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Chapter21_Sensitivity_specificity_risks_rates_odds.html
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<!DOCTYPE html>
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<title>Chapter21_Sensitivity_specificity_risks_rates_odds</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>
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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="c">#from scipy.stats import bayes_mvs</span>
<span class="c">#from math import factorial</span>
<span class="c">#import scikits.bootstrap as bs</span>
<span class="c">#import matplotlib.pyplot as plt</span>
<span class="c">#import seaborn as sns</span>
<span class="kn">from</span> <span class="nn">warnings</span> <span class="k">import</span> <span class="n">filterwarnings</span>
<span class="c">#%matplotlib inline</span>
<span class="n">filterwarnings</span><span class="p">(</span><span class="s">'ignore'</span><span class="p">)</span>
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<p><p>When considering the use of special investigations or even clinical findings, it is important to understand how accurate these are and how this impacts on their use and the clinical decisions we make that depend on them.<p></p>
<p><p>Whereas the sensitivity and specificity are independent of the prevalence of a condition, predictive values are and incidence profoundly affects the use of their reported values. Fortunately, there are equations that can translate reported predictive values to a relevant prevalence.<p></p>
<p><p>Predictive values make use of a form of Bayesian statistics. Bayesian inference makes use of past knowledge (as gained from research) to predict, in essence, the future.<p>
In this chapter I will look into the calculations and use of sensitivity, specificity, predictive values, incidence, cumulative risk, prevalence and some ratios.</p>
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<h1 id="Sensitivity-and-specificity">Sensitivity and specificity<a class="anchor-link" href="#Sensitivity-and-specificity">¶</a></h1>
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<h2 id="Introduction">Introduction<a class="anchor-link" href="#Introduction">¶</a></h2>
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<p><p>Let's start by short explanations for sensitivity and specificity in the context of healthcare. <em>Sensitivity</em> expresses the probability that a test will return a positive result given the existence of a (disease) condition. In contrast to this, <em>specificity</em> expresses the probability that the same test will return a negative result given the absence of that disease condition.<p></p>
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<h2 id="Calculating-sensitivity-and-specifcity">Calculating sensitivity and specifcity<a class="anchor-link" href="#Calculating-sensitivity-and-specifcity">¶</a></h2>
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<h3 id="A-simple-example">A simple example<a class="anchor-link" href="#A-simple-example">¶</a></h3>
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<p>Consider a research study in which, for instance, 1000 volunteers are divided into two groups. In group one, a specific disease is absolutely shown to be present. A good example of this would be tissue taken for histological examination and proved to be diseased. This histological examination will be taken as the gold standard. In group two, this disease is absolutely known to be absent.</p>
<p>Study participants are then investigated with a new test. Results for this test are either *positive* or *negative*. Four results follow from this.
* Group with disease
* A positive result
* A negative result
* Group B
* A positive result
* A negative result
<p>We use specific words to desrcibe these results. A positive result in a participant with the disease is termed a *true positive*. A negative result in a participant with the disease is a *false negative*.</p>
<p>A negative result in a participant without the disese is a *true negative* and a positive in this group would be a *false positive*.</p>
<p>*Sensitivity* is then a simple ratio between the true positive incidence and the total number of participants with the disease. *Specificity* is the ratio between the true negative rate and the number of participants without the disease.</p>
<p>Sensitivity and specificity should be used and understood from the point of view of the decision to use a test in the work-up of a condition. If a test has a high sensitivity it will diagnose a high percentage of patient who have the disease. Likewise, if a test has a high specificity it will indicate a large percentage of patients who do not have that disease.</p>
<p>Note also, how crucial it is to have a gold standard to divide the groups with.</p>
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<p>In this simple example of 1000 participants let's suppose that 10 individuals have the disease and an overwhelming 990 do not. Of the 10 with the disease, nine tests positive (true positive) and tests negative (false negative). Of the 990 without the disease, 90 test positive (false positive) and 900 test negative (true negative).</p>
<p>Sensitivity (*S<sub>n</sub>*) is a simple fraction with true positive (*P<sub>T</sub>*) in the numerator and the sum of the true positives (*P<sub>T</sub>*) and the false negatives (*N<sub>F</sub>*) (in other words, all (10) with the disease) in the denominator.</p>
<p>Specificity (*S<sub>p</sub>*) has true negative (*N<sub>T</sub>*) in the numerator and the sum of the true negative (*N<sub>T</sub>*) and false positive (*P<sub>F</sub>*) (sum of all without the disease) in the denominator.</p>
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$$ { S }_{ N }=\frac { { P }_{ T } }{ { P }_{ T }+{ F }_{ N } } $$
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$$ {{S}_{P} = \frac {{N}_{T}}{{{N}_{T}} + {{P}_{F}}}} $$
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<p>Our code could look something like this:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">true_positive</span> <span class="o">=</span> <span class="mi">9</span> <span class="c"># Create a computer variable called true_positive and give it the value 9</span>
<span class="n">false_negative</span> <span class="o">=</span> <span class="mi">1</span> <span class="c"># True and False are python code words, so don't use these as computer</span>
<span class="c"># variable names</span>
<span class="n">false_positive</span> <span class="o">=</span> <span class="mi">90</span>
<span class="n">true_negative</span> <span class="o">=</span> <span class="mi">900</span>
<span class="n">sensitivity</span> <span class="o">=</span> <span class="n">true_positive</span> <span class="o">/</span> <span class="p">(</span><span class="n">true_positive</span> <span class="o">+</span> <span class="n">false_negative</span><span class="p">)</span> <span class="c"># A simple equation</span>
<span class="n">specificity</span> <span class="o">=</span> <span class="n">true_negative</span> <span class="o">/</span> <span class="p">(</span><span class="n">true_negative</span> <span class="o">+</span> <span class="n">false_positive</span><span class="p">)</span> <span class="c"># Note to add denominator values</span>
<span class="c"># in brackets to make the algebra (order of arithmetic execution) work out</span>
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<div class=" highlight hl-ipython3"><pre><span class="nb">print</span><span class="p">(</span><span class="s">'The sensitivity is: '</span><span class="p">,</span> <span class="p">(</span><span class="n">sensitivity</span> <span class="o">*</span> <span class="mi">100</span><span class="p">),</span> <span class="s">'%'</span><span class="p">)</span>
<span class="c"># Using the python 3.x print statement</span>
<span class="c"># Note the use of quotation marks for literal text</span>
<span class="c"># Multiplying the values with 100 to get percentage</span>
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<pre>The sensitivity is: 90.0 %
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<div class=" highlight hl-ipython3"><pre><span class="nb">print</span><span class="p">(</span><span class="s">'The specificity is: '</span><span class="p">,</span> <span class="p">(</span><span class="n">specificity</span> <span class="o">*</span> <span class="mi">100</span><span class="p">),</span> <span class="s">'%'</span><span class="p">)</span>
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<pre>The specificity is: 90.9090909090909 %
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<p><p>We could no say that this new test will pick up 90% of those with the disease with a positive result and 90% of those without, with a negative result.<p></p>
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<h3 id="Example-using-our-mock-data">Example using our mock data<a class="anchor-link" href="#Example-using-our-mock-data">¶</a></h3>
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<p>We'll do another example importing our mock data.</p>
<p>I won't use the powerful * pandas .groupby()* function, instead doing a laborious construction of new *DataFrame* objects, to make it easy to follow along.</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="c"># Imporitng the csv file and attaching it to the computer variable (object) arbitrarily called data</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"># Inspecting the first 3 rows</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>
<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>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>
</tr>
<tr>
<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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<p><p>We will divide our data set into those with and without histologically proven appendicitis. The histological evaluation is our gold standard. We will now look at using a white cell count (WCC) of 12 or more as a positive result.<p></p>
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<div class=" highlight hl-ipython3"><pre><span class="n">appen_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">'Histo'</span><span class="p">]</span> <span class="o">==</span> <span class="s">'Yes'</span><span class="p">]</span> <span class="c"># A new DataFrame including only those with</span>
<span class="c"># histologically proven appendicitis</span>
<span class="n">appen_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">'Histo'</span><span class="p">]</span> <span class="o">==</span> <span class="s">'No'</span><span class="p">]</span>
<span class="n">tp_all</span> <span class="o">=</span> <span class="n">appen_pos</span><span class="p">[</span><span class="n">appen_pos</span><span class="p">[</span><span class="s">'WCC'</span><span class="p">]</span> <span class="o">>=</span> <span class="mi">12</span><span class="p">]</span> <span class="c"># A new DataFrame object from the appen_pos</span>
<span class="c"># DataFrame with a WCC of 12 or more as a positive result, hence true positive</span>
<span class="n">fn_all</span> <span class="o">=</span> <span class="n">appen_pos</span><span class="p">[</span><span class="n">appen_pos</span><span class="p">[</span><span class="s">'WCC'</span><span class="p">]</span> <span class="o"><</span> <span class="mi">12</span><span class="p">]</span>
<span class="n">tn_all</span> <span class="o">=</span> <span class="n">appen_neg</span><span class="p">[</span><span class="n">appen_neg</span><span class="p">[</span><span class="s">'WCC'</span><span class="p">]</span> <span class="o"><</span> <span class="mi">12</span><span class="p">]</span>
<span class="n">fp_all</span> <span class="o">=</span> <span class="n">appen_neg</span><span class="p">[</span><span class="n">appen_neg</span><span class="p">[</span><span class="s">'WCC'</span><span class="p">]</span> <span class="o">>=</span> <span class="mi">12</span><span class="p">]</span>
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<div class=" highlight hl-ipython3"><pre><span class="n">tp_all</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="o">.</span><span class="n">describe</span><span class="p">()</span> <span class="c"># Just interested in the counts here</span>
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<pre>count 87.000000
mean 16.507471
std 3.729035
min 12.000000
25% 13.605000
50% 15.650000
75% 17.720000
max 26.400000
Name: WCC, dtype: float64</pre>
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<div class=" highlight hl-ipython3"><pre><span class="n">fn_all</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="o">.</span><span class="n">describe</span><span class="p">()</span>
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<pre>count 31.000000
mean 9.361935
std 2.010896
min 2.950000
25% 8.195000
50% 9.390000
75% 10.650000
max 11.990000
Name: WCC, dtype: float64</pre>
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<div class=" highlight hl-ipython3"><pre><span class="n">tn_all</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="o">.</span><span class="n">describe</span><span class="p">()</span>
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<pre>count 18.000000
mean 8.375000
std 2.481184
min 4.220000
25% 6.795000
50% 8.155000
75% 10.542500
max 11.700000
Name: WCC, dtype: float64</pre>
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<div class=" highlight hl-ipython3"><pre><span class="n">fp_all</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="o">.</span><span class="n">describe</span><span class="p">()</span>
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<pre>count 11.000000
mean 16.730909
std 3.835021
min 13.000000
25% 14.155000
50% 15.970000
75% 17.780000
max 26.190000
Name: WCC, dtype: float64</pre>
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<div class=" highlight hl-ipython3"><pre><span class="n">tp</span> <span class="o">=</span> <span class="mi">87</span>
<span class="n">fn</span> <span class="o">=</span> <span class="mi">31</span>
<span class="n">tn</span> <span class="o">=</span> <span class="mi">18</span>
<span class="n">fp</span> <span class="o">=</span> <span class="mi">11</span>
<span class="n">sens</span> <span class="o">=</span> <span class="n">tp</span> <span class="o">/</span> <span class="p">(</span><span class="n">tp</span> <span class="o">+</span> <span class="n">fn</span><span class="p">)</span>
<span class="n">spec</span> <span class="o">=</span> <span class="n">tn</span> <span class="o">/</span> <span class="p">(</span><span class="n">tn</span> <span class="o">+</span> <span class="n">fp</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span class="nb">print</span><span class="p">(</span><span class="s">'The sensitivity of a raised WCC in appendicitis is: '</span><span class="p">,</span> <span class="p">(</span><span class="n">sens</span> <span class="o">*</span> <span class="mi">100</span><span class="p">),</span> <span class="s">'%'</span><span class="p">)</span>
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<pre>The sensitivity of a raised WCC in appendicitis is: 73.72881355932203 %
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<div class=" highlight hl-ipython3"><pre><span class="nb">print</span><span class="p">(</span><span class="s">'The specificity of a raised WCC in appendicitis is: '</span><span class="p">,</span> <span class="p">(</span><span class="n">spec</span> <span class="o">*</span> <span class="mi">100</span><span class="p">),</span> <span class="s">'%'</span><span class="p">)</span>
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<pre>The specificity of a raised WCC in appendicitis is: 62.06896551724138 %
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<p>There we have it. A sensitivity and specificity value for a raised WCC in our data set.</p>
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<h1 id="Predicitive-values">Predicitive values<a class="anchor-link" href="#Predicitive-values">¶</a></h1>
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<h2 id="Introduction">Introduction<a class="anchor-link" href="#Introduction">¶</a></h2>
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<p>Predictive values make use of a form of Bayesian statistics. Bayesian inference makes use of past knowledge (as gained from research) to predict, in essence, the future.</p>
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<p>In contrast to sensitivity and specificity, predictive values should be viewed from the point of view that the test was already performed. The values can either be expressed as a *positive* predictive value, which expresses the probability that a patient has a disease once the test result was positive, or as a *negative* predictive value, which expresses the probability that the patient does not have the disease, once the test result is negative.</p>
<p>Predictive values should be viewed only once a clear indication of the prevalence of the disease in the particular study is given, as this greatly influences the result. In order for predictive values to be used properly, they have to be converted to reflect the prevalence of a disease in the patients seen by a practicing clinician, in his or her setting.</p>
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<h2 id="Calculating-some-values">Calculating some values<a class="anchor-link" href="#Calculating-some-values">¶</a></h2>
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<p>Considering the first research study example above, in which (very importantly and for our purposes here), participants were recruited randomly and thus accurately reflect the true prevalence of a disease (in which case the predictive values will be accurate), calculating the predictive values is quite simple.</p>
<p>The positive predictive (*P<sub>P</sub>*) value is the ratio between the true positive results and the sum of the true positive and false negative rates. It expresses the probability that a patient will have the disease in the case of a a positive test result.</p>
<p>The negative predictive (*P<sub>N</sub>*) value is the ratio between the true negative rate and the sum of the true negative and false negative rates. It expresses the probability that a patient does not have the disease, in the case of a negative test result.</p>
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