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Chapter16_Confidence_intervals.html
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<!DOCTYPE html>
<html>
<head><meta charset="utf-8" />
<title>Chapter16_Confidence_intervals</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="c">#import numpy as 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="c">#from math import 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="n">sns</span><span class="o">.</span><span class="n">set_style</span><span class="p">(</span><span class="s">'white'</span><span class="p">)</span>
<span class="n">sns</span><span class="o">.</span><span class="n">set_context</span><span class="p">(</span><span class="s">'paper'</span><span class="p">,</span> <span class="n">font_scale</span> <span class="o">=</span> <span class="mf">2.0</span><span class="p">,</span> <span class="n">rc</span> <span class="o">=</span> <span class="p">{</span><span class="s">'lines.linewidth'</span><span class="p">:</span> <span class="mf">1.5</span><span class="p">,</span> <span class="s">'figure.figsize'</span> <span class="p">:</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="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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<p>Importing the mock dataset and checking if it imported correctly</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">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">'MOOC_Mock.csv'</span><span class="p">)</span>
<span class="n">data</span><span class="o">.</span><span class="n">head</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="c">#Looking at the first 3 rows</span>
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<th></th>
<th>File</th>
<th>Age</th>
<th>Gender</th>
<th>Delay</th>
<th>Stay</th>
<th>ICU</th>
<th>RVD</th>
<th>CD4</th>
<th>HR</th>
<th>Temp</th>
<th>CRP</th>
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<th>HB</th>
<th>Rupture</th>
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<th>Comp</th>
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<th>0</th>
<td>1</td>
<td>38</td>
<td>Female</td>
<td>3</td>
<td>6</td>
<td>No</td>
<td>No</td>
<td>NaN</td>
<td>97</td>
<td>35.2</td>
<td>NaN</td>
<td>10.49</td>
<td>10.4</td>
<td>No</td>
<td>Yes</td>
<td>Yes</td>
<td>5</td>
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<th>1</th>
<td>2</td>
<td>32</td>
<td>Male</td>
<td>6</td>
<td>10</td>
<td>No</td>
<td>Yes</td>
<td>57</td>
<td>109</td>
<td>38.8</td>
<td>45.3</td>
<td>7.08</td>
<td>19.8</td>
<td>No</td>
<td>No</td>
<td>Yes</td>
<td>8</td>
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<th>2</th>
<td>3</td>
<td>19</td>
<td>Female</td>
<td>1</td>
<td>16</td>
<td>No</td>
<td>No</td>
<td>NaN</td>
<td>120</td>
<td>36.3</td>
<td>10.7</td>
<td>13.00</td>
<td>8.7</td>
<td>No</td>
<td>No</td>
<td>No</td>
<td>3</td>
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<h1 id="Confidence-intervals">Confidence intervals<a class="anchor-link" href="#Confidence-intervals">¶</a></h1>
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<h2 id="Introduction">Introduction<a class="anchor-link" href="#Introduction">¶</a></h2>
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<p>When conducting research by scrutinizing a sample set of individuals taken from a larger population, we create statistics. We then use these statistics to infer parameter values for the greater population. It can be used to guide patient treatment and management, by inferring that our results are applicable to that larger (patient) population.</p>
<p>However, there is no way that we can say with absolute confidence that our statistic is exactly what would to be found in the general population. Instead we can place an arbitrary confidence around our test statistics, and infer that we are sure that the real (larger population) parameters are within that arbitrary confidence that we placed in our statistic(s).</p>
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<h2 id="Confidence-interval-around-mean-of-a-sample-variable-set">Confidence interval around mean of a sample variable set<a class="anchor-link" href="#Confidence-interval-around-mean-of-a-sample-variable-set">¶</a></h2>
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<p><p>Most textbooks begin with the chapter on confidence intervals (CI's) by using an example where the population mean and standard deviation is known. This is almost never the case in healthcare research, so let's have a look at what is commonly observed.</p></p>
<p><p>Most of the time we are stuck with a set of variable values from our sample set and we can calculate the mean, standard deviation and standard error of that set.</p></p>
<p><p>In our example dataset we might look at the white cell count (WCC) on admission. We could divide our sample patients into two groups, say those with and without retroviral disease.</p></p>
<p><p>In the code below we make two new DataFrames, with each DataFrame object attached to a new computer variable.<p></p>
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<div class=" highlight hl-ipython3"><pre><span class="n">RVD_neg</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="n">data</span><span class="p">[</span><span class="s">'RVD'</span><span class="p">]</span> <span class="o">==</span> <span class="s">'No'</span><span class="p">]</span> <span class="c"># Only including the rows in the original DataFrame that have a Yes in the RVD column</span>
<span class="n">RVD_neg_appx</span> <span class="o">=</span> <span class="n">RVD_neg</span><span class="p">[</span><span class="n">RVD_neg</span><span class="p">[</span><span class="s">'Histo'</span><span class="p">]</span> <span class="o">==</span> <span class="s">'Yes'</span><span class="p">]</span> <span class="c"># Extracting only those who actually had appendicitis</span>
<span class="n">RVD_pos</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="n">data</span><span class="p">[</span><span class="s">'RVD'</span><span class="p">]</span> <span class="o">==</span> <span class="s">'Yes'</span><span class="p">]</span> <span class="c"># Doing the same but with No in the RVD column</span>
<span class="n">RVD_pos_appx</span> <span class="o">=</span> <span class="n">RVD_pos</span><span class="p">[</span><span class="n">RVD_pos</span><span class="p">[</span><span class="s">'Histo'</span><span class="p">]</span> <span class="o">==</span> <span class="s">'Yes'</span><span class="p">]</span> <span class="c"># Extracting only those who actually had appendicitis</span>
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<p>Let's look at the first rows in each new DataFrame using the *.head()* function. Leaving the argument blank will result in a default of *5* rows.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">RVD_neg_appx</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
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<th>Gender</th>
<th>Delay</th>
<th>Stay</th>
<th>ICU</th>
<th>RVD</th>
<th>CD4</th>
<th>HR</th>
<th>Temp</th>
<th>CRP</th>
<th>WCC</th>
<th>HB</th>
<th>Rupture</th>
<th>Histo</th>
<th>Comp</th>
<th>MASS</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td>1</td>
<td>38</td>
<td>Female</td>
<td>3</td>
<td>6</td>
<td>No</td>
<td>No</td>
<td>NaN</td>
<td>97</td>
<td>35.2</td>
<td>NaN</td>
<td>10.49</td>
<td>10.4</td>
<td>No</td>
<td>Yes</td>
<td>Yes</td>
<td>5</td>
</tr>
<tr>
<th>7</th>
<td>8</td>
<td>19</td>
<td>Male</td>
<td>2</td>
<td>13</td>
<td>No</td>
<td>No</td>
<td>NaN</td>
<td>102</td>
<td>38.5</td>
<td>224.6</td>
<td>8.99</td>
<td>15.5</td>
<td>No</td>
<td>Yes</td>
<td>Yes</td>
<td>5</td>
</tr>
<tr>
<th>8</th>
<td>9</td>
<td>46</td>
<td>Male</td>
<td>2</td>
<td>9</td>
<td>No</td>
<td>No</td>
<td>NaN</td>
<td>107</td>
<td>36.2</td>
<td>385.1</td>
<td>10.26</td>
<td>15.5</td>
<td>No</td>
<td>Yes</td>
<td>No</td>
<td>1</td>
</tr>
<tr>
<th>9</th>
<td>10</td>
<td>19</td>
<td>Male</td>
<td>7</td>
<td>12</td>
<td>Yes</td>
<td>No</td>
<td>NaN</td>
<td>105</td>
<td>36.6</td>
<td>123</td>
<td>22.19</td>
<td>13.7</td>
<td>No</td>
<td>Yes</td>
<td>Yes</td>
<td>5</td>
</tr>
<tr>
<th>10</th>
<td>11</td>
<td>33</td>
<td>Male</td>
<td>2</td>
<td>14</td>
<td>No</td>
<td>No</td>
<td>NaN</td>
<td>104</td>
<td>36.8</td>
<td>NaN</td>
<td>16.20</td>
<td>12.7</td>
<td>No</td>
<td>Yes</td>
<td>No</td>
<td>9</td>
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<div class=" highlight hl-ipython3"><pre><span class="n">RVD_pos_appx</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
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<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>File</th>
<th>Age</th>
<th>Gender</th>
<th>Delay</th>
<th>Stay</th>
<th>ICU</th>
<th>RVD</th>
<th>CD4</th>
<th>HR</th>
<th>Temp</th>
<th>CRP</th>
<th>WCC</th>
<th>HB</th>
<th>Rupture</th>
<th>Histo</th>
<th>Comp</th>
<th>MASS</th>
</tr>
</thead>
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<th>4</th>
<td>5</td>
<td>28</td>
<td>Female</td>
<td>3</td>
<td>3</td>
<td>No</td>
<td>Yes</td>
<td>491</td>
<td>115</td>
<td>37.1</td>
<td>51.6</td>
<td>21.98</td>
<td>13.4</td>
<td>No</td>
<td>Yes</td>
<td>No</td>
<td>7</td>
</tr>
<tr>
<th>6</th>
<td>7</td>
<td>36</td>
<td>Male</td>
<td>6</td>
<td>14</td>
<td>No</td>
<td>Yes</td>
<td>203</td>
<td>95</td>
<td>37.5</td>
<td>NaN</td>
<td>16.21</td>
<td>17.7</td>
<td>Yes</td>
<td>Yes</td>
<td>Yes</td>
<td>7</td>
</tr>
<tr>
<th>19</th>
<td>20</td>
<td>33</td>
<td>Female</td>
<td>1</td>
<td>19</td>
<td>No</td>
<td>Yes</td>
<td>203</td>
<td>123</td>
<td>37.5</td>
<td>46.5</td>
<td>8.00</td>
<td>10.4</td>
<td>No</td>
<td>Yes</td>
<td>Yes</td>
<td>8</td>
</tr>
<tr>
<th>31</th>
<td>32</td>
<td>39</td>
<td>Male</td>
<td>2</td>
<td>10</td>
<td>No</td>
<td>Yes</td>
<td>102</td>
<td>100</td>
<td>36.8</td>
<td>208</td>
<td>20.43</td>
<td>12.9</td>
<td>Yes</td>
<td>Yes</td>
<td>No</td>
<td>1</td>
</tr>
<tr>
<th>32</th>
<td>33</td>
<td>33</td>
<td>Male</td>
<td>0</td>
<td>21</td>
<td>No</td>
<td>Yes</td>
<td>266</td>
<td>100</td>
<td>39.7</td>
<td>194.1</td>
<td>7.31</td>
<td>10.7</td>
<td>Yes</td>
<td>Yes</td>
<td>Yes</td>
<td>3</td>
</tr>
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<p>Note how the two <strong>RVD</strong> columns contain only <em>No</em> and <em>Yes</em> respectively.</p>
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<p>The easiest way to calculate the confidence intervals is using the *scipy.stats* function *bayes_mvs*. It actually gives nine results, three rows of three values each. The first row gives the mean and then the lower as well as the upper bound of the stated confidence interval.</p>
<p>The second row does the same but for the variance. The last row calculates the values for the standard deviation.</p>
<p>Let's first describe our two DataFrames then calculate the CI's.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">RVD_neg_appx</span><span class="p">[</span><span class="s">'WCC'</span><span class="p">]</span><span class="o">.</span><span class="n">describe</span><span class="p">()</span> <span class="c"># Describing the WCC column in the RVD_neg DataFrame</span>
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<pre>count 78.000000
mean 14.044103
std 4.347939
min 6.610000
25% 11.270000
50% 13.655000
75% 16.305000
max 26.400000
Name: WCC, dtype: float64</pre>
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<div class=" highlight hl-ipython3"><pre><span class="n">RVD_pos_appx</span><span class="p">[</span><span class="s">'WCC'</span><span class="p">]</span><span class="o">.</span><span class="n">describe</span><span class="p">()</span> <span class="c"># Describing the WCC column in the RVD_pos DataFrame</span>
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<pre>count 40.000000
mean 15.773250
std 4.934148
min 2.950000
25% 12.475000
50% 15.575000
75% 19.920000
max 24.890000
Name: WCC, dtype: float64</pre>
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<p>Now for the CI's. The *bayes_mvs()* function takes two arguments, the first is the values themselves and the second is the required confidence interval. In this example I want the 95% CI's. I also make sure to discard all non-numerical values.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">bayes_mvs</span><span class="p">(</span><span class="n">RVD_neg_appx</span><span class="p">[</span><span class="s">'WCC'</span><span class="p">]</span><span class="o">.</span><span class="n">dropna</span><span class="p">(),</span> <span class="mf">0.95</span><span class="p">)</span>
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<pre>(Mean(statistic=14.044102564102564, minmax=(13.063793818374444, 15.024411309830684)),
Variance(statistic=19.408694495726504, minmax=(14.110883384251707, 26.648893927036038)),
Std_dev(statistic=4.3908697719337093, minmax=(3.7564455785025959, 5.1622566700074142)))</pre>
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<div class=" highlight hl-ipython3"><pre><span class="n">bayes_mvs</span><span class="p">(</span><span class="n">RVD_pos_appx</span><span class="p">[</span><span class="s">'WCC'</span><span class="p">]</span><span class="o">.</span><span class="n">dropna</span><span class="p">(),</span> <span class="mf">0.95</span><span class="p">)</span>
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