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Polynomial Regression in R | R Tutorial 5.12 | MarinStatsLectures - YouTube
Channel: MarinStatsLectures-R Programming & Statistics
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Hi, I'm Mike Marin and in this video
we'll discuss the idea of polynomial
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regression and how to fit and assess
these models in R. Polynomial
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regression is a special case of linear
regression where the relationship
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between x and y is modeled using a
polynomial rather than a line. It can be
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used when the relationship between x and
y is nonlinear although this is still
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considered to be a special case of
multiple linear regression; we will be
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working with a different version of the
lung capacity data as was used in other
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videos. A link to download this data can
be found in the video description below.
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You can also download the R script used in this video there as well. I've already
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imported the data into RStudio and
attached it. As an example we'll model
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the relationship between lung capacity
and height. Let's begin by looking at a
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scatter plot and let's fit a simple
linear regression model and look at a
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summary of that model. Here we'll model
the relationship between lung capacity
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and height, we can take note that the
r-squared is about 75% and the residual
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standard error is 1.292;
we can also add the line for this model
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to the plot using the abline command (function);
visually we can see that the
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relationship between lung capacity and
height looks a bit curved or non linear;
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a reminder that we can also use residual
plots to help with assessing linearity
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and checking model assumptions. for a
more thorough discussion of this topic
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you can refer to one of our earlier
videos on checking assumptions for
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linear regression. There are many
approaches to dealing with
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nonlinearities, one we will discuss in
this video is including polynomial terms
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in our model we will start with
including height squared in our model
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first let's take a look at the wrong way
to do this while it may seem like
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including height squared directly in the
model call will work, R will not include
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height squared in the model if entered
this way let's take a look at that here
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we can see if we enter height squared
directly in the model call, in the model
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summary you can see height squared is
left out, R has just ignored height
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squared. It is important to take note of
this because R does not give us a
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warning or an error message with this.
Now let's take a look at the right way
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to do this: to do so we'll use a capital
I and then include height squared within
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the parentheses and if we ask for a
summary of the model we can now see that
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height squared has been included in the
model we'll get to talking about this
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model in a moment but first let's take a
look at a few other ways to get the same
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result
instead of using the approach we just
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showed we could instead first create a
new variable called height squared and
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then include this variable in the model
here we can see we're creating this
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height square variable and including
that in our model and if we ask for the
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summary you can see that this produces
the exact same model and results as the
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previous set of commands, we could also
make use of the Poly command (function) in R,
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here we would let R know that we would
like to include polynomial terms for the
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height variable and we set the degree
argument to the degree of polynomial
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we'd like, in this case setting the
degree equal 2 will include height and
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height squared and if instead of setting
the raw argument to TRUE we set it to
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FALSE, R would fit a model that used
orthogonal polynomials so let's fit that
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model and again you can take the time to
verify that this produces the exact same
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results as the earlier two ways we
looked at so let's just give ourselves a
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quick reminder of the model that we fit
we can see here that with this
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polynomial including height squared the
R squared is about 77% and the residual
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standard error is 1.238, you may recall that for the model
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with only height the R squared was about
75% and the residual standard error was
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1.292, it looks like
height squared may be improving the
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model let's take a look at this visually
we can add the polynomial model to the
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plot using the lines command, so let's
add this line using a thick blue line
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subjectively it looks like the model
that includes height squared may provide
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a better fit to the data than the model
that does not include height squared
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let's compare these two models formally
using the partial F test. for a more
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detailed discussion of the partial F
test you can see one of our earlier
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videos where we discuss this test; this
test has a null hypothesis that there is
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no significant difference between the
two models and an alternative hypothesis
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that's a full model, the model that
includes height squared, is significantly
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better we can run the test in R, using
the ANOVA command we can see that with
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such a small p-value we will reject the
null hypothesis and conclude that we
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have evidence to believe the model
including height squared provides a
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statistically significantly better fit
than the model without height squared
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most often we won't want to include
polynomial terms much beyond x squared
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or x cubed let's explore a model that
includes X cubed as well and it's worth
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noting that you must always include all
lower order terms in a model. If we
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include height cubed we must also
include height squared and height in the
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model let's fit this model including
height squared and height cubed and
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let's ask for a summary we can add this
model to the plot using the lines
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command and we will do this as a thick
green dashed line let's also add a
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legend to the plot to help remind
ourselves which model is which we can
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see visually that there's almost no
difference between the model that
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includes height cubed and the model that
does not as before we can use the
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partial F test to help us decide if
height cubed improves the model here
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when we conduct the partial F test we
can see that the p-value is large and
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there is not a statistically significant
difference in the models we can decide
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that including height cubed is not
necessary in our model before finishing
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off we should add a note that there are
other approaches to dealing with
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nonlinearities some of which include
transforming the x or y variable
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converting x to a categorical variable
or factor or using nonlinear regression
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methods instead. All of these different
approaches have their pros and cons!
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