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In this video, I am going to show
you how we can analyze the results

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from the wastewater treatment example
we were considering in the prior module.

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Remember how tedious it was to
analyze the results by hand?

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I am going to show you a really fast
way of using R in today's class.

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In the previous video, I showed you how we used
the software, RStudio, to analyze the results

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for a very small two-factor system.

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Open RStudio now, and start by creating a new
file for this wastewater treatment example.

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And allow me to work backwards again.

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Start by creating a least squares model
called "water", which is a linear model,

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where we predict the outcome
value "y" from several factors.

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Remember that we had three factors in a water
treatment example, "C" the chemical factor,

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"T" the temperature, and "S"
the stirring speed factor.

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We also have several two
factor interactions, CT, CS,

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and ST. And there's also a
three factor interaction: CTS.

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In the water treatment example, recall that we
had eight experiments required for this system.

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We learned that we will always
require at least as many experiments

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as there are parameters being estimated.

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For example, in the popcorn video, we
had four parameters and four experiments.

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In this example, we have eight experiments.

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So we are able to estimate eight parameters.

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However, as before, we see only
seven of them represented here.

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The main effects of C, T and S,
the three two factor interactions,

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and the three factor interaction.

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The eighth parameter, the
intercept, is built in.

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R will automatically calculate that for us.

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So there are actually eight
parameters here, in our linear model.

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We need to define what C, T and S are, and
also provide the outcome variable, "y".

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We can let R automatically create
C, T and S, using the special form

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of the command, shown on the screen.

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The first line of the code,
creates three variables in one line.

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If we inspect the variables, we can
see they are simply -1, followed by +1.

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Next, expand this into the standard order
table, using the code shown here on the screen.

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We need to also extract from
this, the C, T, and S columns.

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These are the columns from
our standard order table.

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Feel free to reuse this code at the start
of every experimental analysis you do

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in R. The last step that we need, is the "y"
vector containing the eight outcome values.

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We can take these directly
from the standard order table.

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Now we are ready to let the
software calculate the model.

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Run this code to create the linear
model; and use the "summary" command

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to display the model on the screen.

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Notice that the parameters in this prediction
model are identical to those we calculated

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in the prior module: 11.25,
6.25, 0.75, and so on.

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I want to show you a quick shortcut that you can
use in R. Instead of writing the linear model

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by hand, where you could make mistakes writing
out all those two and three factor interactions;

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rather use this notation shown on the screen.

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This gives you exactly the
same model as you had before.

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Now it's time for some advice.

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Always perform your analysis using
software code that you write out by hand.

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This is a permanent record of your work and is
especially helpful if you add many comments.

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Devon: Um, that seems like a lot of work.

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Couldn't I use Excel, or
other statistical tools to,

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sort of click a few buttons
and get the same result?

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Kevin: Absolutely, you can use
those other software tools.

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The problem often is, and I've seen
this happen so many times in companies

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that I've worked with, is that you do the
work and then a few months later you have

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to come back to it, and try to
answer questions from your boss.

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Or another colleague continues
on with the project.

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If you only give them an Excel file,

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or some document that doesn't
record the exact steps you took,

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it's very hard to reconstruct
what you were doing,

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and what was going through
your head at the time.

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Writing out the code like this
explicitly creates a very traceable

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and reproducible record of your work.

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This is so important in many companies

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where there are regulatory
requirements for traceability.

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There is one other piece of code
I would like to share with you

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to help you visualize each
of the effects in the model.

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What I mean by this, is that we have eight
parameters and sometimes we would like to know

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which ones are the most influential
on our outcome variable.

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We can get an idea of the important factors
in our system, by examining the equation model

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in R. We pick out the numbers
that are the largest.

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It's easy to visualize this though and
here's some code that will create a barplot.

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The bar plot shows the absolute
value of the model parameters.

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Why should we use the absolute values?

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We do this, because we want to
compare the magnitude of each factor.

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The sign is important for sure, but it is easier
to compare large negatives and large positives

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if they're on the same side of the plot.

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In order to retain the sign information, I've
used light grey for negative coefficients

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and black for positive coefficients.

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This is important: not everyone can
perceive colour, or sometimes you have

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to print a report in black and white.

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You can modify how you use the code to
get alternative colour schemes though.

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R is really flexible in this way.

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Now we need to interpret the plot.

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We can quickly see that the C
times T times S interaction.

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And the CT and TS interactions are
really small compared to the other terms.

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Devon: Why don't you show
the intercept in the plot?

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Kevin: There are several reasons.

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We always keep an intercept term in our models.

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The intent of this Pareto plot,
is for you to compare the effect

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of the various factors against each other.

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But the intercept isn't something
that you can really change.

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These plots are often used to locate variables
that are uninteresting and then remove them.

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But since the intercept will always be important
in our models; we will never remove it,

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so we don't really need to plot it.

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Furthermore, the intercept can
sometimes be a really large value,

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relative to the other bars,
which will distort the plot.

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The plot shows us the parameters, ranked
from largest absolute value at the top,

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to smallest absolute values at the bottom.

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This quickly allows us to find the
most influential factors in our system.

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And this plot is called a Pareto plot.

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The largest magnitude bars, corresponds to those
factors which most strongly affect the outcome.

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In this case, we have S.
This colour here indicates

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that it has a reducing effect on the outcome.

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Remember that our objective was
to reduce the amount of pollution.

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So we can quickly see here,
that increasing S will result

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in less pollution, which is desirable.

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We investigated the interpretation of
this interaction term in the prior class,

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so I'm not going to repeat it over here.

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And similarly, for the bar that
represents the chemical effect,

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C. Just before finishing this example
today, I'd like to quickly share

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with you what the matrix form
of this problem looks like.

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I know that there are those of you that
are more math-oriented, and will like -

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and actually have a better
understanding of - this representation.

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There's certainly something
for everyone in this course.

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So there it is, the "X" matrix
and the "y" vector.

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And what R is doing behind the scenes,

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is finding efficiently the solution
to the least squares problem.

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To end this video, we hope that you are
enjoying using R to solve your linear models

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and to fit your outcome values
to make predictions.

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Please keep using it.

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There are plenty of practice problems in
this module which you should be attempting.

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Those problems provide full
R code in the solutions.