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<h1 class="title toc-ignore">Problem Set 6</h1>
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<div id="regression-with-triathlon-data" class="section level1">
<h1>Regression with Triathlon Data</h1>
<p>Now that we’ve seen linear regression in R applied to MLB batting
averages, let’s practice what we’ve learned in a context we haven’t
worked with before. We will look at past results of the Ironman
Triathlon and investigate relationships between racing splits and
demographic variables. First let us download the <a
href="data/Ironman%20Data.csv">Ironman Data</a> and save it to our
“data” folder.</p>
<p>Let’s read in the data and take a look at the variables.</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>data <span class="ot"><-</span> <span class="fu">read_csv</span>(<span class="st">'data/Ironman Data.csv'</span>)</span></code></pre></div>
<pre><code>## Rows: 22124 Columns: 389
## ── Column specification ─────────────────────────────────────────────────────────────────────────────────────────────────────────────
## Delimiter: ","
## chr (103): Source Table, Name, Country, Gender, Division, Swim, Bike, Run, Overall, Division Rank, Swim Total Pace, Bike 28.2 mi...
## dbl (102): BIB, Gender Rank, Overall Rank, Log Rank, Bike 28.2 mi Distance, Bike 50.4 mi Distance, Bike 68.6 mi Distance, Bike 8...
## time (184): Swim Total Split Time, Swim Total Race Time, T1, Bike 28.2 mi Split Time, Bike 28.2 mi Race Time, Bike 50.4 mi Split ...
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.</code></pre>
<p>This is a large data set with hundreds of variables. Digging deeper,
we can see that many of the variables seem to be repeated and many have
data only in certain rows. This is common in historical racing results,
so it’s important to determine which variables are most important for
our analysis. In our case, this analysis will be looking at the results
from the 2017 Florida race, and we will focus on the swim, bike, run,
and overall finishing times as well as demographics including country,
gender, and division.</p>
<p>Moreover, we note that the finishing times are actually
character-types in the form of ‘minutes:seconds’. Keeping these
variables in this form will pose problems as we are going to be
performing operations that require numeric data. So, we need to convert
these clock times into pure numbers. The best way to work with clock
times within the tidyverse framework is to use the
<code>lubridate</code> package. Within this package, we can parse the
times in the ‘minutes:seconds’ format using the <code>ms()</code>
function, and then convert the result to a numeric using the
<code>period_to_seconds()</code> function (the result will be a time in
seconds, so we can get this back to minutes by dividing by 60).</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="co">#install.packages("lubridate")</span></span>
<span id="cb3-2"><a href="#cb3-2" tabindex="-1"></a><span class="fu">library</span>(lubridate)</span>
<span id="cb3-3"><a href="#cb3-3" tabindex="-1"></a>results_2017 <span class="ot"><-</span> data <span class="sc">%>%</span> </span>
<span id="cb3-4"><a href="#cb3-4" tabindex="-1"></a> <span class="fu">filter</span>(<span class="st">`</span><span class="at">Source Table</span><span class="st">`</span> <span class="sc">==</span> <span class="st">'2017 - Florida'</span>) <span class="sc">%>%</span> </span>
<span id="cb3-5"><a href="#cb3-5" tabindex="-1"></a> <span class="fu">select</span>(Country, Gender, Division, Swim, Bike, Run, Overall) <span class="sc">%>%</span></span>
<span id="cb3-6"><a href="#cb3-6" tabindex="-1"></a> <span class="fu">drop_na</span>() <span class="sc">%>%</span></span>
<span id="cb3-7"><a href="#cb3-7" tabindex="-1"></a> <span class="fu">mutate</span>(<span class="at">Swim =</span> <span class="fu">period_to_seconds</span>(<span class="fu">ms</span>(Swim))<span class="sc">/</span><span class="dv">60</span>, </span>
<span id="cb3-8"><a href="#cb3-8" tabindex="-1"></a> <span class="at">Bike =</span> <span class="fu">period_to_seconds</span>(<span class="fu">ms</span>(Bike))<span class="sc">/</span><span class="dv">60</span>, </span>
<span id="cb3-9"><a href="#cb3-9" tabindex="-1"></a> <span class="at">Run =</span> <span class="fu">period_to_seconds</span>(<span class="fu">ms</span>(Run))<span class="sc">/</span><span class="dv">60</span>, </span>
<span id="cb3-10"><a href="#cb3-10" tabindex="-1"></a> <span class="at">Overall =</span> <span class="fu">period_to_seconds</span>(<span class="fu">ms</span>(Overall))<span class="sc">/</span><span class="dv">60</span>)</span></code></pre></div>
<div id="simple-linear-regression" class="section level2">
<h2>Simple Linear Regression</h2>
<p>Now that we have our tidy dataset, let’s start looking at the
relationships between each leg of the triathlon.</p>
<ol style="list-style-type: decimal">
<li><p>Create scatterplots of swim times vs. bike times, bike times
vs. run times, and swim times vs. run times. Save them as variables
titled ‘plot.swim_bike’, etc.</p></li>
<li><p>To compare the relationships between these three variables, we
can plot all three plots in the same window, rather than having to
scroll through each one individually. To do so, we use the
<code>ggarrange</code> function within the <code>ggpubr</code> package.
As arguments of the function, we select our three plots and can specify
the number of rows and columns to be displayed.</p></li>
</ol>
<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 class="co">#install.packages("ggpubr")</span></span>
<span id="cb4-2"><a href="#cb4-2" tabindex="-1"></a><span class="fu">library</span>(ggpubr)</span>
<span id="cb4-3"><a href="#cb4-3" tabindex="-1"></a><span class="fu">ggarrange</span>(plot.swim_bike, plot.bike_run, plot.swim_run, <span class="at">nrow =</span> <span class="dv">2</span>, <span class="at">ncol =</span> <span class="dv">2</span>)</span></code></pre></div>
<p><img src="ps6_files/figure-html/pt1%20q2-1.png" width="672" style="display: block; margin: auto;" /></p>
<ul>
<li>Can you visually compare these associations by looking at the
scatterplots? Which legs of the race seem to be more highly
correlated?</li>
</ul>
<ol start="3" style="list-style-type: decimal">
<li>Another way to display three continuous variables at once is by
creating a scatterplot for two of the variables and then coloring the
points by the values of the third variable. Try doing so with different
combinations of our variables to create plots like the one below.</li>
</ol>
<p><img src="ps6_files/figure-html/pt1%20q3%20solution-1.png" width="672" style="display: block; margin: auto;" /></p>
<ul>
<li>From this one, we can see that the racers with faster swim and bike
times tended to have faster run times; however, the specific
relationship between Swim vs. Run and Bike vs. Run is less clear than in
the individual scatter plots.</li>
</ul>
<ol start="4" style="list-style-type: decimal">
<li>To numerically evaluate these relationships, compute the correlation
between each of the variables (you should compute 3 different
correlations).</li>
</ol>
<pre><code>## [1] 0.6379414</code></pre>
<pre><code>## [1] 0.642096</code></pre>
<pre><code>## [1] 0.4433809</code></pre>
<ol start="5" style="list-style-type: decimal">
<li>Now, let’s fit our linear model. Right now, we are doing simple
linear regression, which involves one response variable and one
predictor variable. So, let’s start by running a regression with Swim as
our predictor and Bike as our response.</li>
</ol>
<div class="sourceCode" id="cb8"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" tabindex="-1"></a>lm.swim_bike <span class="ot"><-</span> <span class="fu">lm</span>( ... ) <span class="co"># fill this part in</span></span></code></pre></div>
<ul>
<li>We obtain the following slope and intercept for this
regression.</li>
</ul>
<div class="sourceCode" id="cb9"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" tabindex="-1"></a>lm.swim_bike<span class="sc">$</span>coefficients</span></code></pre></div>
<pre><code>## (Intercept) Swim
## 179.129651 2.395295</code></pre>
<ol start="6" style="list-style-type: decimal">
<li>We also want to add the regression line to our plot. In <a
href="lecture6.html">Lecture 6</a>, we saw that there are two ways of
doing this. The first was to take the intercept and slope coefficients
produced by our linear model and put them into
<code>geom_abline()</code> while the second involved creating a
<code>data_grid</code> over the predictor variable, adding the
predictions to the grid, and then drawing a line through the predictions
created on the grid using <code>geom_line</code>. As discussed in
Lecture 6, the first approach only works for simple linear regression,
while the second can be extended for all types of regression. For this
reason, let’s practice using the second approach.</li>
</ol>
<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 class="fu">library</span>(modelr)</span>
<span id="cb11-2"><a href="#cb11-2" tabindex="-1"></a>grid.swim <span class="ot"><-</span> results_2017 <span class="sc">%>%</span> <span class="fu">data_grid</span>(Swim)</span>
<span id="cb11-3"><a href="#cb11-3" tabindex="-1"></a>grid.swim <span class="ot"><-</span> grid.swim <span class="sc">%>%</span> <span class="fu">add_predictions</span>(<span class="at">model =</span> lm.swim_bike, <span class="at">var =</span> <span class="st">'pred.swim_bike'</span>)</span>
<span id="cb11-4"><a href="#cb11-4" tabindex="-1"></a><span class="fu">ggplot</span>(results_2017) <span class="sc">+</span></span>
<span id="cb11-5"><a href="#cb11-5" tabindex="-1"></a> <span class="fu">geom_point</span>(<span class="fu">aes</span>(Swim, Bike)) <span class="sc">+</span></span>
<span id="cb11-6"><a href="#cb11-6" tabindex="-1"></a> <span class="fu">geom_line</span>(<span class="at">data =</span> grid.swim, <span class="fu">aes</span>(<span class="at">x =</span> Swim, <span class="at">y =</span> pred.swim_bike), <span class="at">color =</span> <span class="st">'red'</span>)</span></code></pre></div>
<p><img src="ps6_files/figure-html/pt1%20q6-1.png" width="672" style="display: block; margin: auto;" /></p>
<ol start="7" style="list-style-type: decimal">
<li>Repeat steps 5 and 6 for the two other combinations of the three
race legs. Note that since we’re only investigating associations between
variables (as opposed to causal relations) it doesn’t really matter
which variable we choose to be our predictor and response; the resulting
plots will be the same, except with flipped axes. If, instead, we were
trying to determine causality, we would put the independent/predictor
variable on the x-axis and the dependent/response variable on the
y-axis.</li>
</ol>
</div>
<div id="going-beyond-simple-linear-regression" class="section level2">
<h2>Going beyond simple linear regression</h2>
<p>In the following two lectures we will be looking at other forms of
regression, namely logistic regression and multiple regression. Whereas
simple linear regression deals with two continuous variables that share
a linear relationship, we can extend our ideas of regression to more
than two variables as well as variables that are binary or discrete in
scale, or share nonlinear relationships.</p>
<div id="nonlinear-regression" class="section level3">
<h3>Nonlinear Regression</h3>
<p>What if we had reason to believe the association between the
finishing times of each leg of the triathlon was quadratic, rather than
linear? For instance, imagine higher times on the run leg were
associated with slightly higher times on the bike leg, but not linearly
higher? This could be possible if we had reason to believe that the top
racers in the Ironman are good on the bike but separate themselves with
really fast run times, while the slower racers are equally slow on the
bike and on the run.</p>
<ol style="list-style-type: decimal">
<li>First let’s start by running simple linear regression on Run
vs. Bike, because it’s best to start with simple models before moving
onto more complex models. Similarly as before, we want to draw this
regression line on our plot. To do this, we create a grid with
discretized Run values as the x values, and the predicted values from
our linear model as the y values. Then, we draw a line through these
points, which is our regression line.</li>
</ol>
<div class="sourceCode" id="cb12"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" tabindex="-1"></a>lm.run_bike <span class="ot"><-</span> <span class="fu">lm</span>(<span class="at">data =</span> results_2017, <span class="at">formula =</span> Bike <span class="sc">~</span> Run)</span>
<span id="cb12-2"><a href="#cb12-2" tabindex="-1"></a>lm.run_bike<span class="sc">$</span>coefficients</span></code></pre></div>
<pre><code>## (Intercept) Run
## 205.7610166 0.5147168</code></pre>
<div class="sourceCode" id="cb14"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" tabindex="-1"></a>grid.run <span class="ot"><-</span> results_2017 <span class="sc">%>%</span> <span class="fu">data_grid</span>(Run)</span>
<span id="cb14-2"><a href="#cb14-2" tabindex="-1"></a>grid.run <span class="ot"><-</span> grid.run <span class="sc">%>%</span> <span class="fu">add_predictions</span>(<span class="at">model =</span> lm.run_bike, <span class="at">var=</span> <span class="st">'pred.run_bike'</span>)</span>
<span id="cb14-3"><a href="#cb14-3" tabindex="-1"></a><span class="fu">ggplot</span>(results_2017) <span class="sc">+</span></span>
<span id="cb14-4"><a href="#cb14-4" tabindex="-1"></a> <span class="fu">geom_point</span>(<span class="fu">aes</span>(Run, Bike)) <span class="sc">+</span></span>
<span id="cb14-5"><a href="#cb14-5" tabindex="-1"></a> <span class="fu">geom_line</span>(<span class="at">data =</span> grid.run, <span class="fu">aes</span>(<span class="at">x =</span> Run, <span class="at">y =</span> pred.run_bike), <span class="at">color =</span> <span class="st">'red'</span>)</span></code></pre></div>
<p><img src="ps6_files/figure-html/pt2%20q1-1.png" width="672" style="display: block; margin: auto;" /></p>
<ol start="2" style="list-style-type: decimal">
<li>Now, we can run the quadratic regression. To include the quadratic
relation into our linear model, we will add a quadratic term as an
additional predictor variable (which, in this case, is Run).</li>
</ol>
<div class="sourceCode" id="cb15"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" tabindex="-1"></a>lm.run_bike2 <span class="ot"><-</span> <span class="fu">lm</span>(<span class="at">data =</span> results_2017, <span class="at">formula =</span> Bike <span class="sc">~</span> Run <span class="sc">+</span> <span class="fu">I</span>(Run<span class="sc">^</span><span class="dv">2</span>))</span>
<span id="cb15-2"><a href="#cb15-2" tabindex="-1"></a>lm.run_bike2<span class="sc">$</span>coefficients</span></code></pre></div>
<pre><code>## (Intercept) Run I(Run^2)
## 26.222478107 1.649746436 -0.001732342</code></pre>
<ol start="3" style="list-style-type: decimal">
<li>To draw this regression line on our plot, we can take the grid of
values used on our simple linear regression in step 1, and add another
column with the predicted values from our new quadratic model. This new
column contains the new y values of the regression line.</li>
</ol>
<div class="sourceCode" id="cb17"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" tabindex="-1"></a>grid.run <span class="ot"><-</span> grid.run <span class="sc">%>%</span> <span class="fu">add_predictions</span>(<span class="at">model =</span> lm.run_bike2, <span class="at">var=</span> <span class="st">'pred.run_bike2'</span>)</span>
<span id="cb17-2"><a href="#cb17-2" tabindex="-1"></a><span class="fu">ggplot</span>(results_2017) <span class="sc">+</span></span>
<span id="cb17-3"><a href="#cb17-3" tabindex="-1"></a> <span class="fu">geom_point</span>(<span class="fu">aes</span>(Run, Bike)) <span class="sc">+</span></span>
<span id="cb17-4"><a href="#cb17-4" tabindex="-1"></a> <span class="fu">geom_line</span>(<span class="at">data =</span> grid.run, <span class="fu">aes</span>(<span class="at">x =</span> Run, <span class="at">y =</span> pred.run_bike), <span class="at">color =</span> <span class="st">'red'</span>) <span class="sc">+</span></span>
<span id="cb17-5"><a href="#cb17-5" tabindex="-1"></a> <span class="fu">geom_line</span>(<span class="at">data =</span> grid.run, <span class="fu">aes</span>(<span class="at">x =</span> Run, <span class="at">y =</span> pred.run_bike2), <span class="at">color =</span> <span class="st">'green'</span>)</span></code></pre></div>
<p><img src="ps6_files/figure-html/pt2%20q3-1.png" width="672" style="display: block; margin: auto;" /></p>
<ol start="4" style="list-style-type: decimal">
<li>Now, we can determine if our quadratic model fits the data better
than our linear model. To do this, we can compute the RMSEs of both
models (as discussed in <a href="lecture4.html">Lecture 4</a>) and see
which is lower.</li>
</ol>
<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="fu">summarize</span>(results_2017,</span>
<span id="cb18-2"><a href="#cb18-2" tabindex="-1"></a> <span class="at">rmse.run_bike =</span> <span class="fu">sqrt</span>(<span class="fu">mean</span>(lm.run_bike<span class="sc">$</span>residuals<span class="sc">^</span><span class="dv">2</span>)),</span>
<span id="cb18-3"><a href="#cb18-3" tabindex="-1"></a> <span class="at">rmse.run_bike2 =</span> <span class="fu">sqrt</span>(<span class="fu">mean</span>(lm.run_bike2<span class="sc">$</span>residuals<span class="sc">^</span><span class="dv">2</span>)))</span></code></pre></div>
<pre><code>## # A tibble: 1 × 2
## rmse.run_bike rmse.run_bike2
## <dbl> <dbl>
## 1 37.3 36.5</code></pre>
<ul>
<li>In fact, it does appear that the quadratic model appears to fit our
data better!</li>
</ul>
</div>
<div id="regression-with-multiple-predictors" class="section level3">
<h3>Regression with multiple predictors</h3>
<p>Let’s start with our Run vs. Bike linear model again; what if we now
want to add a gender effect? This would be the case if we think that the
bike times differ depending on a racer’s gender.</p>
<ol style="list-style-type: decimal">
<li>We have already run the linear model without the gender effect. Now
let’s add the gender effect in.</li>
</ol>
<div class="sourceCode" id="cb20"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb20-1"><a href="#cb20-1" tabindex="-1"></a>lm.run_bike_gender <span class="ot"><-</span> <span class="fu">lm</span>(<span class="at">data =</span> results_2017, <span class="at">formula =</span> Bike <span class="sc">~</span> Run <span class="sc">+</span> Gender)</span>
<span id="cb20-2"><a href="#cb20-2" tabindex="-1"></a>lm.run_bike_gender<span class="sc">$</span>coefficients</span></code></pre></div>
<pre><code>## (Intercept) Run GenderMale
## 229.316963 0.502562 -26.143851</code></pre>
<ul>
<li>To interpret these results, we can say that for every increase in
one minute of run times, bike times increase by about half a minute.
Moreover, being a male is associated with bike time that is on average
26 minutes lower than the alternative (being a female).</li>
</ul>
<ol start="2" style="list-style-type: decimal">
<li>Now, let’s plot our regression. Note that there will be two separate
regression lines: one for the association between running and biking for
males, and one for the association between running and biking for
females. Also note that since we now technically have two predictor
variables to go on our x-axis, we need to create a new grid that spans
the values of the Run variable as well as the Gender variable</li>
</ol>
<div class="sourceCode" id="cb22"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb22-1"><a href="#cb22-1" tabindex="-1"></a>grid.run_gender <span class="ot"><-</span> results_2017 <span class="sc">%>%</span> <span class="fu">data_grid</span>(Run, Gender)</span>
<span id="cb22-2"><a href="#cb22-2" tabindex="-1"></a>grid.run_gender <span class="ot"><-</span> grid.run_gender <span class="sc">%>%</span></span>
<span id="cb22-3"><a href="#cb22-3" tabindex="-1"></a> <span class="fu">add_predictions</span>(<span class="at">model =</span> lm.run_bike_gender, <span class="at">var=</span> <span class="st">'pred.run_bike_gender'</span>)</span>
<span id="cb22-4"><a href="#cb22-4" tabindex="-1"></a><span class="fu">ggplot</span>(results_2017) <span class="sc">+</span></span>
<span id="cb22-5"><a href="#cb22-5" tabindex="-1"></a> <span class="fu">geom_point</span>(<span class="fu">aes</span>(Run, Bike)) <span class="sc">+</span></span>
<span id="cb22-6"><a href="#cb22-6" tabindex="-1"></a> <span class="fu">geom_line</span>(<span class="at">data =</span> grid.run, <span class="fu">aes</span>(<span class="at">x =</span> Run, <span class="at">y =</span> pred.run_bike), <span class="at">color =</span> <span class="st">'red'</span>) <span class="sc">+</span></span>
<span id="cb22-7"><a href="#cb22-7" tabindex="-1"></a> <span class="fu">geom_line</span>(<span class="at">data =</span> grid.run_gender, <span class="fu">aes</span>(<span class="at">x =</span> Run, <span class="at">y =</span> pred.run_bike_gender, <span class="at">col =</span> Gender)) <span class="sc">+</span></span>
<span id="cb22-8"><a href="#cb22-8" tabindex="-1"></a> <span class="fu">scale_color_manual</span>(<span class="at">values =</span> <span class="fu">c</span>(<span class="at">Female=</span><span class="st">"blue"</span>, <span class="at">Male=</span><span class="st">"green"</span>))</span></code></pre></div>
<p><img src="ps6_files/figure-html/pt3%20q2-1.png" width="672" style="display: block; margin: auto;" /></p>
<ul>
<li>We can now clearly see that females have slower bike times than
males. Note that the red line in the middle corresponds to our simple
linear regression without the gender effect.</li>
</ul>
<ol start="3" style="list-style-type: decimal">
<li>We can probably guess that the model with the gender effect fits the
data better than our simple linear regression (i.e. knowing the gender
of a racer and their run time allows us to better predict their bike
time). But let’s verify this computationally.</li>
</ol>
<div class="sourceCode" id="cb23"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb23-1"><a href="#cb23-1" tabindex="-1"></a><span class="fu">summarize</span>(results_2017,</span>
<span id="cb23-2"><a href="#cb23-2" tabindex="-1"></a> <span class="at">rmse.run_bike =</span> <span class="fu">sqrt</span>(<span class="fu">mean</span>(lm.run_bike<span class="sc">$</span>residuals<span class="sc">^</span><span class="dv">2</span>)),</span>
<span id="cb23-3"><a href="#cb23-3" tabindex="-1"></a> <span class="at">rmse.run_bike_gender =</span> <span class="fu">sqrt</span>(<span class="fu">mean</span>(lm.run_bike_gender<span class="sc">$</span>residuals<span class="sc">^</span><span class="dv">2</span>)))</span></code></pre></div>
<pre><code>## # A tibble: 1 × 2
## rmse.run_bike rmse.run_bike_gender
## <dbl> <dbl>
## 1 37.3 35.5</code></pre>
<ul>
<li>Indeed! We will have lower residuals on average by making
predictions from our gender-included model than with our simple linear
regression.</li>
</ul>
</div>
</div>
</div>
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