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Chapter20_Linear_regression.html
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<!DOCTYPE html>
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<title>Chapter20_Linear_regression</title>
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<p>Setting up a fancy stylesheet</p>
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<div class=" highlight hl-ipython3"><pre><span class="kn">from</span> <span class="nn">IPython.core.display</span> <span class="k">import</span> <span class="n">HTML</span>
<span class="n">css_file</span> <span class="o">=</span> <span class="s">'style.css'</span>
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<p>Setting up the required python™ environment</p>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="k">import</span> <span class="n">bayes_mvs</span><span class="p">,</span> <span class="n">linregress</span>
<span class="kn">from</span> <span class="nn">math</span> <span class="k">import</span> <span class="n">factorial</span>
<span class="kn">import</span> <span class="nn">scikits.bootstrap</span> <span class="k">as</span> <span class="nn">bs</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
<span class="kn">from</span> <span class="nn">warnings</span> <span class="k">import</span> <span class="n">filterwarnings</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
<span class="n">filterwarnings</span><span class="p">(</span><span class="s">'ignore'</span><span class="p">)</span>
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<h1 id="Linear-regression">Linear regression<a class="anchor-link" href="#Linear-regression">¶</a></h1>
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<h2 id="Introduction">Introduction<a class="anchor-link" href="#Introduction">¶</a></h2>
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<p>In this lecture I will discuss the correlation between variables. *Correlation* between variables refers to the existence of a relationship between them and *linear* refers to the fact that they are directly coupled. As the value of one increases, the other either increases or decreases proportionally. This proportionality is constant, i.e. one or both are not squared or subject to a transcedental transformation.</p>
<p>Many mathematical steps are required to calculate the relationship between two variables. Fortunately these can all be done with a few simple Python™ functions.</p>
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<h2 id="Simple-linear-regeression">Simple linear regeression<a class="anchor-link" href="#Simple-linear-regeression">¶</a></h2>
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<p>With linear regression we are directly comparing two sets of numerical variables as opposed to two sets of numerical variables locked up in two categorical groups.</p>
<p>The best way to start comparing two sets of numerical variables is to plot them on a Cartesian coordinate (*xy*) plane. We choose one variable to appear on the *x*-axis and one on the*y*-axis. We then take a row of data (i.e. from the same patient), find the corresponding entries for the two variables and plot them on our *xy*-plane. For this to happen, we need to make sure that for every row, both values are available.</p>
<p>It is necessary to discard entries when both values are not available. This can happen frequently in clinical data collection.</p>
<p>Let's give it a go with this data set.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">reg_data</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s">'Regression.csv'</span><span class="p">)</span> <span class="c"># Reading in the csv file and saving it as a computer</span>
<span class="c"># variable named reg_data, which is then a DataFrame object</span>
<span class="n">reg_data</span> <span class="c"># Displaying all the data</span>
</pre></div>
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<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>PCT</th>
<th>CRP</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td>2.6</td>
<td>32.6</td>
</tr>
<tr>
<th>1</th>
<td>9.8</td>
<td>88.2</td>
</tr>
<tr>
<th>2</th>
<td>4.9</td>
<td>44.6</td>
</tr>
<tr>
<th>3</th>
<td>7.7</td>
<td>84.2</td>
</tr>
<tr>
<th>4</th>
<td>2.3</td>
<td>14.3</td>
</tr>
<tr>
<th>5</th>
<td>9.7</td>
<td>106.8</td>
</tr>
<tr>
<th>6</th>
<td>NaN</td>
<td>97.4</td>
</tr>
<tr>
<th>7</th>
<td>9.8</td>
<td>91.9</td>
</tr>
<tr>
<th>8</th>
<td>3.4</td>
<td>31.7</td>
</tr>
<tr>
<th>9</th>
<td>3.5</td>
<td>25.0</td>
</tr>
<tr>
<th>10</th>
<td>7.0</td>
<td>64.1</td>
</tr>
<tr>
<th>11</th>
<td>7.8</td>
<td>69.3</td>
</tr>
<tr>
<th>12</th>
<td>8.2</td>
<td>84.7</td>
</tr>
<tr>
<th>13</th>
<td>9.4</td>
<td>NaN</td>
</tr>
<tr>
<th>14</th>
<td>5.4</td>
<td>45.0</td>
</tr>
<tr>
<th>15</th>
<td>8.1</td>
<td>77.9</td>
</tr>
<tr>
<th>16</th>
<td>5.5</td>
<td>55.8</td>
</tr>
<tr>
<th>17</th>
<td>4.7</td>
<td>51.4</td>
</tr>
<tr>
<th>18</th>
<td>5.0</td>
<td>59.3</td>
</tr>
<tr>
<th>19</th>
<td>8.6</td>
<td>92.3</td>
</tr>
<tr>
<th>20</th>
<td>5.9</td>
<td>54.5</td>
</tr>
<tr>
<th>21</th>
<td>NaN</td>
<td>59.0</td>
</tr>
<tr>
<th>22</th>
<td>5.7</td>
<td>52.7</td>
</tr>
<tr>
<th>23</th>
<td>8.9</td>
<td>87.0</td>
</tr>
<tr>
<th>24</th>
<td>6.4</td>
<td>73.5</td>
</tr>
<tr>
<th>25</th>
<td>5.6</td>
<td>53.2</td>
</tr>
<tr>
<th>26</th>
<td>2.5</td>
<td>17.3</td>
</tr>
<tr>
<th>27</th>
<td>6.4</td>
<td>55.7</td>
</tr>
<tr>
<th>28</th>
<td>9.6</td>
<td>94.0</td>
</tr>
</tbody>
</table>
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<p>First off we note that there are two variables in our data set; procalcitonin (PCT) and C-reactive protein (CRP). Both are ratio-type numerical data. Lastly, we note that some of the data entries are missing.<p>
<p>So, before we go ahead and do all of this, let's get rid of all the rows that do not contain any data (even if only one of the two are absent).</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">reg_data_2</span> <span class="o">=</span> <span class="n">reg_data</span><span class="o">.</span><span class="n">dropna</span><span class="p">(</span><span class="n">subset</span> <span class="o">=</span> <span class="p">[</span><span class="s">'PCT'</span><span class="p">,</span> <span class="s">'CRP'</span><span class="p">])</span> <span class="c"># The subset = [] argument lists all the</span>
<span class="c"># columns from which you would like empty entries to be removed</span>
<span class="n">reg_data_2</span>
</pre></div>
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<div class="output_area"><div class="prompt output_prompt">Out[4]:</div>
<div class="output_html rendered_html output_subarea output_execute_result">
<div>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>PCT</th>
<th>CRP</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td>2.6</td>
<td>32.6</td>
</tr>
<tr>
<th>1</th>
<td>9.8</td>
<td>88.2</td>
</tr>
<tr>
<th>2</th>
<td>4.9</td>
<td>44.6</td>
</tr>
<tr>
<th>3</th>
<td>7.7</td>
<td>84.2</td>
</tr>
<tr>
<th>4</th>
<td>2.3</td>
<td>14.3</td>
</tr>
<tr>
<th>5</th>
<td>9.7</td>
<td>106.8</td>
</tr>
<tr>
<th>7</th>
<td>9.8</td>
<td>91.9</td>
</tr>
<tr>
<th>8</th>
<td>3.4</td>
<td>31.7</td>
</tr>
<tr>
<th>9</th>
<td>3.5</td>
<td>25.0</td>
</tr>
<tr>
<th>10</th>
<td>7.0</td>
<td>64.1</td>
</tr>
<tr>
<th>11</th>
<td>7.8</td>
<td>69.3</td>
</tr>
<tr>
<th>12</th>
<td>8.2</td>
<td>84.7</td>
</tr>
<tr>
<th>14</th>
<td>5.4</td>
<td>45.0</td>
</tr>
<tr>
<th>15</th>
<td>8.1</td>
<td>77.9</td>
</tr>
<tr>
<th>16</th>
<td>5.5</td>
<td>55.8</td>
</tr>
<tr>
<th>17</th>
<td>4.7</td>
<td>51.4</td>
</tr>
<tr>
<th>18</th>
<td>5.0</td>
<td>59.3</td>
</tr>
<tr>
<th>19</th>
<td>8.6</td>
<td>92.3</td>
</tr>
<tr>
<th>20</th>
<td>5.9</td>
<td>54.5</td>
</tr>
<tr>
<th>22</th>
<td>5.7</td>
<td>52.7</td>
</tr>
<tr>
<th>23</th>
<td>8.9</td>
<td>87.0</td>
</tr>
<tr>
<th>24</th>
<td>6.4</td>
<td>73.5</td>
</tr>
<tr>
<th>25</th>
<td>5.6</td>
<td>53.2</td>
</tr>
<tr>
<th>26</th>
<td>2.5</td>
<td>17.3</td>
</tr>
<tr>
<th>27</th>
<td>6.4</td>
<td>55.7</td>
</tr>
<tr>
<th>28</th>
<td>9.6</td>
<td>94.0</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
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<p>Note how all the rows where an entry was missing, was removed.</p>
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<p>As per usual, we can describe our data and even get the normal violin plots (or boxplots if you prefere those) going, so that we can get a feel for our data.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">reg_data_2</span><span class="p">[</span><span class="s">'PCT'</span><span class="p">]</span><span class="o">.</span><span class="n">describe</span><span class="p">()</span>
</pre></div>
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<pre>count 26.000000
mean 6.346154
std 2.362749
min 2.300000
25% 4.925000
50% 6.150000
75% 8.175000
max 9.800000
Name: PCT, dtype: float64</pre>
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<div class=" highlight hl-ipython3"><pre><span class="n">reg_data_2</span><span class="p">[</span><span class="s">'CRP'</span><span class="p">]</span><span class="o">.</span><span class="n">describe</span><span class="p">()</span>
</pre></div>
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<pre>count 26.000000
mean 61.807692
std 25.329081
min 14.300000
25% 46.600000
50% 57.550000
75% 84.575000
max 106.800000
Name: CRP, dtype: float64</pre>
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<div class=" highlight hl-ipython3"><pre><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
<span class="n">sns</span><span class="o">.</span><span class="n">violinplot</span><span class="p">(</span><span class="n">reg_data_2</span><span class="p">,</span> <span class="n">groupby</span> <span class="o">=</span> <span class="p">[</span><span class="s">'PCT'</span><span class="p">,</span> <span class="s">'CRP'</span><span class="p">],</span> <span class="n">names</span> <span class="o">=</span> <span class="p">[</span><span class="s">'PCT'</span><span class="p">,</span> <span class="s">'CRP'</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
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