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<h1 class="title toc-ignore">Practical 08 - Generalised Linear Models
for Count Data</h1>
</div>
<script type="text/x-mathjax-config">
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<hr />
<div id="background" class="section level2">
<h2>Background</h2>
<p>We started this course with simple linear regression and we saw how
increasing the number of parameters can soak up variance and improve a
model’s explanatory power, how we can use mixed effects models to
account for hierarchical data structures, how we can modify the variance
structure to account for heteroskedasticity, and how we can modify the
correlation matrix to correct for temporal, spatial, or phylogenetic
autocorrelation. All of these improvements took place in a linear
regression framework, with models of the general form:</p>
</br>
<center>
<span class="math inline">\(y_i = \beta_0 + \beta_1 \times x_{1,i} + ...
\beta_n \times x_{1,n} + \varepsilon_i \quad\quad \varepsilon_i \sim
\mathcal{N}(0,\, V) \quad\quad V = \sigma^2 {\begin{bmatrix}1& 0
&\cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots
&\vdots &\ddots &\vdots \\ 0 & 0 & \cdots &1
\end{bmatrix}}\)</span>
</center>
<p></br></p>
<p>We can get a <strong><em>lot</em></strong> of mileage out of this
formulation, but the problem with this structure is that the range of
the Gaussian distribution is <span class="math inline">\(-\infty,
\infty\)</span>. This means that if we set up our problem this way our
models tell us that our residuals should be normally distributed and
that our response can, theoretically, take any value between <span
class="math inline">\(-\infty, \infty\)</span>. For many datasets, this
assumption is perfectly appropriate. But what if it’s not? Does this
make sense for response variables that can only take positive values? Or
for discrete outcomes? Or if we’re working with proportions that are
bound between 0 and 1?</p>
<p>There are three steps you can take if you think that Gaussian
distributed residuals is not a reasonable assumption for your data:</p>
<ul>
<li><strong>Nothing.</strong> If the residuals are <span
class="math inline">\(\sim\)</span>normally distributed and the spread
isn’t bad, this isn’t a terrible assumption (remember, no model is ever
going to be 100% correct, so this can be a viable option in some
cases).</li>
<li><strong>Transform your data.</strong> Shoehorning your data to fit
the assumptions of normality can work, but it changes the relationship
between your response and your predictors. It’s also not very practical
for discrete response variables because most transformations will not
make these continuous in any meaningful way.</li>
<li><strong>Choose another distribution.</strong> Switching from
applying linear models to generalised linear models (GLMs) can give you
more flexibility when modelling data that do not easily fit into the
standard Gaussian framework.</li>
</ul>
<p>You are asked to complete the following exercises and submit to
Canvas before the deadline. In addition to the points detailed below, 5
points are assigned to the cleanliness of the code and resulting pdf
document. Only knit documents (.pdf, .doc, or .html) will be accepted.
Unknit .Rmd files will not be graded.</p>
<div id="generalised-linear-models-glms" class="section level3">
<h3>Generalised Linear Models (GLMs)</h3>
<p>``<em>In statistics, the generalized linear model (GLM) is a flexible
generalization of ordinary linear regression that allows for response
variables that have error distribution models other than a normal
distribution.</em>’’ — <a
href="https://en.wikipedia.org/wiki/Generalized_linear_model">Wikipedia</a></p>
<p>In 1972, Nelder & Wedderburn (1972) worked out a generalisation
of the linear regression model that extended the models we’ve been
working with so far to any member of the family of exponential
distributions (Gaussian, Poisson, binomial, negative binomial, gamma,
etc.). In particular, they showed how all of these distributions can be
expressed by the general formulation:</p>
</br>
<center>
<span class="math inline">\(f(Y ; \theta, \phi)=e^{\frac{y \times \theta
- b(\theta)}{a(\phi)}+c(y,\theta)}\)</span>
</center>
<p></br></p>
<p>This means that a single set of equations can be used when modelling
random variables drawn from any of the distributions from the
exponential family. Now that we have a general expression for the
stochastic component of our model, we just need to find a way to `link’
the expectation value of our model with the expectation value of the
distribution. To do this we need to carry out 3 steps when fitting
GLMs:</p>
<ul>
<li>Make a distributional assumption on the response variable <span
class="math inline">\(Y_i\)</span>. This also defines the mean and
variance of <span class="math inline">\(Y_i\)</span>.</li>
<li>Specify the deterministic part of the model.</li>
<li>Formally specify the `link’ between the mean of <span
class="math inline">\(Y_i\)</span> and the deterministic part based on
your distributional assumption.</li>
</ul>
<p>Although it can be challenging to work with these complex
distributions, the <code>glm()</code> function in <code>R</code>
streamlines this process for us. Note: This is not the only function for
fitting GLMs, but it’s a good place to start.</p>
<p></br></p>
<p>In this Practical we will:</p>
<ul>
<li>Fit models to generalised linear regression models (GLMs) to count
data using a number of different distributions</li>
<li>Learn how to check a Poisson GLM for overdispersion.</li>
<li>Learn how to plot GLM regression models</li>
<li>Learn why switching to using GLMs is the prefered alternative as
compared to transformations and model manipulations in a standard
Gaussian framework.</li>
</ul>
<hr />
</div>
</div>
<div id="modelling-count-data" class="section level2">
<h2>Modelling Count Data</h2>
<p>As biologists we often find ourselves with count data. Count data
usually range between 0 and <span class="math inline">\(\infty\)</span>.
They’re also usually discrete integers because we don’t count fractions
of things (unless those things were very unlucky…). The Gaussian
distribution is continuous and has support between <span
class="math inline">\(-\infty, \infty\)</span>, so we can already tell
it’s probably not a good option. If we model count data using a linear
regression model, the stochastic part of our model will be misspecified,
so there is a good chance that our model’s predictive power will be low.
This means that if we’re working with count data, we’re probably going
to need to build our models in a GLM framework. To explore this concept
we are going to use data from:</p>
<p><a href="https://science.sciencemag.org/content/179/4076/893">M. P.
Johnson and P. H. Raven (1973) “Species number and endemism: The
Galapagos Archipelago revisited” Science, 179, 893-895</a></p>
<p>These data are openly available as part of the <code>faraway</code>
package.</p>
<p>In Johnson and Raven’s paper, the authors used regression analyses to
explore the relationship between an island’s area and the number of
plant species found on the island. They measured species abundance and
area for 30 Galapagos islands. The motivation of their work was to build
on the theory of <a
href="https://en.wikipedia.org/wiki/Insular_biogeography">Island
Biogeography</a>. Variables in this dataset include:</p>
<ul>
<li><code>Species</code>: The number of plant species found on the
island</li>
<li><code>Endemics</code>: The number of endemic species</li>
<li><code>Area</code>: The area of the island (km<span
class="math inline">\(^2\)</span>)</li>
<li><code>Elevation</code>: The highest elevation of the island (m)</li>
<li><code>Nearest</code>: The distance from the nearest island (km)</li>
<li><code>Scruz</code>: The distance from Santa Cruz island (km)</li>
<li><code>Adjacent</code>: The area of the adjacent island (km<span
class="math inline">\(^2\)</span>)</li>
</ul>
<div id="modelling-count-data-the-wrong-way" class="section level3">
<h3>Modelling Count Data the wrong way</h3>
<p>One of the simples places to start when fitting a model to a datset
is with a simple linear regression model. This can work fine in many
cases, so let’s see if we can get away with this here.</p>
<ol style="list-style-type: decimal">
<li>6 points
<ol></li>
</ol>
<ul>
<li>Install and load the <code>faraway</code> and <code>nlme</code>
package and import the <code>gala</code> dataset. – 0.5 point(s)</li>
<li>Log10 transform the <code>Area</code> data and store this as a new
variable in the same dataset called <code>log.area</code>. – 0.5
point(s)</li>
<li>Create a scatterplot with the log10 transformed <code>Area</code> on
the X and <code>Species</code> on the Y. Do you expect there to be a
significant relationship? Does a Gaussian linear model look appropriate
for these data? – 1 point(s)</li>
<li>Fit a linear model predicting <code>Species</code> from log10
transformed <code>Area</code> using the <code>gls</code> function. – 0.5
point(s)</li>
<li>Inspect the summary overview of the model. – 0.5 point(s)</li>
<li>Create a scatterplot with the log10 transformed <code>Area</code> on
the X and <code>Species</code> on the Y. Use the <code>abline()</code>
function to plot the fitted model. – 0.5 point(s)</li>
<li>Plot the residuals against the fitted values. – 0.5 point(s)</li>
<li>Use the <code>predict()</code> function on this model to predict the
most likely outcomes for the fitted values. Do these numbers seem
plausible? – 1 point(s)</li>
<li>Inspect the fitted model and all of the outputs you just generated.
Do you think that this is a good model for these data? – 1 point(s)
</ol></li>
</ul>
</div>
<div id="attempting-corrections" class="section level3">
<h3>Attempting corrections</h3>
<p>The data look non-linear, so if we knew nothing about GLMs, we could
try to fit a second order polynomial model to correct for the disconnect
between the fitted model and the data.</p>
<ol style="list-style-type: decimal">
<li>3 points
<ol></li>
</ol>
<ul>
<li>Add a second order polynomial term to the first model. – 1
point(s)</li>
<li>Inspect the summary overview of this new model. – 0.5 point(s)</li>
<li>Plot the residuals against the fitted values. – 0.5 point(s)</li>
<li>Inspect the fitted model and all of the outputs you just generated.
Do you think that this correction is sufficient? – 1 point(s)
</ol></li>
</ul>
</div>
<div id="modelling-count-data-using-glms" class="section level3">
<h3>Modelling Count Data using GLMs</h3>
<p>We could keep trying <em>ad hoc</em> corrections to improve the model
(e.g., maybe log-scaling the species abundance data to smooth out some
of the non-linearity), but these are all going to be half measures
because the real issue here is that we are using the wrong distribution
to model these data. Because we’re working with counts of species, what
we’re looking for is a discrete distribution with support between 0 and
<span class="math inline">\(\infty\)</span>. Given these requirements,
the Poisson distribution is a good candidate for modelling these
data.</p>
<p>In order to model these data using a Poisson distribution to describe
the model’s stochastic component, we need to switch to a GLM workflow.
To do this we need to carry out 3 steps before fitting our GLM:</p>
<ul>
<li>Make a distributional assumption on the response variable.</li>
<li>Specify the deterministic part of the model.</li>
<li>Formally specify the `link’ between the deterministic part based on
your distributional assumption.</li>
</ul>
<p>We just said that we think working with a Poisson distribution is a
good place to start, so after step 1 we get:</p>
</br>
<center>
<span class="math inline">\(Y_i \sim Poisson(\lambda =
\mathrm{Species}_i)\)</span>
</center>
<p></br></p>
<p>The second step of a GLM is to specify deterministic part:</p>
</br>
<center>
<span class="math inline">\(\eta = \beta_0 + \beta_1 \times
\mathrm{log.area}_i\)</span>
</center>
<p></br></p>
<p>The last step is to link <span class="math inline">\(\eta\)</span>
and <span class="math inline">\(\mu_i\)</span>. Because our species
abundance data can only be positive, we can’t use an identity link.
Instead, we use a log-link to ensure the fitted values are always
positive:</p>
</br>
<center>
<span class="math inline">\(\log(\mathrm{Species}_i) = \beta_0 + \beta_1
\times \mathrm{log.area}_i \quad \mathrm{or} \quad \mathrm{Species}_i =
e^{\beta_0 + \beta_1 \times \mathrm{log.area}_i}\)</span>
</center>
<p></br></p>
<p>This whole process is streamlined via the <code>glm()</code>
function, which we will now use to try and improve our model</p>
<ol style="list-style-type: decimal">
<li>5 points
<ol></li>
</ol>
<ul>
<li>Fit a generalised linear model predicting <code>Species</code> from
log10 transformed <code>Area</code> using the <code>glm</code> function.
Be sure to specify the distribution you want to model from and the link
function. – 1 point(s)</li>
<li>Inspect the summary overview of the model. – 1 point(s)</li>
<li>Calculate the pseudo R-squared for the fitted GLM model. Does this
seem good? – 1 point(s)</li>
<li>Create a scatterplot with the log10 transformed <code>Area</code> on
the X and <code>Species</code> on the Y. – 1 point(s)</li>
<li>Use <code>abline()</code> to plot the fitted model. What happened? –
1 point(s)
</ol></li>
</ul>
<p>In a GLM framework we have a link function that lies between <span
class="math inline">\(\eta\)</span> and the response variable. If we
simply plot the fitted model on a linear scale without factoring in the
link function, we can not place our fitted model on the appropriate
scale. We need to work around this when visualising GLMs.</p>
<ol style="list-style-type: decimal">
<li>5 points
<ol></li>
</ol>
<ul>
<li>Create a data frame called <code>New_Data</code> that is comprised
of a single column called <code>log.area</code> that is made up of a
sequence of 200 numbers between -2 to 4. – 1 point(s)</li>
<li>Use these ‘data’ combined with the <code>predict()</code> function
to generate predictions from the fitted GLM model. You will need to
correctly specify some of the arguments of this function, so set
<code>newdata = New_Data</code>, <code>type = "link"</code>, and
<code>se = TRUE</code>. Run <code>class()</code> and
<code>names()</code> on the output of <code>predict()</code>. – 1
point(s)</li>
<li>The <code>fit</code> slot of this list contains the model’s
deterministic prediction. The <code>fit.se</code> slot contains the
standard errors of the predictions. Use the <code>fit</code> slot to
plot the fitted GLM overlayed on the original data. Remember <span
class="math inline">\(\mu_i = e^{\eta}\)</span>, so you need to
transform the predictions before plotting them. – 2 point(s)</li>
<li>Use the <code>fit.se</code> slot to plot the 95% confidence bands.
Note: the 95% CIs = <span class="math inline">\(\mu_i \pm SE \times
1.96\)</span>. – 1 point(s)
</ol></li>
</ul>
</div>
<div id="overdispersion" class="section level3">
<h3>Overdispersion</h3>
<p>Remember that the potential problem with Poisson GLMs is
overdispersion. Overdispersion means that the variance is larger than
the mean (when we compare what we would expect from a Poisson
distribution).</p>
<ol style="list-style-type: decimal">
<li>3 points
<ol></li>
</ol>
<ul>
<li>Check for overdispersion by calculating the following: Residual
Deviance / degrees of freedom on the residual deviance. (Large values
are bad in this context) – 1 point(s)</li>
<li>It is also possible to check the significance of this using a
chisquare test. Check if this overdispersion is significant by
calculating the following:
<code>1-pchisq(residual deviance, degrees of freedom on the residual deviance)</code>
– 1 point(s)</li>
<li>Are the data over-dispersed, or is a Poisson assumption appropriate?
– 1 point(s)</li>
</ul>
<p></br></p>
<strong>Note:</strong> You can get the information needed for this
exercise by running summary on the fitted model.
</ol>
</div>
<div id="negative-binomial-glm-on-count-data" class="section level3">
<h3>Negative binomial GLM on count data</h3>
<p>Switching from a Gaussian distribution to a Poisson distribution is
often a good fix for modelling count data, but it’s not always the most
appropriate distribution for count data. One of the primary reasons why
a Poisson won’t work very well on count data is over-dispersion (because
the variance is tied to the mean and therefore less flexible). The
negative binomial distribution is often a viable alternative to the
Poisson distribution as it allows for more heterogeneity because
variance <span class="math inline">\(\neq\)</span> mean.</p>
<ol style="list-style-type: decimal">
<li>5 points
<ol></li>
</ol>
<ul>
<li>Load in the <code>MASS</code> package.</li>
<li>Fit a negative binomial model predicting <code>Species</code> from
log10 transformed <code>Area</code> using the <code>glm.nb()</code>
function from the <code>MASS</code> package. Be sure to specify the link
function. – 1 point(s)</li>
<li>Inspect the summary overview of the model. – 0.5 point(s)</li>
<li>Plot the fitted model and 95% CIs using the same steps as in
Exercise 4. How does this fit compare to the Poisson model? – 3 points
point(s)</li>
<li>Which of the two GLM models would AIC favour? – 0.5 point(s)
</ol></li>
</ul>
<hr />
</div>
</div>
<div id="references" class="section level2">
<h2>References</h2>
<p>Johnson M.P. & Raven, P.H. (1973) Species number and endemism:
The Galapagos Archipelago revisited Science, 179, 893-895.</p>
<p>Nelder, J.A. & Wedderburn, R.W. (1972). Generalized linear
models. Journal of the Royal Statistical Society: Series A (General),
135, 370–384.</p>
<p>More information on plotting GLMs can be found <a
href="https://www.r-bloggers.com/2015/06/confidence-intervals-for-prediction-in-glmms/">here</a></p>
</div>
</div>
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