Okay, we're into the final stretch here.
And let's go back to our 5-factor example.
Remember there, we had the defining relationship that: I = ABD = ACE = BCDE So,
the way we use this defining relationship, is that if we want to find the terms that are going
to be aliased with, for example, factor B, we go and multiply every word
in the defining relationship with that letter B. So let's see what we get:
IB = ABDB = ACEB = BCDEB We can simplify that a bit.
Remember the rule that any two letters can be dropped away when they are the same,
because they are equal to the identity, or a column of ones.
So this becomes: B = AD = ABCE = CDE Let's interpret that quickly.
It tells me that factor B is going to be aliased with the AD interaction,
as well as the fourth order interaction of ABCE and the third order interaction of CDE.
Now typically, we'll ignore interactions equal to third order or higher
because they're almost are never existent.
So really, the only practical interaction confounded with the main effect of B,
is the two factor interaction of AD.
Try this for yourself now, and figure out what factor C is confounded with.
C = ABCD = AE = BDE One last one, and this is a little surprising.
Find out what factor A is confounded with, and before you go further with the video,
try to also figure out in your mind what the practical implication of your answer is.
A = BD = CE = ABCDE The implication is clearer, when I write out the aliasing,
only showing those aliases up to the second order level.
So, ignoring third factor and higher interactions.
So what I learned here, is that I would plan my experiments ahead of time,
to assign to factor A, a factor that I'm not too concerned about estimating clearly.
That's because factor A is confounded with two second order interactions.
The other factors are only confounded with a single second-order interaction.
For example, in a baking experiment I might be curious about the effect of baking temperature.
But I'm only including that temperature factor because I'm curious,
not for actual use in the future.
So then, I'm quite happy to go assign temperature to be factor A.
Because if it is going to be confounded with two of these second-order interactions,
I might not be too concerned about it.
Factors that I really want to estimate clearly, I will assign either to B, C, D,
or E. Another way that I can use this: let's say, I know that there's no physical way
that factor A times B could interact.
Maybe factor A is the baking temperature
and factor B is the stirring speed when I made my recipe.
It's unlikely that there will be an interaction between those two.
Now what I can go do, is I can go recognize, factor D is now aliased
with a term that is guaranteed to be zero.
So this estimate of factor D, is going to be a pure estimate of that factor.
One final item I'd like to draw your attention to, is to notice that many factors are aliased
with two factor interactions, in this example.
We call that the "resolution" of the design.
Resolution gives me an idea of the level of confounding in the design.
And I'm going to talk about that in a minute.
Now you could step back and ask, what if you wanted a design
when main effects are only aliased with 3rd order interactions?
Is that possible?
Well, that is possible.
Such fractional factorial designs are called resolution IV designs.
The three in a three-factor interaction plus a one
from the main effect adds up to give you a four.
I'm going to give you a useful way to remember that rule shortly.
But before we do that, in the prior video, did you try the example where I'd asked you to look
at 16 experiments using six factors?
If you did, you would have recalled that the defining relationship there was:
I = ABCE = BCDF = ADEF Let's quickly go check what factor A, the main effect, is aliased with.
If you do the calculation, you see that A is aliased: A = BCE = ABCDF = DEF Notice
that there are no 2-factor interactions.
The lowest interaction here is a third order interaction.
This is a great design if you want good, clear estimates of the main effect.
Because as we've said several times now, third order or higher interaction seldom exist.
What would a 2-factor interaction be aliased with in this example?
Try that on the CD interaction.
Let's go check: CD = ABDE = BF = ACEF It seems that this 2-factor interaction,
is only aliased with another 2-factor interaction.
If you've got lots of time on your hand, you can go try all possible combinations
to go make sure that this general rule holds.
That two-factor interactions are only aliased
with other two-factor interactions and higher, in this design.
Fortunately other people have gone and done the work and reported it for us.
The answer has actually been here all along in the trade off table.
See the Roman numeral in each cell?
That is the resolution of the experiment.
In the example with 16 experiments and six factors, we had a resolution IV design.
In our prior example, with five factors and eight experiments,
we had what was called a resolution III design.
Because main effects were confounded with two-factor interactions.
The resolution is always equal to the length of the shortest word in the defining relationship.
So, here's one useful way to remember resolution.
If you have a resolution III design, you're going to have confounding
between main effects and two-factor interactions.
If you have a resolution IV design, then you're going to have confounding
between two-factor interactions and other two-factor interactions.
Also, your main effects are going to be confounded with three factor interactions.
And then finally the resolution V design has main effects confounded
with four-factor interactions, and your three-factor interactions are confounded
with two-factor interactions and vice versa.
Resolution III designs are what are called "screening designs".
They are screening to see which of the factors are important.
We know some of them will be.
We just aren't sure which ones yet.
We're willing to confound main effects
and two factor interactions during a screening experiment.
Resolution IV designs are used when you need a bit of, um, resolution or clarity in your model.
These models are perfectly good at making robust predictions and work well for many situations.
They are good for characterizing or describing a system.
The ultimate level of resolution is a level V design.
Such designs have very high fidelity predictions and should be the design you pick
when you have a large budget, and you really need to model
and predict subtle interactions in the system.
In the course textbook we show two examples,
one with seven factors and another with eight factors.
Go through those examples and really see how fractional factorials are used.
All the details on calculating the generators
and the defining relationships are given in those examples.
You can use these quick sequence of steps as a general approach.
Read the generator from the trade off table.
Multiply the generators to express them as I = ....
Take all combinations of the words from the rearranged generators
to form the defining relationship Remember the defining relationship has two to the power
of "p" words, where "p" is the measure of reduction in our experimental efforts.
Use the full defining relationship to compute the aliasing pattern.
Lastly, use the full defining relationship to compute the aliasing pattern.
Make sure the aliasing is not problematic.
If it is, go ahead and reassign your letters, or pick a different design.
Otherwise, you're ready to start your experiments.
The general rule, is to pick a design that has the highest resolution possible.
That means, we should move over to the left of the table but this is counteracted
by our general desire to include as many factors as we can especially for screening designs.
That says we should move over to the right hand side of the table.
That's why this is a tradeoff table, include as many factors as you can afford over here
on the right but get pulled over to the left again, to get the resolution
that matches your objective, and that's the key thing,
what is your objective with these experiments.
I often work with companies, and regularly see them deciding to eliminate factors,
so that they can get a high resolution, or a full factorial design even.
This is before they have run even a single experiment.
If you're just starting out and trying to learn about your system,
it is a good bet that you want to be as far over to the right as you can.
And include as many factors as you can.
It is premature to eliminate factors based on your intuition.
Rather use the evidence from the experimental data, to prove to yourself
and to your colleagues that you can now eliminate that factor.
There's an example to demonstrate this in the next video.
We know this has been a long module; the concepts introduced are critical though
for saving yourself the time, and money, instead of running full factorials.