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Hello and welcome to

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this art tutorial
and in previous tutorials, we already

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implemented a multiple linear regression
model that we fitted to the training set.

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But when we take a step back,
do you think it's actually

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the optimal model that we can make
with the data set that we have here?

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You know,
because when we built this model,

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we actually used
all the independent variables.

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But what if

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among these independent variables
there are some

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that are highly statistically significant,
that is have great impact

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or great effect
on the dependent variable profit,

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and some that are not
statistically significant at all.

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That means that if we removed this non
statistically significant variables

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from the model, well we would still get
some amazing predictions.

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So the goal in this tutorial
is to actually find a team,

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an optimal team of independent variables,
so that each independent

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variable of the team has great impact
on the dependent variable profit.

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That is, each independent
variable of the team

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is a powerful predictor
that is highly statistically significant

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and definitely has an effect
on the dependent variable profit.

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And this effect can be positive.

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That is, for an increase of one
unit of the independent variable,

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the profit will increase
or it can be negative.

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That is, for an increase of one
unit of the independent variable

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the profit will decrease.

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And so we're going to divide
this final step of building

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this optimal model using backward
elimination into two tutorials.

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In this first tutorial that we are having
right now, I'm going to walk you

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through the backward elimination algorithm
or without completing it up to the end.

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That means that at the end of this
tutorial, you'll get a homework

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which will consist of completing
what we started with backward elimination.

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So I'm sure you will have no problem,
because I'm going to walk you through

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the introduction of backward elimination
so that you can understand everything

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and have all the tools
to complete the homework.

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And then in the next tutorial,
I'll give you the solution of this

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homework and we will complete together
the backward elimination.

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So I hope you're excited.

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Let's start right now.

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The backward elimination.

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So for those of you
who follow the Python tutorial,

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you will notice that it's
actually a little more simple here,

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because in the Python tutorial
we had to use another library and another

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multiple linear regression
model to implement backward elimination.

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And this time we will simply
take the model that we created here,

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and we will use this amazing function
summary of R that returns

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a great deal of statistical information
that can help make our model more robust.

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And then same as in Python,
we are going to do some copy paste

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very simply until we get to the final team
of our independent variables.

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So let's do this.

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You're going to see it's really quick
and easy.

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We're going to do that very efficiently.

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And the first step of doing
that is to actually take our model

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because as I just mentioned,

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we are going to use this same model
to implement backward elimination.

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So here I just copied the model
and I'm going to paste it here.

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And now we're just going to change
two things.

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The first thing is
instead of having this dot here,

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you know this dot that represents
all the independent variables,

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we we're going to write all the
independent variables separated by a plus

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because you know,
the principle of backward elimination

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is that we will remove
each independent variable

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that is not statistically
significant one by one.

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So we need here to write
each of the independent variables

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so that when we copy paste this model here
we will just need to remove

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the non statistically significant variable
from this equation here okay.

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So let's first do this

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I'm going to take our data set here
to look at the independent variables.

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So the first independent
variable is already spent.

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So let's add it here.

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So r dot d.

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So you know as a reminder
there is a dot here

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because the original name
for this independent variable is r space

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d and r just replaced the space
by a dot here.

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So it's good to know
that if you're working with R

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and if you have some data
sets with spaces in your column names,

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and then okay, so what was the name.

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Dot another dot.

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So that means another space dot and spend.

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All right.
So that's the first independent variable.

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And now let's add the second one.

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So we need to separate them by a plus
okay.

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And what is the second one.

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The second one is administration okay.

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So here there is no dot.

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Everything is fine
just as it's spelled right.

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Administration plus.

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Marketing spend.

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Plus.

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And we have one one last
independent variable which is the state.

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So here we don't need to create the dummy
variables as we did in Python.

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Because remember we used here this amazing

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factor function that encoded this state
categorical variable

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into factors that are one two, three.

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And there is no relational order
between those categories.

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So everything is fine.

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We don't need to create
any dummy variables.

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So that's the beauty of R.

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And so here same we don't need to sum
two separate dummy variables.

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We can take the original independent
variable state okay.

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So as I mentioned there are two things
that we would like to change here.

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The first thing was to replace the dot
by all this independent

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variable separated by a plus.

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And now

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the second thing that we would like to do,
but it's not compulsory, is just because

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I would like to use all the data
set to see the correlations

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is to replace here training set by

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simply our data set.

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So that's not compulsory.

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We can actually do backward elimination
using the training set.

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But we're just taking the whole data set
in order to have complete information

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about which independent variables
are statistically significant

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and which independent variables are not.

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Okay. And now actually we're almost ready.

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We just need to use the summary function,
which we actually already used before.

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And there is nothing more simple
than using the summary function.

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We just need to take the summary function
here.

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And then in parenthesis
we input our regressor.

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Here it is. And now that's actually ready.

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We're actually ready to start the
first steps of our backward elimination.

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Well speaking of backward elimination
let's have a look at the slide.

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You saw with Kirill in the intuition
tutorial.

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And here is the slide.

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So let's have a quick reminder
about the five steps here.

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So the first step is to select

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a significance level
that is a threshold for our p value

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such that if the p value of an independent
variable is below

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the significance level, then this
independent variable stays in the model.

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And if the p value of the independent
variable is higher

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than the significance level,
then it will not stay in the model.

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We will remove it.

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So. But the first step is just to select
a significance level.

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We don't have to do anything here
with the independent variables.

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We just need to choose one.

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And we're going to choose 5% 0.05 okay.

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And now step to step two is to fit the
full model with all possible predictors.

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So that's actually what we've just done.

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You know
by taking all our independent variables

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in our regressor using the LM function
that actually fits the full model

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with all the possible predictors, that is
with all the independent variables.

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Okay. So done.

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And now what is step three.

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Step three is to look at the predictor
that has the highest p value.

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So we will find it
thanks to our summary function.

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And if the p value is higher
than the significance level

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that is if it's higher than 5%,
then we'll go to step four.

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And if that's not the case
our model is actually ready.

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But don't worry,
it will not be that quick.

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So actually.

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So let's suppose

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we found the highest p value higher
than the significance level of 5%.

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Then we need to move on to step four.

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And the step four is actually to remove

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this independent variable
that has the highest p value.

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And once we remove the predictor
we're ready to move on to step five

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which is to fit the model
without this variable.

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So that's why, you know,
we wrote all the independent variables

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one by one separated by a plus.

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Because, you know, once we reached step
five here, we will just copy paste

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the regressor and the summary function
and remove this independent variable

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that had the highest p value
from the regressor

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to build a new regressor
without this variable.

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And that will fit this model
without the variable.

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And once it's done
we go back to step three here

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to repeat this same pathway that is.

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Once again we're going to look
for the independent variables

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among the new team
of independent variables

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without the independent variable
that we just removed.

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So we're going to look for the independent
variable that has a highest p value.

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And same

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if the p value is higher than the
significance level we'll go to step four.

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And otherwise our model is ready.

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So let's do this.

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We already completed step one
by choosing a significance level of 5%.

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And same for step two.

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We actually fitted the model
with all possible predictors.

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Well we need
of course to execute the code.

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And now we will move on to step three
which will consist

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of looking for the independent variable
that has the highest p value.

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So let's do this right now

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okay.

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So as
I just mentioned we need to execute this

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to build our regressor
with all the independent variables.

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Well actually we don't really need
to execute this because we actually

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executed this code section here
that builds exactly the same regressor.

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But just to complete all these steps in
this tutorial, let's execute that again.

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That will not cause any issue.

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So I'm going to press Command and Control
plus enter to execute.

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And here we go.

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Same regressor created again
with a different syntax.

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And that's because we want to remove
the non-significant independent

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variable one by one.

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Great.

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So that actually completes step two.

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And now let's move on to step three
which was to look

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for the independent variable
that has the highest p value.

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And to do this we are going to select this

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summary function with the regressor
inside and press Command and Control Plus.

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And to execute.

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Let's move that up a little bit.

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So these informations are very important

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informations
when we want to build a robust model.

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And it's not only thanks to
the informations about the P values here

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that will help select
the optimal team of independent variables.

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Because below
we also have this multiple r squared.

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And this adjusted r squared.

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That will help us build
even more robust model than the one

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we are going to make
in the next two tutorials.

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Because at the end of this part there
is this section called Evaluating models

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performance
to actually improve the model performance.

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And in this part we will actually use
the multiple r squared and the adjusted

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R squared to finalize our journey
towards the most robust model.

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And you will perfectly understand
why at the end of this part.

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But for now, let's focus on the p values.

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So the p values are actually
in this column.

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And in R there's actually a shortcut
to look at the statistical significance.

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It's this last column here.

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Well this last column doesn't have a name.

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But you need to look at the stars here
because as a reminder,

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the more the p value is below
5% our significance level,

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then the more the independent variable
will be statistically significant.

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For the dependent variable profit,
and the more the p value

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is higher than the significance level 5%,

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then the less statistically significant
the independent variable will be.

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So in short, the lower is the p value, 
the more your independent

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variable will have high impact
on your dependent variable,

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and the higher is the p value,
the less effect.

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In fact, your independent variable is
going to have on the dependent variable.

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And there is this reminder here
that says that if the p value

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is between zero and 0.1 percent,

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then it's three stars, meaning that it's
highly statistically significant.

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Then if the p value is between
0.1% and 1%,

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then it's two stars, meaning
that it's very statistically significant

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but less significant
than when there is three stars.

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Then if the p value is between 1% and 5%,

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then it's
simply statistically significant.

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That is, your independent variable

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still have some good impact
on the dependent variable.

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And then if the p value is between
5% and 10%, then it's a dot, meaning

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that there is definitely a certain level
of statistical significance.

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That is, your independent variable has a
certain effect on your dependent variable,

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but definitely not as much as your other
independent variables

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that are in these categories here,
especially for this one.

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And finally,
if your p value is between 10% and one,

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well, there's absolutely no
statistical significance.

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00:13:02,400 --> 00:13:05,266
So that means that
with what we first observe here,

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well we can see that the R&D spend
is highly statistically significant.

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But the rest seems to be not significant.

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But let's wait for the backward
elimination to be completed to find out

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if our final team is actually
only composed of already spent,

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because by removing some independent
variables here, that will remove

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some possible bias, that once
some independent variables are removed,

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we can actually find
an independent variable that is more

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statistically significant than what
it appeared to be at the beginning.

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That is at the first step here.

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So let's find out about that.

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And actually you will find out

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about that yourself because this will be
the subject of the homework.

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But don't worry I'm going to walk you
through the first steps of

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backward elimination
and you will complete it yourself.

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And in the next tutorial,
of course, we'll have the solution

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and we'll complete it together.

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So I look forward to seeing
if we get the same results.

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00:13:59,466 --> 00:14:02,733
Okay, so now let's carry on with backward
elimination.

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So remember we are at step three.

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And step three is actually to look

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for the independent variable
that has the highest p value.

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And we can find it very easily.

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It's actually this one
because indeed its p value is 0.999.

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That is 99%.

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So that's actually a very high p value.

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And we are way above the significance
level of 5%.

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So this dummy variable for state state

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two is definitely not
statistically significant at all.

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It has absolutely no effect
on the dependent variable profit.

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And by the way we also observe
that state three here has a 94% p value.

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And there is no way
that if we remove state two

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well this p value will decrease below
the 5% significance level.

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So we can actually remove this state three
independent variable as well,

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because clearly the state has no effect or
impact on the dependent variable profit.

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So we will actually make some kind
of a shortcut here.

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And instead of removing the independent
variable that has the highest p value

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that is state two, we will actually remove
both these dummy variables for state

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00:15:13,466 --> 00:15:17,100
because definitely the state
is not statistically significant.

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So let's do this.

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00:15:18,766 --> 00:15:21,766
Let's remove
the state variable from our equation.

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00:15:21,766 --> 00:15:23,533
So I'm going to put that down

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00:15:25,166 --> 00:15:25,500
okay.

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00:15:25,500 --> 00:15:27,000
So as I told you it's very simple.

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00:15:27,000 --> 00:15:31,733
We're just going to copy this
and paste it here.

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00:15:32,300 --> 00:15:34,733
And then so what do we have to do. Now.

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00:15:34,733 --> 00:15:39,200
We just need to remove the state
independent variable from our equation.

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00:15:39,200 --> 00:15:41,366
Here.

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00:15:41,366 --> 00:15:44,800
And by doing that we complete step four.

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00:15:44,866 --> 00:15:49,400
Because if we go back to our slide, step
four is to actually remove the predictor.

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00:15:49,800 --> 00:15:50,333
Great.

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00:15:50,333 --> 00:15:53,800
And now we can move on to step five
which is to fit the multiple linear

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00:15:53,800 --> 00:15:58,733
regression model without the independent
variable state that we just removed okay.

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00:15:58,733 --> 00:16:00,033
So we removed state.

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00:16:00,033 --> 00:16:02,533
So step four completed.

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00:16:02,533 --> 00:16:05,900
And now step
five is to actually fit the model

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00:16:05,900 --> 00:16:09,133
without this independent variable state
that we just removed.

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00:16:09,233 --> 00:16:10,200
So let's do this.

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00:16:10,200 --> 00:16:13,433
We just need to select this press command
and control.

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00:16:13,433 --> 00:16:14,500
Press enter to execute.

305
00:16:15,466 --> 00:16:16,500
And here it is.

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00:16:16,500 --> 00:16:20,700
Our new regressor is ready
without the state independent variable.

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00:16:20,900 --> 00:16:23,700
So now we have a team of three
independent variables.

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00:16:23,700 --> 00:16:27,400
Wait and see which independent variable
is going to be kicked out of the team.

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00:16:27,900 --> 00:16:32,200
And speaking of that, this is where I'm
going to leave you alone for the homework.

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00:16:32,466 --> 00:16:35,566
But don't worry, you will have
the solution in the next tutorial.

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00:16:35,733 --> 00:16:38,666
But really try to implement this yourself.

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00:16:38,666 --> 00:16:41,566
Complete backward
elimination up to the end,

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00:16:41,566 --> 00:16:44,133
and it will be fun
to see if we get the same results.

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00:16:44,133 --> 00:16:45,633
And there is actually kind of a decision

315
00:16:45,633 --> 00:16:47,933
to make at the end of backward
elimination.

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00:16:47,933 --> 00:16:51,633
So I'm curious to see how you make that
decision, make that call,

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00:16:51,666 --> 00:16:54,966
because both solutions are actually great

318
00:16:55,333 --> 00:16:58,300
and we will talk about that
in the solution.

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00:16:58,300 --> 00:17:01,466
So good luck for the homework
you're going to see.

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00:17:01,466 --> 00:17:02,100
It's going to be fine.

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00:17:02,100 --> 00:17:06,866
So basically what you only have to do is
to follow this backward elimination slide.

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00:17:07,200 --> 00:17:10,933
And so together in this tutorial
we went up to this step five here.

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00:17:11,233 --> 00:17:11,700
And now.

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00:17:11,700 --> 00:17:15,833
As you can see you have to go back
to step three and redo the steps

325
00:17:15,833 --> 00:17:18,833
three four, five exactly like we just did.

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00:17:18,833 --> 00:17:24,266
Until you find that the highest p value
is not higher than the significance level,

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00:17:24,533 --> 00:17:27,533
and when it's the case,
your model will be ready.

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00:17:28,233 --> 00:17:31,000
Okay, so I look forward to seeing you
in the next tutorial.

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00:17:31,000 --> 00:17:34,833
I look forward to comparing with you
your results to mine,

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00:17:35,266 --> 00:17:37,466
and I'm sure everything will be okay.

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00:17:37,466 --> 00:17:39,566
Backward elimination is very practical

332
00:17:39,566 --> 00:17:43,000
and it will actually be fun
and easy to complete this.

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00:17:43,633 --> 00:17:46,866
So thank you for watching this tutorial
and I look forward to seeing you

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00:17:46,866 --> 00:17:49,000
in the next one for the solution.

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00:17:49,000 --> 00:17:50,733
Until then, enjoy machine learning.