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<h1 class="title toc-ignore">Exercises</h1>
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<p> </p>
<div id="exercise-5-basic-statistics-in-r" class="section level2">
<h2>Exercise 5: Basic statistics in R</h2>
<p> </p>
<p>Read <a href="https://intro2r.com/stats_r.html">Chapter 6</a> to help
you complete the questions in this exercise.</p>
<p> </p>
<p>Although this short course is primarily focussed on introducing you
to R, it wouldn’t be complete if we didn’t have a peek at some of R’s
statistical roots. Having said that, this will be a very brief overview
with very little in the way of theory so don’t worry if you get a little
lost - this is just a taster, the main course is still to come!</p>
<p> </p>
<p>1. Download the datafile <em>‘prawnGR.CSV’</em> from the <strong><a
href="data.html"><i class="fa fa-download"></i> Data</a></strong> link
and save it to the <code>data</code> directory. Import these data into R
and assign to a variable with an appropriate name. These data were
collected from an experiment to investigate the difference in growth
rate of the <a
href="https://en.wikipedia.org/wiki/Penaeus_monodon">giant tiger
prawn</a> (<em>Penaeus monodon</em>) fed either an artificial or natural
diet. Have a quick look at the structure of this dataset and plot the
growth rate versus the diet using an appropriate plot. How many
observations are there in each diet treatment?</p>
<p> </p>
<p>2. You want to compare the difference in growth rate between the two
diets using a two sample t-test. Before you conduct the test, you need
to assess the normality and equal variance assumptions. Use the function
<code>shapiro.test()</code> to assess normality of growth rate for each
diet separately (Hint: use your indexing skills to extract the growth
rate for each diet <code>GRate[diet=='Natural']</code> first). Use the
function <code>var.test()</code> to test for equal variance (see
<code>?var.test</code> for more information or <a
href="https://intro2r.com/one-and-two-sample-tests.html#one-and-two-sample-tests">Section
6.1</a> of the book for more details). Are your data normally
distributed and have equal variances? Note: We don’t really advocate
using these ‘approaches’ for assessing the normality and equal variance
assumptions assumptions but include them here as many people will still
want to use them. A much better way to assess assumptions is to use
diagnostic plots of the residuals (see Q6 for an example).</p>
<p> </p>
<p>3. Conduct a two sample t-test using the <code>t.test()</code>
function (<a
href="https://intro2r.com/one-and-two-sample-tests.html#one-and-two-sample-tests">Section
6.1</a> of the book). Use the argument <code>var.equal = TRUE</code> to
perform the t-test assuming equal variances. What is the null hypothesis
you want to test? Do you reject or fail to reject the null hypothesis?
What is the value of the t statistic, degrees of freedom and p value?
How would you summarise these summary statistics in a report?</p>
<p> </p>
<p>4. An alternative (but equivalent) way to compare the mean growth
rate between diets is to use a linear model. Use the <code>lm()</code>
function to fit a linear model with <code>GRate</code> as the response
variable and <code>diet</code> as an explanatory variable (see <a
href="https://intro2r.com/simple_lm.html#simple_lm">Section 6.3</a> for
a very brief introduction to linear modelling). Assign
(<code><-</code>) the results of the linear model to a variable with
an appropriate name (i.e. <code>growth.lm</code>).</p>
<p> </p>
<p>5. Produce an ANOVA table for the fitted model using the
<code>anova()</code> function i.e. <code>anova(growth.lm)</code>.
Compare the ANOVA p value to the p value obtained using a t-test. What
do you notice? What is the value of the F statistics and degrees of
freedom? How would you summarise these results in a report?</p>
<p> </p>
<p>6. Assess the normality and equal variance assumptions by plotting
the residuals of the fitted model (see <a
href="https://intro2r.com/simple_lm.html#simple_lm">Section 6.3</a> for
more details). Split the plotting device into 2 rows and 2 columns using
<code>par(mfrow=c(2,2))</code> so you can fit four plots on a single
device. Use the <code>plot()</code> function on your fitted model
(<code>plot(growth.lm)</code>) to plot the graphs. Discuss with an
instructor how to interpret these plots. What are your conclusions?</p>
<p> </p>
<p>7. Download the datafile <em>‘Gigartina.CSV’</em> from the <strong><a
href="data.html"><i class="fa fa-download"></i> Data</a></strong> link
and save it to the <code>data</code> directory. Import the dataset into
R and assign the dataframe an appropriate name. These data were
collected from a study to examine the change in <code>diameter</code> of
red algae <a
href="https://en.wikipedia.org/wiki/Mastocarpus_stellatus"><em>Mastocarpus
stellatus</em></a> spores grown in three different diatom cultures and a
control group grown in artificial seawater (<code>diatom.treat</code>
variable). Use the function <code>str()</code> to examine the dataframe.
How many replicates are there per diatom treatment? Use an appropriate
plot to examine whether there are any obvious differences in diameter
between the treatments.</p>
<p> </p>
<p>8. You wish to compare the mean diameter of <em>Metacarpus</em> grown
in the four treatment groups (<code>ASGM</code>, <code>Sdecl</code>,
<code>Sexpo</code>, <code>Sstat</code>) using a one-way analysis of
variance (ANOVA). What is your null hypothesis?</p>
<p> </p>
<p>9. We will conduct the ANOVA using the linear model function
<code>lm()</code> once again. Make sure you know which of the variables
is your response variable and which is your explanatory variable (ask an
instructor if in doubt). Fit the linear model and assign the model
output to a variable with an appropriate name
(i.e. <code>gigartina.lm</code>).</p>
<p> </p>
<p>10. Produce an ANOVA table using the <code>anova()</code> function.
What is the value of the p value? Do you reject or fail to reject the
null hypothesis? What is the value of the <em>F</em> statistic and
degrees of freedom? How would you report these summary statistics in a
report?</p>
<p> </p>
<p>11. Assess the assumptions of normality and equal variance of the
residuals by producing the residual plots as before. Don’t forget to
split the plotting device into 2 rows and 2 columns before plotting.
Discuss with an instructor whether the residuals meet these assumptions.
Do you accept this model as acceptable?</p>
<p> </p>
<p>12. Let’s compare the treatment group means to determine which
treatment group is different from other treatment groups. In general,
you should be careful with these types of post-hoc comparisons,
especially if you have a large number of groups (There are much better
ways to do this, but that’s for another course!). In this case we only
have 4 groups, and therefore we will use Tukey’s Honest significant
difference to perform the comparisons and control for type 1 error rate
(rejecting a true null hypothesis).</p>
<p>We will use the function <code>TukeyHSD()</code> from the
<code>mosaic</code> package to perform these comparisons (you will need
to install this package first and then use <code>library(mosaic)</code>
to make the function available). Which groups are different from each
other if we use the p-value cutoff (alpha) of p < 0.05?</p>
<p> </p>
<p>13. We can also produce a plot of the comparisons to help us
interpret the table of comparisons. Use the <code>plot()</code> function
with the <code>TukeyHSD(gigartina.lm)</code>. Ask if you get stuck (or
Google it!).</p>
<p> </p>
<p>14. Download the <em>‘TemoraBR.csv’</em> file from the <strong><a
href="data.html"><i class="fa fa-download"></i> Data</a></strong> link
and save it to the <code>data</code> directory. Import the dataset into
R and as usual assign it to a variable. These data are from an
experiment that was conducted to investigate the relationship between
temperature (<code>temp</code>) and the beat rate (Hz)
<code>beat_rate</code> of the copepod <a
href="https://en.wikipedia.org/wiki/Temora_longicornis"><em>Temora
longicornis</em></a> which had been acclimatised at three different
temperature regimes (<code>acclimitisation_temp</code>). Examine the
structure of the dataset. How many variables are there? What type of
variables are they? Which is the response (dependent) variable, and
which are the explanatory (independent) variables?</p>
<p> </p>
<p>15. What type of variable is <code>acclimitisation_temp</code>? Is it
a factor? Convert <code>acclimitisation_temp</code> to a factor and
store the result in a new column in your dataframe called
<code>Facclimitisation_temp</code>. Hint: use the function
<code>factor()</code>. Use an appropriate plot to visualise these data
(perhaps a coplot or similar?).</p>
<p> </p>
<p>16. We will analyse these data using an Analysis of Covariance
(ANCOVA) to compare the slopes and the intercepts of the relationship
between <code>beat_rate</code> and <code>temp</code> for each level of
<code>Facclimatisation_temp</code>. Take a look at the plot you produced
in Q16, do you think the relationships are different?</p>
<p> </p>
<p>17. As usual we will fit the model using the <code>lm()</code>
function. You will need to fit the main effects of <code>temp</code> and
<code>Facclimatisation_temp</code> and the interaction between
<code>temp</code> and <code>Facclimatisation_temp</code>. You can do
this using either of the equivalent specifications:</p>
<p><code>temp + Facclimatisation_temp + temp:Facclimatisation_temp</code>
or</p>
<p><code>temp * Facclimatisation_temp</code></p>
<p> </p>
<p>18. Produce the summary ANOVA table as usual. Is the interaction
between <code>temp</code> and <code>Facclimatisation_temp</code>
significant? What is the interpretation of the interaction term? Should
we interpret the main effects of <code>temp</code> and
<code>Facclimatisation_temp</code> as well?</p>
<p> </p>
<p>19. Assess the assumptions of normality and equal variance of the
residuals in the usual way. Do the residuals meet these assumptions?
Discuss with a instructor.</p>
<p> </p>
<p>20. Write a short summary in you R script (don’t forget to comment
this out with <code>#</code>) describing the interpretation of this
model. Report the appropriate summary statistics such as <em>F</em>
values, degrees of freedom and p values.</p>
<p> </p>
<p>21. (Optional) refit the model using the square root transformed
<code>beat_rate</code> as the response variable. Does the interpretation
of the model change? Do the validation plots of the residuals look
better?</p>
<p> </p>
<p>End of Exercise 5</p>
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