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Chapter18_Comparing_means.html
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<!DOCTYPE html>
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<head><meta charset="utf-8" />
<title>Chapter18_Comparing_means</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>
<span class="n">HTML</span><span class="p">(</span><span class="nb">open</span><span class="p">(</span><span class="n">css_file</span><span class="p">,</span> <span class="s">'r'</span><span class="p">)</span><span class="o">.</span><span class="n">read</span><span class="p">())</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</span> <span class="k">import</span> <span class="n">stats</span>
<span class="c">#from scipy.stats import bayes_mvs</span>
<span class="kn">from</span> <span class="nn">math</span> <span class="k">import</span> <span class="n">factorial</span>
<span class="c">#import scikits.bootstrap as 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">sns</span><span class="o">.</span><span class="n">set_style</span><span class="p">(</span><span class="s">'whitegrid'</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="n">filterwarnings</span><span class="p">(</span><span class="s">'ignore'</span><span class="p">)</span>
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<h1 id="Comparing-the-means-of-two-groups">Comparing the means of two groups<a class="anchor-link" href="#Comparing-the-means-of-two-groups">¶</a></h1>
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<h2 id="Introduction">Introduction<a class="anchor-link" href="#Introduction">¶</a></h2>
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<p>One of the most common statistical tasks is to compare the means of two groups and to decide if they differ from each other in a statistically significant way.</p>
<p>For now we are going to look at very specific groups. In essence the two groups are of nominal, categorical type. In the first example below I want to see if there is a statistically significant difference between the admission white cell count (WCC) between those patients with and without histologically proven appendicitis. Group A is then those without appendicitis and group B then include those with. Groups A and B are categorical, but the actual data values are of ratio numerical type. Later we will have a look at comparing two groups of numerical data. This is called regression analysis.</p>
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<h2 id="Importing-and-examining-the-dataset">Importing and examining the dataset<a class="anchor-link" href="#Importing-and-examining-the-dataset">¶</a></h2>
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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>
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<div class=" highlight hl-ipython3"><pre><span class="n">data</span><span class="o">.</span><span class="n">tail</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="c"># This time looking at the bottom three entries using the .tail() function</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>147</th>
<td>148</td>
<td>54</td>
<td>Male</td>
<td>5</td>
<td>11</td>
<td>Yes</td>
<td>Yes</td>
<td>NaN</td>
<td>115</td>
<td>38.2</td>
<td>NaN</td>
<td>17.64</td>
<td>15.7</td>
<td>Yes</td>
<td>Yes</td>
<td>No</td>
<td>1</td>
</tr>
<tr>
<th>148</th>
<td>149</td>
<td>29</td>
<td>Male</td>
<td>4</td>
<td>6</td>
<td>No</td>
<td>Yes</td>
<td>45</td>
<td>133</td>
<td>41.2</td>
<td>163.5</td>
<td>15.50</td>
<td>19.2</td>
<td>Yes</td>
<td>Yes</td>
<td>Yes</td>
<td>9</td>
</tr>
<tr>
<th>149</th>
<td>150</td>
<td>55</td>
<td>Female</td>
<td>5</td>
<td>2</td>
<td>Yes</td>
<td>Yes</td>
<td>78</td>
<td>95</td>
<td>40.6</td>
<td>NaN</td>
<td>24.22</td>
<td>16.5</td>
<td>Yes</td>
<td>Yes</td>
<td>No</td>
<td>4</td>
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<h2 id="Creating-the-two-groups">Creating the two groups<a class="anchor-link" href="#Creating-the-two-groups">¶</a></h2>
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<p>*Pandas* has a very powerful function called *.groupby()*. In stead, though, I will simply create two new DataFrames here.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">appx_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">appx_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>
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<p>Let's have a quick descriptive look at our data.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">appx_neg</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>
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<pre>count 29.000000
mean 11.544483
std 5.100563
min 4.220000
25% 7.700000
50% 10.910000
75% 14.530000
max 26.190000
Name: WCC, dtype: float64</pre>
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<div class=" highlight hl-ipython3"><pre><span class="n">appx_pos</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>
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<pre>count 118.000000
mean 14.630254
std 4.607867
min 2.950000
25% 11.975000
50% 14.275000
75% 16.760000
max 26.400000
Name: WCC, dtype: float64</pre>
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<p>Using the *.describe()* function for the **WCC** column we note that there are 29 values in the negative group and 118 in the positive group. We simply want an answer to the question: Is there a significant difference in admission WCC between the groups?</p>
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<h2 id="Step-1---Hypotheses">Step 1 - Hypotheses<a class="anchor-link" href="#Step-1---Hypotheses">¶</a></h2>
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<p><p>This (and step 2) must happen before any data collection occurs!!! Let's imagine for a moment that this is what happened!</p></p>
<p><p>Our null hypothesis is that there is no difference between the admission WCC in those with and without histologically proven acute appendicitis.</p></p>
<p><p>Our alternate or test hypothesis therefor will be that there is a difference.<p></p>
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<h2 id="Step-2---Set-alpha-value">Step 2 - Set alpha value<a class="anchor-link" href="#Step-2---Set-alpha-value">¶</a></h2>
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<p><p>As per usual, we'll go for 0.05.</p></p>
<p><p>This means we will construct a t distribution, convert the difference between the two means into a t-statistic and calculate the area under the curve from that point out (what was the probability of finding a value that extreme). The alpha value (area) of 0.05 is split in two, with 0.025 on either side (by our null hypotheses, a two-tail test).</p>
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<h2 id="Step-3---Collect-data">Step 3 - Collect data<a class="anchor-link" href="#Step-3---Collect-data">¶</a></h2>
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<p>OK, done!</p>
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<h2 id="Step-4---Choose-an-appropriate-statistical-test">Step 4 - Choose an appropriate statistical test<a class="anchor-link" href="#Step-4---Choose-an-appropriate-statistical-test">¶</a></h2>
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<p>Our sample values are of the ratio, numerical type. We should therefore choose a type of t-test, for the fact that we don't know the population standard deviation or mean and have to construct a t distribution.</p>
<p>From the lecture on parametric and non-parametric tests you will remember that there are two types of t-tests.</p>
<p>A parametric t-test is used when the sample data is taken from a normal distribution and a non-parametric test is used when our sample data is not taken from a normal distribution.</p>
<p>Let's use the QQ plot to see if our data values for both groups come from normally distributed data.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">stats</span><span class="o">.</span><span class="n">probplot</span><span class="p">(</span><span class="n">appx_neg</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="n">dist</span> <span class="o">=</span> <span class="s">'norm'</span><span class="p">,</span> <span class="n">plot</span> <span class="o">=</span> <span class="n">plt</span><span class="p">);</span>
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