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    <h2 id="toc-title">Contents</h2>

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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="#loan-analysis." id="toc-loan-analysis." class="nav-link" data-scroll-target="#loan-analysis."><span class="header-section-number">1</span> Loan analysis.</a>
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  <li><a href="#explore-the-database." id="toc-explore-the-database." class="nav-link" data-scroll-target="#explore-the-database."><span class="header-section-number">1.1</span> Explore the database.</a></li>
  <li><a href="#logistic-models." id="toc-logistic-models." class="nav-link" data-scroll-target="#logistic-models."><span class="header-section-number">1.2</span> Logistic models.</a></li>
  <li><a href="#prediction-and-model-evaluation." id="toc-prediction-and-model-evaluation." class="nav-link" data-scroll-target="#prediction-and-model-evaluation."><span class="header-section-number">1.3</span> Prediction and model evaluation.</a></li>
  <li><a href="#the-bank-strategy." id="toc-the-bank-strategy." class="nav-link" data-scroll-target="#the-bank-strategy."><span class="header-section-number">1.4</span> The bank strategy.</a></li>
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<div class="quarto-title-block"><div><h1 class="title">Credit risk with <i class="fa-brands fa-r-project" aria-label="r-project"></i> <span style="color:white">R</span></h1><button type="button" class="btn code-tools-button dropdown-toggle" id="quarto-code-tools-menu" data-bs-toggle="dropdown" aria-expanded="false"><i class="bi"></i> Code</button><ul class="dropdown-menu dropdown-menu-end" aria-labelelledby="quarto-code-tools-menu"><li><a id="quarto-show-all-code" class="dropdown-item" href="javascript:void(0)" role="button">Show All Code</a></li><li><a id="quarto-hide-all-code" class="dropdown-item" href="javascript:void(0)" role="button">Hide All Code</a></li></ul></div></div>
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    <div>
    <div class="quarto-title-meta-heading">Author</div>
    <div class="quarto-title-meta-contents">
             <p>Dr.&nbsp;Martín Lozano. </p>
          </div>
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    <div>
    <div class="quarto-title-meta-heading">Modified</div>
    <div class="quarto-title-meta-contents">
      <p class="date-modified">December 25, 2024, 02:55:43 am.</p>
    </div>
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  </div>

<div>
  <div class="abstract">
    <div class="block-title">Abstract</div>
    This material relies on <span class="citation" data-cites="Hull">Hull (<a href="#ref-Hull" role="doc-biblioref">2015</a>)</span> credit risk chapters and Lore Dirick credit risk DataCamp course. Some mathematical background is skipped to emphasize the data analysis, model logic, discussion, graphical approach and R coding <span class="citation" data-cites="Rref">(<a href="#ref-Rref" role="doc-biblioref">R Core Team 2024</a>)</span>. As in the philosophy of <span class="citation" data-cites="knuth1984literate">Knuth (<a href="#ref-knuth1984literate" role="doc-biblioref">1984</a>)</span>, the objective of this document is to explain to human beings what we want a computer to do as literate programming. This is a work in progress and it is under revision.
  </div>
</div>


</header>


<section id="introduction." class="level1 unnumbered">
<h1 class="unnumbered">Introduction.</h1>
<p>Credit risk models play a pivotal role in the financial landscape, providing institutions with a systematic framework to assess and manage the risks associated with lending and investment activities. One crucial component of these models is the estimation of the probability of default, which quantifies the likelihood that a borrower will fail to meet their debt obligations. The relevance of credit risk models lies in their ability to enhance decision-making processes by offering a comprehensive understanding of the potential creditworthiness of individuals, companies, or financial instruments. By utilizing statistical methods, historical data, and various financial indicators, these models enable financial institutions to make informed decisions about lending, pricing, and portfolio management. The estimation of the probability of default not only aids in risk assessment but also supports regulatory compliance, allowing institutions to maintain a healthy balance between risk and return in their credit portfolios. In an ever-evolving financial landscape, the continual refinement and application of credit risk models are essential for fostering stability and resilience within the financial system.</p>
</section>
<section id="loan-analysis." class="level1" data-number="1">
<h1 data-number="1"><span class="header-section-number">1</span> Loan analysis.</h1>
<p>The main objective of this section is to predict whether a person or client applying for a loan will repay it or not. This issue can be considered a classification problem, where based on information about the client and the characteristics of their loan application, the client is classified into one of two categories: whether they will default or not. There are various ways to approach this classification problem; in this section, we will use a simple machine learning model called logistic regression. The model is first estimated or trained and then evaluated using new data to determine how well it performs the classification.</p>
<p>Let’s load the required R packages first.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1"></a><span class="co"># Logistic models</span></span>
<span id="cb1-2"><a href="#cb1-2"></a><span class="fu">library</span>(gmodels) <span class="co"># CrossTable()</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stderr">
<pre><code>Warning: package 'gmodels' was built under R version 4.4.2</code></pre>
</div>
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1"></a><span class="fu">library</span>(ggplot2)</span>
<span id="cb3-2"><a href="#cb3-2"></a><span class="fu">library</span>(tidyr) <span class="co"># gather()</span></span>
<span id="cb3-3"><a href="#cb3-3"></a><span class="fu">library</span>(dplyr)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stderr">
<pre><code>
Adjuntando el paquete: 'dplyr'</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>The following objects are masked from 'package:stats':

    filter, lag</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>The following objects are masked from 'package:base':

    intersect, setdiff, setequal, union</code></pre>
</div>
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1"></a><span class="fu">library</span>(pROC) <span class="co"># roc</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stderr">
<pre><code>Warning: package 'pROC' was built under R version 4.4.2</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>Type 'citation("pROC")' for a citation.</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>
Adjuntando el paquete: 'pROC'</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>The following object is masked from 'package:gmodels':

    ci</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>The following objects are masked from 'package:stats':

    cov, smooth, var</code></pre>
</div>
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1"></a><span class="fu">library</span>(vembedr)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
</div>
<p>This section relies on the DataCamp course <em>Credit Risk Modeling in R</em> by Lore Dirick. However, we incorporate a slightly different database and an extended analysis.</p>
<section id="explore-the-database." class="level2" data-number="1.1">
<h2 data-number="1.1" class="anchored" data-anchor-id="explore-the-database."><span class="header-section-number">1.1</span> Explore the database.</h2>
<p>Let’s load the data called <tt><code>loan_data_ARF.rds</code></tt> and then understand its structure before conducting any further analysis. This database is available <a href="https://github.com/mlozanoqf/tutorial_arf/blob/main/loan_data_ARF.rds">here</a>.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1"></a>dat <span class="ot">&lt;-</span> <span class="fu">readRDS</span>(<span class="st">"loan_data_ARF.rds"</span>)</span>
<span id="cb14-2"><a href="#cb14-2"></a><span class="fu">str</span>(dat)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stdout">
<pre><code>'data.frame':   29092 obs. of  10 variables:
 $ loan_status   : int  0 0 0 0 0 0 1 0 1 0 ...
 $ loan_amnt     : int  5000 2400 10000 5000 3000 12000 9000 3000 10000 1000 ...
 $ int_rate      : num  10.6 11 13.5 11 11 ...
 $ grade         : Factor w/ 7 levels "A","B","C","D",..: 2 3 3 1 5 2 3 2 2 4 ...
 $ emp_length    : int  10 25 13 3 9 11 0 3 3 0 ...
 $ home_ownership: Factor w/ 4 levels "MORTGAGE","OTHER",..: 4 4 4 4 4 3 4 4 4 4 ...
 $ annual_inc    : num  24000 12252 49200 36000 48000 ...
 $ age           : int  33 31 24 39 24 28 22 22 28 22 ...
 $ sex           : Factor w/ 2 levels "0","1": 1 1 1 1 2 2 2 2 1 2 ...
 $ region        : Factor w/ 4 levels "E","N","S","W": 1 1 3 3 2 2 2 2 4 1 ...</code></pre>
</div>
</div>
<p>This dataset could represent a typical collection of data from a financial institution, such as a bank or a company that uses credit channels to sell its products or services. It contains 29,092 observations across 10 variables. Each observation corresponds to the personal and loan characteristics of an individual loan.</p>
<p>A key variable, our dependent variable, is <code>loan_st</code>, which indicates loan status. A value of 0 represents no default, while a value of 1 represents default. A default occurs when a borrower fails to make timely payments, misses payments, or ceases payments on the interest or principal owed. The definition of default can vary depending on the goals and interests of the analysis. For the purposes of this study, we classify loans simply as either default or no default.</p>
<p>The variable <code>loan_st</code> is dichotomous, or categorical. Our primary interest lies in predicting whether a new loan application will result in default or not.</p>
<p>Clearly, <code>loan_data_ARF.rds</code> represents past information, as we know with certainty whether an individual defaulted (1) or not (0). Historical data is valuable for understanding how likely an individual is to default based on their personal and loan characteristics. It is also essential for training quantitative models to predict outcomes for new applicants and for evaluating the performance of our classifications.</p>
<p>A variable name is considered too long when a shorter name can convey the same purpose just as effectively. Let’s rename some of them.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb16"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb16-1"><a href="#cb16-1"></a>old_names <span class="ot">&lt;-</span> <span class="fu">colnames</span>(dat)</span>
<span id="cb16-2"><a href="#cb16-2"></a><span class="fu">colnames</span>(dat) <span class="ot">&lt;-</span> <span class="fu">c</span>(<span class="st">"loan_st"</span>, <span class="st">"l_amnt"</span>, <span class="st">"int"</span>, <span class="st">"grade"</span>, <span class="st">"emp_len"</span>, </span>
<span id="cb16-3"><a href="#cb16-3"></a>                   <span class="st">"home"</span>, <span class="st">"income"</span>, <span class="st">"age"</span>, <span class="st">"sex"</span>, <span class="st">"region"</span>)</span>
<span id="cb16-4"><a href="#cb16-4"></a><span class="fu">data.frame</span>(old_names, <span class="st">"new_names"</span> <span class="ot">=</span> <span class="fu">colnames</span>(dat))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stdout">
<pre><code>        old_names new_names
1     loan_status   loan_st
2       loan_amnt    l_amnt
3        int_rate       int
4           grade     grade
5      emp_length   emp_len
6  home_ownership      home
7      annual_inc    income
8             age       age
9             sex       sex
10         region    region</code></pre>
</div>
</div>
<p>New variable names have now been assigned.</p>
<p>We can take a look at the information in different ways. For example, look at the first 10 rows out of 29,092.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb18"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb18-1"><a href="#cb18-1"></a><span class="fu">head</span>(dat, <span class="dv">10</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stdout">
<pre><code>   loan_st l_amnt   int grade emp_len home income age sex region
1        0   5000 10.65     B      10 RENT  24000  33   0      E
2        0   2400 10.99     C      25 RENT  12252  31   0      E
3        0  10000 13.49     C      13 RENT  49200  24   0      S
4        0   5000 10.99     A       3 RENT  36000  39   0      S
5        0   3000 10.99     E       9 RENT  48000  24   1      N
6        0  12000 12.69     B      11  OWN  75000  28   1      N
7        1   9000 13.49     C       0 RENT  30000  22   1      N
8        0   3000  9.91     B       3 RENT  15000  22   1      N
9        1  10000 10.65     B       3 RENT 100000  28   0      W
10       0   1000 16.29     D       0 RENT  28000  22   1      E</code></pre>
</div>
</div>
<p>Note that <tt><code>sex</code></tt> is 1 for female and 0 for male. Now, instead of looking the details of the first 10 rows, we can summarize with respect to <tt><code>home</code></tt> with the <tt><code>CrossTable()</code></tt> function.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb20"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb20-1"><a href="#cb20-1"></a><span class="fu">CrossTable</span>(dat<span class="sc">$</span>home)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stdout">
<pre><code>

   Cell Contents
|-------------------------|
|                       N |
|         N / Table Total |
|-------------------------|


Total Observations in Table:  29092


          |  MORTGAGE |     OTHER |       OWN |      RENT |
          |-----------|-----------|-----------|-----------|
          |     12002 |        97 |      2301 |     14692 |
          |     0.413 |     0.003 |     0.079 |     0.505 |
          |-----------|-----------|-----------|-----------|


 </code></pre>
</div>
</div>
<p>We can also use <tt><code>CrossTable()</code></tt> to summarize two variables. In particular, instead of counting for home ownership we can add a second dimension like <tt><code>loan_st</code></tt>. This allows us to create more informative tables.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb22-1"><a href="#cb22-1"></a><span class="fu">CrossTable</span>(dat<span class="sc">$</span>home, dat<span class="sc">$</span>loan_st, <span class="at">prop.r =</span> <span class="cn">TRUE</span>,</span>
<span id="cb22-2"><a href="#cb22-2"></a>           <span class="at">prop.c =</span> <span class="cn">FALSE</span>, <span class="at">prop.t =</span> <span class="cn">FALSE</span>, <span class="at">prop.chisq =</span> <span class="cn">FALSE</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stdout">
<pre><code>

   Cell Contents
|-------------------------|
|                       N |
|           N / Row Total |
|-------------------------|


Total Observations in Table:  29092


             | dat$loan_st
    dat$home |         0 |         1 | Row Total |
-------------|-----------|-----------|-----------|
    MORTGAGE |     10821 |      1181 |     12002 |
             |     0.902 |     0.098 |     0.413 |
-------------|-----------|-----------|-----------|
       OTHER |        80 |        17 |        97 |
             |     0.825 |     0.175 |     0.003 |
-------------|-----------|-----------|-----------|
         OWN |      2049 |       252 |      2301 |
             |     0.890 |     0.110 |     0.079 |
-------------|-----------|-----------|-----------|
        RENT |     12915 |      1777 |     14692 |
             |     0.879 |     0.121 |     0.505 |
-------------|-----------|-----------|-----------|
Column Total |     25865 |      3227 |     29092 |
-------------|-----------|-----------|-----------|

 </code></pre>
</div>
</div>
<p>This table reveals defaults by home ownership. We can use histograms to see one variable distribution. In this case we have the interest rate distribution.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb24"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb24-1"><a href="#cb24-1"></a><span class="fu">ggplot</span>(dat, <span class="fu">aes</span>(<span class="at">x =</span> int)) <span class="sc">+</span> </span>
<span id="cb24-2"><a href="#cb24-2"></a>  <span class="fu">geom_histogram</span>(<span class="fu">aes</span>(<span class="at">y=</span>..density..), <span class="at">binwidth =</span> <span class="fl">0.5</span>, <span class="at">colour =</span> <span class="st">"black"</span>, </span>
<span id="cb24-3"><a href="#cb24-3"></a>                 <span class="at">fill =</span> <span class="st">"white"</span>) <span class="sc">+</span></span>
<span id="cb24-4"><a href="#cb24-4"></a>  <span class="fu">labs</span>(<span class="at">y =</span> <span class="st">"Density"</span>, <span class="at">x =</span> <span class="st">"Interest rate"</span>) <span class="sc">+</span></span>
<span id="cb24-5"><a href="#cb24-5"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">"bottom"</span>, <span class="at">legend.title =</span> <span class="fu">element_blank</span>())</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stderr">
<pre><code>Warning: The dot-dot notation (`..density..`) was deprecated in ggplot2 3.4.0.
ℹ Please use `after_stat(density)` instead.</code></pre>
</div>
<div class="cell-output-display">
<div id="fig-irh" class="quarto-float quarto-figure quarto-figure-center anchored">
<figure class="quarto-float quarto-float-fig figure">
<div aria-describedby="fig-irh-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<img src="Logistic_files/figure-html/fig-irh-1.png" class="img-fluid figure-img" width="672">
</div>
<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-irh-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure&nbsp;1.1: Interest rate histogram.
</figcaption>
</figure>
</div>
</div>
</div>
<p>The following is a dotplot figure for the annual income.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb26"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb26-1"><a href="#cb26-1"></a><span class="fu">ggplot</span>(dat, <span class="fu">aes</span>(income)) <span class="sc">+</span> </span>
<span id="cb26-2"><a href="#cb26-2"></a>  <span class="fu">geom_dotplot</span>(<span class="at">binwidth =</span> <span class="dv">100000</span>, <span class="at">fill =</span> <span class="st">"blue"</span>) <span class="sc">+</span></span>
<span id="cb26-3"><a href="#cb26-3"></a>  <span class="fu">labs</span>(<span class="at">y =</span> <span class="st">"Density"</span>,</span>
<span id="cb26-4"><a href="#cb26-4"></a>       <span class="at">x =</span> <span class="st">"Annual income"</span>) <span class="sc">+</span></span>
<span id="cb26-5"><a href="#cb26-5"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">"bottom"</span>, <span class="at">legend.title =</span> <span class="fu">element_blank</span>())</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output-display">
<div id="fig-aib" class="quarto-float quarto-figure quarto-figure-center anchored">
<figure class="quarto-float quarto-float-fig figure">
<div aria-describedby="fig-aib-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<img src="Logistic_files/figure-html/fig-aib-1.png" class="img-fluid figure-img" width="672">
</div>
<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-aib-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure&nbsp;1.2: Annual income dotplot.
</figcaption>
</figure>
</div>
</div>
</div>
<p>The <a href="#fig-aib" class="quarto-xref">Figure&nbsp;1.2</a> above appears suspicious. The horizontal axis shows a very large annual income value (6,000,000). Additionally, there are only a few individuals with extremely high incomes. We should investigate further to determine whether these are valid observations or errors in the original dataset.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb27"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb27-1"><a href="#cb27-1"></a>high_income <span class="ot">&lt;-</span> dat[(dat<span class="sc">$</span>income <span class="sc">&gt;</span> <span class="dv">1000000</span>), ]</span>
<span id="cb27-2"><a href="#cb27-2"></a>high_income</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stdout">
<pre><code>      loan_st l_amnt   int grade emp_len     home  income age sex region
4861        0  12025 14.27     C      13     RENT 1782000  63   0      E
13931       0  10000  6.54     A      16      OWN 1200000  36   0      W
15386       0   1500 10.99     A       5 MORTGAGE 1900000  60   1      N
16713       0  12000  7.51     A       1 MORTGAGE 1200000  32   0      E
19486       0   5000 12.73     C      12 MORTGAGE 6000000 144   1      E
22811       0  10000 10.99     A       1 MORTGAGE 1200000  40   1      E
23361       0   6400  7.40     A       7 MORTGAGE 1440000  44   1      E
23683       0   6600  7.74     A       9 MORTGAGE 1362000  47   0      E
28468       0   8450 12.29     C       0     RENT 2039784  42   0      E</code></pre>
</div>
</div>
<p>One individual (ID 19486) is not only extremely wealthy but also 144 years old. As a result, I have decided to remove these nine observations. Data cleaning is a common task when working with large datasets, and it is acceptable as long as the integrity of the data remains intact.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb29"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb29-1"><a href="#cb29-1"></a>high_income_index <span class="ot">&lt;-</span> <span class="fu">data.frame</span>(<span class="at">value =</span> <span class="fu">as.integer</span>(<span class="fu">rownames</span>(high_income)))</span>
<span id="cb29-2"><a href="#cb29-2"></a>dat <span class="ot">&lt;-</span> dat[<span class="sc">-</span>high_income_index<span class="sc">$</span>value,]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
</div>
<p>Remember, the original dataset contains 29,092 rows. After removing 9 observations, we are left with 29,083 rows. Let’s take a look at the result.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb30"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb30-1"><a href="#cb30-1"></a><span class="fu">ggplot</span>(dat, <span class="fu">aes</span>(income)) <span class="sc">+</span> </span>
<span id="cb30-2"><a href="#cb30-2"></a>  <span class="fu">geom_dotplot</span>(<span class="at">binwidth =</span> <span class="dv">10000</span>, <span class="at">fill =</span> <span class="st">"blue"</span>) <span class="sc">+</span></span>
<span id="cb30-3"><a href="#cb30-3"></a>  <span class="fu">labs</span>(<span class="at">y =</span> <span class="st">"Density"</span>, <span class="at">x =</span> <span class="st">"Annual income"</span>) <span class="sc">+</span></span>
<span id="cb30-4"><a href="#cb30-4"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">"bottom"</span>, <span class="at">legend.title =</span> <span class="fu">element_blank</span>())</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output-display">
<div id="fig-aiwev" class="quarto-float quarto-figure quarto-figure-center anchored">
<figure class="quarto-float quarto-float-fig figure">
<div aria-describedby="fig-aiwev-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<img src="Logistic_files/figure-html/fig-aiwev-1.png" class="img-fluid figure-img" width="672">
</div>
<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-aiwev-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure&nbsp;1.3: Annual income without extreme values.
</figcaption>
</figure>
</div>
</div>
</div>
<p>Somewhat better now.</p>
<p>We could also explore some gender issues.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb31"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb31-1"><a href="#cb31-1"></a>summary_table <span class="ot">&lt;-</span> dat <span class="sc">|&gt;</span></span>
<span id="cb31-2"><a href="#cb31-2"></a>  <span class="fu">group_by</span>(sex) <span class="sc">|&gt;</span></span>
<span id="cb31-3"><a href="#cb31-3"></a>  <span class="fu">summarize</span>(</span>
<span id="cb31-4"><a href="#cb31-4"></a>    <span class="at">Mean =</span> <span class="fu">mean</span>(income),</span>
<span id="cb31-5"><a href="#cb31-5"></a>    <span class="at">Median =</span> <span class="fu">median</span>(income),</span>
<span id="cb31-6"><a href="#cb31-6"></a>    <span class="at">Min =</span> <span class="fu">min</span>(income),</span>
<span id="cb31-7"><a href="#cb31-7"></a>    <span class="at">Max =</span> <span class="fu">max</span>(income),</span>
<span id="cb31-8"><a href="#cb31-8"></a>    <span class="at">Count =</span> <span class="fu">n</span>()</span>
<span id="cb31-9"><a href="#cb31-9"></a>  ) <span class="sc">|&gt;</span></span>
<span id="cb31-10"><a href="#cb31-10"></a>  <span class="fu">mutate</span>(</span>
<span id="cb31-11"><a href="#cb31-11"></a>    <span class="at">Gender =</span> <span class="fu">case_when</span>(</span>
<span id="cb31-12"><a href="#cb31-12"></a>      sex <span class="sc">==</span> <span class="dv">0</span> <span class="sc">~</span> <span class="st">"Male"</span>,</span>
<span id="cb31-13"><a href="#cb31-13"></a>      sex <span class="sc">==</span> <span class="dv">1</span> <span class="sc">~</span> <span class="st">"Female"</span></span>
<span id="cb31-14"><a href="#cb31-14"></a>    )</span>
<span id="cb31-15"><a href="#cb31-15"></a> )</span>
<span id="cb31-16"><a href="#cb31-16"></a>  </span>
<span id="cb31-17"><a href="#cb31-17"></a>summary_table</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stdout">
<pre><code># A tibble: 2 × 7
  sex     Mean Median   Min    Max Count Gender
  &lt;fct&gt;  &lt;dbl&gt;  &lt;dbl&gt; &lt;dbl&gt;  &lt;dbl&gt; &lt;int&gt; &lt;chr&gt;
1 0     66021.  56000  4000 900000 13877 Male
2 1     67064.  57000  4200 948000 15206 Female</code></pre>
</div>
</div>
<p>The table shows that females tend to have slightly higher average and median incomes compared to males. Additionally, the income distribution for females appears more variable, with a wider range of incomes. While the sample sizes for both groups are similar, females are slightly more represented. Overall, the data suggests a modest income advantage for females and greater variability in their earnings.</p>
<p>The better we understand the data, the better positioned we are to analyze it, gain deeper insights into the problem, and interpret the solutions more effectively. However, there is a risk of falling into the trap of overexploration, losing sight of the main problem and failing to address the objectives. Data cleaning, for instance, can be an ongoing task. Removing outliers with excessively high incomes is just one example of how data cleaning can be approached. For now, we pause the exploration and cleaning process and move on to the next section, where we estimate models that will help us achieve the outlined objectives.</p>
</section>
<section id="logistic-models." class="level2" data-number="1.2">
<h2 data-number="1.2" class="anchored" data-anchor-id="logistic-models."><span class="header-section-number">1.2</span> Logistic models.</h2>
<p>Logistic models allows us to make predictions about loan defaults. Logistic regression is a statistical model that in its basic form uses a logistic function to model a binary dependent variable like <tt><code>loan_st</code></tt>. In this case, the binary dependent variable is default (1) or no default (0). A good reference for this section is <span class="citation" data-cites="hull2020machine">Hull (<a href="#ref-hull2020machine" role="doc-biblioref">2020</a>)</span>.</p>
<p>First, load the data and split it into two sets: (1) training and (2) test. The training set is for building and estimate models, and the test set is used to evaluate our model predictions with new data. When estimating models, it is common practice to separate the available data into two parts, <em>training</em> and <em>test</em> data, where the training data is used to estimate parameters (in-sample) and the test data is used to evaluate its accuracy (out of sample). Because the test data is not used in determining the estimation, it should provide a reliable indication of how well the model is likely to estimate or forecast on new data. In sum, we train the model, we test the model, and once we are OK with the model performance on new data, we are ready to use it in real-life applications. If we ignore this split and use the whole database to estimate our models, we may succeed at explaining defaults in our database but we may fail to explain defaults for new loan applications.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb33"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb33-1"><a href="#cb33-1"></a><span class="co"># It is convenient to set the loan status as factor.</span></span>
<span id="cb33-2"><a href="#cb33-2"></a>dat<span class="sc">$</span>loan_st <span class="ot">&lt;-</span> <span class="fu">as.factor</span>(dat<span class="sc">$</span>loan_st)</span>
<span id="cb33-3"><a href="#cb33-3"></a><span class="fu">set.seed</span>(<span class="dv">567</span>)</span>
<span id="cb33-4"><a href="#cb33-4"></a>index_train <span class="ot">&lt;-</span> <span class="fu">cbind</span>(<span class="fu">runif</span>(<span class="dv">1</span> <span class="sc">:</span> <span class="fu">nrow</span>(dat), <span class="dv">0</span> , <span class="dv">1</span>), <span class="fu">c</span>(<span class="dv">1</span> <span class="sc">:</span> <span class="fu">nrow</span>(dat)))</span>
<span id="cb33-5"><a href="#cb33-5"></a>index_train <span class="ot">&lt;-</span> <span class="fu">order</span>(index_train[, <span class="dv">1</span>])</span>
<span id="cb33-6"><a href="#cb33-6"></a>index_train <span class="ot">&lt;-</span> index_train[<span class="dv">1</span><span class="sc">:</span> (<span class="dv">2</span><span class="sc">/</span><span class="dv">3</span> <span class="sc">*</span> <span class="fu">nrow</span>(dat))]</span>
<span id="cb33-7"><a href="#cb33-7"></a><span class="co"># Create training set</span></span>
<span id="cb33-8"><a href="#cb33-8"></a>train <span class="ot">&lt;-</span> dat[index_train, ]</span>
<span id="cb33-9"><a href="#cb33-9"></a><span class="co"># Create test set</span></span>
<span id="cb33-10"><a href="#cb33-10"></a>test <span class="ot">&lt;-</span> dat[<span class="sc">-</span>index_train, ]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
</div>
<p>Variables as factors are useful for model estimation and data visualization. Factors are variables in R which take on a limited number of different values; such variables are often referred to as categorical variables.</p>
<p>We have 29,083 observations in <tt><code>dat</code></tt>. The code above randomly selects <span class="math inline">\(29083 \times (2/3)=19388\)</span> rows to form the <tt><code>train</code></tt>. The <tt><code>test</code></tt> are the remaining <span class="math inline">\(29083-19388=9695\)</span> rows. The random selection is highly recommended as the <tt><code>dat</code></tt> may have some structure or sorting that could bias our model estimation and negatively impact our model test. For example, imagine that for some weird reason the database is sorted in such a way that the first observations are all no default. If that is the case, then the training and the test set would not have portions of default and no default cases and we may distort the whole analysis. The random selection allows us to replicate a real situation in which our database is unsorted, with different characteristics.</p>
<p>Let’s see if the recent created <tt><code>train</code></tt> and <tt><code>test</code></tt> have the same basic properties than <tt><code>dat</code></tt>.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb34"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb34-1"><a href="#cb34-1"></a><span class="co"># Combine data into a list for processing</span></span>
<span id="cb34-2"><a href="#cb34-2"></a>data_list <span class="ot">&lt;-</span> <span class="fu">list</span>(<span class="at">dat =</span> dat<span class="sc">$</span>loan_st, <span class="at">train =</span> train<span class="sc">$</span>loan_st, <span class="at">test =</span> test<span class="sc">$</span>loan_st)</span>
<span id="cb34-3"><a href="#cb34-3"></a></span>
<span id="cb34-4"><a href="#cb34-4"></a><span class="co"># Compute proportions and combine into a data frame</span></span>
<span id="cb34-5"><a href="#cb34-5"></a>prop <span class="ot">&lt;-</span> <span class="fu">do.call</span>(rbind, <span class="fu">lapply</span>(data_list, <span class="cf">function</span>(x) <span class="fu">prop.table</span>(<span class="fu">table</span>(x))))</span>
<span id="cb34-6"><a href="#cb34-6"></a><span class="fu">colnames</span>(prop) <span class="ot">&lt;-</span> <span class="fu">c</span>(<span class="st">"no defaults"</span>, <span class="st">"defaults"</span>)</span>
<span id="cb34-7"><a href="#cb34-7"></a><span class="fu">row.names</span>(prop) <span class="ot">&lt;-</span> <span class="fu">c</span>(<span class="st">"dat_prop"</span>, <span class="st">"train_prop"</span>, <span class="st">"test_prop"</span>)</span>
<span id="cb34-8"><a href="#cb34-8"></a>prop</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stdout">
<pre><code>           no defaults  defaults
dat_prop     0.8890417 0.1109583
train_prop   0.8882298 0.1117702
test_prop    0.8906653 0.1093347</code></pre>
</div>
</div>
<p>The proportions of “defaults” and “no defaults” in the train and test datasets are very similar to those in the original dataset, indicating that both samples are representative. Specifically, the proportion of “defaults” in the train and test datasets closely matches that of the original dataset, as does the proportion of “no defaults.” This suggests that the random sampling process has preserved the overall distribution of the target variable, ensuring that the train and test sets reflect the characteristics of the full dataset.</p>
<p>Assume we think that the <tt><code>loan_st</code></tt> depends on the age of the individual. We can estimate a simple logistic model to learn about the relationship between age and loan status.</p>
<p><span class="math inline">\(\log\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1 \times \text{age}\)</span>.</p>
<p>Where:</p>
<ul>
<li><p><span class="math inline">\(p\)</span> is the probability that <code>loan_st = 1</code> (e.g., the probability of default).</p></li>
<li><p><span class="math inline">\(\beta_0\)</span> is the intercept, and <span class="math inline">\(\beta_1\)</span> is the coefficient for the predictor variable <code>age</code>.</p></li>
</ul>
<p>The dependent variable in logistic regression is expressed as: <span class="math inline">\(\log\left(\frac{p}{1-p}\right)\)</span>. This transformation, known as the <em>logit function</em>, maps probabilities <span class="math inline">\(p\)</span> which range between 0 and 1 to the entire real number line <span class="math inline">\(-\infty\)</span> to <span class="math inline">\(+\infty\)</span>. This allows the model to establish a linear relationship between the independent variable(s) (e.g., age) and the transformed dependent variable, making estimation feasible. The fraction <span class="math inline">\(\frac{p}{1-p}\)</span>, called the <em>odds</em>, represents the ratio of the probability of an event happening <span class="math inline">\(p\)</span> to the probability of it not happening <span class="math inline">\(1-p\)</span>, making it a natural choice for modeling binary outcomes.</p>
<p>Let’s estimate the age model.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb36"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb36-1"><a href="#cb36-1"></a><span class="co"># Fitting a simple logistic model.</span></span>
<span id="cb36-2"><a href="#cb36-2"></a>logi_age <span class="ot">&lt;-</span> <span class="fu">glm</span>(loan_st <span class="sc">~</span> age, <span class="at">family =</span> <span class="st">"binomial"</span>, <span class="at">data =</span> train)</span>
<span id="cb36-3"><a href="#cb36-3"></a>logi_age</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stdout">
<pre><code>
Call:  glm(formula = loan_st ~ age, family = "binomial", data = train)

Coefficients:
(Intercept)          age
   -1.90097     -0.00623

Degrees of Freedom: 19387 Total (i.e. Null);  19386 Residual
Null Deviance:      13580
Residual Deviance: 13580    AIC: 13580</code></pre>
</div>
</div>
<p>The estimated model is: <span class="math inline">\(\log\left(\frac{p}{1-p}\right) = -1.90097 - 0.00623 \times \text{age}\)</span>.</p>
<p>The model suggests there is a negative relationship between age and loan status.</p>
<p>We can solve for <span class="math inline">\(p\)</span> to find the estimate of the probability of default according to the age model: <span class="math inline">\(p = \frac{\exp(-1.90097 - 0.00623 \times \text{age})}{1 + \exp(-1.90097 - 0.00623 \times \text{age})}\)</span>. Let’s evaluate <span class="math inline">\(p\)</span> for a couple of age values.</p>
<p>Substituting <code>age = 18</code>: <span class="math inline">\(p = \frac{\exp(-1.90097 - 0.00623 \times 18)}{1 + \exp(-1.90097 - 0.00623 \times 18)}\)</span>,</p>
<p><span class="math inline">\(p = \frac{\exp(-2.01311)}{1 + \exp(-2.01311)} \approx \frac{0.1333}{1 + 0.1333} \approx 0.1176\)</span>.</p>
<p>Substituting <code>age = 60</code>: <span class="math inline">\(p = \frac{\exp(-1.90097 - 0.00623 \times 60)}{1 + \exp(-1.90097 - 0.00623 \times 60)}\)</span>,</p>
<p><span class="math inline">\(p = \frac{\exp(-2.27477)}{1 + \exp(-2.27477)} \approx \frac{0.1037}{1 + 0.1037} \approx 0.0939\)</span>.</p>
<p>The results of the default probability estimation evaluated at ages 18 and 60 are not very promising because the difference in probabilities is very small. This means that the model does not generate significantly differentiated default probabilities, even when evaluated over a wide age range. This can be problematic because it suggests that this is a poor model.</p>
<p>The following is a graphical representation of the logistic regression model’s output.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb38"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb38-1"><a href="#cb38-1"></a><span class="co"># Define the function for the logistic regression model</span></span>
<span id="cb38-2"><a href="#cb38-2"></a>logistic_function <span class="ot">&lt;-</span> <span class="cf">function</span>(age) {</span>
<span id="cb38-3"><a href="#cb38-3"></a>  beta0 <span class="ot">&lt;-</span> <span class="sc">-</span><span class="fl">1.90097</span></span>
<span id="cb38-4"><a href="#cb38-4"></a>  beta1 <span class="ot">&lt;-</span> <span class="sc">-</span><span class="fl">0.00623</span></span>
<span id="cb38-5"><a href="#cb38-5"></a>  p <span class="ot">&lt;-</span> <span class="fu">exp</span>(beta0 <span class="sc">+</span> beta1 <span class="sc">*</span> age) <span class="sc">/</span> (<span class="dv">1</span> <span class="sc">+</span> <span class="fu">exp</span>(beta0 <span class="sc">+</span> beta1 <span class="sc">*</span> age))</span>
<span id="cb38-6"><a href="#cb38-6"></a>  <span class="fu">return</span>(p)</span>
<span id="cb38-7"><a href="#cb38-7"></a>}</span>
<span id="cb38-8"><a href="#cb38-8"></a></span>
<span id="cb38-9"><a href="#cb38-9"></a><span class="co"># Generate a sequence of ages</span></span>
<span id="cb38-10"><a href="#cb38-10"></a>ages <span class="ot">&lt;-</span> <span class="fu">seq</span>(<span class="sc">-</span><span class="dv">1000</span>, <span class="dv">500</span>, <span class="at">by =</span> <span class="dv">1</span>)</span>
<span id="cb38-11"><a href="#cb38-11"></a></span>
<span id="cb38-12"><a href="#cb38-12"></a><span class="co"># Apply the logistic function to each age</span></span>
<span id="cb38-13"><a href="#cb38-13"></a>p_values <span class="ot">&lt;-</span> <span class="fu">sapply</span>(ages, logistic_function)</span>
<span id="cb38-14"><a href="#cb38-14"></a></span>
<span id="cb38-15"><a href="#cb38-15"></a><span class="co"># Create the plot</span></span>
<span id="cb38-16"><a href="#cb38-16"></a><span class="fu">library</span>(ggplot2)</span>
<span id="cb38-17"><a href="#cb38-17"></a><span class="fu">ggplot</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(<span class="at">age =</span> ages, <span class="at">p =</span> p_values), </span>
<span id="cb38-18"><a href="#cb38-18"></a>       <span class="fu">aes</span>(<span class="at">x =</span> age, <span class="at">y =</span> p)) <span class="sc">+</span></span>
<span id="cb38-19"><a href="#cb38-19"></a>  <span class="fu">geom_line</span>(<span class="at">color =</span> <span class="st">"blue"</span>, <span class="at">size =</span> <span class="dv">2</span>) <span class="sc">+</span></span>
<span id="cb38-20"><a href="#cb38-20"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">"Age"</span>, <span class="at">y =</span> <span class="st">"Probability (p)"</span>) <span class="sc">+</span></span>
<span id="cb38-21"><a href="#cb38-21"></a>  <span class="fu">theme_minimal</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stderr">
<pre><code>Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.</code></pre>
</div>
<div class="cell-output-display">
<div id="fig-scolcam" class="quarto-float quarto-figure quarto-figure-center anchored">
<figure class="quarto-float quarto-float-fig figure">
<div aria-describedby="fig-scolcam-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<img src="Logistic_files/figure-html/fig-scolcam-1.png" class="img-fluid figure-img" width="672">
</div>
<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-scolcam-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure&nbsp;1.4: Sigmoid curve or logistic curve: age model.
</figcaption>
</figure>
</div>
</div>
</div>
<p>The logistic curve derived from this model is unreasonable because the probability <span class="math inline">\(p\)</span> changes very slowly with respect to age. This implies that, to observe the full range of probabilities from 0 to 1, we would require age values that are unrealistic or implausibly extreme. Such behavior suggests that the model is poorly specified, as it fails to generate meaningful distinctions in probabilities across a realistic range of ages.</p>
<p>The AIC value of the age model (13,580) is useful when comparing models. The Akaike information criterion (AIC) is a mathematical method for evaluating how well a model fits the data it was generated from. In statistics, AIC is used to compare different possible models and determine which one is the best fit for the data. At the moment we cannot interpret the AIC simply because we only have one model and we cannot compare it with another AIC.</p>
<p>Let’s estimate another simple model where the interest rate category is used as a predictor of the <tt><code>loan_st</code></tt>. Remember we are not conducting any prediction at all at this moment, we are only estimating models using the training set.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb40"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb40-1"><a href="#cb40-1"></a><span class="co"># Build a glm model with variable interest rate as a predictor.</span></span>
<span id="cb40-2"><a href="#cb40-2"></a>logi_int <span class="ot">&lt;-</span> <span class="fu">glm</span>(<span class="at">formula =</span> loan_st <span class="sc">~</span> int, <span class="at">family =</span> <span class="st">"binomial"</span>, <span class="at">data =</span> train)</span>
<span id="cb40-3"><a href="#cb40-3"></a><span class="co"># Print the parameter estimates.</span></span>
<span id="cb40-4"><a href="#cb40-4"></a>logi_int</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stdout">
<pre><code>
Call:  glm(formula = loan_st ~ int, family = "binomial", data = train)

Coefficients:
(Intercept)          int
     -3.710        0.142

Degrees of Freedom: 19387 Total (i.e. Null);  19386 Residual
Null Deviance:      13580
Residual Deviance: 13210    AIC: 13220</code></pre>
</div>
</div>
<p>The AIC is a lower (13,220 versus 13,580), so we have a better model now.</p>
<p>Using one single predictor as age or interest rate is clearly a limited approach. Let’s add some more predictors. Also, let’s introduce the <tt><code>summary()</code></tt> function to extract more information about the model estimation results. The <tt><code>logi_multi</code></tt> below assumes that the loan status depend on the age, interest rate, grade, loan amount, and annual income.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb42"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb42-1"><a href="#cb42-1"></a><span class="co"># Multiple variables in a logistic regression model.</span></span>
<span id="cb42-2"><a href="#cb42-2"></a>logi_multi <span class="ot">&lt;-</span> <span class="fu">glm</span>(loan_st <span class="sc">~</span> age <span class="sc">+</span> int <span class="sc">+</span> grade <span class="sc">+</span> <span class="fu">log</span>(l_amnt) <span class="sc">+</span> </span>
<span id="cb42-3"><a href="#cb42-3"></a>                  <span class="fu">log</span>(income) , <span class="at">family =</span> <span class="st">"binomial"</span>, <span class="at">data =</span> train)</span>
<span id="cb42-4"><a href="#cb42-4"></a><span class="co"># Obtain significance levels using summary().</span></span>
<span id="cb42-5"><a href="#cb42-5"></a><span class="fu">summary</span>(logi_multi)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stdout">
<pre><code>
Call:
glm(formula = loan_st ~ age + int + grade + log(l_amnt) + log(income),
    family = "binomial", data = train)

Coefficients:
             Estimate Std. Error z value Pr(&gt;|z|)
(Intercept)  1.996240   0.477911   4.177 2.95e-05 ***
age         -0.002302   0.003825  -0.602   0.5473
int          0.038767   0.017249   2.247   0.0246 *
gradeB       0.503409   0.087435   5.758 8.54e-09 ***
gradeC       0.748229   0.117765   6.354 2.10e-10 ***
gradeD       0.964343   0.147283   6.548 5.85e-11 ***
gradeE       1.033442   0.190817   5.416 6.10e-08 ***
gradeF       1.619470   0.257900   6.279 3.40e-10 ***
gradeG       1.867494   0.440232   4.242 2.21e-05 ***
log(l_amnt)  0.015718   0.036341   0.433   0.6654
log(income) -0.470748   0.046423 -10.140  &lt; 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 13579  on 19387  degrees of freedom
Residual deviance: 13028  on 19377  degrees of freedom
AIC: 13050

Number of Fisher Scoring iterations: 5</code></pre>
</div>
</div>
<p>Our multi-factor model works well. In <tt><code>logi_multi</code></tt>, the AIC value is the lowest so far (13,050 versus 13,220), so this should be considered as the best in-sample model at the moment. The <tt><code>summary()</code></tt> function shows the significance levels of the estimators, but we are currently more interested in the goodness of fit of the models because we want to conduct predictions about the <tt><code>loan_st</code></tt>. This is, we are interested to use a model to find out whether new applicants in the test set are expected to default or not, rather than in the applicants’ credit risk factors. This is why we are concentrated in AIC now.</p>
<p>When a customer fill out a credit application form, we collect information but we do not know for sure whether she or he will eventually default. A credit risk model can help us in this task.</p>
</section>
<section id="prediction-and-model-evaluation." class="level2" data-number="1.3">
<h2 data-number="1.3" class="anchored" data-anchor-id="prediction-and-model-evaluation."><span class="header-section-number">1.3</span> Prediction and model evaluation.</h2>
<p>Let’s take our three models: <tt><code>logi_age</code></tt>, <tt><code>logi_int</code></tt> and <tt><code>logi_multi</code></tt> from the previous subsection to carry out a simple prediction exercise. We start by identifying one observation in the test set and ask the models to predict the <tt><code>loan_st</code></tt>. This is, we take the first guy age, then we apply the <tt><code>logi_age</code></tt> model, and compare the predicted <tt><code>loan_st</code></tt> with respect to what really happened. Remember we know what really happened with this guy because we have the information in the test set. Every model is expected to produce different <tt><code>loan_st</code></tt> predictions. If the model is good, then the predicted <tt><code>loan_st</code></tt> will match what really happened.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb44"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb44-1"><a href="#cb44-1"></a><span class="co"># Define one single observation in test_set.</span></span>
<span id="cb44-2"><a href="#cb44-2"></a>John_Doe <span class="ot">&lt;-</span> <span class="fu">as.data.frame</span>(test[<span class="dv">1</span>, ])</span>
<span id="cb44-3"><a href="#cb44-3"></a>John_Doe</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stdout">
<pre><code>  loan_st l_amnt   int grade emp_len home income age sex region
1       0   5000 10.65     B      10 RENT  24000  33   0      E</code></pre>
</div>
</div>
<p>We know in advance that the <tt><code>loan_st</code></tt> of this observation taken from the test set is 0. However, the models cannot know this simply because we did not use the test set to estimate the logistic models. Our models were estimated using the training set. A good credit risk model should predict a no default given this new applicant.</p>
<p>The values of <tt><code>loan_st</code></tt> in the test set is either 0 or 1. However, the logistic models estimate the <tt><code>loan_st</code></tt> as values in the range of 0 to 1. This mean that we would expect the estimated <tt><code>loan_st</code></tt> to be close to 0. <em>But, how close?</em> We will deal with this issue later.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb46"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb46-1"><a href="#cb46-1"></a><span class="co"># Predict the loan status.</span></span>
<span id="cb46-2"><a href="#cb46-2"></a>logi_age_pred <span class="ot">&lt;-</span> <span class="fu">predict</span>(logi_age, <span class="at">newdata =</span> John_Doe, <span class="at">type =</span> <span class="st">"response"</span>)</span>
<span id="cb46-3"><a href="#cb46-3"></a>logi_int_pred <span class="ot">&lt;-</span> <span class="fu">predict</span>(logi_int, <span class="at">newdata =</span> John_Doe, <span class="at">type =</span> <span class="st">"response"</span>)</span>
<span id="cb46-4"><a href="#cb46-4"></a>logi_multi_pred <span class="ot">&lt;-</span> <span class="fu">predict</span>(logi_multi, <span class="at">newdata =</span> John_Doe, <span class="at">type =</span> <span class="st">"response"</span>)</span>
<span id="cb46-5"><a href="#cb46-5"></a><span class="co"># Collect all.</span></span>
<span id="cb46-6"><a href="#cb46-6"></a>pred_John <span class="ot">&lt;-</span> <span class="fu">rbind</span>(<span class="st">"logi_age"</span> <span class="ot">=</span> logi_age_pred, </span>
<span id="cb46-7"><a href="#cb46-7"></a>                     <span class="st">"logi_int"</span> <span class="ot">=</span> logi_int_pred, </span>
<span id="cb46-8"><a href="#cb46-8"></a>                     <span class="st">"logi_multi"</span> <span class="ot">=</span> logi_multi_pred)</span>
<span id="cb46-9"><a href="#cb46-9"></a><span class="co"># Prepare a table.</span></span>
<span id="cb46-10"><a href="#cb46-10"></a><span class="fu">colnames</span>(pred_John) <span class="ot">&lt;-</span> <span class="st">"Loan status predictions for John Doe."</span></span>
<span id="cb46-11"><a href="#cb46-11"></a>pred_John</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stdout">
<pre><code>           Loan status predictions for John Doe.
logi_age                              0.10846236
logi_int                              0.09996702
logi_multi                            0.14461840</code></pre>
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<p>These values are low as they are close to 0. We could interpret this as a certain ability of the models to predict this single case from the test set. However, several questions remains unanswered and requires further analysis. For example: <em>How can we determine if the prediction is low enough to consider it as a non-default?</em> We may need a cut-off value to decide. We will explore this issue later.</p>
<p>Another aspect is: <em>What about the rest of the cases in the test set?</em> We have 9,695 observations in the test set and in the example above we only test for the first one. We are interested in the entire test set, not only for John Doe. Fortunately, this issue is easy to address as we only need to change the <tt><code>newdata</code></tt> parameter in the <tt><code>predict()</code></tt> function. In particular, instead of <tt><code>newdata = John_Doe</code></tt>, which is one observation, and can change it to <tt><code>newdata = test</code></tt>, which is the entire 9,695 test set.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb48"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb48-1"><a href="#cb48-1"></a><span class="co"># Predict the loan status with the three models.</span></span>
<span id="cb48-2"><a href="#cb48-2"></a>pred_logi_age <span class="ot">&lt;-</span> <span class="fu">predict</span>(logi_age, <span class="at">newdata =</span> test, <span class="at">type =</span> <span class="st">"response"</span>)</span>
<span id="cb48-3"><a href="#cb48-3"></a>pred_logi_int <span class="ot">&lt;-</span> <span class="fu">predict</span>(logi_int, <span class="at">newdata =</span> test, <span class="at">type =</span> <span class="st">"response"</span>)</span>
<span id="cb48-4"><a href="#cb48-4"></a>pred_logi_multi <span class="ot">&lt;-</span> <span class="fu">predict</span>(logi_multi, <span class="at">newdata =</span> test, </span>
<span id="cb48-5"><a href="#cb48-5"></a>                           <span class="at">type =</span> <span class="st">"response"</span>)</span>
<span id="cb48-6"><a href="#cb48-6"></a></span>
<span id="cb48-7"><a href="#cb48-7"></a>pred_range <span class="ot">&lt;-</span> <span class="fu">rbind</span>(<span class="st">"logi_age"</span> <span class="ot">=</span> <span class="fu">range</span>(pred_logi_age), </span>
<span id="cb48-8"><a href="#cb48-8"></a>                     <span class="st">"logi_int"</span> <span class="ot">=</span> <span class="fu">range</span>(pred_logi_int),</span>
<span id="cb48-9"><a href="#cb48-9"></a>                     <span class="st">"logi_multi"</span> <span class="ot">=</span> <span class="fu">range</span>(pred_logi_multi))</span>
<span id="cb48-10"><a href="#cb48-10"></a>aic <span class="ot">&lt;-</span> <span class="fu">rbind</span>(logi_age<span class="sc">$</span>aic, logi_int<span class="sc">$</span>aic, logi_multi<span class="sc">$</span>aic)</span>
<span id="cb48-11"><a href="#cb48-11"></a>pred_range <span class="ot">&lt;-</span> <span class="fu">cbind</span>(pred_range, aic)</span>
<span id="cb48-12"><a href="#cb48-12"></a><span class="fu">colnames</span>(pred_range) <span class="ot">&lt;-</span> <span class="fu">c</span>(<span class="st">"min(loan_st)"</span>, <span class="st">"max(loan_st)"</span>, <span class="st">"AIC"</span>)</span>
<span id="cb48-13"><a href="#cb48-13"></a>pred_range</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stdout">
<pre><code>           min(loan_st) max(loan_st)      AIC
logi_age     0.08417892    0.1165453 13580.65
logi_int     0.05019756    0.3982977 13215.61
logi_multi   0.01739107    0.4668004 13050.30</code></pre>
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<p>Now, instead of the prediction of one single applicant we have conducted a prediction of all 9,695 applicants in the test set. The lower value column corresponds to the lower predicted <tt><code>loan_st</code></tt> for each model. The logistic models produce values in the range of zero to one, and in this case the ranges are rather narrow.</p>
<p>Narrow ranges (the difference between higher and lower <tt><code>loan_st</code></tt> predicted values) could be problematic because the model could not be able to discriminate between defaults (predictions closer to 1) and no-defaults (predictions closer to 0). The higher AIC corresponds to the worst in-sample model and the lower AIC to the best in-sample model. Here, we can see some in and out of sample consistency because the best model according to the AIC, corresponds to the model with the higher prediction range.</p>
<p>Let’s explore all the predicted <tt><code>loan_st</code></tt> values for the <tt><code>logi_age</code></tt>:</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb50"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb50-1"><a href="#cb50-1"></a><span class="fu">ggplot</span>(<span class="fu">data.frame</span>(pred_logi_age), <span class="fu">aes</span>(<span class="at">x =</span> pred_logi_age)) <span class="sc">+</span> </span>
<span id="cb50-2"><a href="#cb50-2"></a>  <span class="fu">geom_density</span>(<span class="at">fill =</span> <span class="st">"red"</span>) <span class="sc">+</span></span>
<span id="cb50-3"><a href="#cb50-3"></a>  <span class="fu">labs</span>(<span class="at">y =</span> <span class="st">"Density"</span>, <span class="at">x =</span> <span class="st">"Default prediction"</span>) <span class="sc">+</span></span>
<span id="cb50-4"><a href="#cb50-4"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">"bottom"</span>, <span class="at">legend.title =</span> <span class="fu">element_blank</span>())</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output-display">
<div id="fig-amph" class="quarto-float quarto-figure quarto-figure-center anchored">
<figure class="quarto-float quarto-float-fig figure">
<div aria-describedby="fig-amph-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<img src="Logistic_files/figure-html/fig-amph-1.png" class="img-fluid figure-img" width="672">
</div>
<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-amph-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure&nbsp;1.5: Age model prediction histogram.
</figcaption>
</figure>
</div>
</div>
</div>
<p>The <tt><code>logi_age</code></tt> fails to predict values ranging from 0 to 1. In fact, these values are quite concentrated in a very small range of values. As a consequence, this model fails to differentiate between default and no default predictions.</p>
<p>Let’s visualize the predictions of the <tt><code>logi_int</code></tt> and <tt><code>logi_multi</code></tt>. We first collect all predictions in a single data frame just for convenience.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb51"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb51-1"><a href="#cb51-1"></a>pred_logi <span class="ot">&lt;-</span> <span class="fu">data.frame</span>(<span class="fu">cbind</span>(pred_logi_age, pred_logi_int,</span>
<span id="cb51-2"><a href="#cb51-2"></a>                                  pred_logi_multi))</span>
<span id="cb51-3"><a href="#cb51-3"></a>pred_logi <span class="ot">&lt;-</span> <span class="fu">gather</span>(pred_logi, <span class="at">key =</span> <span class="st">"model"</span>,  <span class="at">value =</span> <span class="st">"pred"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
</div>
<p>Now we plot the <tt><code>logi_int</code></tt> and <tt><code>logi_multi</code></tt>.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb52"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb52-1"><a href="#cb52-1"></a><span class="fu">ggplot</span>(pred_logi[pred_logi<span class="sc">$</span>model <span class="sc">!=</span> <span class="st">"pred_logi_age"</span>,], </span>
<span id="cb52-2"><a href="#cb52-2"></a>       <span class="fu">aes</span>(<span class="at">x =</span> pred, <span class="at">fill =</span> model)) <span class="sc">+</span> </span>
<span id="cb52-3"><a href="#cb52-3"></a>  <span class="fu">geom_density</span>(<span class="at">alpha =</span> <span class="fl">0.4</span>) <span class="sc">+</span></span>
<span id="cb52-4"><a href="#cb52-4"></a>  <span class="fu">labs</span>(<span class="at">y =</span> <span class="st">"Density"</span>, <span class="at">x =</span> <span class="st">"Default prediction"</span>) <span class="sc">+</span></span>
<span id="cb52-5"><a href="#cb52-5"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">"bottom"</span>, <span class="at">legend.title =</span> <span class="fu">element_blank</span>())</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output-display">
<div id="fig-irammph" class="quarto-float quarto-figure quarto-figure-center anchored">
<figure class="quarto-float quarto-float-fig figure">
<div aria-describedby="fig-irammph-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<img src="Logistic_files/figure-html/fig-irammph-1.png" class="img-fluid figure-img" width="672">
</div>
<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-irammph-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure&nbsp;1.6: Interest rate and multi models predictions histograms.
</figcaption>
</figure>
</div>
</div>
</div>
<p>Let’s add the <tt><code>age</code></tt> model as well.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb53"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb53-1"><a href="#cb53-1"></a><span class="fu">ggplot</span>(pred_logi, <span class="fu">aes</span>(<span class="at">x =</span> pred, <span class="at">y =</span> model, <span class="at">fill =</span> model)) <span class="sc">+</span> </span>
<span id="cb53-2"><a href="#cb53-2"></a>  <span class="fu">geom_boxplot</span>() <span class="sc">+</span></span>
<span id="cb53-3"><a href="#cb53-3"></a>  <span class="fu">labs</span>(<span class="at">y =</span> <span class="st">"Model"</span>, <span class="at">x =</span> <span class="st">"Default prediction"</span>) <span class="sc">+</span></span>
<span id="cb53-4"><a href="#cb53-4"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">"none"</span>, <span class="at">legend.title =</span> <span class="fu">element_blank</span>())</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output-display">
<div id="fig-aairaammpb" class="quarto-float quarto-figure quarto-figure-center anchored">
<figure class="quarto-float quarto-float-fig figure">
<div aria-describedby="fig-aairaammpb-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<img src="Logistic_files/figure-html/fig-aairaammpb-1.png" class="img-fluid figure-img" width="672">
</div>
<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-aairaammpb-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure&nbsp;1.7: Age, Interest rate and multi models predictions boxplot.
</figcaption>
</figure>
</div>
</div>
</div>
<p>Presumably, a model which considers all available predictors could be better for predicting the <tt><code>loan_st</code></tt>.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb54"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb54-1"><a href="#cb54-1"></a><span class="co"># Logistic regression model using all available predictors in the data set.</span></span>
<span id="cb54-2"><a href="#cb54-2"></a>logi_full <span class="ot">&lt;-</span> <span class="fu">glm</span>(loan_st <span class="sc">~</span> age <span class="sc">+</span> int <span class="sc">+</span> grade <span class="sc">+</span> <span class="fu">log</span>(l_amnt) <span class="sc">+</span> </span>
<span id="cb54-3"><a href="#cb54-3"></a>                   <span class="fu">log</span>(income) <span class="sc">+</span> emp_len <span class="sc">+</span> home <span class="sc">+</span> sex <span class="sc">+</span></span>
<span id="cb54-4"><a href="#cb54-4"></a>                   region, <span class="at">family =</span> <span class="st">"binomial"</span>, <span class="at">data =</span> train)</span>
<span id="cb54-5"><a href="#cb54-5"></a><span class="co"># Loan status predictions for all test set elements.</span></span>
<span id="cb54-6"><a href="#cb54-6"></a>pred_logi_full <span class="ot">&lt;-</span> <span class="fu">predict</span>(logi_full, <span class="at">newdata =</span> test, <span class="at">type =</span> <span class="st">"response"</span>)</span>
<span id="cb54-7"><a href="#cb54-7"></a><span class="co"># Look at the predictions range.</span></span>
<span id="cb54-8"><a href="#cb54-8"></a><span class="fu">range</span>(pred_logi_full)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 1.422469e-09 8.544424e-01</code></pre>
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<p>Now, the <tt><code>pred_logi_full</code></tt> prediction range is wider. A wider range means that the <tt><code>loan_st</code></tt> predictions are now closer to 1. This is good because we need the model to be able to predict both no-defaults (0) and defaults (1). Let’s see a prediction comparison with respect to the rest of the models.</p>
<div class="cell">
<details class="code-fold">
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb56"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb56-1"><a href="#cb56-1"></a>pred_range <span class="ot">&lt;-</span> <span class="fu">rbind</span>(<span class="st">"logi_age"</span> <span class="ot">=</span> <span class="fu">range</span>(pred_logi_age), </span>
<span id="cb56-2"><a href="#cb56-2"></a>                     <span class="st">"logi_int"</span> <span class="ot">=</span> <span class="fu">range</span>(pred_logi_int),</span>
<span id="cb56-3"><a href="#cb56-3"></a>                     <span class="st">"logi_multi"</span> <span class="ot">=</span> <span class="fu">range</span>(pred_logi_multi),</span>