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<h1 id="methods-for-survival-and-duration-analysis"><span id="duration"></span><span id="module-statsmodels.duration"></span>Methods for Survival and Duration Analysis<a class="headerlink" href="#methods-for-survival-and-duration-analysis" title="Link to this heading">¶</a></h1>
<p><a class="reference internal" href="#module-statsmodels.duration" title="statsmodels.duration: Models for durations"><code class="xref py py-mod docutils literal notranslate"><span class="pre">statsmodels.duration</span></code></a> implements several standard methods for
working with censored data. These methods are most commonly used when
the data consist of durations between an origin time point and the
time at which some event of interest occurred. A typical example is a
medical study in which the origin is the time at which a subject is
diagnosed with some condition, and the event of interest is death (or
disease progression, recovery, etc.).</p>
<p>Currently only right-censoring is handled. Right censoring occurs
when we know that an event occurred after a given time <cite>t</cite>, but we do
not know the exact event time.</p>
<h2 id="survival-function-estimation-and-inference">Survival function estimation and inference<a class="headerlink" href="#survival-function-estimation-and-inference" title="Link to this heading">¶</a></h2>
<p>The <code class="xref py py-class docutils literal notranslate"><span class="pre">statsmodels.api.SurvfuncRight</span></code> class can be used to
estimate a survival function using data that may be right censored.
<code class="docutils literal notranslate"><span class="pre">SurvfuncRight</span></code> implements several inference procedures including
confidence intervals for survival distribution quantiles, pointwise
and simultaneous confidence bands for the survival function, and
plotting procedures. The <code class="docutils literal notranslate"><span class="pre">duration.survdiff</span></code> function provides
testing procedures for comparing survival distributions.</p>
<p>Here we create a <code class="docutils literal notranslate"><span class="pre">SurvfuncRight</span></code> object using data from the
<cite>flchain</cite> study, which is available through the R datasets repository.
We fit the survival distribution only for the female subjects.</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [1]: </span><span class="kn">import</span><span class="w"> </span><span class="nn">statsmodels.api</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">sm</span>
<span class="gp">In [2]: </span><span class="n">data</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">datasets</span><span class="o">.</span><span class="n">get_rdataset</span><span class="p">(</span><span class="s2">"flchain"</span><span class="p">,</span> <span class="s2">"survival"</span><span class="p">)</span><span class="o">.</span><span class="n">data</span>
<span class="gp">In [3]: </span><span class="n">df</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">data</span><span class="o">.</span><span class="n">sex</span> <span class="o">==</span> <span class="s2">"F"</span><span class="p">,</span> <span class="p">:]</span>
<span class="gp">In [4]: </span><span class="n">sf</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">SurvfuncRight</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">"futime"</span><span class="p">],</span> <span class="n">df</span><span class="p">[</span><span class="s2">"death"</span><span class="p">])</span>
</code></pre></div>
</div>
<p>The main features of the fitted survival distribution can be seen by
calling the <code class="docutils literal notranslate"><span class="pre">summary</span></code> method:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [5]: </span><span class="n">sf</span><span class="o">.</span><span class="n">summary</span><span class="p">()</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
<span class="gh">Out[5]: </span>
<span class="go"> Surv prob Surv prob SE num at risk num events</span>
<span class="go">Time </span>
<span class="go">0 0.999310 0.000398 4350 3.0</span>
<span class="go">1 0.998851 0.000514 4347 2.0</span>
<span class="go">2 0.998621 0.000563 4343 1.0</span>
<span class="go">3 0.997931 0.000689 4342 3.0</span>
<span class="go">4 0.997471 0.000762 4338 2.0</span>
</code></pre></div>
</div>
<p>We can obtain point estimates and confidence intervals for quantiles
of the survival distribution. Since only around 30% of the subjects
died during this study, we can only estimate quantiles below the 0.3
probability point:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [6]: </span><span class="n">sf</span><span class="o">.</span><span class="n">quantile</span><span class="p">(</span><span class="mf">0.25</span><span class="p">)</span>
<span class="gh">Out[6]: </span><span class="go">np.int64(3995)</span>
<span class="gp">In [7]: </span><span class="n">sf</span><span class="o">.</span><span class="n">quantile_ci</span><span class="p">(</span><span class="mf">0.25</span><span class="p">)</span>
<span class="gh">Out[7]: </span><span class="go">(np.int64(3776), np.int64(4166))</span>
</code></pre></div>
</div>
<p>To plot a single survival function, call the <code class="docutils literal notranslate"><span class="pre">plot</span></code> method:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [8]: </span><span class="n">sf</span><span class="o">.</span><span class="n">plot</span><span class="p">()</span>
<span class="gh">Out[8]: </span><span class="go"><Figure size 640x480 with 1 Axes></span>
</code></pre></div>
</div>
<img alt="_images/duration_survival_plot1.png" src="_images/duration_survival_plot1.png" />
<p>Since this is a large dataset with a lot of censoring, we may wish
to not plot the censoring symbols:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [9]: </span><span class="n">fig</span> <span class="o">=</span> <span class="n">sf</span><span class="o">.</span><span class="n">plot</span><span class="p">()</span>
<span class="gp">In [10]: </span><span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">get_axes</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
<span class="gp">In [11]: </span><span class="n">pt</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">get_lines</span><span class="p">()[</span><span class="mi">1</span><span class="p">]</span>
<span class="gp">In [12]: </span><span class="n">pt</span><span class="o">.</span><span class="n">set_visible</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
</code></pre></div>
</div>
<img alt="_images/duration_survival_nocensor_plot.png" src="_images/duration_survival_nocensor_plot.png" />
<p>We can also add a 95% simultaneous confidence band to the plot.
Typically these bands only plotted for central part of the
distribution.</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [13]: </span><span class="n">fig</span> <span class="o">=</span> <span class="n">sf</span><span class="o">.</span><span class="n">plot</span><span class="p">()</span>
<span class="gp">In [14]: </span><span class="n">lcb</span><span class="p">,</span> <span class="n">ucb</span> <span class="o">=</span> <span class="n">sf</span><span class="o">.</span><span class="n">simultaneous_cb</span><span class="p">()</span>
<span class="gp">In [15]: </span><span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">get_axes</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
<span class="gp">In [16]: </span><span class="n">ax</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span><span class="n">sf</span><span class="o">.</span><span class="n">surv_times</span><span class="p">,</span> <span class="n">lcb</span><span class="p">,</span> <span class="n">ucb</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'lightgrey'</span><span class="p">)</span>
<span class="gh">Out[16]: </span><span class="go"><matplotlib.collections.FillBetweenPolyCollection at 0x7f2e16dd8730></span>
<span class="gp">In [17]: </span><span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="mi">365</span><span class="p">,</span> <span class="mi">365</span><span class="o">*</span><span class="mi">10</span><span class="p">)</span>
<span class="gh">Out[17]: </span><span class="go">(365.0, 3650.0)</span>
<span class="gp">In [18]: </span><span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="mf">0.7</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="gh">Out[18]: </span><span class="go">(0.7, 1.0)</span>
<span class="gp">In [19]: </span><span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"Proportion alive"</span><span class="p">)</span>
<span class="gh">Out[19]: </span><span class="go">Text(0, 0.5, 'Proportion alive')</span>
<span class="gp">In [20]: </span><span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"Days since enrollment"</span><span class="p">)</span>
<span class="gh">Out[20]: </span><span class="go">Text(0.5, 0, 'Days since enrollment')</span>
</code></pre></div>
</div>
<img alt="_images/duration_survival_95ci_plot.png" src="_images/duration_survival_95ci_plot.png" />
<p>Here we plot survival functions for two groups (females and males) on
the same axes:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [21]: </span><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
<span class="gp">In [22]: </span><span class="n">gb</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s2">"sex"</span><span class="p">)</span>
<span class="gp">In [23]: </span><span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">axes</span><span class="p">()</span>
<span class="gp">In [24]: </span><span class="n">sexes</span> <span class="o">=</span> <span class="p">[]</span>
<span class="gp">In [25]: </span><span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">gb</span><span class="p">:</span>
<span class="gp"> ....: </span> <span class="n">sexes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="gp"> ....: </span> <span class="n">sf</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">SurvfuncRight</span><span class="p">(</span><span class="n">g</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="s2">"futime"</span><span class="p">],</span> <span class="n">g</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="s2">"death"</span><span class="p">])</span>
<span class="gp"> ....: </span> <span class="n">sf</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="p">)</span>
<span class="gp"> ....: </span>
<span class="gp">In [26]: </span><span class="n">li</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">get_lines</span><span class="p">()</span>
<span class="gp">In [27]: </span><span class="n">li</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_visible</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="gp">In [28]: </span><span class="n">li</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span><span class="o">.</span><span class="n">set_visible</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="gp">In [29]: </span><span class="n">plt</span><span class="o">.</span><span class="n">figlegend</span><span class="p">((</span><span class="n">li</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">li</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span> <span class="n">sexes</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="s2">"center right"</span><span class="p">)</span>
<span class="gh">Out[29]: </span><span class="go"><matplotlib.legend.Legend at 0x7f2e16e4e620></span>
<span class="gp">In [30]: </span><span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="mf">0.6</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="gh">Out[30]: </span><span class="go">(0.6, 1.0)</span>
<span class="gp">In [31]: </span><span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"Proportion alive"</span><span class="p">)</span>
<span class="gh">Out[31]: </span><span class="go">Text(0, 0.5, 'Proportion alive')</span>