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So now let's build an equation
to predict points scored

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using some common
basketball statistics.

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So our dependent variable
would now be points,

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and our independent
variables would

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be some of the common
basketball statistics

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that we have in our data set.

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So for example, the number of
two-point field goal attempts,

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the number of three-point
field goal attempts,

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offensive rebounds,
defensive rebounds,

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assists, steals,
blocks, turnovers,

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free throw attempts--
we can use all of these.

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So let's build this regression
and call it PointsReg.

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And that will just be equal
to lm of PTS regressing

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on all those variables
we just talked about.

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So X2PA for two-point
attempts, plus X3PA

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for three-point attempts, plus
FTA for free throw attempts,

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AST for assists, ORB
offensive rebounds,

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DRB for defensive rebounds,
TOV for turnovers,

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and STL for steals.

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And let's also throw
in blocks, BLK.

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Okay.

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And as always, the data
is from the NBA data set.

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So we can go ahead
and run this command.

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So now that we've created
our points regression model,

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let's take a look
at the summary.

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This is always the first
thing you should do.

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Okay, so taking a
look at this, we

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can see that some of our
variables are indeed very,

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very, significant.

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Others are less significant.

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For example, steals only
has one significance star.

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And some don't seem to
be significant at all.

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For example, defensive
rebounds, turnovers, and blocks.

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We do have a pretty good
R-squared value, 0.8992,

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so it shows that there really
is a linear relationship

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between points and all of
these basketball statistics.

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Let's compute the
residuals here.

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We can just do that by
directly calling them,

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so that's PointsReg$residuals.

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So there's this
giant list of them,

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and we'll use this to compute
the sum of squared errors.

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SSE, standing for sum
of squared errors,

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is just equal to the
sum(PointsReg$residuals^2).

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So what is the sum of
squared errors here?

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It's quite a lot-- 28,394,314.

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So the sum of
squared errors number

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is not really a very
interpretable quantity.

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But remember, we
can also calculate

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the root mean
squared error, which

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is much more interpretable.

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It's more like the average error
we make in our predictions.

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So the root mean squared error,
RMSE-- let's calculate it here.

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So RMSE is just equal
to the square root

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of the sum of squared
errors divided by n,

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where n here is the number
of rows in our data set.

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So the RMSE in
our case is 184.4.

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So on average, we make an
error of about 184.4 points.

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That seems like quite a
lot, until you remember that

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the average number of points
in a season is, let's see,

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mean(NBA$PTS).

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This will give us the average
number of points in a season,

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and it's 8,370.

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So, okay, if we have an average
number of points of 8,370,

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being off by about 184.4
points is really not so bad.

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But I think we still have room
for improvement in this model.

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If you recall, not all the
variables were significant.

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Let's see if we can remove some
of the insignificant variables

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one at a time.

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We'll take a look
again at our model,

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summary(PointsReg),
in order to figure out

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which variable we
should remove first.

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The first variable we
would want to remove

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is probably turnovers.

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And why do I say turnovers?

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It's because the p value for
turnovers, which you see here

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in this column, 0.6859, is the
highest of all of the p values.

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So that means that turnovers
is the least statistically

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significant variable
in our model.

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So let's create a new regression
model without turnovers.

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An easy way to do this
is just to use your up

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arrow on your
keyboard, and scroll up

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through all of your
previous commands

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until you find the command
where you defined the regression

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model.

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Okay, so this is the command
where we defined the model.

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Now let's delete turnovers, and
then we can rename the model,

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and we'll call this
one just PointsReg2.

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So in our first regression
model, PointsReg,

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we had an R-squared of 0.8992.

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Let's take a look at the
R-squared of PointsReg2.

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And we see that it's 0.8991.

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So almost exactly identical.

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It does go down, as we
would expect, but very, very

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slightly.

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So it seems that we're
justified in removing turnovers.

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Let's see if we can
remove another one

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of the insignificant variables.

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The next one, based on p-value,
that we would want to remove

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is defensive rebounds.

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So again, let's
create our model,

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taking out defensive rebounds,
and calling this PointsReg3.

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We'll just scroll up again.

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Take out DRB, for
defensive rebounds,

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and change the name
of this to PointsReg3

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so we don't
overwrite PointsReg2.

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Let's look at the
summary again to see

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if the R-squared has changed.

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And it's the same, it's 0.8991.

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So I think we're justified again
in removing defensive rebounds.

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Let's try this one more time
and see if we can remove blocks.

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So we'll remove blocks,
and call it PointsReg4.

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Take a look at the
summary of PointsReg4.

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And again, the R-squared
value stayed the same.

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So now we've gotten down to a
model which is a bit simpler.

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All the variables
are significant.

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We've still got an
R-squared 0.899.

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And let's take a look now
at the sum of squared errors

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and the root mean
square error, just

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to make sure we
didn't inflate those

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too much by removing
a few variables.

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So, remember that the sum
of squared errors that we

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had in the original model was
this giant number, 28,394,314.

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And the root mean squared error,
the much more interpretable

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number, was 184.4.

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So those are the numbers we'll
be comparing against when

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we calculate the new
sum of squared errors

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of the new model, PointsReg4.

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So let's call this SSE_4.

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And that will just be equal to
sum(PointsReg4$residuals^2).

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And then RMSE_4 will just be
the square root of SSE_4 divided

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by n, the number of
rows in our data set.

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Okay, so let's take
a look at these.

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The new sum of squared
errors is now 28,421,465.

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Again, I find this very
difficult to interpret.

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I like to look at the root
mean squared error instead.

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So the root mean squared
error here is just RMSE_4,

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and so it's 184.5.

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So although we've increased
the root mean squared error

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a little bit by removing
those variables,

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it's really a very,
very, small amount.

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Essentially, we've kept the root
mean squared error the same.

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So it seems like we've narrowed
down on a much better model

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because it's simpler,
it's more interpretable,

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and it's got just about
the same amount of error.