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Chapter14_Z_and_t_distributions.html
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<!DOCTYPE html>
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<title>Chapter14_Z_and_t_distributions</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="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="c">#from math import 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">filterwarnings</span><span class="p">(</span><span class="s">'ignore'</span><span class="p">)</span>
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<h1 id="The-Z-and-t-distributions">The Z and t distributions<a class="anchor-link" href="#The-Z-and-t-distributions">¶</a></h1>
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<h2 id="Introduction">Introduction<a class="anchor-link" href="#Introduction">¶</a></h2>
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<p>It is of utmost importance to understand the difference between the distribution of your dataset (per variable) as opposed to the two most common statistical distributions, the Z and t distributions. Let's take white cell count (WCC) as an example. Imagine for a moment that you could measure the WCC of 1 000 000 healthy individuals. What would the ditribution of values look like?</p>
<p>Fist of all, let's be clear that by distribution I refer to the fact that we start counting how many times a specific WCC value comes up. Certainly some values will come up more commonly than others. Something deeper is going on here. You might have guessed it. Yes, WCC is a continuous variable. For all practical purposes, the WCC is infinitely divisible. If we had a very powerful WCC counting apparatus, it could measure the WCC of an individual as 5.5 or 5.56 or 5.564, or 5.5648. Would we count the occurence of all of these as seperate values? Not at all. They form part of a seemless mass. We can only make sense of ranges of values, say lump (bin) all values between 5.50 and 5.55 in a group (called a bin). To be sure, there are many, many possible individual values between these two bounds.</p>
<p>Again, we would have to ask: Are all of these arbitrary range bins equally likely to occur?</p>
<p>Random variables come in patterns, which we call distributions. Most variables pertaining to healthcare come in the normal form. This means if we were to get a histogram of the variables, it would be bell-shaped, or have a normal curve. There are many other continous distributions and also discrete distributions. The geometric distribution is and examples of a discrete random variable distribution.</p>
<p>In the first section below I'll show you how python™ (using the *scipy.stats* library can generate random variables for you to play with. We'll use the plotting libraries of *pandas*, *matplotlib*, and *seaborn*, to image these variables.</p>
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<h2 id="Generating-random-normal-variables">Generating random normal variables<a class="anchor-link" href="#Generating-random-normal-variables">¶</a></h2>
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<p><p>In this first section I will create a computer variable called <em>rv1</em> and attach to it the normal distribution. In actual fact, it will be the standard normal distribution. The standard normal distribution has a mean of zero and a standard deviation of one. It doesn't do much, but just to show you I'll call the <em>.mean()</em>, <em>.median()</em>, and <em>.std()</em> functions.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">rv1</span> <span class="o">=</span> <span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span>
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<div class=" highlight hl-ipython3"><pre><span class="n">rv1</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
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<pre>0.0</pre>
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<div class=" highlight hl-ipython3"><pre><span class="n">rv1</span><span class="o">.</span><span class="n">median</span><span class="p">()</span>
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<pre>0.0</pre>
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<div class=" highlight hl-ipython3"><pre><span class="n">rv1</span><span class="o">.</span><span class="n">std</span><span class="p">()</span>
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<pre>1.0</pre>
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<p>Now that wasn't too exiting. I can ask the *stats* library to generate a whole lot of random variables from the normal distribution. Here I attach it to a *pandas Series* object and attach it to the computer variable name *rv1_set*. I then use the *.describe()* function to find out a little more about my new dataset.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">rv1_values</span> <span class="o">=</span> <span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">size</span> <span class="o">=</span> <span class="mi">2000</span><span class="p">)</span> <span class="c"># I create 2000 values, all taken from a normal distribution</span>
<span class="n">rv1_set</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">(</span><span class="n">rv1_values</span><span class="p">)</span> <span class="c"># Making a pandas Series object</span>
<span class="n">rv1_set</span><span class="o">.</span><span class="n">describe</span><span class="p">()</span> <span class="c"># Every time you run this cell, a new set of variables will be created</span>
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<pre>count 2000.000000
mean 0.005339
std 1.001829
min -3.632323
25% -0.645921
50% -0.032196
75% 0.658415
max 4.241615
dtype: float64</pre>
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<p>Now for the really interesting bit. Graphing (plotting) the dataset. I'll use *seaborn*'s *distplot* plot. It will generate a histogram, which will autogenerate (unless I give it a specific value in the arguments) a bunch of bins (each describes a range of values) and then gives the number of times that a value in my dataset occurs in that range, but expressing it as a probability. You'll clearly see that we have a bell-shaped curve.</p>
<p>The *displot* goes further and constructs what is called a kernel denisty estimate. A mathematical equation is created to derive a smooth curve of the dataset.</p>
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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">distplot</span><span class="p">(</span><span class="n">rv1_set</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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<h3 id="The-probability-density-function">The probability density function<a class="anchor-link" href="#The-probability-density-function">¶</a></h3>
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<p>The kernel density function above is a type of probability density function (see below for Z and t distributions). A probability density function stretches from negative infinity to infinity on the x-axis and includes all of the possible values in a set, with the probability of any value occuring on the y-axis. The python™ function that I use is *.cdf()*. It calculates the area under the curve from the left up until a point on the x-axis that I want, which I add as an argument. Say I want to ask the question: Hwo likely was it to get a value of *-2* or less from a standard normal distribution? That would be the area under the curve from negative infinity (from the left) up until the *-2* mark. Here we go!</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">rv1</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
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<pre>0.022750131948179195</pre>
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<p>The value given is the probability of getting a value in your random dataset of less than *-2*. Let's ask how likely it is to get a value of more than *0.7*. Not so easy. Remember that we calculate the area from the left, so if we plug in *0.7* as an argument, we will get the probability of getting a value of less than *0.7*. To solve this little conundrum, remember that the total area under the curve is *1.0*. So we can subtract the *0.7* area value from *1.0* and *voila!* we have the answer.</p>
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<div class=" highlight hl-ipython3"><pre><span class="mi">1</span> <span class="o">-</span> <span class="n">rv1</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="mf">0.7</span><span class="p">)</span>
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<pre>0.24196365222307303</pre>
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<h3 id="Other-normal-distributions">Other normal distributions<a class="anchor-link" href="#Other-normal-distributions">¶</a></h3>
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<p>I needn't be stuck with the standard normal distribution. I can generate and play with a set of variables from a normal distribution, with any given mean and standard deviation.</p>
<p>Here I take 200 random sample from a normal distribution with a mean of *10* and a standard deviation of *4* and construct the sample values as a *pandas Series* object that I attach to the computer variable *rv2_set*. I'll then desribe and plot the dataset.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">rv2_set</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">(</span><span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">loc</span> <span class="o">=</span> <span class="mi">10</span><span class="p">,</span> <span class="n">scale</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span> <span class="n">size</span> <span class="o">=</span> <span class="mi">200</span><span class="p">))</span> <span class="c"># The argument loc is the eman and scale is the standard deviation</span>
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