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<h1 class="title toc-ignore">Practical 09 - Logistic Regression:
Generalised Linear Models for 0,1 and Proportion Data</h1>
</div>
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<hr />
<div id="background" class="section level2">
<h2>Background</h2>
<p>Last practical we saw how generalised linear models (GLMs) can give
you more flexibility when modelling count data that do not easily fit
into the standard Gaussian framework. We learned that to fit a GLM we
need to carry out 3 steps beyond those required by conventional
regressional analyses:</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>Because <code>R</code> functions such as <code>glm()</code>
streamline the process of fitting GLMs, the key step that’s left in your
hands is knowing when you will need to switch from a Gaussian model to a
GLM, and how to identify the correct distribution for your dataset. In
this practical we are going to learn how to extend what we covered last
practical about modelling count data to two other common data types in
biology:</p>
<ul>
<li>0-1 data (e.g., presence-absence, infected or not, alive vs. dead,
etc.)</li>
<li>Proportional/percentage data (e.g.,proportion of population
infected, percent forest cover, proportion of population with a
mutation, etc.)</li>
</ul>
<p>Applying GLMs to these data is also commonly referred to as ‘logistic
regression’. The term ‘logistic regression’ comes from the fact that the
link function we use fits a logistic curve to the relationship between
<span class="math inline">\(x\)</span> and <span
class="math inline">\(y\)</span>. The three step process we covered last
practical is identical for these data types, we just need to familiarise
ourselves with a new set of distributions, and a new link function.</p>
<div id="logistic-regression-generalised-linear-models-glms-for-01-data"
class="section level3">
<h3>Logistic Regression: Generalised Linear Models (GLMs) for 0,1
data</h3>
<p>Logistic regression is a method for fitting a regression curve, <span
class="math inline">\(y = f(x)\)</span>, when <span
class="math inline">\(y\)</span> consists of proportions, probabilities,
or binary coded (0,1–failure,success) data (i.e., anything bound between
0 and 1). Like linear regression, logistic regression makes a number of
key assumptions:</p>
<ul>
<li>The true conditional probabilities are a logistic function of the
independent variables (i.e., correct model specification).</li>
<li>No important variables are omitted & no extra one included.</li>
<li>The independent variables are measured without error.</li>
<li>The observations are independent.</li>
<li>There is no collinearity in the independent variables.</li>
</ul>
<p>The first step of fitting a GLM is to make a distributional
assumption on our 0,1 or proportion data. A good candidate for data that
scale between 0 and 1 is the binomial distribution. The binomial
distribution describes the probability of obtaining <span
class="math inline">\(k\)</span> yes/no successes in a sample of size
<span class="math inline">\(n\)</span>, or in other words, the
distribution of the number of successful trials among a defined number
of trials. The Probability Mass Function (PMF) of the binomial
distribution is given by:</p>
</br>
<center>
<span class="math inline">\(\binom {n}{k}p^{k}(1-p)^{n-k}\)</span>,
<span class="math inline">\(\quad\)</span> where <span
class="math inline">\(\quad {\binom {n}{k}}={\frac
{n!}{k!(n-k)!}}\)</span>
</center>
<p></br></p>
<p>The second step is to specify the deterministic model. This is no
different from fitting other regressional models.</p>
</br>
<center>
<span class="math inline">\(\pi_i = \beta_0 + \beta_1 X +
\ldots\)</span>
</center>
<p></br></p>
<p>The last step is to specify a link function that maps the values
between 0 and 1. The ‘logit’ link is a link function that maps values
between <span class="math inline">\(0, 1\)</span> and the most routinely
used link function for modelling <span class="math inline">\(0,
1\)</span> data.</p>
</br>
<center>
<span class="math inline">\(\mu = \frac{e^{\beta_0 + \beta_1 X +
\ldots}} {1 + e^{\beta_0 + \beta_1 X + \ldots}}\)</span>
</center>
<p></br></p>
These are the three pieces the pieces we need for fitting a GLM to these
data: </br>
<center>
<p><span class="math inline">\(Y_i \sim Binomial(1, \pi_i) \quad \quad
E(Y_i) = \pi_i \quad \mathrm{and} \quad \mathrm{var}(Y_i) = \pi_i
\times(1-\pi_i)\)</span></p>
<span class="math inline">\(\pi_i = \frac{e^{\beta_0 + \beta_1 X +
\ldots}} {1 + e^{\beta_0 + \beta_1 X + \ldots}}\)</span>
</center>
<p></br></p>
<p>Although it can be challenging to work with these complex
distributions and link functions, the <code>glm()</code> function in
<code>R</code> streamlines this process for us. Again, 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>Explore ways for visualising 0,1 data.</li>
<li>Fit models to generalised linear regression models (GLMs) to 0,1 and
proportion data using logistic regression.</li>
<li>Learn how to plot and interpret fitted logistic regression
models.</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>
<hr />
</div>
</div>
<div id="modelling-discrete-01-data" class="section level2">
<h2>Modelling discrete 0,1 Data</h2>
<p>As biologists we often find ourselves with discrete data with values
that be either 0 or 1. The Gaussian distribution is continuous and has
support between <span class="math inline">\(-\infty, \infty\)</span>, so
it’s clearly not a good option when have this kind of data. If we model
these data using a linear regression model, the stochastic part of our
model will be misspecified, so our model’s predictive power will be low,
the residuals will almost always look terrible and, typically, no amount
of transformations will help us. We need a deterministic function that
maps the values between 0 and 1, and a distribution that makes more
sense. In other words, if we’re working with discrete, 0,1 data, we’re
probably going to need to build our models in a logistic regression
framework. To explore this concept we are going to use data from:</p>
<p><a href="https://doi.org/10.1890/08-0219.1">Ozgul, A., Oli, M.K.,
Bolker, B.M. and Perez-Heydrich, C. (2009), Upper respiratory tract
disease, force of infection, and effects on survival of gopher
tortoises. Ecological Applications, 19: 786-798.</a></p>
<p>These data are openly available <a
href="https://github.com/bbolker/krakow_2018/find/master">here</a>.</p>
<p>In Ozgul et al’s paper, the authors explored factors influencing the
prevalence of upper respiratory tract disease (URTD) caused by
<em>Mycoplasma agassizii</em> in gopher tortoises (<em>Gopherus
polyphemus</em>). Variables in this dataset include:</p>
<ul>
<li><code>TortID</code>: Tortoise identification number</li>
<li><code>Date</code>: Date of capture and sampling</li>
<li><code>YEAR</code>: Year of capture and sampling</li>
<li><code>CL</code>: Carapace length in millimeters</li>
<li><code>Sex</code>: Sex of tortoise (M: Male, F: Female, Juv:
Juvenile)</li>
<li><code>ELISA</code>: Result of ELISA test (POSITIVE, negative)</li>
<li><code>SITE</code>: Study site ID (all data from CF site)</li>
<li><code>status</code>: Result of ELISA test (1: <em>M. agassizii</em>
Positive, 0: Negative)</li>
<li><code>age</code>: Equivalent to carapace length</li>
</ul>
<div id="visualising-01-data" class="section level3">
<h3>Visualising 0,1 Data</h3>
<p>The first thing to do when working with a new dataset is to carry
exploratory data visualisation. Visualising binary data can be
challenging, however, and many standard data visualisation tools will
not result in readily interpretable figures. Here we will compare
traditional scatter plots and box plots with two new approaches.</p>
<ol style="list-style-type: decimal">
<li>4 points
<ol></li>
</ol>
<ul>
<li>Import the <code>gophertortoise</code> dataset. Make sure that sex
is imported as a factor.</li>
<li>Create a scatter plot of infection status as a function of carapace
length (<code>CL</code>). Add a regression line to this as a visual aid.
Do you expect there to be a relationship? Do you expect it to be strong?
– 1 point(s)</li>
<li>Another option for visualising 0,1 data are conditional density
plots. These display smoothed proportions of each category (0 or 1)
within various levels of a continuous variable. Use the
<code>cdplot()</code> function to create a conditional density plot of
infection status as a function of carapace length (<code>CL</code>). If
this more or less clear than the scatter plot? Note: For plotting
purposes, <code>status</code> will have to be a factor here. – 0.5
point(s)</li>
<li>Create a boxplot of infection status as a function of sex. Is this
an informative figure? Do you expect there to be a relationship between
these variables? – 0.5 point(s)</li>
<li>Create a frequency table of the number of infected/non-infected
individuals as a function of sex. – 0.5 point(s)</li>
<li>Create a barplot with frequencies of infected vs. non infected on
the Y axis and sex category on the X. Do you see any differences in the
proportion infected or not in any of the three categories? – 0.5
point(s)</li>
<li>Visually, do you expect to see a relationship between carapace
length, sex, and infection status? – 1 point(s)
</ol></li>
</ul>
</div>
<div id="fitting-and-selecting-logistic-regression-models"
class="section level3">
<h3>Fitting and selecting Logistic Regression Models</h3>
<p>After getting a feel for the data, the next step is to fit a logistic
regression model and perform model selection to identify the best fit
model for the data.</p>
<ol style="list-style-type: decimal">
<li>6 points
<ol></li>
</ol>
<ul>
<li>Fit a logistic regression model predicting <code>status</code> from
<code>CL</code>, <code>Sex</code>, and the interaction between these
using the <code>glm</code> function. Be sure to specify the correct
distribution and link function for these data. – 1 point(s)</li>
<li>Inspect the summary overview of the full model. – 0.5 point(s)</li>
<li>Carry out AICc based model selection. What terms should you keep?
which can be dropped? Refit the selected model. – 1 point(s)</li>
<li>Plot the residuals of the selected model against the fitted values.
– 0.5 point(s)</li>
<li>Pseudo R<span class="math inline">\(^2\)</span> values are not very
useful in a logistic regression framework and the residuals are usually
difficult to interpret. Use the <code>CVbinary()</code> function from
the <code>DAAG</code> package to cross-validate the model. – 1
point(s)</li>
<li>What was the <span class="math inline">\(\Delta\)</span>AICc between
the selected model and the intercept only model. Calculate the evidence
ratio to quantify how much the best fit model was an improvement over an
intercept only model? – 1 point(s)</li>
<li>Briefly describe the best fit model and the cross validation
results. – 1 point(s)
</ol></li>
</ul>
</div>
<div id="visualising-a-logistic-regression-model"
class="section level3">
<h3>Visualising a Logistic Regression Model</h3>
<p>In a GLM framework we have a link function that lies between the
fitted model and the response variable. As we saw last practical, 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. We also have
the complication here that we have two parameters to visualise, carapace
length, and sex.</p>
<ol style="list-style-type: decimal">
<li>6 points
<ol></li>
</ol>
<ul>
<li>Create a data frame called <code>New_Data</code> that is comprised
of two columns. The first should be called <code>CL</code> that is made
up of a sequence of numbers between the minimum and maximum values of
<code>CL</code> by steps of 0.1, the second should be called
<code>Sex</code>, and should be a column filled with the factor
<code>"F"</code>. – 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>, and <code>type = "response"</code>. – 1
point(s)</li>
<li>The output of this prediction contains the model’s deterministic
prediction for the infection probability of females on a logit
scale.</li>
<li>Repeat this process for males and juveniles. – 2 point(s)</li>
<li>Use these predictions to plot the fitted GLM overlaid on the
original data for each sex. – 2 point(s)</li>
</ul>
</br> Note: If we wanted to add confidence intervals the process would
be the same as we saw in Practical 8, but with 9 lines the plot would be
difficult to interpret so we will not do that here.
</ol>
<hr />
</div>
</div>
<div id="logistic-regression-on-proportion-data" class="section level2">
<h2>Logistic Regression on Proportion Data</h2>
<p>As many as ~15% of papers in ecology include some kind of
proportional data. Proportions scale between 0-1 (or 0 and 100 for
percentages), but they can also take any value between these limits.
Most of the time, ecologists model proportion data using an <span
class="math inline">\(\arcsin(\sqrt{p})\)</span> transformation, but
this is not an ideal solution (if you’re interested in knowing why this
is the case, I encourage you to read <a
href="https://esajournals.onlinelibrary.wiley.com/doi/10.1890/10-0340.1">this
paper</a>)</p>
<p>Fitting a logistic regression to proportion data is very similar to
fitting a logistic regression to 0,1 data. To learn how to do this we
will use data collected as part of a series of laboratory experiments on
the density- and size-dependent predation rate of an African reed frog,
<em>Hyperolius spinigularis</em>, and used in the following
publication:</p>
<p><a
href="https://esajournals.onlinelibrary.wiley.com/doi/abs/10.1890/04-0535">Vonesh
and Bolker (2005) Compensatory larval responses shift trade-offs
associated with predator-induced hatching plasticity. Ecology
86:1580-1591</a></p>
<p>These data are openly available as part of the <code>emdbook</code>
package. Variables in this dataset include:</p>
<ul>
<li><code>density</code>: initial tadpole density (number of tadpoles in
a 1.2 x 0.8 x 0.4 m tank) [experiment 1]</li>
<li><code>pred</code>: factor: predators present or absent [experiment
1]</li>
<li><code>size</code>: factor: big or small tadpoles [experiment 1]</li>
<li><code>surv</code>: number surviving</li>
<li><code>propsurv</code>: proportion surviving (=surv/density)
[experiment 1]</li>
</ul>
<ol style="list-style-type: decimal">
<li>5 points
<ol></li>
</ol>
<ul>
<li>Load in the <code>emdbook</code> package and import the
<code>ReedfrogPred</code> dataset.</li>
<li>Create two box plots, one depicting the proportion surviving as a
function of body size, the second the proportion surviving as a function
of predator treatment. – 0.5 point(s)</li>
<li>Do you expect to see a relationship between either of these factors
and the proportion surviving? – 0.5 point(s)</li>
<li>Fit a logistic regression model predicting <code>propsurv</code>
from <code>size</code>, <code>pred</code>, and the interaction between
these using the <code>glm</code> function. – 1 point(s)</li>
<li>Inspect the summary overview of the full model. – 0.5 point(s)</li>
<li>Carry out AICc based model selection. What terms should you keep?
which can be dropped? Refit the selected model. – 1 point(s)</li>
<li>What was the <span class="math inline">\(\Delta\)</span>AICc between
the selected model and the intercept only model. Calculate the evidence
ratio to quantify how much the best fit model was an improvement over an
intercept only model? – 0.5 point(s)</li>
<li>Briefly describe the best fit model. – 1 point(s)
</ol></li>
</ul>
<hr />
</div>
<div id="references" class="section level2">
<h2>References</h2>
<p>Ozgul, A., Oli, M.K., Bolker, B.M. and Perez-Heydrich, C. (2009),
Upper respiratory tract disease, force of infection, and effects on
survival of gopher tortoises. Ecological Applications, 19: 786-798.</p>
<p>Vonesh and Bolker (2005) Compensatory larval responses shift
trade-offs associated with predator-induced hatching plasticity. Ecology
86:1580-1591</p>
</div>
</div>
</div>
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