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<h1 class="title toc-ignore">Exercises</h1>
</div>
<p> </p>
<div
id="exercise-linear-model-with-single-categorical-explanatory-variable"
class="section level2">
<h2>Exercise: Linear model with single categorical explanatory
variable</h2>
<p> </p>
<p>1. As in previous exercises, either create a new R script (perhaps
call it ‘linear_model_2’) 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. Once again import the data file ‘loyn.txt’ into R and take a look
at the structure of this dataframe using the <code>str()</code>
function. In this exercise you will investigate whether the abundance of
birds (<code>ABUND</code>) is different in areas with different grazing
intensities (<code>GRAZE</code>). Remember, the <code>GRAZE</code>
variable is an index of livestock grazing intensity. Level 1 = low
grazing intensity and level 5 = high grazing intensity.</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" tabindex="-1"></a>loyn <span class="ot"><-</span> <span class="fu">read.table</span>(<span class="st">"data/loyn.txt"</span>, <span class="at">header =</span> <span class="cn">TRUE</span>, </span>
<span id="cb1-2"><a href="#cb1-2" tabindex="-1"></a> <span class="at">stringsAsFactors =</span> <span class="cn">TRUE</span>)</span>
<span id="cb1-3"><a href="#cb1-3" tabindex="-1"></a><span class="fu">str</span>(loyn)</span>
<span id="cb1-4"><a href="#cb1-4" tabindex="-1"></a><span class="do">## 'data.frame': 67 obs. of 8 variables:</span></span>
<span id="cb1-5"><a href="#cb1-5" tabindex="-1"></a><span class="do">## $ SITE : int 1 60 2 3 61 4 5 6 7 8 ...</span></span>
<span id="cb1-6"><a href="#cb1-6" tabindex="-1"></a><span class="do">## $ ABUND : num 5.3 10 2 1.5 13 17.1 13.8 14.1 3.8 2.2 ...</span></span>
<span id="cb1-7"><a href="#cb1-7" tabindex="-1"></a><span class="do">## $ AREA : num 0.1 0.2 0.5 0.5 0.6 1 1 1 1 1 ...</span></span>
<span id="cb1-8"><a href="#cb1-8" tabindex="-1"></a><span class="do">## $ DIST : int 39 142 234 104 191 66 246 234 467 284 ...</span></span>
<span id="cb1-9"><a href="#cb1-9" tabindex="-1"></a><span class="do">## $ LDIST : int 39 142 234 311 357 66 246 285 467 1829 ...</span></span>
<span id="cb1-10"><a href="#cb1-10" tabindex="-1"></a><span class="do">## $ YR.ISOL: int 1968 1961 1920 1900 1957 1966 1918 1965 1955 1920 ...</span></span>
<span id="cb1-11"><a href="#cb1-11" tabindex="-1"></a><span class="do">## $ GRAZE : int 2 2 5 5 2 3 5 3 5 5 ...</span></span>
<span id="cb1-12"><a href="#cb1-12" tabindex="-1"></a><span class="do">## $ ALT : int 160 180 60 140 185 160 140 130 90 60 ...</span></span></code></pre></div>
<p> </p>
<p>3. As we discussed in the graphical data exploration exercise the
<code>GRAZE</code> variable was originally coded as a numeric (i.e. 1,
2, 3, 4, 5). In this exercise we actually want to treat
<code>GRAZE</code> as a categorical variable with five levels (aka a
factor). So the first thing we need to do is create a new variable in
the <code>loyn</code> dataframe called <code>FGRAZE</code> in which we
store the <code>GRAZE</code> variable coerced to be a categorical
variable with the <code>factor()</code> function (you can also use the
<code>as.factor()</code> function if you prefer).</p>
<div class="sourceCode" id="cb2"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" tabindex="-1"></a><span class="co"># create factor GRAZE as it was originally coded as an integer</span></span>
<span id="cb2-2"><a href="#cb2-2" tabindex="-1"></a>loyn<span class="sc">$</span>FGRAZE <span class="ot"><-</span> <span class="fu">factor</span>(loyn<span class="sc">$</span>GRAZE)</span>
<span id="cb2-3"><a href="#cb2-3" tabindex="-1"></a></span>
<span id="cb2-4"><a href="#cb2-4" tabindex="-1"></a><span class="co"># check this</span></span>
<span id="cb2-5"><a href="#cb2-5" tabindex="-1"></a><span class="fu">class</span>(loyn<span class="sc">$</span>FGRAZE)</span>
<span id="cb2-6"><a href="#cb2-6" tabindex="-1"></a><span class="do">## [1] "factor"</span></span></code></pre></div>
<p> </p>
<p>4. Explore any potential differences in bird abundance between each
level of <code>FGRAZE</code> graphically using an appropriate plot
(hint: a boxplot might be useful here). How would you interpret this
plot? What might you expect to see in your analysis? Write your
predictions in your R script as a comment. What is the mean number of
birds for each level of <code>FGRAZE</code>?</p>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" tabindex="-1"></a><span class="fu">boxplot</span>(ABUND <span class="sc">~</span> FGRAZE, <span class="at">xlab =</span> <span class="st">"Grazing level"</span>, <span class="at">ylab =</span> <span class="st">"Bird abundance"</span>, <span class="at">data =</span> loyn)</span></code></pre></div>
<p><img src="linear_model_2_exercise_solutions_files/figure-html/Q4-1.png" width="672" /></p>
<div class="sourceCode" id="cb4"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" tabindex="-1"></a></span>
<span id="cb4-2"><a href="#cb4-2" tabindex="-1"></a><span class="co"># mean bird abundance for each level of FGRAZE</span></span>
<span id="cb4-3"><a href="#cb4-3" tabindex="-1"></a><span class="fu">tapply</span>(loyn<span class="sc">$</span>ABUND, loyn<span class="sc">$</span>FGRAZE, mean, <span class="at">na.rm =</span> <span class="cn">TRUE</span>)</span>
<span id="cb4-4"><a href="#cb4-4" tabindex="-1"></a><span class="do">## 1 2 3 4 5 </span></span>
<span id="cb4-5"><a href="#cb4-5" tabindex="-1"></a><span class="do">## 28.623077 19.418182 20.164706 18.961538 6.292308</span></span>
<span id="cb4-6"><a href="#cb4-6" tabindex="-1"></a></span>
<span id="cb4-7"><a href="#cb4-7" tabindex="-1"></a><span class="co"># it looks from this plot and the table of means that the bird abundance is lowest for FGRAZE level 5 and </span></span>
<span id="cb4-8"><a href="#cb4-8" tabindex="-1"></a><span class="co"># highest for level 1. The bird abundance for levels 2, 3 and 4 all look similar.</span></span>
<span id="cb4-9"><a href="#cb4-9" tabindex="-1"></a><span class="co"># so in terms of differences in ABUND between groups we might expect FGRAZE level 5 to be different from</span></span>
<span id="cb4-10"><a href="#cb4-10" tabindex="-1"></a><span class="co"># the other grazing intensity group and possibly FGRAZE level 1 to be different from graze level 2,3 and 4</span></span>
<span id="cb4-11"><a href="#cb4-11" tabindex="-1"></a><span class="co"># but this is not particularly clear. We might also expect there to be no differences between grazing </span></span>
<span id="cb4-12"><a href="#cb4-12" tabindex="-1"></a><span class="co"># levels 2,3 and 4. </span></span></code></pre></div>
<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 variable
<code>FGRAZE</code>. Remember to use the <code>data =</code> argument.
Assign this linear model to an appropriately named object
(<code>birds_lm</code> if your imagination fails you!).</p>
<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" tabindex="-1"></a>birds_lm <span class="ot"><-</span> <span class="fu">lm</span>(ABUND <span class="sc">~</span> FGRAZE, <span class="at">data =</span> loyn)</span></code></pre></div>
<p> </p>
<p>6. Produce the ANOVA table using the <code>anova()</code> function on
the model object. What null hypothesis is being tested? Do you reject or
fail to reject the null hypothesis? What summary statistics would you
report? Summarise in your R script as a comment.</p>
<div class="sourceCode" id="cb6"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" tabindex="-1"></a><span class="fu">anova</span>(birds_lm)</span>
<span id="cb6-2"><a href="#cb6-2" tabindex="-1"></a><span class="do">## Analysis of Variance Table</span></span>
<span id="cb6-3"><a href="#cb6-3" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb6-4"><a href="#cb6-4" tabindex="-1"></a><span class="do">## Response: ABUND</span></span>
<span id="cb6-5"><a href="#cb6-5" tabindex="-1"></a><span class="do">## Df Sum Sq Mean Sq F value Pr(>F) </span></span>
<span id="cb6-6"><a href="#cb6-6" tabindex="-1"></a><span class="do">## FGRAZE 4 3324.2 831.06 14.985 1.272e-08 ***</span></span>
<span id="cb6-7"><a href="#cb6-7" tabindex="-1"></a><span class="do">## Residuals 62 3438.6 55.46 </span></span>
<span id="cb6-8"><a href="#cb6-8" tabindex="-1"></a><span class="do">## ---</span></span>
<span id="cb6-9"><a href="#cb6-9" tabindex="-1"></a><span class="do">## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</span></span>
<span id="cb6-10"><a href="#cb6-10" tabindex="-1"></a></span>
<span id="cb6-11"><a href="#cb6-11" tabindex="-1"></a><span class="co"># null hypothesis : There is no difference in the mean bird abundance between the </span></span>
<span id="cb6-12"><a href="#cb6-12" tabindex="-1"></a><span class="co"># five levels of grazing.</span></span>
<span id="cb6-13"><a href="#cb6-13" tabindex="-1"></a><span class="co"># the p value is very small therefore reject this null hypothesis. In other words</span></span>
<span id="cb6-14"><a href="#cb6-14" tabindex="-1"></a><span class="co"># there is a difference in the mean bird abundance between grazing intensity levels.</span></span>
<span id="cb6-15"><a href="#cb6-15" tabindex="-1"></a></span>
<span id="cb6-16"><a href="#cb6-16" tabindex="-1"></a><span class="co"># for a report you might write something like:</span></span>
<span id="cb6-17"><a href="#cb6-17" tabindex="-1"></a><span class="co"># there was a significant difference in the mean abundance of birds between the five levels</span></span>
<span id="cb6-18"><a href="#cb6-18" tabindex="-1"></a><span class="co"># of grazing intensity (F_4,62 = 14.98, p < 0.0001)</span></span></code></pre></div>
<p> </p>
<p>7. Use the <code>summary()</code> function on the model object to
produce the table of parameter estimates (remember these are called
coefficients in R). Using this output what is the estimate of the
intercept and what does this represent? What is the null hypothesis
associated with the intercept? do you reject or fail to reject this
hypothesis?</p>
<p>Next we move onto the the <code>FGRAZE2</code> parameter, how do you
interpret this parameter? (remember they are contrasts). Again, what is
the null hypothesis associated with the <code>FGRAZE2</code> parameter?
do you reject or fail to reject this hypothesis?</p>
<p>Repeat this interpretation for the <code>FGRAZE3</code>,
<code>FGRAZE4</code> and <code>FGRAZE5</code> parameters. Summarise this
as a comment in your R script.</p>
<div class="sourceCode" id="cb7"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" tabindex="-1"></a><span class="fu">summary</span>(birds_lm)</span>
<span id="cb7-2"><a href="#cb7-2" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb7-3"><a href="#cb7-3" tabindex="-1"></a><span class="do">## Call:</span></span>
<span id="cb7-4"><a href="#cb7-4" tabindex="-1"></a><span class="do">## lm(formula = ABUND ~ FGRAZE, data = loyn)</span></span>
<span id="cb7-5"><a href="#cb7-5" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb7-6"><a href="#cb7-6" tabindex="-1"></a><span class="do">## Residuals:</span></span>
<span id="cb7-7"><a href="#cb7-7" tabindex="-1"></a><span class="do">## Min 1Q Median 3Q Max </span></span>
<span id="cb7-8"><a href="#cb7-8" tabindex="-1"></a><span class="do">## -17.2647 -4.3269 -0.0182 5.0948 17.5353 </span></span>
<span id="cb7-9"><a href="#cb7-9" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb7-10"><a href="#cb7-10" tabindex="-1"></a><span class="do">## Coefficients:</span></span>
<span id="cb7-11"><a href="#cb7-11" tabindex="-1"></a><span class="do">## Estimate Std. Error t value Pr(>|t|) </span></span>
<span id="cb7-12"><a href="#cb7-12" tabindex="-1"></a><span class="do">## (Intercept) 28.623 2.065 13.858 < 2e-16 ***</span></span>
<span id="cb7-13"><a href="#cb7-13" tabindex="-1"></a><span class="do">## FGRAZE2 -9.205 3.051 -3.017 0.00370 ** </span></span>
<span id="cb7-14"><a href="#cb7-14" tabindex="-1"></a><span class="do">## FGRAZE3 -8.458 2.744 -3.083 0.00306 ** </span></span>
<span id="cb7-15"><a href="#cb7-15" tabindex="-1"></a><span class="do">## FGRAZE4 -9.662 2.921 -3.308 0.00157 ** </span></span>
<span id="cb7-16"><a href="#cb7-16" tabindex="-1"></a><span class="do">## FGRAZE5 -22.331 2.921 -7.645 1.64e-10 ***</span></span>
<span id="cb7-17"><a href="#cb7-17" tabindex="-1"></a><span class="do">## ---</span></span>
<span id="cb7-18"><a href="#cb7-18" tabindex="-1"></a><span class="do">## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</span></span>
<span id="cb7-19"><a href="#cb7-19" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb7-20"><a href="#cb7-20" tabindex="-1"></a><span class="do">## Residual standard error: 7.447 on 62 degrees of freedom</span></span>
<span id="cb7-21"><a href="#cb7-21" tabindex="-1"></a><span class="do">## Multiple R-squared: 0.4915, Adjusted R-squared: 0.4587 </span></span>
<span id="cb7-22"><a href="#cb7-22" tabindex="-1"></a><span class="do">## F-statistic: 14.98 on 4 and 62 DF, p-value: 1.272e-08</span></span>
<span id="cb7-23"><a href="#cb7-23" tabindex="-1"></a></span>
<span id="cb7-24"><a href="#cb7-24" tabindex="-1"></a><span class="co"># Here the intercept (baseline) is the mean abundance of birds for FGRAZE level 1.</span></span>
<span id="cb7-25"><a href="#cb7-25" tabindex="-1"></a><span class="co"># the null hypothesis for the intercept is that the intercept = 0.</span></span>
<span id="cb7-26"><a href="#cb7-26" tabindex="-1"></a><span class="co"># As the p value (p < 2e-16) is very small we reject this null hypothesis and conclude that the</span></span>
<span id="cb7-27"><a href="#cb7-27" tabindex="-1"></a><span class="co"># intercept is significantly different from 0. However, from a biological perspective this</span></span>
<span id="cb7-28"><a href="#cb7-28" tabindex="-1"></a><span class="co"># is not a particularly informative hypothesis to test.</span></span>
<span id="cb7-29"><a href="#cb7-29" tabindex="-1"></a></span>
<span id="cb7-30"><a href="#cb7-30" tabindex="-1"></a><span class="co"># the remaining estimates are differences (contrasts) between each level and the </span></span>
<span id="cb7-31"><a href="#cb7-31" tabindex="-1"></a><span class="co"># baseline. For example the FGRAZE2 estimate is - 9.2 and therefore there are 9.2 fewer </span></span>
<span id="cb7-32"><a href="#cb7-32" tabindex="-1"></a><span class="co"># birds on average in graze level 2 compared to graze level 1. This difference is </span></span>
<span id="cb7-33"><a href="#cb7-33" tabindex="-1"></a><span class="co"># significantly different from zero (p = 0.003). </span></span>
<span id="cb7-34"><a href="#cb7-34" tabindex="-1"></a></span>
<span id="cb7-35"><a href="#cb7-35" tabindex="-1"></a><span class="co"># The difference between graze level 3 (FGRAZE3) and graze level 1 (intercept) is </span></span>
<span id="cb7-36"><a href="#cb7-36" tabindex="-1"></a><span class="co"># -8.45 (8.45 fewer birds in graze 3 compared to graze 1). This difference is significantly </span></span>
<span id="cb7-37"><a href="#cb7-37" tabindex="-1"></a><span class="co"># different from 0 (p = 0.003) and therefore the mean abundance of birds in graze level 1 is</span></span>
<span id="cb7-38"><a href="#cb7-38" tabindex="-1"></a><span class="co"># significantly different from graze level 1.</span></span>
<span id="cb7-39"><a href="#cb7-39" tabindex="-1"></a></span>
<span id="cb7-40"><a href="#cb7-40" tabindex="-1"></a><span class="co"># The difference between graze level 4 (FGRAZE4) and graze level 1 (intercept) is </span></span>
<span id="cb7-41"><a href="#cb7-41" tabindex="-1"></a><span class="co"># -9.66 (9.66 fewer birds in graze 4 compared to graze 1). This difference is significantly </span></span>
<span id="cb7-42"><a href="#cb7-42" tabindex="-1"></a><span class="co"># different from 0 (p = 0.001) and therefore the mean abundance of birds in graze level 1 is</span></span>
<span id="cb7-43"><a href="#cb7-43" tabindex="-1"></a><span class="co"># significantly different from graze level 4.</span></span>
<span id="cb7-44"><a href="#cb7-44" tabindex="-1"></a></span>
<span id="cb7-45"><a href="#cb7-45" tabindex="-1"></a><span class="co"># The difference between graze level 5 (FGRAZE5) and graze level 1 (intercept) is </span></span>
<span id="cb7-46"><a href="#cb7-46" tabindex="-1"></a><span class="co"># -22.33 (22.33 fewer birds in graze 5 compared to graze 1). This difference is significantly </span></span>
<span id="cb7-47"><a href="#cb7-47" tabindex="-1"></a><span class="co"># different from 0 (p = 1.64e-10) and therefore the mean abundance of birds in graze level 1 is</span></span>
<span id="cb7-48"><a href="#cb7-48" tabindex="-1"></a><span class="co"># significantly different from graze level 5.</span></span></code></pre></div>
<p> </p>
<p>8. Now that you have interpreted all the contrasts with
<code>FGRAZE</code> level 1 as the intercept, set the intercept to
<code>FGRAZE</code> level 2 using the <code>relevel()</code> function,
refit the model, produce the new table of parameter estimates using the
<code>summary()</code> function again and interpret.</p>
<p>Repeat this for <code>FGRAZE</code> levels 3, 4 and 5. Can you
summarise which levels of <code>FGRAZE</code> are different from each
other?</p>
<div class="sourceCode" id="cb8"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" tabindex="-1"></a><span class="co"># Set FGRAZE level 2 to be the intercept</span></span>
<span id="cb8-2"><a href="#cb8-2" tabindex="-1"></a></span>
<span id="cb8-3"><a href="#cb8-3" tabindex="-1"></a>loyn<span class="sc">$</span>FGRAZE <span class="ot"><-</span> <span class="fu">relevel</span>(loyn<span class="sc">$</span>FGRAZE, <span class="at">ref =</span> <span class="st">"2"</span>)</span>
<span id="cb8-4"><a href="#cb8-4" tabindex="-1"></a>birds_lm2 <span class="ot"><-</span> <span class="fu">lm</span>(ABUND <span class="sc">~</span> FGRAZE, <span class="at">data =</span> loyn)</span>
<span id="cb8-5"><a href="#cb8-5" tabindex="-1"></a><span class="fu">summary</span>(birds_lm2)</span>
<span id="cb8-6"><a href="#cb8-6" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb8-7"><a href="#cb8-7" tabindex="-1"></a><span class="do">## Call:</span></span>
<span id="cb8-8"><a href="#cb8-8" tabindex="-1"></a><span class="do">## lm(formula = ABUND ~ FGRAZE, data = loyn)</span></span>
<span id="cb8-9"><a href="#cb8-9" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb8-10"><a href="#cb8-10" tabindex="-1"></a><span class="do">## Residuals:</span></span>
<span id="cb8-11"><a href="#cb8-11" tabindex="-1"></a><span class="do">## Min 1Q Median 3Q Max </span></span>
<span id="cb8-12"><a href="#cb8-12" tabindex="-1"></a><span class="do">## -17.2647 -4.3269 -0.0182 5.0948 17.5353 </span></span>
<span id="cb8-13"><a href="#cb8-13" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb8-14"><a href="#cb8-14" tabindex="-1"></a><span class="do">## Coefficients:</span></span>
<span id="cb8-15"><a href="#cb8-15" tabindex="-1"></a><span class="do">## Estimate Std. Error t value Pr(>|t|) </span></span>
<span id="cb8-16"><a href="#cb8-16" tabindex="-1"></a><span class="do">## (Intercept) 19.4182 2.2454 8.648 3.00e-12 ***</span></span>
<span id="cb8-17"><a href="#cb8-17" tabindex="-1"></a><span class="do">## FGRAZE1 9.2049 3.0509 3.017 0.0037 ** </span></span>
<span id="cb8-18"><a href="#cb8-18" tabindex="-1"></a><span class="do">## FGRAZE3 0.7465 2.8817 0.259 0.7965 </span></span>
<span id="cb8-19"><a href="#cb8-19" tabindex="-1"></a><span class="do">## FGRAZE4 -0.4566 3.0509 -0.150 0.8815 </span></span>
<span id="cb8-20"><a href="#cb8-20" tabindex="-1"></a><span class="do">## FGRAZE5 -13.1259 3.0509 -4.302 6.11e-05 ***</span></span>
<span id="cb8-21"><a href="#cb8-21" tabindex="-1"></a><span class="do">## ---</span></span>
<span id="cb8-22"><a href="#cb8-22" tabindex="-1"></a><span class="do">## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</span></span>
<span id="cb8-23"><a href="#cb8-23" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb8-24"><a href="#cb8-24" tabindex="-1"></a><span class="do">## Residual standard error: 7.447 on 62 degrees of freedom</span></span>
<span id="cb8-25"><a href="#cb8-25" tabindex="-1"></a><span class="do">## Multiple R-squared: 0.4915, Adjusted R-squared: 0.4587 </span></span>
<span id="cb8-26"><a href="#cb8-26" tabindex="-1"></a><span class="do">## F-statistic: 14.98 on 4 and 62 DF, p-value: 1.272e-08</span></span>
<span id="cb8-27"><a href="#cb8-27" tabindex="-1"></a></span>
<span id="cb8-28"><a href="#cb8-28" tabindex="-1"></a><span class="co"># The intercept is now FGRAZE level 2, we can now compare between levels '2 and 3', '2 and 4', and '2 and 5'</span></span>
<span id="cb8-29"><a href="#cb8-29" tabindex="-1"></a><span class="co"># Also note that the rest of the model output (R^2, F, DF etc) is the same as the previous model (i.e. its </span></span>
<span id="cb8-30"><a href="#cb8-30" tabindex="-1"></a><span class="co"># the same model we have just changed the intercept and therefore the contrasts).</span></span>
<span id="cb8-31"><a href="#cb8-31" tabindex="-1"></a></span>
<span id="cb8-32"><a href="#cb8-32" tabindex="-1"></a>loyn<span class="sc">$</span>FGRAZE <span class="ot"><-</span> <span class="fu">relevel</span>(loyn<span class="sc">$</span>FGRAZE, <span class="at">ref =</span> <span class="st">"3"</span>)</span>
<span id="cb8-33"><a href="#cb8-33" tabindex="-1"></a>birds_lm3 <span class="ot"><-</span> <span class="fu">lm</span>(ABUND <span class="sc">~</span> FGRAZE, <span class="at">data =</span> loyn)</span>
<span id="cb8-34"><a href="#cb8-34" tabindex="-1"></a><span class="fu">summary</span>(birds_lm3)</span>
<span id="cb8-35"><a href="#cb8-35" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb8-36"><a href="#cb8-36" tabindex="-1"></a><span class="do">## Call:</span></span>
<span id="cb8-37"><a href="#cb8-37" tabindex="-1"></a><span class="do">## lm(formula = ABUND ~ FGRAZE, data = loyn)</span></span>
<span id="cb8-38"><a href="#cb8-38" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb8-39"><a href="#cb8-39" tabindex="-1"></a><span class="do">## Residuals:</span></span>
<span id="cb8-40"><a href="#cb8-40" tabindex="-1"></a><span class="do">## Min 1Q Median 3Q Max </span></span>
<span id="cb8-41"><a href="#cb8-41" tabindex="-1"></a><span class="do">## -17.2647 -4.3269 -0.0182 5.0948 17.5353 </span></span>
<span id="cb8-42"><a href="#cb8-42" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb8-43"><a href="#cb8-43" tabindex="-1"></a><span class="do">## Coefficients:</span></span>
<span id="cb8-44"><a href="#cb8-44" tabindex="-1"></a><span class="do">## Estimate Std. Error t value Pr(>|t|) </span></span>
<span id="cb8-45"><a href="#cb8-45" tabindex="-1"></a><span class="do">## (Intercept) 20.1647 1.8062 11.164 < 2e-16 ***</span></span>
<span id="cb8-46"><a href="#cb8-46" tabindex="-1"></a><span class="do">## FGRAZE2 -0.7465 2.8817 -0.259 0.79645 </span></span>
<span id="cb8-47"><a href="#cb8-47" tabindex="-1"></a><span class="do">## FGRAZE1 8.4584 2.7438 3.083 0.00306 ** </span></span>
<span id="cb8-48"><a href="#cb8-48" tabindex="-1"></a><span class="do">## FGRAZE4 -1.2032 2.7438 -0.438 0.66255 </span></span>
<span id="cb8-49"><a href="#cb8-49" tabindex="-1"></a><span class="do">## FGRAZE5 -13.8724 2.7438 -5.056 4.06e-06 ***</span></span>
<span id="cb8-50"><a href="#cb8-50" tabindex="-1"></a><span class="do">## ---</span></span>
<span id="cb8-51"><a href="#cb8-51" tabindex="-1"></a><span class="do">## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</span></span>
<span id="cb8-52"><a href="#cb8-52" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb8-53"><a href="#cb8-53" tabindex="-1"></a><span class="do">## Residual standard error: 7.447 on 62 degrees of freedom</span></span>
<span id="cb8-54"><a href="#cb8-54" tabindex="-1"></a><span class="do">## Multiple R-squared: 0.4915, Adjusted R-squared: 0.4587 </span></span>
<span id="cb8-55"><a href="#cb8-55" tabindex="-1"></a><span class="do">## F-statistic: 14.98 on 4 and 62 DF, p-value: 1.272e-08</span></span>
<span id="cb8-56"><a href="#cb8-56" tabindex="-1"></a></span>
<span id="cb8-57"><a href="#cb8-57" tabindex="-1"></a><span class="co"># The intercept is now FGRAZE level 3, we can now compare between levels '3 and 4', 'and 3 and 5'</span></span>
<span id="cb8-58"><a href="#cb8-58" tabindex="-1"></a></span>
<span id="cb8-59"><a href="#cb8-59" tabindex="-1"></a>loyn<span class="sc">$</span>FGRAZE <span class="ot"><-</span> <span class="fu">relevel</span>(loyn<span class="sc">$</span>FGRAZE, <span class="at">ref =</span> <span class="st">"4"</span>)</span>
<span id="cb8-60"><a href="#cb8-60" tabindex="-1"></a>birds_lm4 <span class="ot"><-</span> <span class="fu">lm</span>(ABUND <span class="sc">~</span> FGRAZE, <span class="at">data =</span> loyn)</span>
<span id="cb8-61"><a href="#cb8-61" tabindex="-1"></a><span class="fu">summary</span>(birds_lm4)</span>
<span id="cb8-62"><a href="#cb8-62" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb8-63"><a href="#cb8-63" tabindex="-1"></a><span class="do">## Call:</span></span>
<span id="cb8-64"><a href="#cb8-64" tabindex="-1"></a><span class="do">## lm(formula = ABUND ~ FGRAZE, data = loyn)</span></span>
<span id="cb8-65"><a href="#cb8-65" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb8-66"><a href="#cb8-66" tabindex="-1"></a><span class="do">## Residuals:</span></span>
<span id="cb8-67"><a href="#cb8-67" tabindex="-1"></a><span class="do">## Min 1Q Median 3Q Max </span></span>
<span id="cb8-68"><a href="#cb8-68" tabindex="-1"></a><span class="do">## -17.2647 -4.3269 -0.0182 5.0948 17.5353 </span></span>
<span id="cb8-69"><a href="#cb8-69" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb8-70"><a href="#cb8-70" tabindex="-1"></a><span class="do">## Coefficients:</span></span>
<span id="cb8-71"><a href="#cb8-71" tabindex="-1"></a><span class="do">## Estimate Std. Error t value Pr(>|t|) </span></span>
<span id="cb8-72"><a href="#cb8-72" tabindex="-1"></a><span class="do">## (Intercept) 18.9615 2.0655 9.180 3.65e-13 ***</span></span>
<span id="cb8-73"><a href="#cb8-73" tabindex="-1"></a><span class="do">## FGRAZE3 1.2032 2.7438 0.438 0.66255 </span></span>
<span id="cb8-74"><a href="#cb8-74" tabindex="-1"></a><span class="do">## FGRAZE2 0.4566 3.0509 0.150 0.88151 </span></span>
<span id="cb8-75"><a href="#cb8-75" tabindex="-1"></a><span class="do">## FGRAZE1 9.6615 2.9210 3.308 0.00157 ** </span></span>
<span id="cb8-76"><a href="#cb8-76" tabindex="-1"></a><span class="do">## FGRAZE5 -12.6692 2.9210 -4.337 5.41e-05 ***</span></span>
<span id="cb8-77"><a href="#cb8-77" tabindex="-1"></a><span class="do">## ---</span></span>
<span id="cb8-78"><a href="#cb8-78" tabindex="-1"></a><span class="do">## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</span></span>
<span id="cb8-79"><a href="#cb8-79" tabindex="-1"></a><span class="do">## </span></span>
<span id="cb8-80"><a href="#cb8-80" tabindex="-1"></a><span class="do">## Residual standard error: 7.447 on 62 degrees of freedom</span></span>
<span id="cb8-81"><a href="#cb8-81" tabindex="-1"></a><span class="do">## Multiple R-squared: 0.4915, Adjusted R-squared: 0.4587 </span></span>
<span id="cb8-82"><a href="#cb8-82" tabindex="-1"></a><span class="do">## F-statistic: 14.98 on 4 and 62 DF, p-value: 1.272e-08</span></span>
<span id="cb8-83"><a href="#cb8-83" tabindex="-1"></a></span>
<span id="cb8-84"><a href="#cb8-84" tabindex="-1"></a><span class="co"># The intercept is now FGRAZE level 4, we can now compare between levels '4 and 5'</span></span></code></pre></div>
<p> </p>
<p>9. Staying with the summary table of parameter estimates, how much of
the variation in bird abundance does the explanatory variable
<code>FGRAZE</code> explain?</p>
<div class="sourceCode" id="cb9"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" tabindex="-1"></a><span class="co"># The multiple R-squared value is 0.491 and therefore 49.1% of</span></span>
<span id="cb9-2"><a href="#cb9-2" tabindex="-1"></a><span class="co"># the variation in ABUND is explained by FGRAZE</span></span></code></pre></div>
<p> </p>
<p>10. Now onto a really important part of the model fitting process.
Let’s check the assumptions of your linear model by creating plots of
the residuals from the model. Remember, you can easily create these
plots by using the <code>plot()</code> function on your model object.
Also remember that if you want to see all plots at once then you should
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 in your R script.</p>
<div class="sourceCode" id="cb10"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" tabindex="-1"></a><span class="co"># first split the plotting device into 2 rows and 2 columns</span></span>
<span id="cb10-2"><a href="#cb10-2" tabindex="-1"></a><span class="fu">par</span>(<span class="at">mfrow =</span> <span class="fu">c</span>(<span class="dv">2</span>,<span class="dv">2</span>))</span>
<span id="cb10-3"><a href="#cb10-3" tabindex="-1"></a></span>
<span id="cb10-4"><a href="#cb10-4" tabindex="-1"></a><span class="co"># now create the residuals plots</span></span>
<span id="cb10-5"><a href="#cb10-5" tabindex="-1"></a><span class="fu">plot</span>(birds_lm)</span></code></pre></div>
<p><img src="linear_model_2_exercise_solutions_files/figure-html/Q10-1.png" width="672" /></p>
<div class="sourceCode" id="cb11"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" tabindex="-1"></a></span>
<span id="cb11-2"><a href="#cb11-2" tabindex="-1"></a><span class="co"># To test the normality of residuals assumption we use the Normal Q-Q plot. Although the majority of the residuals </span></span>
<span id="cb11-3"><a href="#cb11-3" tabindex="-1"></a><span class="co"># lie along the 1:1 line there are five residuals which are all below the line resulting in reasonably substantial </span></span>
<span id="cb11-4"><a href="#cb11-4" tabindex="-1"></a><span class="co"># negative residuals. This suggest that the model does not fit these observation very well.</span></span>
<span id="cb11-5"><a href="#cb11-5" tabindex="-1"></a></span>
<span id="cb11-6"><a href="#cb11-6" tabindex="-1"></a><span class="co"># Looking at the homogeneity of variance assumption (Residuals vs Fitted and Scale-Location plot) you can see the </span></span>
<span id="cb11-7"><a href="#cb11-7" tabindex="-1"></a><span class="co"># five columns of residuals corresponding to the fitted values for the five grazing levels. Again, things don't look </span></span>
<span id="cb11-8"><a href="#cb11-8" tabindex="-1"></a><span class="co"># great. The spread for the lower fitted values (left side of the plot) is much narrower when compared to the other groups.</span></span>
<span id="cb11-9"><a href="#cb11-9" tabindex="-1"></a><span class="co"># This suggests that the homogeneity of variance assumption is not met (i.e. the variances are not the same). The same cluster</span></span>
<span id="cb11-10"><a href="#cb11-10" tabindex="-1"></a><span class="co"># of negative residuals we spotted in the Normal Q-Q plot also appears in the Residuals vs Fitted plot suggesting that it is</span></span>
<span id="cb11-11"><a href="#cb11-11" tabindex="-1"></a><span class="co"># these residuals that are responsible.</span></span>
<span id="cb11-12"><a href="#cb11-12" tabindex="-1"></a></span>
<span id="cb11-13"><a href="#cb11-13" tabindex="-1"></a><span class="co"># The only real good news is that there doesn't appear to be any influential or unusual residuals as indicated in the </span></span>
<span id="cb11-14"><a href="#cb11-14" tabindex="-1"></a><span class="co"># Residuals vs Leverage plot. </span></span>
<span id="cb11-15"><a href="#cb11-15" tabindex="-1"></a></span>
<span id="cb11-16"><a href="#cb11-16" tabindex="-1"></a><span class="co"># So what to do? You could go back and check the original field notebook data to see if a</span></span>
<span id="cb11-17"><a href="#cb11-17" tabindex="-1"></a><span class="co"># transcribing mistake has been made (seems unlikely and you dont have this luxury anyway). </span></span>
<span id="cb11-18"><a href="#cb11-18" tabindex="-1"></a><span class="co"># You could also try applying a transformation (log or square root) on the ABUND variable, refit the model and </span></span>
<span id="cb11-19"><a href="#cb11-19" tabindex="-1"></a><span class="co"># see if this improves things. </span></span>
<span id="cb11-20"><a href="#cb11-20" tabindex="-1"></a><span class="co"># for example </span></span>
<span id="cb11-21"><a href="#cb11-21" tabindex="-1"></a></span>
<span id="cb11-22"><a href="#cb11-22" tabindex="-1"></a>loyn<span class="sc">$</span>ABUND.SQRT <span class="ot"><-</span> <span class="fu">sqrt</span>(loyn<span class="sc">$</span>ABUND)</span>
<span id="cb11-23"><a href="#cb11-23" tabindex="-1"></a>birds_lm_sqrt <span class="ot"><-</span> <span class="fu">lm</span>(ABUND.SQRT <span class="sc">~</span> FGRAZE, <span class="at">data =</span> loyn)</span>
<span id="cb11-24"><a href="#cb11-24" tabindex="-1"></a><span class="fu">par</span>(<span class="at">mfrow =</span> <span class="fu">c</span>(<span class="dv">2</span>,<span class="dv">2</span>))</span>
<span id="cb11-25"><a href="#cb11-25" tabindex="-1"></a><span class="fu">plot</span>(birds_lm_sqrt)</span></code></pre></div>
<p><img src="linear_model_2_exercise_solutions_files/figure-html/Q10-2.png" width="672" /></p>
<div class="sourceCode" id="cb12"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" tabindex="-1"></a></span>
<span id="cb12-2"><a href="#cb12-2" tabindex="-1"></a><span class="co"># Sadly this doesn't seemed to have improved things!</span></span>
<span id="cb12-3"><a href="#cb12-3" tabindex="-1"></a></span>
<span id="cb12-4"><a href="#cb12-4" tabindex="-1"></a><span class="co"># Or finally, you can relax the assumption of equal variance and estimate a separate variance for each group using </span></span>
<span id="cb12-5"><a href="#cb12-5" tabindex="-1"></a><span class="co"># generalised least squares. This is not something we will do on this course but will cover in a more advanced statistics course!</span></span></code></pre></div>
<p> </p>
<p>11. This is an optional question and really just for information.
I’ll give you the code in the solutions so don’t overly stress about
this! Use Google (yep, this is OK!) to figure out how to plot your
fitted values and 95% confidence intervals. Try Googling the
<code>gplots</code> package or the <code>effects</code> package.</p>
<p>Alternatively, have a go at using our old trusty
<code>predict()</code> function to calculate the fitted values and
standard errors. Add the fitted values and 95% confidence intervals to a
plot of bird abundance and graze level (to add your upper and lower
confidence intervals will need to use either the <code>segments()</code>
or <code>arrows()</code> function).</p>
<p>Or we can even use the <code>ggplot2</code> package. Check out the
solutions code if you’re thoroughly confused!</p>
<div class="sourceCode" id="cb13"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" tabindex="-1"></a><span class="co"># Using the gplots package, you may need to install this package first</span></span>
<span id="cb13-2"><a href="#cb13-2" tabindex="-1"></a><span class="co"># install.packages('gplots')</span></span>
<span id="cb13-3"><a href="#cb13-3" tabindex="-1"></a></span>
<span id="cb13-4"><a href="#cb13-4" tabindex="-1"></a>loyn<span class="sc">$</span>FGRAZE <span class="ot"><-</span> <span class="fu">relevel</span>(loyn<span class="sc">$</span>FGRAZE, <span class="at">ref =</span> <span class="st">"1"</span>)</span>
<span id="cb13-5"><a href="#cb13-5" tabindex="-1"></a><span class="fu">library</span>(gplots)</span>
<span id="cb13-6"><a href="#cb13-6" tabindex="-1"></a><span class="fu">plotmeans</span>(ABUND <span class="sc">~</span> FGRAZE, <span class="at">xlab =</span> <span class="st">"grazing level"</span>,</span>
<span id="cb13-7"><a href="#cb13-7" tabindex="-1"></a> <span class="at">ylab =</span> <span class="st">"bird abundance"</span>, <span class="at">data =</span> loyn, <span class="at">connect =</span> <span class="cn">FALSE</span>)</span></code></pre></div>
<p><img src="linear_model_2_exercise_solutions_files/figure-html/Q11a-1.png" width="672" /></p>
<div class="sourceCode" id="cb14"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" tabindex="-1"></a><span class="co"># Using the effects package, you may need to install this package first</span></span>
<span id="cb14-2"><a href="#cb14-2" tabindex="-1"></a><span class="co"># install.packages('effects')</span></span>
<span id="cb14-3"><a href="#cb14-3" tabindex="-1"></a></span>
<span id="cb14-4"><a href="#cb14-4" tabindex="-1"></a><span class="fu">library</span>(effects)</span>
<span id="cb14-5"><a href="#cb14-5" tabindex="-1"></a>loyn_effects <span class="ot"><-</span> <span class="fu">allEffects</span>(birds_lm)</span>
<span id="cb14-6"><a href="#cb14-6" tabindex="-1"></a><span class="fu">plot</span>(loyn_effects,<span class="st">"FGRAZE"</span>, <span class="at">lty =</span> <span class="dv">0</span>)</span></code></pre></div>
<p><img src="linear_model_2_exercise_solutions_files/figure-html/Q11b-1.png" width="672" /></p>
<div class="sourceCode" id="cb15"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" tabindex="-1"></a><span class="co"># and finally using old faithful the predict function and base R graphics</span></span>
<span id="cb15-2"><a href="#cb15-2" tabindex="-1"></a><span class="co"># with the segments function</span></span>
<span id="cb15-3"><a href="#cb15-3" tabindex="-1"></a></span>
<span id="cb15-4"><a href="#cb15-4" tabindex="-1"></a>my_data <span class="ot"><-</span> <span class="fu">data.frame</span>(<span class="at">FGRAZE =</span> <span class="fu">c</span>(<span class="st">"1"</span>, <span class="st">"2"</span>, <span class="st">"3"</span>, <span class="st">"4"</span>, <span class="st">"5"</span>))</span>
<span id="cb15-5"><a href="#cb15-5" tabindex="-1"></a>pred_vals <span class="ot"><-</span> <span class="fu">predict</span>(birds_lm, <span class="at">newdata =</span> my_data, <span class="at">se.fit =</span> <span class="cn">TRUE</span>)</span>
<span id="cb15-6"><a href="#cb15-6" tabindex="-1"></a></span>
<span id="cb15-7"><a href="#cb15-7" tabindex="-1"></a><span class="co"># now plot these values</span></span>
<span id="cb15-8"><a href="#cb15-8" tabindex="-1"></a></span>
<span id="cb15-9"><a href="#cb15-9" tabindex="-1"></a><span class="fu">plot</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">5</span>, <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">50</span>, <span class="at">length=</span><span class="dv">5</span>), <span class="at">type =</span> <span class="st">"n"</span>, <span class="at">xlab =</span> <span class="st">"Graze intensity"</span>, <span class="at">ylab =</span> <span class="st">"Bird Abundance"</span>)</span>
<span id="cb15-10"><a href="#cb15-10" tabindex="-1"></a><span class="fu">points</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">5</span>, pred_vals<span class="sc">$</span>fit)</span>
<span id="cb15-11"><a href="#cb15-11" tabindex="-1"></a><span class="fu">segments</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">5</span>, pred_vals<span class="sc">$</span>fit, <span class="dv">1</span><span class="sc">:</span><span class="dv">5</span>, pred_vals<span class="sc">$</span>fit <span class="sc">-</span> <span class="fl">1.96</span> <span class="sc">*</span> pred_vals<span class="sc">$</span>se.fit)</span>
<span id="cb15-12"><a href="#cb15-12" tabindex="-1"></a><span class="fu">segments</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">5</span>, pred_vals<span class="sc">$</span>fit, <span class="dv">1</span><span class="sc">:</span><span class="dv">5</span>, pred_vals<span class="sc">$</span>fit <span class="sc">+</span> <span class="fl">1.96</span> <span class="sc">*</span> pred_vals<span class="sc">$</span>se.fit)</span></code></pre></div>
<p><img src="linear_model_2_exercise_solutions_files/figure-html/Q11c-1.png" width="672" /></p>
<div class="sourceCode" id="cb16"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb16-1"><a href="#cb16-1" tabindex="-1"></a><span class="co"># using old faithful the predict function and base R graphics </span></span>
<span id="cb16-2"><a href="#cb16-2" tabindex="-1"></a><span class="co"># with the arrows function </span></span>
<span id="cb16-3"><a href="#cb16-3" tabindex="-1"></a></span>
<span id="cb16-4"><a href="#cb16-4" tabindex="-1"></a>my_data <span class="ot"><-</span> <span class="fu">data.frame</span>(<span class="at">FGRAZE =</span> <span class="fu">c</span>(<span class="st">"1"</span>, <span class="st">"2"</span>, <span class="st">"3"</span>, <span class="st">"4"</span>, <span class="st">"5"</span>))</span>
<span id="cb16-5"><a href="#cb16-5" tabindex="-1"></a>pred_vals <span class="ot"><-</span> <span class="fu">predict</span>(birds_lm, <span class="at">newdata =</span> my_data, <span class="at">se.fit =</span> <span class="cn">TRUE</span>)</span>
<span id="cb16-6"><a href="#cb16-6" tabindex="-1"></a></span>
<span id="cb16-7"><a href="#cb16-7" tabindex="-1"></a><span class="co"># now plot these values</span></span>
<span id="cb16-8"><a href="#cb16-8" tabindex="-1"></a></span>
<span id="cb16-9"><a href="#cb16-9" tabindex="-1"></a><span class="fu">plot</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">5</span>, <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">50</span>, <span class="at">length=</span><span class="dv">5</span>), <span class="at">type =</span> <span class="st">"n"</span>, <span class="at">xlab =</span> <span class="st">"Graze intensity level"</span>, <span class="at">ylab =</span> <span class="st">"Bird Abundance"</span>)</span>
<span id="cb16-10"><a href="#cb16-10" tabindex="-1"></a><span class="fu">arrows</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">5</span>, pred_vals<span class="sc">$</span>fit, <span class="dv">1</span><span class="sc">:</span><span class="dv">5</span>, pred_vals<span class="sc">$</span>fit <span class="sc">-</span> <span class="fl">1.96</span> <span class="sc">*</span> pred_vals<span class="sc">$</span>se.fit,</span>
<span id="cb16-11"><a href="#cb16-11" tabindex="-1"></a> <span class="at">angle =</span> <span class="dv">90</span>, <span class="at">code =</span> <span class="dv">2</span>, <span class="at">length =</span> <span class="fl">0.05</span>, <span class="at">col =</span> <span class="st">"blue"</span>)</span>
<span id="cb16-12"><a href="#cb16-12" tabindex="-1"></a><span class="fu">arrows</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">5</span>, pred_vals<span class="sc">$</span>fit, <span class="dv">1</span><span class="sc">:</span><span class="dv">5</span>, pred_vals<span class="sc">$</span>fit <span class="sc">+</span> <span class="fl">1.96</span> <span class="sc">*</span> pred_vals<span class="sc">$</span>se.fit,</span>
<span id="cb16-13"><a href="#cb16-13" tabindex="-1"></a> <span class="at">angle =</span> <span class="dv">90</span>, <span class="at">code =</span> <span class="dv">2</span>, <span class="at">length =</span> <span class="fl">0.05</span>, <span class="at">col =</span> <span class="st">"blue"</span>)</span>
<span id="cb16-14"><a href="#cb16-14" tabindex="-1"></a><span class="fu">points</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">5</span>, pred_vals<span class="sc">$</span>fit, <span class="at">pch =</span> <span class="dv">16</span>)</span></code></pre></div>
<p><img src="linear_model_2_exercise_solutions_files/figure-html/Q11d-1.png" width="672" /></p>
<div class="sourceCode" id="cb17"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" tabindex="-1"></a><span class="co"># or using the ggplot2 package</span></span>
<span id="cb17-2"><a href="#cb17-2" tabindex="-1"></a><span class="fu">library</span>(ggplot2) <span class="co"># make the functions in ggplot2 available</span></span>
<span id="cb17-3"><a href="#cb17-3" tabindex="-1"></a></span>
<span id="cb17-4"><a href="#cb17-4" tabindex="-1"></a></span>
<span id="cb17-5"><a href="#cb17-5" tabindex="-1"></a><span class="co"># This plot will plot the means for each level of FGRAZE </span></span>
<span id="cb17-6"><a href="#cb17-6" tabindex="-1"></a><span class="co"># and also the 95% confidence intervals</span></span>
<span id="cb17-7"><a href="#cb17-7" tabindex="-1"></a></span>
<span id="cb17-8"><a href="#cb17-8" tabindex="-1"></a><span class="fu">ggplot</span>(loyn, <span class="fu">aes</span>(<span class="at">x =</span> FGRAZE, <span class="at">y =</span> ABUND)) <span class="sc">+</span> </span>
<span id="cb17-9"><a href="#cb17-9" tabindex="-1"></a> <span class="fu">stat_summary</span>(<span class="at">fun =</span> mean, <span class="at">geom =</span> <span class="st">"point"</span>, <span class="at">color =</span> <span class="st">"firebrick"</span>,</span>
<span id="cb17-10"><a href="#cb17-10" tabindex="-1"></a> <span class="at">size =</span> <span class="dv">3</span>, <span class="at">position=</span><span class="fu">position_nudge</span>(<span class="at">x =</span> <span class="fl">0.15</span>)) <span class="sc">+</span></span>
<span id="cb17-11"><a href="#cb17-11" tabindex="-1"></a> <span class="fu">stat_summary</span>(<span class="at">fun.data =</span> mean_cl_normal, <span class="at">geom =</span> <span class="st">"errorbar"</span>, </span>
<span id="cb17-12"><a href="#cb17-12" tabindex="-1"></a> <span class="at">width =</span> <span class="fl">0.1</span>, <span class="at">position=</span><span class="fu">position_nudge</span>(<span class="at">x =</span> <span class="fl">0.15</span>))</span></code></pre></div>
<p><img src="linear_model_2_exercise_solutions_files/figure-html/Q11e-1.png" width="672" /></p>
<div class="sourceCode" id="cb18"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb18-1"><a href="#cb18-1" tabindex="-1"></a><span class="co"># and as an added bonus, if you wanted to also plot the </span></span>
<span id="cb18-2"><a href="#cb18-2" tabindex="-1"></a><span class="co"># raw data along with the means for FGRAZE</span></span>
<span id="cb18-3"><a href="#cb18-3" tabindex="-1"></a></span>
<span id="cb18-4"><a href="#cb18-4" tabindex="-1"></a><span class="fu">ggplot</span>(loyn, <span class="fu">aes</span>(<span class="at">x =</span> FGRAZE, <span class="at">y =</span> ABUND)) <span class="sc">+</span> </span>
<span id="cb18-5"><a href="#cb18-5" tabindex="-1"></a> <span class="fu">geom_point</span>(<span class="at">color =</span> <span class="st">"firebrick"</span>, <span class="at">size =</span> <span class="dv">3</span>, <span class="at">alpha =</span> <span class="fl">0.6</span>) <span class="sc">+</span> </span>
<span id="cb18-6"><a href="#cb18-6" tabindex="-1"></a> <span class="fu">stat_summary</span>(<span class="at">fun =</span> mean, <span class="at">geom =</span> <span class="st">"point"</span>, <span class="at">color =</span> <span class="st">"firebrick"</span>,</span>
<span id="cb18-7"><a href="#cb18-7" tabindex="-1"></a> <span class="at">size =</span> <span class="dv">3</span>, <span class="at">position=</span><span class="fu">position_nudge</span>(<span class="at">x =</span> <span class="fl">0.15</span>)) <span class="sc">+</span></span>
<span id="cb18-8"><a href="#cb18-8" tabindex="-1"></a> <span class="fu">stat_summary</span>(<span class="at">fun.data =</span> mean_cl_normal, <span class="at">geom =</span> <span class="st">"errorbar"</span>, </span>
<span id="cb18-9"><a href="#cb18-9" tabindex="-1"></a> <span class="at">width =</span> <span class="fl">0.1</span>, <span class="at">position=</span><span class="fu">position_nudge</span>(<span class="at">x =</span> <span class="fl">0.15</span>))</span></code></pre></div>
<p><img src="linear_model_2_exercise_solutions_files/figure-html/Q11e-2.png" width="672" /></p>
<p> </p>
<p>End of the linear model with single categorical explanatory variable
exercise</p>
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