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Okay, we're into the final stretch here.

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And let's go back to our 5-factor example.

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Remember there, we had the defining
relationship that: I = ABD = ACE = BCDE So,

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the way we use this defining relationship, is
that if we want to find the terms that are going

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to be aliased with, for example,
factor B, we go and multiply every word

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in the defining relationship with that
letter B. So let's see what we get:

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IB = ABDB = ACEB = BCDEB
We can simplify that a bit.

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Remember the rule that any two letters can
be dropped away when they are the same,

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because they are equal to the
identity, or a column of ones.

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So this becomes: B = AD = ABCE =
CDE Let's interpret that quickly.

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It tells me that factor B is going to
be aliased with the AD interaction,

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as well as the fourth order interaction of
ABCE and the third order interaction of CDE.

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Now typically, we'll ignore interactions
equal to third order or higher

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because they're almost are never existent.

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So really, the only practical interaction
confounded with the main effect of B,

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is the two factor interaction of AD.

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Try this for yourself now, and figure
out what factor C is confounded with.

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C = ABCD = AE = BDE One last one,
and this is a little surprising.

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Find out what factor A is confounded with,
and before you go further with the video,

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try to also figure out in your mind what
the practical implication of your answer is.

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A = BD = CE = ABCDE The implication is
clearer, when I write out the aliasing,

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only showing those aliases
up to the second order level.

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So, ignoring third factor
and higher interactions.

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So what I learned here, is that I would
plan my experiments ahead of time,

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to assign to factor A, a factor that I'm
not too concerned about estimating clearly.

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That's because factor A is confounded
with two second order interactions.

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The other factors are only confounded
with a single second-order interaction.

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For example, in a baking experiment I might be
curious about the effect of baking temperature.

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But I'm only including that
temperature factor because I'm curious,

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not for actual use in the future.

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So then, I'm quite happy to go
assign temperature to be factor A.

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Because if it is going to be confounded
with two of these second-order interactions,

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I might not be too concerned about it.

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Factors that I really want to estimate
clearly, I will assign either to B, C, D,

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or E. Another way that I can use this: let's
say, I know that there's no physical way

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that factor A times B could interact.

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Maybe factor A is the baking temperature

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and factor B is the stirring
speed when I made my recipe.

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It's unlikely that there will be
an interaction between those two.

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Now what I can go do, is I can go
recognize, factor D is now aliased

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with a term that is guaranteed to be zero.

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So this estimate of factor D, is going
to be a pure estimate of that factor.

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One final item I'd like to draw your attention
to, is to notice that many factors are aliased

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with two factor interactions, in this example.

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We call that the "resolution" of the design.

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Resolution gives me an idea of the
level of confounding in the design.

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And I'm going to talk about that in a minute.

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Now you could step back and
ask, what if you wanted a design

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when main effects are only aliased
with 3rd order interactions?

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Is that possible?

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Well, that is possible.

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Such fractional factorial designs
are called resolution IV designs.

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The three in a three-factor
interaction plus a one

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from the main effect adds up to give you a four.

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I'm going to give you a useful
way to remember that rule shortly.

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But before we do that, in the prior video, did
you try the example where I'd asked you to look

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at 16 experiments using six factors?

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If you did, you would have recalled that
the defining relationship there was:

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I = ABCE = BCDF = ADEF Let's quickly go check
what factor A, the main effect, is aliased with.

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If you do the calculation, you see that A
is aliased: A = BCE = ABCDF = DEF Notice

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that there are no 2-factor interactions.

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The lowest interaction here
is a third order interaction.

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This is a great design if you want good,
clear estimates of the main effect.

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Because as we've said several times now, third
order or higher interaction seldom exist.

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What would a 2-factor interaction
be aliased with in this example?

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Try that on the CD interaction.

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Let's go check: CD = ABDE = BF = ACEF
It seems that this 2-factor interaction,

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is only aliased with another
2-factor interaction.

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If you've got lots of time on your hand,
you can go try all possible combinations

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to go make sure that this general rule holds.

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That two-factor interactions are only aliased

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with other two-factor interactions
and higher, in this design.

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Fortunately other people have gone and
done the work and reported it for us.

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The answer has actually been here
all along in the trade off table.

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See the Roman numeral in each cell?

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That is the resolution of the experiment.

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In the example with 16 experiments and six
factors, we had a resolution IV design.

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In our prior example, with five
factors and eight experiments,

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we had what was called a resolution III design.

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Because main effects were confounded
with two-factor interactions.

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The resolution is always equal to the length of
the shortest word in the defining relationship.

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So, here's one useful way
to remember resolution.

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If you have a resolution III design,
you're going to have confounding

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between main effects and
two-factor interactions.

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If you have a resolution IV design,
then you're going to have confounding

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between two-factor interactions
and other two-factor interactions.

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Also, your main effects are going to be
confounded with three factor interactions.

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And then finally the resolution V
design has main effects confounded

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with four-factor interactions, and your
three-factor interactions are confounded

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with two-factor interactions and vice versa.

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Resolution III designs are what
are called "screening designs".

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They are screening to see which
of the factors are important.

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We know some of them will be.

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We just aren't sure which ones yet.

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We're willing to confound main effects

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and two factor interactions
during a screening experiment.

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Resolution IV designs are used when you need a
bit of, um, resolution or clarity in your model.

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These models are perfectly good at making robust
predictions and work well for many situations.

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They are good for characterizing
or describing a system.

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The ultimate level of resolution
is a level V design.

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Such designs have very high fidelity
predictions and should be the design you pick

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when you have a large budget,
and you really need to model

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and predict subtle interactions in the system.

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In the course textbook we show two examples,

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one with seven factors and
another with eight factors.

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Go through those examples and really
see how fractional factorials are used.

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All the details on calculating the generators

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and the defining relationships
are given in those examples.

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You can use these quick sequence
of steps as a general approach.

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Read the generator from the trade off table.

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Multiply the generators to
express them as I = ....

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Take all combinations of the words
from the rearranged generators

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to form the defining relationship Remember
the defining relationship has two to the power

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of "p" words, where "p" is the measure
of reduction in our experimental efforts.

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Use the full defining relationship
to compute the aliasing pattern.

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Lastly, use the full defining relationship
to compute the aliasing pattern.

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Make sure the aliasing is not problematic.

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If it is, go ahead and reassign your
letters, or pick a different design.

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Otherwise, you're ready to
start your experiments.

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The general rule, is to pick a design
that has the highest resolution possible.

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That means, we should move over to the
left of the table but this is counteracted

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by our general desire to include as many factors
as we can especially for screening designs.

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That says we should move over to
the right hand side of the table.

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That's why this is a tradeoff table, include
as many factors as you can afford over here

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on the right but get pulled over to
the left again, to get the resolution

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that matches your objective,
and that's the key thing,

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what is your objective with these experiments.

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I often work with companies, and regularly
see them deciding to eliminate factors,

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so that they can get a high resolution,
or a full factorial design even.

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This is before they have run
even a single experiment.

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If you're just starting out and
trying to learn about your system,

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it is a good bet that you want to be
as far over to the right as you can.

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And include as many factors as you can.

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It is premature to eliminate
factors based on your intuition.

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Rather use the evidence from the
experimental data, to prove to yourself

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and to your colleagues that you
can now eliminate that factor.

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There's an example to demonstrate
this in the next video.

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We know this has been a long module; the
concepts introduced are critical though

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for saving yourself the time, and money,
instead of running full factorials.