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<h1 class="title toc-ignore">Exercises</h1>
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<p> </p>
<div
id="linear-model-with-an-interaction-between-a-continuous-and-a-categorical-variable"
class="section level2">
<h2>Linear model with an interaction between a continuous and a
categorical variable</h2>
<p> </p>
<p>This exercise builds on the previous exercises where you fitted a
linear model with either a single continuous explanatory variable or a
single categorical explanatory variable. In this exercise you will fit a
model with both a continuous and categorical explanatory variable and
allow their effects to interact (i.e. the effect of one explanatory
variable on the response variable changes with the value of the another
explanatory variable). This is the third of four complementary
exercises, based on the <code>loyn</code> data set.</p>
<p> </p>
<p>1. As in previous exercises, either create a new R script (perhaps
call it ‘linear_model_3’) or continue with your previous R script in
your RStudio Project. Again, make sure you include any metadata you feel
is appropriate (title, description of task, date of creation etc) and
don’t forget to comment out your metadata with a <code>#</code> at the
beginning of the line.</p>
<p> </p>
<p>2. Import the data file ‘loyn.txt’ into R and take a look at the
structure of this dataframe using the <code>str()</code> function. We
know that the abundance of birds <code>ABUND</code> increases with the
log<sub>10</sub> transformed area of the forest patch
(<code>LOGAREA</code> variable). We also know that bird abundance
changes with the grazing intensity (<code>FGRAZE</code> variable) with
forest patches with higher grazing intensity having fewer birds on
average. But how do these effects combine together? Would a small patch
with low grazing intensity have more birds than a larger patch with high
grazing intensity? Could the fit of the <code>ABUND ~ LOGAREA</code>
model for the large patches be improved if we accounted for grazing
intensity in the patches?</p>
<p> </p>
<p>3. As previously we want to treat <code>AREA</code> as a
log<sub>10</sub>-transformed area to limit the influence of the couple
of disproportionately large forest patches, and <code>GRAZE</code> as a
categorical variable with five levels. So the first thing we need to do
is create the corresponding variables in the <code>loyn</code>
dataframe, called <code>LOGAREA</code> and <code>FGRAZE</code>.</p>
<p> </p>
<p>4. Explore the relationship between bird abundance and log
transformed forest patch area for each level of grazing. You could
manually create a separate plot for each graze level but it’s more
efficient to use a conditional scatterplot (aka coplot, see <a
href="https://intro2r.com/simple-base-r-plots.html#coplots">section
4.2.6</a> of the R book or the help page for the function
<code>coplot()</code>). A coplot lets you visualise the relationship
between <code>ABUND</code> and <code>LOGAREA</code> for each level of
<code>FGRAZE</code>, with <code>FGRAZE</code> levels increasing from the
bottom-left panel (Graze level 1) to the top-right panel (Graze level
5). What patterns do you see? Is it okay to assume that the relationship
between <code>ABUND</code> and <code>LOGAREA</code> is the same for all
grazing levels or does the relationship change? This is effectively
asking if the slopes of the relationship between <code>ABUND</code> and
<code>LOGAREA</code> are different for each <code>FGRAZE</code> level -
this is called an interaction.</p>
<p> </p>
<p>5. Fit an appropriate linear model in R to explain the variation in
the response variable <code>ABUND</code> with the explanatory variables
<code>LOGAREA</code> and <code>FGRAZE</code>. Also include the
interaction between <code>LOGAREA</code> and <code>FGRAZE</code>. Hint:
<code>:</code> is the interaction symbol! Remember to use the
<code>data =</code> argument. Assign this linear model to an
appropriately named object, like <code>birds_inter1</code>. Optional:
Can you remember how to specify the model using the ‘shortcut’ version
(with <code>*</code>) instead?</p>
<p> </p>
<p>6. As conscientious researchers, let’s first check the assumptions of
our linear model by creating plots of the residuals. Remember, that you
can split your plotting device into 2 rows and 2 columns using the
<code>par()</code> function before you create the plots. Check each of
the assumptions using these plots and report whether your model meets
these assumptions.</p>
<p> </p>
<p>7. Use the <code>anova()</code> function to produce the ANOVA table
of your linear model. Remember, this ANOVA table is based on sequential
sums of squares (the order of the explanatory variables matters). The
only P value that we should interpret is the one that occurs in the
second to last row of the table (the one above ‘Residuals’). This is the
interaction term between <code>FGRAZE</code> and <code>LOGAREA</code>
(<code>FGRAZE:LOGAREA</code>). What is the null hypothesis associated
with this interaction term?</p>
<p>Do we reject or fail to reject this hypothesis?</p>
<p>What is the biological interpretation of this interaction term? Can
we say anything about the hypotheses for the main effects of
<code>FGRAZE</code> or <code>LOGAREA</code>?</p>
<p> </p>
<p>8. OK, now for the part you all know and love! Use the
<code>summary()</code> function on your model object to produce the
table of parameter estimates. Using this output, take each line in turn
and answer the following questions:</p>
<ul>
<li>what does this parameter estimate?</li>
<li>What is the biological interpretation of the corresponding
estimate?</li>
<li>What is the null hypothesis associated with it?</li>
<li>Do you reject or fail to reject this hypothesis?</li>
</ul>
<p>Also compare the multiple R<sup>2</sup> from this model the models
you created in the previous two exercises.</p>
<p>Please ask one of the course instructors to take you through this if
you’re confused :).</p>
<p> </p>
<p>9. Right, now hang onto your hat! Let’s plot the predictions from
your model to figure out how it really fits the data (and help us
understand the output from the <code>summary()</code> function :).
Here’s a general recipe, using the <code>predict()</code> function.</p>
<ul>
<li><p>plot the raw data, using a different colour for points from each
<code>FGRAZE</code> level</p></li>
<li><p>for each <code>FGRAZE</code> level in turn:</p>
<ul>
<li>create a sequence of <code>LOGAREA</code> from the minimum value to
the maximum within the grazing level (unless you wish to predict outside
the range of observed values, probably best not too!).</li>
<li>store it in a data frame (i.e. <code>dat4pred</code>) containing the
variables <code>FGRAZE</code> and <code>LOGAREA.</code> Remember that
<code>FGRAZE</code> is a factor, so its value need to be placed in
quotes (i.e. <code>FGRAZE = "1"</code>).</li>
<li>Create a vector of predicted bird abundances using our new dataframe
(<code>dat4pred</code>) using the <code>predict()</code> function.</li>
<li>Add the predicted values to the plot using the <code>lines()</code>
function with the appropriate colour (same points colour for each
<code>FGRAZE</code> level above).</li>
</ul></li>
</ul>
<p>See the solutions for one of many ways of doing this. Now this might
seem like a huge amount of code just to plot your predicted values (and
admittedly it is!) but most of the code is just repeated for each level
of graze. Just work through it slowly and logically and hopefully it
will make sense (please ask if you are confused). I will also show you
some alternative (easier?) ways to do this in the solutions.</p>
<p><strong>(Optional, for the geeks)</strong> Alternative method given
in the solutions using a<code>for</code> loop. Just keep this code in
case you ever want to do something like this in the future.</p>
<p><strong>(Optional, for the lazy!)</strong> And, if you want an even
easier way, then we can use the <code>ggolot2</code> package (code in
the solutions). Note: you will need to install the <code>ggplot2</code>
package first if you don’t already have it.</p>
<p> </p>
<p>End of the Linear model with continuous and categorical explanatory
variables exercise</p>
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