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In the previous video, I showed you a
process where we selected half the number

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of experiments, and I showed you
that I used a rule where C = A*B.

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Now we shouldn't automatically use these rules.

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Let me show you where I obtained it from.

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What I'm showing you here is a table that
describes how you can do less work in a way

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that you can still recover
most of the information.

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This table doesn't really have a name that
I'm aware of, so I call it a 'trade-off table'

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because it allows me to plan and
budget for a set of experiments

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where I can trade off the costs with
the information I hope to obtain.

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You're going to see this table several times
again throughout the rest of the course

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but for now: remember how in the
previous class we did four experiments

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and we had three factors in our system?

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On this table, we can locate ourselves
over here in the top left corner,

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with four runs and three factors.

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In this entry in the table, we are told
to generate C as the product of AB.

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Now remember in the wastewater example,
where we had factors C, T and S?

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That would correspond to
setting S as the product of C*T

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if we were using those factor names.

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When you're dealing with your case and your
factor names are different, simply replace them

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with A, B, Cs temporarily
to use this trade-off table.

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Now I guess you're curious about
that plus and minus sign here.

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If I use the minus, it says to
generate -C as the product of AB.

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In our wastewater example, that would
translate to -S is the product of CT.

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If you created your design following
the rule with the minus sign,

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you'd notice that you would
end up with the closed circles

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on the original cube, rather
than the open circles.

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But notice the symmetry.

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The closed circles still have

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that same collapsibility we noticed
earlier with the open circles.

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Now with so many learners from a wide
range of backgrounds in this course,

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I know that some of you might have
difficulty with the math that follows.

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Be patient.

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Watch it all unfold here on the screen.

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And I'll explain it in the
end in plain language.

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We're hoping to appeal to those of
you with a technical background,

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and also those of you with
a non-technical background.

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So let's get started with that math.

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Start by writing out the standard order table.

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Three factors means that
there are eight experiments.

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But in the last video, I showed
you that the best four experiments

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to pick are either the open
circles or the closed circles.

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Let's work with the open circles, they
corresponded to rows two, three, five,

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and eight from the original
standard order table.

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So I'm going to remove those other four
rows, the experiments we didn't run,

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and leave only the four behind
that we actually ran.

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Now I'm going to slightly
rearrange them for you.

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Put row five first, and then
original row two and three,

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and then finally the original
eighth row becomes our fourth row.

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Notice that columns A and B are in standard
order and that column C is the product of A

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and B. In a prior video I showed you the
matrix form for a three factor system,

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in the context of the wastewater
treatment example.

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The matrix form adds extra columns for the
two factor interactions and an extra column

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for the three factor interaction.

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There were eight unknowns in that
example, in that vector b. So I'm going

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to remove those other four rows,
the experiments we didn't run,

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and leave only the four behind
that we actually ran.

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Now I'm going to slightly
rearrange them for you.

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Put row five first, and then
the original row two and three,

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and then finally the original
row eight becomes our fourth row.

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Examine any patterns you
notice in the eight columns.

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Did you pick up on the similar
columns in the matrix?

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For example, notice that the
column AB matches the column for C.

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And that the B column has the same recurring
pattern as the AC column, minus minus plus plus.

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The BC column matches A. And finally, the
ABC column matches the intercept column.

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What you need to take away
from this explanation is

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that you notice the same pattern
reoccurring in certain columns.

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What this means is we won't be able to
tell those columns apart from each other.

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Telling columns apart is critical.

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In the prior four factorials,
perhaps you noticed

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that each column was unique and independent.

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This helps us know the unique contribution of
that factor to the outcome, the y variable.

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What does it mean if we cannot
tell the columns apart?

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Let's use A and the BC columns as an example.

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By doing this fractional factorial,
from a mathematical perspective,

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those two columns appear to be the same.

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We use the word alias to describe the situation.

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Now, you may be familiar with the word
alias as used in the English language.

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For example, if I asked you,
who is Jorge Mario Bergoglio?

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You may not necessarily know.

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But you do know his alias, Pope Francis.

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We all have aliases.

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In the real world, my family and friends
know me by my full name, Kevin George Dunn.

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But, online, my alias is my email
address or my username for a website.

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So an alias simply means we have
another name for something else.

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Back to these experiments, where the alias for
A is B times C. There are three other aliases,

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B is the alias for A times C,
C is the alias for A times B,

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and that one was intentional, remember.

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We chose to set C as the
product of A and B in our design.

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Finally, the last alias is the three
factor interaction between A, B,

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and C, is aliased with the intercept.

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Now when I talked about not being able to
distinguish between A and the BC interaction,

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or not being able to tell the columns
apart, that is a consequence of aliasing.

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The effects of A and the effects

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of the BC interaction are aliased
or mixed up with each other.

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So if there's a large effect size, it
might be due to A, or it might be due

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to the BC two factor interaction.

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We won't know until we run more experiments.

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The term confounding is used to
describe what is happening here.

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That means, after doing the
reduced set of four experiments,

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you will never really be sure whether it
was A that caused the change in the outcome,

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or whether it was the two factor interaction BC.

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Confounded is a word that means confused, it
means the effect of A is confused or mixed

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up with or confounded with the BC interaction.

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We cannot tell them apart, A is
an alias for BC and BC is an alias

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for A. That's the price we pay for doing
half the experiments, and in some cases,

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it is a price that is worth paying.

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Let's review the math.

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Those with a background in this sort of thing
will recognize that one way to solve a set

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of underdetermined equations where
there are more unknowns than equations,

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is to collapse these columns together to
achieve a square system that can be solved.

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Now you can see algebraically
how this confounding develops.

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The coefficient for A in the model is a
combination of the original A, plus BC.

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Similarly, the coefficient for B is the
sum of the original B plus AC, and so on.

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These four entries here are our aliases.

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It also explains why you see only four
coefficients in the R software outputs

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and NA values for the remainder of them.

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R notices that there are aliased terms in the
full system that we requested a model for,

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but it will only report the
coefficient for one of the aliases.

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To end off the class, you might be
concerned that you've lost quite a bit

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of information by using a fractional factorial.

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In the next video we see the
situation isn't quite so bad.

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Using prior knowledge about our process we can
actually benefit from knowing about the aliases.

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Now hopefully this video has not
left you too confused or confounded.

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Just to summarize, there are three
new terms that you must be comfortable

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with from today's class, half
fractions, aliasing, and confounding.

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We're going to use them a
lot more in the next class.