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  <li><a href="#introduction" id="toc-introduction" class="nav-link active" data-scroll-target="#introduction">Introduction</a></li>
  <li><a href="#preprocessing" id="toc-preprocessing" class="nav-link" data-scroll-target="#preprocessing">Preprocessing</a></li>
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  <li><a href="#heart-disease-distribution" id="toc-heart-disease-distribution" class="nav-link" data-scroll-target="#heart-disease-distribution">Heart Disease Distribution</a></li>
  <li><a href="#continuous-predictors" id="toc-continuous-predictors" class="nav-link" data-scroll-target="#continuous-predictors">Continuous Predictors</a></li>
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<h1 class="title">Heart Failure Prediction Case Study</h1>
</div>


<div class="quarto-title-meta">

    <div>
    <div class="quarto-title-meta-heading">Author</div>
    <div class="quarto-title-meta-contents">
             <p>Arvon Clemons II </p>
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    <div>
    <div class="quarto-title-meta-heading">Published</div>
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      <p class="date">January 9, 2026</p>
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</header>


<section id="introduction" class="level2">
<h2 class="anchored" data-anchor-id="introduction">Introduction</h2>
<p>The goal of this case study is to create a heart failure prediction model with at least 90% accuracy using the <a href="https://www.kaggle.com/datasets/fedesoriano/heart-failure-prediction/data?select=heart.csv">Heart Failure Prediction Dataset</a> by <code>fedesoriano</code>.</p>
<p>This case study will follow several common steps of Data Analysis process such as preprocessing, Exploratory Data Analysis (EDA), model building, model evaluation, and model selection.</p>
</section>
<section id="preprocessing" class="level2">
<h2 class="anchored" data-anchor-id="preprocessing">Preprocessing</h2>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="annotated-cell-1"><pre class="sourceCode r code-annotation-code code-with-copy code-annotated"><code class="sourceCode r"><span id="annotated-cell-1-1"><a href="#annotated-cell-1-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(pacman)</span>
<a class="code-annotation-anchor" data-target-cell="annotated-cell-1" data-target-annotation="1" onclick="event.preventDefault();" href="">1</a><span id="annotated-cell-1-2" class="code-annotation-target"><a href="#annotated-cell-1-2" aria-hidden="true" tabindex="-1"></a><span class="fu">p_load</span>(tidymodels,</span>
<a class="code-annotation-anchor" data-target-cell="annotated-cell-1" data-target-annotation="2" onclick="event.preventDefault();" href="">2</a><span id="annotated-cell-1-3" class="code-annotation-target"><a href="#annotated-cell-1-3" aria-hidden="true" tabindex="-1"></a>       psych,</span>
<a class="code-annotation-anchor" data-target-cell="annotated-cell-1" data-target-annotation="3" onclick="event.preventDefault();" href="">3</a><span id="annotated-cell-1-4" class="code-annotation-target"><a href="#annotated-cell-1-4" aria-hidden="true" tabindex="-1"></a>       corrr,</span>
<a class="code-annotation-anchor" data-target-cell="annotated-cell-1" data-target-annotation="4" onclick="event.preventDefault();" href="">4</a><span id="annotated-cell-1-5" class="code-annotation-target"><a href="#annotated-cell-1-5" aria-hidden="true" tabindex="-1"></a>       car)</span><div class="code-annotation-gutter-bg"></div><div class="code-annotation-gutter"></div></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-annotation">
<dl class="code-annotation-container-grid">
<dt data-target-cell="annotated-cell-1" data-target-annotation="1">1</dt>
<dd>
<span data-code-cell="annotated-cell-1" data-code-lines="2" data-code-annotation="1">This is a popular suite of packages used for modeling in the Tidy data syntax.</span>
</dd>
<dt data-target-cell="annotated-cell-1" data-target-annotation="2">2</dt>
<dd>
<span data-code-cell="annotated-cell-1" data-code-lines="3" data-code-annotation="2">Some useful functions for Exploratory Data Analysis</span>
</dd>
<dt data-target-cell="annotated-cell-1" data-target-annotation="3">3</dt>
<dd>
<span data-code-cell="annotated-cell-1" data-code-lines="4" data-code-annotation="3">Required for some data visualizations</span>
</dd>
<dt data-target-cell="annotated-cell-1" data-target-annotation="4">4</dt>
<dd>
<span data-code-cell="annotated-cell-1" data-code-lines="5" data-code-annotation="4">Useful functions for model diagnostics</span>
</dd>
</dl>
</div>
</div>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a>hf_data <span class="ot">&lt;-</span> readr<span class="sc">::</span><span class="fu">read_csv</span>(<span class="st">"archive/heart.csv"</span>,</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>                           <span class="at">col_names =</span> T)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-stderr">
<pre><code>Rows: 918 Columns: 12
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (5): Sex, ChestPainType, RestingECG, ExerciseAngina, ST_Slope
dbl (7): Age, RestingBP, Cholesterol, FastingBS, MaxHR, Oldpeak, HeartDisease

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>
</div>
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb3"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="fu">describeData</span>(hf_data)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-stdout">
<pre><code>n.obs =  918 of which  918   are complete cases.   Number of variables =  12  of which all are numeric  FALSE
                variable # n.obs type     H1     H2  H3     H4     T1     T2
Age*                     1   918    4     40     49  37     48     68     57
Sex*                     2   918    4      M      F   M      F      M      M
ChestPainType*           3   918    4    ATA    NAP ATA    ASY    ASY    ASY
RestingBP*               4   918    4    140    160 130    138    144    130
Cholesterol*             5   918    4    289    180 283    214    193    131
FastingBS*               6   918    4      0      0   0      0      1      0
RestingECG*              7   918    4 Normal Normal  ST Normal Normal Normal
MaxHR*                   8   918    4    172    156  98    108    141    115
ExerciseAngina*          9   918    4      N      N   N      Y      N      Y
Oldpeak*                10   918    4    0.0    1.0 0.0    1.5    3.4    1.2
ST_Slope*               11   918    4     Up   Flat  Up   Flat   Flat   Flat
HeartDisease*           12   918    4      0      1   0      1      1      1
                  T3     T4
Age*              57     38
Sex*               F      M
ChestPainType*   ATA    NAP
RestingBP*       130    138
Cholesterol*     236    175
FastingBS*         0      0
RestingECG*      LVH Normal
MaxHR*           174    173
ExerciseAngina*    N      N
Oldpeak*         0.0    0.0
ST_Slope*       Flat     Up
HeartDisease*      1      0</code></pre>
</div>
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="fu">describeFast</span>(hf_data)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-stdout">
<pre><code>
 Number of observations =  918 of which  918   are complete cases.   Number of variables =  12  of which  0  are numeric and  0  are factors

 To list the items and their counts, print with short = FALSE</code></pre>
</div>
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="fu">glimpse</span>(hf_data)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-stdout">
<pre><code>Rows: 918
Columns: 12
$ Age            &lt;dbl&gt; 40, 49, 37, 48, 54, 39, 45, 54, 37, 48, 37, 58, 39, 49,…
$ Sex            &lt;chr&gt; "M", "F", "M", "F", "M", "M", "F", "M", "M", "F", "F", …
$ ChestPainType  &lt;chr&gt; "ATA", "NAP", "ATA", "ASY", "NAP", "NAP", "ATA", "ATA",…
$ RestingBP      &lt;dbl&gt; 140, 160, 130, 138, 150, 120, 130, 110, 140, 120, 130, …
$ Cholesterol    &lt;dbl&gt; 289, 180, 283, 214, 195, 339, 237, 208, 207, 284, 211, …
$ FastingBS      &lt;dbl&gt; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ RestingECG     &lt;chr&gt; "Normal", "Normal", "ST", "Normal", "Normal", "Normal",…
$ MaxHR          &lt;dbl&gt; 172, 156, 98, 108, 122, 170, 170, 142, 130, 120, 142, 9…
$ ExerciseAngina &lt;chr&gt; "N", "N", "N", "Y", "N", "N", "N", "N", "Y", "N", "N", …
$ Oldpeak        &lt;dbl&gt; 0.0, 1.0, 0.0, 1.5, 0.0, 0.0, 0.0, 0.0, 1.5, 0.0, 0.0, …
$ ST_Slope       &lt;chr&gt; "Up", "Flat", "Up", "Flat", "Up", "Up", "Up", "Up", "Fl…
$ HeartDisease   &lt;dbl&gt; 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1…</code></pre>
</div>
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb9"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>janitor<span class="sc">::</span><span class="fu">get_dupes</span>(hf_data) <span class="sc">%&gt;%</span> <span class="co"># determine if we have duplicate rows</span></span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">nrow</span>()</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-stderr">
<pre><code>No variable names specified - using all columns.

No duplicate combinations found of: Age, Sex, ChestPainType, RestingBP, Cholesterol, FastingBS, RestingECG, MaxHR, ExerciseAngina, ... and 3 other variables</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0</code></pre>
</div>
</div>
<p>This dataset isn’t missing any values and has no duplicate rows. Based on the provided context for this dataset, we will need to convert several variables into <code>factors</code> such as <code>Sex</code>, <code>FastingsBS</code> and <code>ExerciseAngina</code>.</p>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb12"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a>hf_data <span class="ot">&lt;-</span> hf_data <span class="sc">%&gt;%</span> <span class="fu">mutate</span>(<span class="fu">across</span>(<span class="at">.cols =</span> <span class="fu">c</span>(Sex,</span>
<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a>                                    ChestPainType,</span>
<span id="cb12-3"><a href="#cb12-3" aria-hidden="true" tabindex="-1"></a>                                    FastingBS,</span>
<span id="cb12-4"><a href="#cb12-4" aria-hidden="true" tabindex="-1"></a>                                    RestingECG,</span>
<span id="cb12-5"><a href="#cb12-5" aria-hidden="true" tabindex="-1"></a>                                    ExerciseAngina,</span>
<span id="cb12-6"><a href="#cb12-6" aria-hidden="true" tabindex="-1"></a>                                    ST_Slope,</span>
<span id="cb12-7"><a href="#cb12-7" aria-hidden="true" tabindex="-1"></a>                                    HeartDisease),</span>
<span id="cb12-8"><a href="#cb12-8" aria-hidden="true" tabindex="-1"></a>                          <span class="at">.fns =</span> as.factor))</span>
<span id="cb12-9"><a href="#cb12-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb12-10"><a href="#cb12-10" aria-hidden="true" tabindex="-1"></a><span class="fu">glimpse</span>(hf_data)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-stdout">
<pre><code>Rows: 918
Columns: 12
$ Age            &lt;dbl&gt; 40, 49, 37, 48, 54, 39, 45, 54, 37, 48, 37, 58, 39, 49,…
$ Sex            &lt;fct&gt; M, F, M, F, M, M, F, M, M, F, F, M, M, M, F, F, M, F, M…
$ ChestPainType  &lt;fct&gt; ATA, NAP, ATA, ASY, NAP, NAP, ATA, ATA, ASY, ATA, NAP, …
$ RestingBP      &lt;dbl&gt; 140, 160, 130, 138, 150, 120, 130, 110, 140, 120, 130, …
$ Cholesterol    &lt;dbl&gt; 289, 180, 283, 214, 195, 339, 237, 208, 207, 284, 211, …
$ FastingBS      &lt;fct&gt; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ RestingECG     &lt;fct&gt; Normal, Normal, ST, Normal, Normal, Normal, Normal, Nor…
$ MaxHR          &lt;dbl&gt; 172, 156, 98, 108, 122, 170, 170, 142, 130, 120, 142, 9…
$ ExerciseAngina &lt;fct&gt; N, N, N, Y, N, N, N, N, Y, N, N, Y, N, Y, N, N, N, N, N…
$ Oldpeak        &lt;dbl&gt; 0.0, 1.0, 0.0, 1.5, 0.0, 0.0, 0.0, 0.0, 1.5, 0.0, 0.0, …
$ ST_Slope       &lt;fct&gt; Up, Flat, Up, Flat, Up, Up, Up, Up, Flat, Up, Up, Flat,…
$ HeartDisease   &lt;fct&gt; 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1…</code></pre>
</div>
</div>
<p>Now we can get a better picture of our dataset through Exploratory Data Analysis.</p>
</section>
<section id="exploratory-data-analysis" class="level2">
<h2 class="anchored" data-anchor-id="exploratory-data-analysis">Exploratory Data Analysis</h2>
<p>For our analysis we will select <code>HeartDisease</code> as the outcome (dependent) variable and explore the remaining variables in relation to it.</p>
<section id="heart-disease-distribution" class="level3">
<h3 class="anchored" data-anchor-id="heart-disease-distribution">Heart Disease Distribution</h3>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb14"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a>p <span class="ot">&lt;-</span> <span class="fu">ggplot</span>(<span class="at">data =</span> hf_data)</span>
<span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb14-3"><a href="#cb14-3" aria-hidden="true" tabindex="-1"></a>p <span class="sc">+</span> </span>
<span id="cb14-4"><a href="#cb14-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_bar</span>() <span class="sc">+</span></span>
<span id="cb14-5"><a href="#cb14-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">aes</span>(<span class="at">x =</span> HeartDisease, </span>
<span id="cb14-6"><a href="#cb14-6" aria-hidden="true" tabindex="-1"></a>      <span class="at">fill =</span> HeartDisease) <span class="sc">+</span></span>
<span id="cb14-7"><a href="#cb14-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_fill_discrete</span>(<span class="at">palette =</span> <span class="st">"Accent"</span>,</span>
<span id="cb14-8"><a href="#cb14-8" aria-hidden="true" tabindex="-1"></a>                      <span class="at">name =</span> <span class="st">"Status"</span>,</span>
<span id="cb14-9"><a href="#cb14-9" aria-hidden="true" tabindex="-1"></a>                      <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">"0"</span> <span class="ot">=</span> <span class="st">"Negative"</span>, <span class="st">"1"</span> <span class="ot">=</span> <span class="st">"Positive"</span>)) <span class="sc">+</span></span>
<span id="cb14-10"><a href="#cb14-10" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">"Heart Disease Status"</span>,</span>
<span id="cb14-11"><a href="#cb14-11" aria-hidden="true" tabindex="-1"></a>       <span class="at">title =</span> <span class="st">"Distribution of Heart Disease Outcome"</span>)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="Heart-Failure-Prediction_files/figure-html/Distribution%20of%20Dependent%20Variable-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
</div>
<p>As we can see our outcome variable is binary with sufficient counts in each category. Our initial assumption would be that a Logistic Regression model would be best for creating a predictive model.</p>
<p>There several other assumptions which must be met to perform a Logistic Regression model:</p>
<ol type="1">
<li><p>Our observations must be independent, based on the data source this appears to be true.</p></li>
<li><p>There is low multicollinearity among our independent (predictor) variables. This we will be testing to confirm.</p></li>
<li><p>Linearity of the log-odds of the dependent variable and the continuous independent variables.</p></li>
<li><p>A large enough sample size, with our sample size of 918 observations we’re confident that we meet this requirement. Using <a href="https://pubmed.ncbi.nlm.nih.gov/8970487/">Peduzzi Formula</a> <span class="math inline">\(N =\frac {10 \times k}{p}\)</span> confirms that a sample size of at least 200 is enough.</p></li>
</ol>
</section>
<section id="continuous-predictors" class="level3">
<h3 class="anchored" data-anchor-id="continuous-predictors">Continuous Predictors</h3>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb15"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a>corr_data <span class="ot">&lt;-</span> hf_data <span class="sc">%&gt;%</span> </span>
<span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">correlate</span>() <span class="sc">%&gt;%</span> </span>
<span id="cb15-3"><a href="#cb15-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">rearrange</span>() <span class="sc">%&gt;%</span> </span>
<span id="cb15-4"><a href="#cb15-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">shave</span>()</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-stderr">
<pre><code>Non-numeric variables removed from input: `Sex`, `ChestPainType`, `FastingBS`, `RestingECG`, `ExerciseAngina`, `ST_Slope`, and `HeartDisease`
Correlation computed with
• Method: 'pearson'
• Missing treated using: 'pairwise.complete.obs'</code></pre>
</div>
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb17"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a>corr_data <span class="sc">%&gt;%</span> </span>
<span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">rplot</span>(<span class="at">print_cor =</span> T,</span>
<span id="cb17-3"><a href="#cb17-3" aria-hidden="true" tabindex="-1"></a>        <span class="at">colors =</span> <span class="fu">c</span>(<span class="st">'violetred2'</span>, <span class="st">'snow2'</span>, <span class="st">'dodgerblue2'</span>)</span>
<span id="cb17-4"><a href="#cb17-4" aria-hidden="true" tabindex="-1"></a>        )</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-stderr">
<pre><code>Warning: `aes_string()` was deprecated in ggplot2 3.0.0.
ℹ Please use tidy evaluation idioms with `aes()`.
ℹ See also `vignette("ggplot2-in-packages")` for more information.
ℹ The deprecated feature was likely used in the corrr package.
  Please report the issue at &lt;https://github.com/tidymodels/corrr/issues&gt;.</code></pre>
</div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="Heart-Failure-Prediction_files/figure-html/Correlation%20of%20Continuous%20Predictors-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb19"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a>corr_data <span class="sc">%&gt;%</span> </span>
<span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">fashion</span>()</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-stdout">
<pre><code>         term  Age Oldpeak RestingBP Cholesterol MaxHR
1         Age
2     Oldpeak  .26
3   RestingBP  .25     .16
4 Cholesterol -.10     .05       .10
5       MaxHR -.38    -.16      -.11         .24      </code></pre>
</div>
</div>
<p>The above correlation matrix information suggests that there is little collinearity between the predictors. We will further confirm this by obtaining the <strong>Variance Inflation Factor</strong> (VIF) after fitting our model.</p>
<p>Next we will visualize the distribution of our continuous predictors by the outcome.</p>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb21"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a>hf_data <span class="sc">%&gt;%</span></span>
<span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">select</span>(HeartDisease, <span class="fu">where</span>(is.numeric)) <span class="sc">%&gt;%</span></span>
<span id="cb21-3"><a href="#cb21-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="fu">where</span>(is.numeric), <span class="at">names_to =</span> <span class="st">"variable"</span>, <span class="at">values_to =</span> <span class="st">"value"</span>) <span class="sc">%&gt;%</span></span>
<span id="cb21-4"><a href="#cb21-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x=</span> HeartDisease, <span class="at">y =</span> value, <span class="at">fill =</span> HeartDisease)) <span class="sc">+</span></span>
<span id="cb21-5"><a href="#cb21-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_boxplot</span>() <span class="sc">+</span></span>
<span id="cb21-6"><a href="#cb21-6" aria-hidden="true" tabindex="-1"></a>  <span class="fu">facet_wrap</span>(<span class="sc">~</span> variable, <span class="at">scales =</span> <span class="st">"free"</span>) <span class="sc">+</span></span>
<span id="cb21-7"><a href="#cb21-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">"Heart Disease Status"</span>,</span>
<span id="cb21-8"><a href="#cb21-8" aria-hidden="true" tabindex="-1"></a>       <span class="at">y =</span> <span class="st">"value"</span>,</span>
<span id="cb21-9"><a href="#cb21-9" aria-hidden="true" tabindex="-1"></a>       <span class="at">title =</span> <span class="st">"Distribution of Continuous Predictors by Heart Disease Status"</span>) <span class="sc">+</span></span>
<span id="cb21-10"><a href="#cb21-10" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_fill_discrete</span>(<span class="at">palette =</span> <span class="st">"Accent"</span>,</span>
<span id="cb21-11"><a href="#cb21-11" aria-hidden="true" tabindex="-1"></a>                      <span class="at">name =</span> <span class="st">"Status"</span>,</span>
<span id="cb21-12"><a href="#cb21-12" aria-hidden="true" tabindex="-1"></a>                      <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">"0"</span> <span class="ot">=</span> <span class="st">"Negative"</span>, <span class="st">"1"</span> <span class="ot">=</span> <span class="st">"Positive"</span>))</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="Heart-Failure-Prediction_files/figure-html/Distribution%20of%20Continuous%20Predictors-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
</div>
<p>We can see that for those who do not have heart disease:</p>
<ol type="1">
<li><p>Tend to be younger</p></li>
<li><p>Most serum cholesterol levels ~227 mm/dl</p></li>
<li><p>A higher maximum heart rate</p></li>
<li><p>Lower ST Depression (Old Peak)</p></li>
</ol>
<p>Noticeably the resting blood pressure seems to be very similar between the groups.</p>
</section>
<section id="categorical-predictors" class="level3">
<h3 class="anchored" data-anchor-id="categorical-predictors">Categorical Predictors</h3>
<p>Finally are the Categorical Predictors</p>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb22"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a>hf_data <span class="sc">%&gt;%</span></span>
<span id="cb22-2"><a href="#cb22-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">select</span>(<span class="fu">where</span>(is.factor)) <span class="sc">%&gt;%</span></span>
<span id="cb22-3"><a href="#cb22-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span>HeartDisease, <span class="at">names_to =</span> <span class="st">"variable"</span>, <span class="at">values_to =</span> <span class="st">"value"</span>) <span class="sc">%&gt;%</span> </span>
<span id="cb22-4"><a href="#cb22-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> value, <span class="at">fill =</span> HeartDisease)) <span class="sc">+</span></span>
<span id="cb22-5"><a href="#cb22-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_bar</span>(<span class="at">position =</span> <span class="st">"dodge2"</span>) <span class="sc">+</span></span>
<span id="cb22-6"><a href="#cb22-6" aria-hidden="true" tabindex="-1"></a>  <span class="fu">facet_wrap</span>(<span class="sc">~</span> variable, <span class="at">scales =</span> <span class="st">"free"</span>) <span class="sc">+</span></span>
<span id="cb22-7"><a href="#cb22-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">""</span>,</span>
<span id="cb22-8"><a href="#cb22-8" aria-hidden="true" tabindex="-1"></a>       <span class="at">title =</span> <span class="st">"Distribution of Categorical Predictors by Heart Disease Status"</span>) <span class="sc">+</span></span>
<span id="cb22-9"><a href="#cb22-9" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_fill_discrete</span>(<span class="at">palette =</span> <span class="st">"Accent"</span>,</span>
<span id="cb22-10"><a href="#cb22-10" aria-hidden="true" tabindex="-1"></a>                      <span class="at">name =</span> <span class="st">"Status"</span>,</span>
<span id="cb22-11"><a href="#cb22-11" aria-hidden="true" tabindex="-1"></a>                      <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">"0"</span> <span class="ot">=</span> <span class="st">"Negative"</span>, <span class="st">"1"</span> <span class="ot">=</span> <span class="st">"Positive"</span>))</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="Heart-Failure-Prediction_files/figure-html/Distribution%20of%20Categorical%20Predictors-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
</div>
<p>We can see that for those who do not have heart disease:</p>
<ol type="1">
<li><p>They are mostly non-asymptomatic for chest pain.</p></li>
<li><p>The majority lack exercise-induced angina.</p></li>
<li><p>Few have a fasting blood sugar level greater than 120 mg/dl</p></li>
<li><p>Fewer resting ECG with probable or definite left ventricular hypertrophy</p></li>
<li><p>Demonstrate major upsloping of the peak exercise ST segment, in contrast those who are positive show a flatter slope.</p></li>
</ol>
<p>Notably most cases who are positive for heart disease are overwhelmingly male. Furthermore Fasting Blood Sugar levels are a big vague as a categorical variable in contrast to a continuous variable as there are similar amounts of participants within both outcome groups with blood sugar equal or lesser than 120 mg/dl.</p>
</section>
</section>
<section id="model-building" class="level2">
<h2 class="anchored" data-anchor-id="model-building">Model Building</h2>
<p>In regression analysis there are many things to consider when building a model such as whether or not there is interaction between the predictive variables. In the interest of keeping things simple for this case study, I will be assuming <strong>no interaction</strong>.</p>
<p>Our Exploratory Data Analysis has revealed two predictors that may not be useful for determining heart disease. Fasting blood sugar and resting blood pressure.</p>
<p>We build four different models:</p>
<ol type="1">
<li><p>A most restricted model without either the <code>FastingBS</code> or <code>RestingBP</code></p></li>
<li><p>A less restricted model without <code>FastingBS</code></p></li>
<li><p>A less restricted model without <code>RestingBP</code></p></li>
<li><p>A least restricted model with all predictors.</p></li>
</ol>
<p>First we must split our data between a testing and training set.</p>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb23"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">01022026</span>)</span>
<span id="cb23-2"><a href="#cb23-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb23-3"><a href="#cb23-3" aria-hidden="true" tabindex="-1"></a><span class="co"># 70/30 training/testing split</span></span>
<span id="cb23-4"><a href="#cb23-4" aria-hidden="true" tabindex="-1"></a>hf_split <span class="ot">&lt;-</span> <span class="fu">initial_split</span>(hf_data, <span class="at">prop =</span> <span class="fl">0.70</span>)</span>
<span id="cb23-5"><a href="#cb23-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb23-6"><a href="#cb23-6" aria-hidden="true" tabindex="-1"></a>hf_split</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-stdout">
<pre><code>&lt;Training/Testing/Total&gt;
&lt;642/276/918&gt;</code></pre>
</div>
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb25"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a>hf_train <span class="ot">&lt;-</span> <span class="fu">training</span>(hf_split) <span class="co"># training dataset</span></span>
<span id="cb25-2"><a href="#cb25-2" aria-hidden="true" tabindex="-1"></a>hf_test <span class="ot">&lt;-</span> <span class="fu">testing</span>(hf_split) <span class="co"># testing dataset</span></span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
</div>
<p>We now have a Training/Testing data split of 642/276.</p>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="annotated-cell-16"><pre class="sourceCode r code-annotation-code code-with-copy"><code class="sourceCode r"><span id="annotated-cell-16-1"><a href="#annotated-cell-16-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Define logistic regression model</span></span>
<span id="annotated-cell-16-2"><a href="#annotated-cell-16-2" aria-hidden="true" tabindex="-1"></a>glm_spec <span class="ot">&lt;-</span> <span class="fu">logistic_reg</span>() <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-16-3"><a href="#annotated-cell-16-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">set_engine</span>(<span class="st">"glm"</span>) <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-16-4"><a href="#annotated-cell-16-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">set_mode</span>(<span class="st">"classification"</span>)</span>
<span id="annotated-cell-16-5"><a href="#annotated-cell-16-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="annotated-cell-16-6"><a href="#annotated-cell-16-6" aria-hidden="true" tabindex="-1"></a><span class="co"># Create recipes for each model</span></span>
<span id="annotated-cell-16-7"><a href="#annotated-cell-16-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="annotated-cell-16-8"><a href="#annotated-cell-16-8" aria-hidden="true" tabindex="-1"></a>least_restricted <span class="ot">&lt;-</span> <span class="fu">recipe</span>(HeartDisease <span class="sc">~</span> .,</span>
<span id="annotated-cell-16-9"><a href="#annotated-cell-16-9" aria-hidden="true" tabindex="-1"></a>                          <span class="at">data =</span> hf_train) <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-16-10"><a href="#annotated-cell-16-10" aria-hidden="true" tabindex="-1"></a>  <span class="fu">step_dummy</span>(<span class="fu">all_nominal_predictors</span>()) <span class="co"># create dummy variables</span></span>
<span id="annotated-cell-16-11"><a href="#annotated-cell-16-11" aria-hidden="true" tabindex="-1"></a></span>
<span id="annotated-cell-16-12"><a href="#annotated-cell-16-12" aria-hidden="true" tabindex="-1"></a>rm_fasting <span class="ot">&lt;-</span> least_restricted <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-16-13"><a href="#annotated-cell-16-13" aria-hidden="true" tabindex="-1"></a>  <span class="fu">step_rm</span>(<span class="fu">contains</span>(<span class="st">"FastingBS"</span>))</span>
<span id="annotated-cell-16-14"><a href="#annotated-cell-16-14" aria-hidden="true" tabindex="-1"></a></span>
<span id="annotated-cell-16-15"><a href="#annotated-cell-16-15" aria-hidden="true" tabindex="-1"></a>rm_resting <span class="ot">&lt;-</span> least_restricted <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-16-16"><a href="#annotated-cell-16-16" aria-hidden="true" tabindex="-1"></a>  <span class="fu">step_rm</span>(RestingBP)</span>
<span id="annotated-cell-16-17"><a href="#annotated-cell-16-17" aria-hidden="true" tabindex="-1"></a></span>
<span id="annotated-cell-16-18"><a href="#annotated-cell-16-18" aria-hidden="true" tabindex="-1"></a>most_restricted <span class="ot">&lt;-</span> least_restricted <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-16-19"><a href="#annotated-cell-16-19" aria-hidden="true" tabindex="-1"></a>  <span class="fu">step_rm</span>(RestingBP, <span class="fu">contains</span>(<span class="st">"FastingBS"</span>))</span>
<span id="annotated-cell-16-20"><a href="#annotated-cell-16-20" aria-hidden="true" tabindex="-1"></a></span>
<span id="annotated-cell-16-21"><a href="#annotated-cell-16-21" aria-hidden="true" tabindex="-1"></a>preproc <span class="ot">&lt;-</span> <span class="fu">list</span>(</span>
<span id="annotated-cell-16-22"><a href="#annotated-cell-16-22" aria-hidden="true" tabindex="-1"></a>  <span class="at">least =</span> least_restricted,</span>
<span id="annotated-cell-16-23"><a href="#annotated-cell-16-23" aria-hidden="true" tabindex="-1"></a>  <span class="at">fasting =</span> rm_fasting,</span>
<span id="annotated-cell-16-24"><a href="#annotated-cell-16-24" aria-hidden="true" tabindex="-1"></a>  <span class="at">resting =</span> rm_resting,</span>
<span id="annotated-cell-16-25"><a href="#annotated-cell-16-25" aria-hidden="true" tabindex="-1"></a>  <span class="at">most =</span> most_restricted</span>
<span id="annotated-cell-16-26"><a href="#annotated-cell-16-26" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="annotated-cell-16-27"><a href="#annotated-cell-16-27" aria-hidden="true" tabindex="-1"></a></span>
<span id="annotated-cell-16-28"><a href="#annotated-cell-16-28" aria-hidden="true" tabindex="-1"></a>glm_models <span class="ot">&lt;-</span> <span class="fu">workflow_set</span>(preproc, </span>
<span id="annotated-cell-16-29"><a href="#annotated-cell-16-29" aria-hidden="true" tabindex="-1"></a>                           <span class="fu">list</span>(<span class="at">glm =</span> glm_spec),</span>
<span id="annotated-cell-16-30"><a href="#annotated-cell-16-30" aria-hidden="true" tabindex="-1"></a>                           <span class="at">cross =</span> T)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
</div>
<p>In <code>tidymodels</code> there is a consistent and simpler syntax for building models, one such example is the use of the <code>set_engine()</code> and <code>logistic_regression()</code> functions to build a Logistic Regression model. Instead of creating multiple models at once, <code>tidymodels</code> allows us to instead create several <code>recipes</code> for each model and then a <code>workflow_set</code> to coordinate the construction.]</p>
<p>More can be learned from the <a href="https://www.tmwr.org/">Tidy Modeling With R</a> online textbook.</p>
<ol type="1">
<li><p>The least restricted model with all predictors included</p></li>
<li><p>All <code>FastBS</code> predictors removed</p></li>
<li><p><code>RestingBP</code> predictor removed</p></li>
<li><p>The most restricted model with <code>FastBS</code> and <code>RestingBP</code> predictors removed.</p></li>
</ol>
</section>
<section id="model-evaluation" class="level2">
<h2 class="anchored" data-anchor-id="model-evaluation">Model Evaluation</h2>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb26"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">01032026</span>)</span>
<span id="cb26-2"><a href="#cb26-2" aria-hidden="true" tabindex="-1"></a>hf_folds <span class="ot">&lt;-</span> <span class="fu">vfold_cv</span>(hf_train, <span class="at">v =</span> <span class="dv">10</span>) <span class="co"># cross-validation, 10-fold)</span></span>
<span id="cb26-3"><a href="#cb26-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb26-4"><a href="#cb26-4" aria-hidden="true" tabindex="-1"></a>keep_pred <span class="ot">&lt;-</span> <span class="fu">control_resamples</span>(<span class="at">save_pred =</span> <span class="cn">TRUE</span>, <span class="co"># store predictions upon fitting</span></span>
<span id="cb26-5"><a href="#cb26-5" aria-hidden="true" tabindex="-1"></a>                               <span class="at">save_workflow =</span> <span class="cn">TRUE</span>)</span>
<span id="cb26-6"><a href="#cb26-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb26-7"><a href="#cb26-7" aria-hidden="true" tabindex="-1"></a>glm_models <span class="ot">&lt;-</span> glm_models <span class="sc">%&gt;%</span> </span>
<span id="cb26-8"><a href="#cb26-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">workflow_map</span>(<span class="st">"fit_resamples"</span>, <span class="co"># model fitting</span></span>
<span id="cb26-9"><a href="#cb26-9" aria-hidden="true" tabindex="-1"></a>               <span class="at">seed =</span> <span class="dv">2026</span>,</span>
<span id="cb26-10"><a href="#cb26-10" aria-hidden="true" tabindex="-1"></a>               <span class="at">verbose =</span> T,</span>
<span id="cb26-11"><a href="#cb26-11" aria-hidden="true" tabindex="-1"></a>               <span class="at">resamples =</span> hf_folds,</span>
<span id="cb26-12"><a href="#cb26-12" aria-hidden="true" tabindex="-1"></a>               <span class="at">control =</span> keep_pred)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-stderr">
<pre><code>i 1 of 4 resampling: least_glm</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>✔ 1 of 4 resampling: least_glm (950ms)</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>i 2 of 4 resampling: fasting_glm</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>✔ 2 of 4 resampling: fasting_glm (6.7s)</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>i 3 of 4 resampling: resting_glm</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>✔ 3 of 4 resampling: resting_glm (3.2s)</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>i 4 of 4 resampling: most_glm</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>✔ 4 of 4 resampling: most_glm (3.1s)</code></pre>
</div>
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb35"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb35-1"><a href="#cb35-1" aria-hidden="true" tabindex="-1"></a>glm_models <span class="sc">%&gt;%</span> </span>
<span id="cb35-2"><a href="#cb35-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">collect_metrics</span>() <span class="sc">%&gt;%</span> <span class="co"># useful function to autogenerate evaluation metrics</span></span>
<span id="cb35-3"><a href="#cb35-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">select</span>(wflow_id, .metric, mean) <span class="sc">%&gt;%</span> </span>
<span id="cb35-4"><a href="#cb35-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">arrange</span>(.metric, <span class="fu">desc</span>(mean))</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-stdout">
<pre><code># A tibble: 12 × 3
   wflow_id    .metric       mean
   &lt;chr&gt;       &lt;chr&gt;        &lt;dbl&gt;
 1 resting_glm accuracy    0.883
 2 least_glm   accuracy    0.882
 3 fasting_glm accuracy    0.871
 4 most_glm    accuracy    0.869
 5 fasting_glm brier_class 0.0944
 6 most_glm    brier_class 0.0940
 7 least_glm   brier_class 0.0912
 8 resting_glm brier_class 0.0908
 9 least_glm   roc_auc     0.941
10 resting_glm roc_auc     0.940
11 most_glm    roc_auc     0.938
12 fasting_glm roc_auc     0.937 </code></pre>
</div>
</div>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb37"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb37-1"><a href="#cb37-1" aria-hidden="true" tabindex="-1"></a><span class="fu">autoplot</span>(</span>
<span id="cb37-2"><a href="#cb37-2" aria-hidden="true" tabindex="-1"></a>   glm_models,</span>
<span id="cb37-3"><a href="#cb37-3" aria-hidden="true" tabindex="-1"></a>   <span class="at">rank_metric =</span> <span class="st">"accuracy"</span>,  <span class="co"># &lt;- how to order models</span></span>
<span id="cb37-4"><a href="#cb37-4" aria-hidden="true" tabindex="-1"></a>   <span class="at">metric =</span> <span class="st">"accuracy"</span>,       <span class="co"># &lt;- which metric to visualize</span></span>
<span id="cb37-5"><a href="#cb37-5" aria-hidden="true" tabindex="-1"></a>) <span class="sc">+</span></span>
<span id="cb37-6"><a href="#cb37-6" aria-hidden="true" tabindex="-1"></a>   <span class="fu">geom_text</span>(<span class="fu">aes</span>(<span class="at">label =</span> wflow_id,</span>
<span id="cb37-7"><a href="#cb37-7" aria-hidden="true" tabindex="-1"></a>                 <span class="at">colour =</span> wflow_id),</span>
<span id="cb37-8"><a href="#cb37-8" aria-hidden="true" tabindex="-1"></a>             <span class="at">angle =</span> <span class="dv">90</span>,</span>
<span id="cb37-9"><a href="#cb37-9" aria-hidden="true" tabindex="-1"></a>             <span class="at">vjust =</span> <span class="dv">1</span>) <span class="sc">+</span></span>
<span id="cb37-10"><a href="#cb37-10" aria-hidden="true" tabindex="-1"></a>   <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">"none"</span>) <span class="sc">+</span></span>
<span id="cb37-11"><a href="#cb37-11" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">title =</span> <span class="st">"Accuracy"</span>,</span>
<span id="cb37-12"><a href="#cb37-12" aria-hidden="true" tabindex="-1"></a>       <span class="at">x =</span> <span class="st">"Rank"</span>,</span>
<span id="cb37-13"><a href="#cb37-13" aria-hidden="true" tabindex="-1"></a>       <span class="at">y =</span> <span class="st">"Accuracy"</span>)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="Heart-Failure-Prediction_files/figure-html/Model%20Evaluation%20-%20Accuracy-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
</div>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb38"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb38-1"><a href="#cb38-1" aria-hidden="true" tabindex="-1"></a><span class="fu">autoplot</span>(</span>
<span id="cb38-2"><a href="#cb38-2" aria-hidden="true" tabindex="-1"></a>   glm_models,</span>
<span id="cb38-3"><a href="#cb38-3" aria-hidden="true" tabindex="-1"></a>   <span class="at">rank_metric =</span> <span class="st">"brier_class"</span>,  <span class="co"># &lt;- how to order models</span></span>
<span id="cb38-4"><a href="#cb38-4" aria-hidden="true" tabindex="-1"></a>   <span class="at">metric =</span> <span class="st">"brier_class"</span>,       <span class="co"># &lt;- which metric to visualize</span></span>
<span id="cb38-5"><a href="#cb38-5" aria-hidden="true" tabindex="-1"></a>) <span class="sc">+</span></span>
<span id="cb38-6"><a href="#cb38-6" aria-hidden="true" tabindex="-1"></a>   <span class="fu">geom_text</span>(<span class="fu">aes</span>(<span class="at">label =</span> wflow_id,</span>
<span id="cb38-7"><a href="#cb38-7" aria-hidden="true" tabindex="-1"></a>                 <span class="at">colour =</span> wflow_id),</span>
<span id="cb38-8"><a href="#cb38-8" aria-hidden="true" tabindex="-1"></a>             <span class="at">angle =</span> <span class="dv">90</span>,</span>
<span id="cb38-9"><a href="#cb38-9" aria-hidden="true" tabindex="-1"></a>             <span class="at">vjust =</span> <span class="dv">1</span>) <span class="sc">+</span></span>
<span id="cb38-10"><a href="#cb38-10" aria-hidden="true" tabindex="-1"></a>   <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">"none"</span>) <span class="sc">+</span></span>
<span id="cb38-11"><a href="#cb38-11" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">title =</span> <span class="st">"Brier Classification"</span>,</span>
<span id="cb38-12"><a href="#cb38-12" aria-hidden="true" tabindex="-1"></a>       <span class="at">x =</span> <span class="st">"Rank"</span>,</span>
<span id="cb38-13"><a href="#cb38-13" aria-hidden="true" tabindex="-1"></a>       <span class="at">y =</span> <span class="st">"Brier Classification"</span>)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="Heart-Failure-Prediction_files/figure-html/Model%20Evaluation%20-%20Brier%20Classification-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
</div>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb39"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb39-1"><a href="#cb39-1" aria-hidden="true" tabindex="-1"></a><span class="fu">autoplot</span>(</span>
<span id="cb39-2"><a href="#cb39-2" aria-hidden="true" tabindex="-1"></a>   glm_models,</span>
<span id="cb39-3"><a href="#cb39-3" aria-hidden="true" tabindex="-1"></a>   <span class="at">rank_metric =</span> <span class="st">"roc_auc"</span>,  <span class="co"># &lt;- how to order models</span></span>
<span id="cb39-4"><a href="#cb39-4" aria-hidden="true" tabindex="-1"></a>   <span class="at">metric =</span> <span class="st">"roc_auc"</span>,       <span class="co"># &lt;- which metric to visualize</span></span>
<span id="cb39-5"><a href="#cb39-5" aria-hidden="true" tabindex="-1"></a>) <span class="sc">+</span></span>
<span id="cb39-6"><a href="#cb39-6" aria-hidden="true" tabindex="-1"></a>   <span class="fu">geom_text</span>(<span class="fu">aes</span>(<span class="at">label =</span> wflow_id,</span>
<span id="cb39-7"><a href="#cb39-7" aria-hidden="true" tabindex="-1"></a>                 <span class="at">colour =</span> wflow_id),</span>
<span id="cb39-8"><a href="#cb39-8" aria-hidden="true" tabindex="-1"></a>             <span class="at">angle =</span> <span class="dv">90</span>,</span>
<span id="cb39-9"><a href="#cb39-9" aria-hidden="true" tabindex="-1"></a>             <span class="at">vjust =</span> <span class="dv">1</span>) <span class="sc">+</span></span>
<span id="cb39-10"><a href="#cb39-10" aria-hidden="true" tabindex="-1"></a>   <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">"none"</span>) <span class="sc">+</span></span>
<span id="cb39-11"><a href="#cb39-11" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">title =</span> <span class="st">"Area Under The ROC Curve"</span>,</span>
<span id="cb39-12"><a href="#cb39-12" aria-hidden="true" tabindex="-1"></a>       <span class="at">x =</span> <span class="st">"Rank"</span>,</span>
<span id="cb39-13"><a href="#cb39-13" aria-hidden="true" tabindex="-1"></a>       <span class="at">y =</span> <span class="st">"Area Under The ROC Curve"</span>)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="Heart-Failure-Prediction_files/figure-html/Model%20Evaluation%20-%20Area%20Under%20The%20ROC%20Curve-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
</div>
<p>Based on the above metric evaluations it appears that either the <code>least_glm</code> or <code>resting_glm</code> models would be the best selection, however there is significant overlap in the confidence intervals for each metric across all models.</p>
<p>A general rule-of-thumb is to select the most parsimonious (simple) model in such a case and thus for the final model I select the <code>most_glm</code> model which is defined as:</p>
<p><span class="math display">\[
\begin{equation}
P(Y = 1)= \beta_0 + \beta_1 Age + \beta_2 Cholesterol + \beta_3 MaxHR + \beta_4 Oldpeak + \beta_5 Sex_{M} +\beta_6 ChestPainType_{ATA} + \beta_7 ChestPainType_{NA}P + \beta_8 ChestPainType_{TA} + \varepsilon
\end{equation}
\]</span> Where <span class="math inline">\(\beta_0\)</span> is the intercept, the value of log-odds (logit) of the outcome <span class="math inline">\(\mathrm{Y}\)</span> when all predictors are at zero or baseline levels.</p>
<p>While <span class="math inline">\(\beta_{x+i}\)</span> are the Log Odds Ratio coefficients for each predictor variable and <span class="math inline">\(\varepsilon\)</span> is the random error.</p>
</section>
<section id="final-model-build" class="level2">
<h2 class="anchored" data-anchor-id="final-model-build">Final Model Build</h2>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="annotated-cell-22"><pre class="sourceCode r code-annotation-code code-with-copy code-annotated"><code class="sourceCode r"><span id="annotated-cell-22-1"><a href="#annotated-cell-22-1" aria-hidden="true" tabindex="-1"></a>final_workflow <span class="ot">&lt;-</span> glm_models <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-22-2"><a href="#annotated-cell-22-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">extract_workflow_set_result</span>(<span class="st">"most_glm"</span>) <span class="sc">%&gt;%</span> <span class="co"># extract 'most_glm' formulation</span></span>
<span id="annotated-cell-22-3"><a href="#annotated-cell-22-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">select_best</span>(<span class="at">metric =</span> <span class="st">"brier_class"</span>) <span class="co"># metric choice doesn't matter for Logit Reg</span></span>
<span id="annotated-cell-22-4"><a href="#annotated-cell-22-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="annotated-cell-22-5"><a href="#annotated-cell-22-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Model Fit</span></span>
<span id="annotated-cell-22-6"><a href="#annotated-cell-22-6" aria-hidden="true" tabindex="-1"></a>final_fit <span class="ot">&lt;-</span> glm_models <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-22-7"><a href="#annotated-cell-22-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">extract_workflow</span>(<span class="st">"most_glm"</span>) <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-22-8"><a href="#annotated-cell-22-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">finalize_workflow</span>(final_workflow) <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-22-9"><a href="#annotated-cell-22-9" aria-hidden="true" tabindex="-1"></a>  <span class="fu">last_fit</span>(hf_split)</span>
<span id="annotated-cell-22-10"><a href="#annotated-cell-22-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="annotated-cell-22-11"><a href="#annotated-cell-22-11" aria-hidden="true" tabindex="-1"></a><span class="co"># Extract Model</span></span>
<span id="annotated-cell-22-12"><a href="#annotated-cell-22-12" aria-hidden="true" tabindex="-1"></a>final_fit <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-22-13"><a href="#annotated-cell-22-13" aria-hidden="true" tabindex="-1"></a>  <span class="fu">extract_fit_engine</span>() <span class="sc">%&gt;%</span> <span class="co"># allows extraction of base R model info</span></span>
<span id="annotated-cell-22-14"><a href="#annotated-cell-22-14" aria-hidden="true" tabindex="-1"></a>  <span class="fu">tidy</span>() <span class="sc">%&gt;%</span></span>
<span id="annotated-cell-22-15"><a href="#annotated-cell-22-15" aria-hidden="true" tabindex="-1"></a>  <span class="fu">rename</span>(<span class="at">p_value =</span> p.value) <span class="sc">%&gt;%</span> </span>
<a class="code-annotation-anchor" data-target-cell="annotated-cell-22" data-target-annotation="1" onclick="event.preventDefault();" href="">1</a><span id="annotated-cell-22-16" class="code-annotation-target"><a href="#annotated-cell-22-16" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">p_value =</span> <span class="fu">signif</span>(p_value, <span class="dv">2</span>))</span>
<span id="annotated-cell-22-17"><a href="#annotated-cell-22-17" aria-hidden="true" tabindex="-1"></a></span>
<span id="annotated-cell-22-18"><a href="#annotated-cell-22-18" aria-hidden="true" tabindex="-1"></a>final_fit <span class="sc">%&gt;%</span> </span>
<a class="code-annotation-anchor" data-target-cell="annotated-cell-22" data-target-annotation="2" onclick="event.preventDefault();" href="">2</a><span id="annotated-cell-22-19" class="code-annotation-target"><a href="#annotated-cell-22-19" aria-hidden="true" tabindex="-1"></a>  <span class="fu">collect_metrics</span>() <span class="sc">%&gt;%</span></span>
<span id="annotated-cell-22-20"><a href="#annotated-cell-22-20" aria-hidden="true" tabindex="-1"></a>  <span class="fu">select</span>(<span class="sc">-</span><span class="fu">c</span>(.estimator, .config)) <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-22-21"><a href="#annotated-cell-22-21" aria-hidden="true" tabindex="-1"></a>  <span class="fu">rename</span>(<span class="at">Metric =</span> .metric,</span>
<span id="annotated-cell-22-22"><a href="#annotated-cell-22-22" aria-hidden="true" tabindex="-1"></a>         <span class="at">Value =</span> .estimate)</span><div class="code-annotation-gutter-bg"></div><div class="code-annotation-gutter"></div></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-annotation">
<dl class="code-annotation-container-grid">
<dt data-target-cell="annotated-cell-22" data-target-annotation="1">1</dt>
<dd>
<span data-code-cell="annotated-cell-22" data-code-lines="16" data-code-annotation="1">Under the <code>estimate</code> column we can see the <span class="math inline">\(\beta_{x+1}\)</span> values for each predictor. Under <code>p_value</code> we get some information telling us whether the predictor has a statistically significant effect on the outcome (i.e.&nbsp;is the coefficient actually 0)</span>
</dd>
<dt data-target-cell="annotated-cell-22" data-target-annotation="2">2</dt>
<dd>
<span data-code-cell="annotated-cell-22" data-code-lines="19" data-code-annotation="2">We get some basic model evaluation metrics such as Accuracy, the Area Under the ROC Curve score and Brier Score. For both Accuracy and AUC the closer to 1 the better the model fit, while for Brier Score the closer to 0 means a better fit.</span>
</dd>
</dl>
</div>
<div class="cell-output cell-output-stdout">
<pre><code># A tibble: 14 × 5
   term              estimate std.error statistic    p_value
   &lt;chr&gt;                &lt;dbl&gt;     &lt;dbl&gt;     &lt;dbl&gt;      &lt;dbl&gt;
 1 (Intercept)       -1.26      1.63       -0.769 0.44
 2 Age                0.0465    0.0165      2.81  0.0049
 3 Cholesterol       -0.00606   0.00137    -4.44  0.0000091
 4 MaxHR             -0.00381   0.00632    -0.603 0.55
 5 Oldpeak            0.371     0.156       2.38  0.018
 6 Sex_M              1.21      0.334       3.62  0.00029
 7 ChestPainType_ATA -1.86      0.388      -4.79  0.0000016
 8 ChestPainType_NAP -1.69      0.337      -5.00  0.00000056
 9 ChestPainType_TA  -1.60      0.514      -3.11  0.0019
10 RestingECG_Normal -0.0888    0.341      -0.261 0.79
11 RestingECG_ST      0.271     0.464       0.585 0.56
12 ExerciseAngina_Y   1.01      0.324       3.12  0.0018
13 ST_Slope_Flat      1.34      0.526       2.54  0.011
14 ST_Slope_Up       -1.36      0.555      -2.44  0.015
# A tibble: 3 × 2
  Metric      Value
  &lt;chr&gt;       &lt;dbl&gt;
1 accuracy    0.812
2 roc_auc     0.880
3 brier_class 0.140</code></pre>
</div>
</div>
<p>Here we can see that our model seems to fit pretty well.</p>
<section id="metrics" class="level3">
<h3 class="anchored" data-anchor-id="metrics">Metrics</h3>
<p>Now we will do a final evaluation of our model using several additional metrics based on the predictions as well as check for collinearity between the predictors.</p>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="annotated-cell-23"><pre class="sourceCode r code-annotation-code code-with-copy code-annotated"><code class="sourceCode r"><span id="annotated-cell-23-1"><a href="#annotated-cell-23-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Check for collinearity</span></span>
<span id="annotated-cell-23-2"><a href="#annotated-cell-23-2" aria-hidden="true" tabindex="-1"></a>final_fit <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-23-3"><a href="#annotated-cell-23-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">extract_fit_engine</span>() <span class="sc">%&gt;%</span> </span>
<a class="code-annotation-anchor" data-target-cell="annotated-cell-23" data-target-annotation="1" onclick="event.preventDefault();" href="">1</a><span id="annotated-cell-23-4" class="code-annotation-target"><a href="#annotated-cell-23-4" aria-hidden="true" tabindex="-1"></a>  car<span class="sc">::</span><span class="fu">vif</span>()</span>
<span id="annotated-cell-23-5"><a href="#annotated-cell-23-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="annotated-cell-23-6"><a href="#annotated-cell-23-6" aria-hidden="true" tabindex="-1"></a><span class="co"># Data frame of predictions</span></span>
<span id="annotated-cell-23-7"><a href="#annotated-cell-23-7" aria-hidden="true" tabindex="-1"></a>final_df <span class="ot">&lt;-</span> final_fit <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-23-8"><a href="#annotated-cell-23-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">collect_predictions</span>() <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-23-9"><a href="#annotated-cell-23-9" aria-hidden="true" tabindex="-1"></a>  <span class="fu">select</span>(<span class="fu">contains</span>(<span class="st">".pred_"</span>), <span class="at">truth =</span> HeartDisease)</span>
<span id="annotated-cell-23-10"><a href="#annotated-cell-23-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="annotated-cell-23-11"><a href="#annotated-cell-23-11" aria-hidden="true" tabindex="-1"></a><span class="co"># Confusion Matrix</span></span>
<span id="annotated-cell-23-12"><a href="#annotated-cell-23-12" aria-hidden="true" tabindex="-1"></a><span class="fu">conf_mat</span>(final_df, <span class="at">truth =</span> truth, <span class="at">estimate =</span> .pred_class)</span>
<span id="annotated-cell-23-13"><a href="#annotated-cell-23-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="annotated-cell-23-14"><a href="#annotated-cell-23-14" aria-hidden="true" tabindex="-1"></a><span class="co"># More metrics</span></span>
<span id="annotated-cell-23-15"><a href="#annotated-cell-23-15" aria-hidden="true" tabindex="-1"></a>multi_metric<span class="ot">&lt;-</span> <span class="fu">metric_set</span>(mcc,</span>
<span id="annotated-cell-23-16"><a href="#annotated-cell-23-16" aria-hidden="true" tabindex="-1"></a>                          f_meas,</span>
<span id="annotated-cell-23-17"><a href="#annotated-cell-23-17" aria-hidden="true" tabindex="-1"></a>                          sens,</span>
<span id="annotated-cell-23-18"><a href="#annotated-cell-23-18" aria-hidden="true" tabindex="-1"></a>                          spec,</span>
<span id="annotated-cell-23-19"><a href="#annotated-cell-23-19" aria-hidden="true" tabindex="-1"></a>                          precision,</span>
<span id="annotated-cell-23-20"><a href="#annotated-cell-23-20" aria-hidden="true" tabindex="-1"></a>                          kap)</span>
<span id="annotated-cell-23-21"><a href="#annotated-cell-23-21" aria-hidden="true" tabindex="-1"></a></span>
<a class="code-annotation-anchor" data-target-cell="annotated-cell-23" data-target-annotation="2" onclick="event.preventDefault();" href="">2</a><span id="annotated-cell-23-22" class="code-annotation-target"><a href="#annotated-cell-23-22" aria-hidden="true" tabindex="-1"></a><span class="fu">multi_metric</span>(final_df,</span>
<span id="annotated-cell-23-23"><a href="#annotated-cell-23-23" aria-hidden="true" tabindex="-1"></a>             <span class="at">truth =</span> truth,</span>
<span id="annotated-cell-23-24"><a href="#annotated-cell-23-24" aria-hidden="true" tabindex="-1"></a>             <span class="at">estimate =</span> .pred_class,</span>
<span id="annotated-cell-23-25"><a href="#annotated-cell-23-25" aria-hidden="true" tabindex="-1"></a>             <span class="at">event_level =</span> <span class="st">"first"</span>,</span>
<span id="annotated-cell-23-26"><a href="#annotated-cell-23-26" aria-hidden="true" tabindex="-1"></a>             .pred_0) <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-23-27"><a href="#annotated-cell-23-27" aria-hidden="true" tabindex="-1"></a>  <span class="fu">select</span>(<span class="sc">-</span>.estimator) <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-23-28"><a href="#annotated-cell-23-28" aria-hidden="true" tabindex="-1"></a>  <span class="fu">rename</span>(<span class="at">Metric =</span> .metric,</span>
<span id="annotated-cell-23-29"><a href="#annotated-cell-23-29" aria-hidden="true" tabindex="-1"></a>         <span class="at">Value =</span> .estimate)</span>
<span id="annotated-cell-23-30"><a href="#annotated-cell-23-30" aria-hidden="true" tabindex="-1"></a></span>
<span id="annotated-cell-23-31"><a href="#annotated-cell-23-31" aria-hidden="true" tabindex="-1"></a>final_fit <span class="sc">%&gt;%</span> </span>
<span id="annotated-cell-23-32"><a href="#annotated-cell-23-32" aria-hidden="true" tabindex="-1"></a>  <span class="fu">extract_fit_engine</span>() <span class="sc">%&gt;%</span> </span>
<a class="code-annotation-anchor" data-target-cell="annotated-cell-23" data-target-annotation="3" onclick="event.preventDefault();" href="">3</a><span id="annotated-cell-23-33" class="code-annotation-target"><a href="#annotated-cell-23-33" aria-hidden="true" tabindex="-1"></a>  car<span class="sc">::</span><span class="fu">outlierTest</span>()</span><div class="code-annotation-gutter-bg"></div><div class="code-annotation-gutter"></div></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-annotation">
<dl class="code-annotation-container-grid">
<dt data-target-cell="annotated-cell-23" data-target-annotation="1">1</dt>
<dd>
<span data-code-cell="annotated-cell-23" data-code-lines="4" data-code-annotation="1">Using <code>car::vif()</code> we can see there are only two predictors of some moderate concern for collinearity within our model, <code>ST_Slope_Flat</code> and <code>ST_Slope_Up</code> with values ~ 4. Conventionally values of 4 - 5 are considered acceptable in prediction models.</span>
</dd>
<dt data-target-cell="annotated-cell-23" data-target-annotation="2">2</dt>
<dd>
<span data-code-cell="annotated-cell-23" data-code-lines="22" data-code-annotation="2">Additional metrics used shows that our model is good.</span>
</dd>
<dt data-target-cell="annotated-cell-23" data-target-annotation="3">3</dt>
<dd>
<span data-code-cell="annotated-cell-23" data-code-lines="33" data-code-annotation="3">Outlier data points which could improve our model if removed.</span>
</dd>
</dl>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>              Age       Cholesterol             MaxHR           Oldpeak
         1.237698          1.183375          1.303443          1.339915
            Sex_M ChestPainType_ATA ChestPainType_NAP  ChestPainType_TA
         1.074764          1.172856          1.219346          1.190734
RestingECG_Normal     RestingECG_ST  ExerciseAngina_Y     ST_Slope_Flat
         1.572531          1.596112          1.228986          3.918000
      ST_Slope_Up
         4.249506
          Truth
Prediction   0   1
         0  96  19
         1  33 128
# A tibble: 6 × 2
  Metric    Value
  &lt;chr&gt;     &lt;dbl&gt;
1 mcc       0.622
2 f_meas    0.787
3 sens      0.744
4 spec      0.871
5 precision 0.835
6 kap       0.619
No Studentized residuals with Bonferroni p &lt; 0.05
Largest |rstudent|:
     rstudent unadjusted p-value Bonferroni p
373 -2.938527          0.0032978           NA</code></pre>
</div>
</div>
</section>
<section id="visualizations" class="level3">
<h3 class="anchored" data-anchor-id="visualizations">Visualizations</h3>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb42"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb42-1"><a href="#cb42-1" aria-hidden="true" tabindex="-1"></a>final_fit <span class="sc">%&gt;%</span> </span>
<span id="cb42-2"><a href="#cb42-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">extract_fit_engine</span>() <span class="sc">%&gt;%</span> </span>
<span id="cb42-3"><a href="#cb42-3" aria-hidden="true" tabindex="-1"></a>  car<span class="sc">::</span><span class="fu">influenceIndexPlot</span>() <span class="co"># identify outliers</span></span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="Heart-Failure-Prediction_files/figure-html/Final%20Model%20Visualizations-1.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb43"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb43-1"><a href="#cb43-1" aria-hidden="true" tabindex="-1"></a>final_fit <span class="sc">%&gt;%</span> </span>
<span id="cb43-2"><a href="#cb43-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">extract_fit_engine</span>() <span class="sc">%&gt;%</span> </span>
<span id="cb43-3"><a href="#cb43-3" aria-hidden="true" tabindex="-1"></a>  car<span class="sc">::</span><span class="fu">influencePlot</span>() <span class="co"># Influential Points</span></span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="Heart-Failure-Prediction_files/figure-html/Final%20Model%20Visualizations-2.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>      StudRes         Hat       CookD
220  2.081002 0.058948140 0.029395346
255 -2.864008 0.003861187 0.014861413
373 -2.938527 0.004679786 0.021455026
416 -1.207421 0.160628460 0.014681669
640  1.042947 0.117022856 0.006969127</code></pre>
</div>
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb45"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb45-1"><a href="#cb45-1" aria-hidden="true" tabindex="-1"></a><span class="co"># final_fit %&gt;% </span></span>
<span id="cb45-2"><a href="#cb45-2" aria-hidden="true" tabindex="-1"></a><span class="co">#   extract_fit_engine() %&gt;% </span></span>
<span id="cb45-3"><a href="#cb45-3" aria-hidden="true" tabindex="-1"></a><span class="co">#   car::marginalModelPlots()</span></span>
<span id="cb45-4"><a href="#cb45-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb45-5"><a href="#cb45-5" aria-hidden="true" tabindex="-1"></a><span class="fu">roc_curve</span>(final_df,</span>
<span id="cb45-6"><a href="#cb45-6" aria-hidden="true" tabindex="-1"></a>          <span class="at">truth =</span> truth,</span>
<span id="cb45-7"><a href="#cb45-7" aria-hidden="true" tabindex="-1"></a>          <span class="at">event_level =</span> <span class="st">"first"</span>,</span>
<span id="cb45-8"><a href="#cb45-8" aria-hidden="true" tabindex="-1"></a>          .pred_0) <span class="sc">%&gt;%</span></span>
<span id="cb45-9"><a href="#cb45-9" aria-hidden="true" tabindex="-1"></a>  <span class="fu">autoplot</span>() <span class="co"># ROC Curve Plot</span></span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><img src="Heart-Failure-Prediction_files/figure-html/Final%20Model%20Visualizations-3.png" class="img-fluid figure-img" width="672"></p>
</figure>
</div>
</div>
</div>
<p>Our visualizations provide several outliers or influential observations that we should consider removing before refitting our model.</p>
</section>
</section>
<section id="next-steps" class="level2">
<h2 class="anchored" data-anchor-id="next-steps">Next Steps</h2>
<p>Unfortunately our final model does not meet the 90% accuracy goal we have set at the beginning of this Case Study. A suggestion would be to use another classification model such as a Regularized Logistic Regression model, Support Vector Machine, k-Nearest Neighbors and Decision Trees.</p>
<p>We could also remove data points with high influence or outliers to see if that improves our fit.</p>
<p>We will approach this again in another Case Study.</p>
</section>

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    if (window.document.getElementById('quarto-embedded-source-code-modal')) {
      const clipboardModal = new window.ClipboardJS('.code-copy-button[data-in-quarto-modal]', {
        text: getTextToCopy,
        container: window.document.getElementById('quarto-embedded-source-code-modal')
      });
      clipboardModal.on('success', onCopySuccess);
    }
      var localhostRegex = new RegExp(/^(?:http|https):\/\/localhost\:?[0-9]*\//);
      var mailtoRegex = new RegExp(/^mailto:/);
        var filterRegex = new RegExp('/' + window.location.host + '/');
      var isInternal = (href) => {
          return filterRegex.test(href) || localhostRegex.test(href) || mailtoRegex.test(href);
      }
      // Inspect non-navigation links and adorn them if external
     var links = window.document.querySelectorAll('a[href]:not(.nav-link):not(.navbar-brand):not(.toc-action):not(.sidebar-link):not(.sidebar-item-toggle):not(.pagination-link):not(.no-external):not([aria-hidden]):not(.dropdown-item):not(.quarto-navigation-tool):not(.about-link)');
      for (var i=0; i<links.length; i++) {
        const link = links[i];
        if (!isInternal(link.href)) {
          // undo the damage that might have been done by quarto-nav.js in the case of
          // links that we want to consider external
          if (link.dataset.originalHref !== undefined) {
            link.href = link.dataset.originalHref;
          }
        }
      }
    function tippyHover(el, contentFn, onTriggerFn, onUntriggerFn) {
      const config = {
        allowHTML: true,
        maxWidth: 500,
        delay: 100,
        arrow: false,
        appendTo: function(el) {
            return el.parentElement;
        },
        interactive: true,
        interactiveBorder: 10,
        theme: 'quarto',
        placement: 'bottom-start',
      };
      if (contentFn) {
        config.content = contentFn;
      }
      if (onTriggerFn) {
        config.onTrigger = onTriggerFn;
      }
      if (onUntriggerFn) {
        config.onUntrigger = onUntriggerFn;
      }
      window.tippy(el, config);
    }
    const noterefs = window.document.querySelectorAll('a[role="doc-noteref"]');
    for (var i=0; i<noterefs.length; i++) {
      const ref = noterefs[i];
      tippyHover(ref, function() {
        // use id or data attribute instead here
        let href = ref.getAttribute('data-footnote-href') || ref.getAttribute('href');
        try { href = new URL(href).hash; } catch {}
        const id = href.replace(/^#\/?/, "");
        const note = window.document.getElementById(id);
        if (note) {
          return note.innerHTML;
        } else {
          return "";
        }
      });
    }
    const xrefs = window.document.querySelectorAll('a.quarto-xref');
    const processXRef = (id, note) => {
      // Strip column container classes
      const stripColumnClz = (el) => {
        el.classList.remove("page-full", "page-columns");
        if (el.children) {
          for (const child of el.children) {
            stripColumnClz(child);