also choose C equals negative AB.
So let's just kind of summarize
what we concluded with the half fractional design
two to the three minus one.
In the table column A equals column B
and similarly B equals AC and C equals AB,
and ABC equals I.
We can thus say that A and BC are aliases
as are B and AC and C and AB as I discussed earlier.
As a result, the effects of A and BC
cannot be differentiated using this design
and same with the other equalities.
Again, typically binary interactions like BC
are significantly less than main effects.
And so if we are getting a significant effect
due to the combination of A and BC,
it's likely that most of that
is caused by the main effect of A.
Also, whenever we have a column equal to I
like ABC equals I,
we cannot determine the effect for that column.
In order to determine effects,
you have to be able to calculate the contrast
and the contrast you need both positive and negative values.
So if we have all ones in a column
that means we cannot calculate the contrast
nor effect of that particular interaction.
So for this design we cannot determine
the ternary effect ABC.
So what's gained in a half fractional design?
You only have to run four experiments
to identify likely important main effects.
So if these experiments are very expensive
maybe you're having to test something for weeks
just to get one replicate,
then a half fractional design might be a good idea.
What's lost?
The ability to distinguish main and binary effects is lost.
In other words,
we lose resolution in elucidating the effects.
So there's kind of a trade off, less money, less time,
but you're not able to distinguish
between the main and binary effects quite as much.