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<h1 id="contingency-tables"><span id="id1"></span>Contingency tables<a class="headerlink" href="#contingency-tables" title="Link to this heading">¶</a></h1>
<p>statsmodels supports a variety of approaches for analyzing contingency
tables, including methods for assessing independence, symmetry,
homogeneity, and methods for working with collections of tables from a
stratified population.</p>
<p>The methods described here are mainly for two-way tables.  Multi-way
tables can be analyzed using log-linear models.  statsmodels does not
currently have a dedicated API for loglinear modeling, but Poisson
regression in <code class="xref py py-class docutils literal notranslate"><span class="pre">statsmodels.genmod.GLM</span></code> can be used for this
purpose.</p>
<p>A contingency table is a multi-way table that describes a data set in
which each observation belongs to one category for each of several
variables.  For example, if there are two variables, one with
<span class="math notranslate nohighlight">\(r\)</span> levels and one with <span class="math notranslate nohighlight">\(c\)</span> levels, then we have a
<span class="math notranslate nohighlight">\(r \times c\)</span> contingency table.  The table can be described in
terms of the number of observations that fall into a given cell of the
table, e.g. <span class="math notranslate nohighlight">\(T_{ij}\)</span> is the number of observations that have
level <span class="math notranslate nohighlight">\(i\)</span> for the first variable and level <span class="math notranslate nohighlight">\(j\)</span> for the
second variable.  Note that each variable must have a finite number of
levels (or categories), which can be either ordered or unordered.  In
different contexts, the variables defining the axes of a contingency
table may be called <strong>categorical variables</strong> or <strong>factor variables</strong>.
They may be either <strong>nominal</strong> (if their levels are unordered) or
<strong>ordinal</strong> (if their levels are ordered).</p>
<p>The underlying population for a contingency table is described by a
<strong>distribution table</strong> <span class="math notranslate nohighlight">\(P_{i, j}\)</span>.  The elements of <span class="math notranslate nohighlight">\(P\)</span>
are probabilities, and the sum of all elements in <span class="math notranslate nohighlight">\(P\)</span> is 1.
Methods for analyzing contingency tables use the data in <span class="math notranslate nohighlight">\(T\)</span> to
learn about properties of <span class="math notranslate nohighlight">\(P\)</span>.</p>
<p>The <code class="xref py py-class docutils literal notranslate"><span class="pre">statsmodels.stats.Table</span></code> is the most basic class for
working with contingency tables.  We can create a <code class="docutils literal notranslate"><span class="pre">Table</span></code> object
directly from any rectangular array-like object containing the
contingency table cell counts:</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">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>

<span class="gp">In [2]: </span><span class="kn">import</span><span class="w"> </span><span class="nn">pandas</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">pd</span>

<span class="gp">In [3]: </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 [4]: </span><span class="n">df</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">&quot;Arthritis&quot;</span><span class="p">,</span> <span class="s2">&quot;vcd&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">data</span>

<span class="gp">In [5]: </span><span class="n">df</span><span class="o">.</span><span class="n">fillna</span><span class="p">({</span><span class="s2">&quot;Improved&quot;</span><span class="p">:</span><span class="s2">&quot;None&quot;</span><span class="p">},</span> <span class="n">inplace</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>

<span class="gp">In [6]: </span><span class="n">tab</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">crosstab</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;Treatment&#39;</span><span class="p">],</span> <span class="n">df</span><span class="p">[</span><span class="s1">&#39;Improved&#39;</span><span class="p">])</span>

<span class="gp">In [7]: </span><span class="n">tab</span> <span class="o">=</span> <span class="n">tab</span><span class="o">.</span><span class="n">loc</span><span class="p">[:,</span> <span class="p">[</span><span class="s2">&quot;None&quot;</span><span class="p">,</span> <span class="s2">&quot;Some&quot;</span><span class="p">,</span> <span class="s2">&quot;Marked&quot;</span><span class="p">]]</span>

<span class="gp">In [8]: </span><span class="n">table</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">Table</span><span class="p">(</span><span class="n">tab</span><span class="p">)</span>
</code></pre></div>
</div>
<p>Alternatively, we can pass the raw data and let the Table class
construct the array of cell counts for us:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [9]: </span><span class="n">data</span> <span class="o">=</span> <span class="n">df</span><span class="p">[[</span><span class="s2">&quot;Treatment&quot;</span><span class="p">,</span> <span class="s2">&quot;Improved&quot;</span><span class="p">]]</span>

<span class="gp">In [10]: </span><span class="n">table</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">Table</span><span class="o">.</span><span class="n">from_data</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
</code></pre></div>
</div>
<h2 id="independence">Independence<a class="headerlink" href="#independence" title="Link to this heading">¶</a></h2>
<p><strong>Independence</strong> is the property that the row and column factors occur
independently. <strong>Association</strong> is the lack of independence.  If the
joint distribution is independent, it can be written as the outer
product of the row and column marginal distributions:</p>
<div class="math notranslate nohighlight">
\[P_{ij} = \sum_k P_{ij} \cdot \sum_k P_{kj} \quad \text{for all} \quad  i, j\]</div>
<p>We can obtain the best-fitting independent distribution for our
observed data, and then view residuals which identify particular cells
that most strongly violate independence:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [11]: </span><span class="nb">print</span><span class="p">(</span><span class="n">table</span><span class="o">.</span><span class="n">table_orig</span><span class="p">)</span>
<span class="go">Improved   Marked  None  Some</span>
<span class="go">Treatment                    </span>
<span class="go">Placebo         7    29     7</span>
<span class="go">Treated        21    13     7</span>

<span class="gp">In [12]: </span><span class="nb">print</span><span class="p">(</span><span class="n">table</span><span class="o">.</span><span class="n">fittedvalues</span><span class="p">)</span>
<span class="go">Improved      Marked  None      Some</span>
<span class="go">Treatment                           </span>
<span class="go">Placebo    14.333333  21.5  7.166667</span>
<span class="go">Treated    13.666667  20.5  6.833333</span>

<span class="gp">In [13]: </span><span class="nb">print</span><span class="p">(</span><span class="n">table</span><span class="o">.</span><span class="n">resid_pearson</span><span class="p">)</span>
<span class="go">Improved     Marked      None      Some</span>
<span class="go">Treatment                              </span>
<span class="go">Placebo   -1.936992  1.617492 -0.062257</span>
<span class="go">Treated    1.983673 -1.656473  0.063758</span>
</code></pre></div>
</div>
<p>In this example, compared to a sample from a population in which the
rows and columns are independent, we have too many observations in the
placebo/no improvement and treatment/marked improvement cells, and too
few observations in the placebo/marked improvement and treated/no
improvement cells.  This reflects the apparent benefits of the
treatment.</p>
<p>If the rows and columns of a table are unordered (i.e. are nominal
factors), then the most common approach for formally assessing
independence is using Pearson’s <span class="math notranslate nohighlight">\(\chi^2\)</span> statistic.  It’s often
useful to look at the cell-wise contributions to the <span class="math notranslate nohighlight">\(\chi^2\)</span>
statistic to see where the evidence for dependence is coming from.</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [14]: </span><span class="n">rslt</span> <span class="o">=</span> <span class="n">table</span><span class="o">.</span><span class="n">test_nominal_association</span><span class="p">()</span>

<span class="gp">In [15]: </span><span class="nb">print</span><span class="p">(</span><span class="n">rslt</span><span class="o">.</span><span class="n">pvalue</span><span class="p">)</span>
<span class="go">0.0014626434089526352</span>

<span class="gp">In [16]: </span><span class="nb">print</span><span class="p">(</span><span class="n">table</span><span class="o">.</span><span class="n">chi2_contribs</span><span class="p">)</span>
<span class="go">Improved     Marked      None      Some</span>
<span class="go">Treatment                              </span>
<span class="go">Placebo    3.751938  2.616279  0.003876</span>
<span class="go">Treated    3.934959  2.743902  0.004065</span>
</code></pre></div>
</div>
<p>For tables with ordered row and column factors, we can us the <strong>linear
by linear</strong> association test to obtain more power against alternative
hypotheses that respect the ordering.  The test statistic for the
linear by linear association test is</p>
<div class="math notranslate nohighlight">
\[\sum_k r_i c_j T_{ij}\]</div>
<p>where <span class="math notranslate nohighlight">\(r_i\)</span> and <span class="math notranslate nohighlight">\(c_j\)</span> are row and column scores.  Often
these scores are set to the sequences 0, 1, ….  This gives the
‘Cochran-Armitage trend test’.</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [17]: </span><span class="n">rslt</span> <span class="o">=</span> <span class="n">table</span><span class="o">.</span><span class="n">test_ordinal_association</span><span class="p">()</span>

<span class="gp">In [18]: </span><span class="nb">print</span><span class="p">(</span><span class="n">rslt</span><span class="o">.</span><span class="n">pvalue</span><span class="p">)</span>
<span class="go">0.023644578093923983</span>
</code></pre></div>
</div>
<p>We can assess the association in a <span class="math notranslate nohighlight">\(r\times x\)</span> table by
constructing a series of <span class="math notranslate nohighlight">\(2\times 2\)</span> tables and calculating
their odds ratios.  There are two ways to do this.  The <strong>local odds
ratios</strong> construct <span class="math notranslate nohighlight">\(2\times 2\)</span> tables from adjacent row and
column categories.</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [19]: </span><span class="nb">print</span><span class="p">(</span><span class="n">table</span><span class="o">.</span><span class="n">local_oddsratios</span><span class="p">)</span>
<span class="go">Improved     Marked      None  Some</span>
<span class="go">Treatment                          </span>
<span class="go">Placebo    0.149425  2.230769   NaN</span>
<span class="go">Treated         NaN       NaN   NaN</span>

<span class="gp">In [20]: </span><span class="n">taloc</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">Table2x2</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">29</span><span class="p">],</span> <span class="p">[</span><span class="mi">21</span><span class="p">,</span> <span class="mi">13</span><span class="p">]]))</span>

<span class="gp">In [21]: </span><span class="nb">print</span><span class="p">(</span><span class="n">taloc</span><span class="o">.</span><span class="n">oddsratio</span><span class="p">)</span>
<span class="go">0.14942528735632185</span>

<span class="gp">In [22]: </span><span class="n">taloc</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">Table2x2</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([[</span><span class="mi">29</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="p">[</span><span class="mi">13</span><span class="p">,</span> <span class="mi">7</span><span class="p">]]))</span>

<span class="gp">In [23]: </span><span class="nb">print</span><span class="p">(</span><span class="n">taloc</span><span class="o">.</span><span class="n">oddsratio</span><span class="p">)</span>
<span class="go">2.230769230769231</span>
</code></pre></div>
</div>
<p>The <strong>cumulative odds ratios</strong> construct <span class="math notranslate nohighlight">\(2\times 2\)</span> tables by
dichotomizing the row and column factors at each possible point.</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [24]: </span><span class="nb">print</span><span class="p">(</span><span class="n">table</span><span class="o">.</span><span class="n">cumulative_oddsratios</span><span class="p">)</span>
<span class="go">Improved     Marked      None  Some</span>
<span class="go">Treatment                          </span>
<span class="go">Placebo    0.185185  1.058824   NaN</span>
<span class="go">Treated         NaN       NaN   NaN</span>

<span class="gp">In [25]: </span><span class="n">tab1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">29</span> <span class="o">+</span> <span class="mi">7</span><span class="p">],</span> <span class="p">[</span><span class="mi">21</span><span class="p">,</span> <span class="mi">13</span> <span class="o">+</span> <span class="mi">7</span><span class="p">]])</span>

<span class="gp">In [26]: </span><span class="n">tacum</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">Table2x2</span><span class="p">(</span><span class="n">tab1</span><span class="p">)</span>

<span class="gp">In [27]: </span><span class="nb">print</span><span class="p">(</span><span class="n">tacum</span><span class="o">.</span><span class="n">oddsratio</span><span class="p">)</span>
<span class="go">0.18518518518518517</span>

<span class="gp">In [28]: </span><span class="n">tab1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([[</span><span class="mi">7</span> <span class="o">+</span> <span class="mi">29</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="p">[</span><span class="mi">21</span> <span class="o">+</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">7</span><span class="p">]])</span>

<span class="gp">In [29]: </span><span class="n">tacum</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">Table2x2</span><span class="p">(</span><span class="n">tab1</span><span class="p">)</span>

<span class="gp">In [30]: </span><span class="nb">print</span><span class="p">(</span><span class="n">tacum</span><span class="o">.</span><span class="n">oddsratio</span><span class="p">)</span>
<span class="go">1.0588235294117647</span>
</code></pre></div>
</div>
<p>A mosaic plot is a graphical approach to informally assessing
dependence in two-way tables.</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [31]: </span><span class="kn">from</span><span class="w"> </span><span class="nn">statsmodels.graphics.mosaicplot</span><span class="w"> </span><span class="kn">import</span> <span class="n">mosaic</span>

<span class="gp">In [32]: </span><span class="n">fig</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">mosaic</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">index</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;Treatment&quot;</span><span class="p">,</span> <span class="s2">&quot;Improved&quot;</span><span class="p">])</span>
</code></pre></div>
</div>
<h2 id="symmetry-and-homogeneity">Symmetry and homogeneity<a class="headerlink" href="#symmetry-and-homogeneity" title="Link to this heading">¶</a></h2>
<p><strong>Symmetry</strong> is the property that <span class="math notranslate nohighlight">\(P_{i, j} = P_{j, i}\)</span> for
every <span class="math notranslate nohighlight">\(i\)</span> and <span class="math notranslate nohighlight">\(j\)</span>.  <strong>Homogeneity</strong> is the property that
the marginal distribution of the row factor and the column factor are
identical, meaning that</p>
<div class="math notranslate nohighlight">
\[\sum_j P_{ij} = \sum_j P_{ji} \forall i\]</div>
<p>Note that for these properties to be applicable the table <span class="math notranslate nohighlight">\(P\)</span>
(and <span class="math notranslate nohighlight">\(T\)</span>) must be square, and the row and column categories must
be identical and must occur in the same order.</p>
<p>To illustrate, we load a data set, create a contingency table, and
calculate the row and column margins.  The <a class="reference internal" href="generated/statsmodels.stats.contingency_tables.Table.html#statsmodels.stats.contingency_tables.Table" title="statsmodels.stats.contingency_tables.Table (Python class) — A two-way contingency table."><code class="xref py py-class docutils literal notranslate"><span class="pre">Table</span></code></a> class
contains methods for analyzing <span class="math notranslate nohighlight">\(r \times c\)</span> contingency tables.
The data set loaded below contains assessments of visual acuity in
people’s left and right eyes.  We first load the data and create a
contingency table.</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [33]: </span><span class="n">df</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">&quot;VisualAcuity&quot;</span><span class="p">,</span> <span class="s2">&quot;vcd&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">data</span>

<span class="gp">In [34]: </span><span class="n">df</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">gender</span> <span class="o">==</span> <span class="s2">&quot;female&quot;</span><span class="p">,</span> <span class="p">:]</span>

<span class="gp">In [35]: </span><span class="n">tab</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">set_index</span><span class="p">([</span><span class="s1">&#39;left&#39;</span><span class="p">,</span> <span class="s1">&#39;right&#39;</span><span class="p">])</span>

<span class="gp">In [36]: </span><span class="k">del</span> <span class="n">tab</span><span class="p">[</span><span class="s2">&quot;gender&quot;</span><span class="p">]</span>

<span class="gp">In [37]: </span><span class="n">tab</span> <span class="o">=</span> <span class="n">tab</span><span class="o">.</span><span class="n">unstack</span><span class="p">()</span>

<span class="gp">In [38]: </span><span class="n">tab</span><span class="o">.</span><span class="n">columns</span> <span class="o">=</span> <span class="n">tab</span><span class="o">.</span><span class="n">columns</span><span class="o">.</span><span class="n">get_level_values</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>

<span class="gp">In [39]: </span><span class="nb">print</span><span class="p">(</span><span class="n">tab</span><span class="p">)</span>
<span class="go">right     1     2     3    4</span>
<span class="go">left                        </span>
<span class="go">1      1520   234   117   36</span>
<span class="go">2       266  1512   362   82</span>
<span class="go">3       124   432  1772  179</span>
<span class="go">4        66    78   205  492</span>
</code></pre></div>
</div>
<p>Next we create a <a class="reference internal" href="generated/statsmodels.stats.contingency_tables.SquareTable.html#statsmodels.stats.contingency_tables.SquareTable" title="statsmodels.stats.contingency_tables.SquareTable (Python class) — Methods for analyzing a square contingency table."><code class="xref py py-class docutils literal notranslate"><span class="pre">SquareTable</span></code></a> object from the contingency
table.</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [40]: </span><span class="n">sqtab</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">SquareTable</span><span class="p">(</span><span class="n">tab</span><span class="p">)</span>

<span class="gp">In [41]: </span><span class="n">row</span><span class="p">,</span> <span class="n">col</span> <span class="o">=</span> <span class="n">sqtab</span><span class="o">.</span><span class="n">marginal_probabilities</span>

<span class="gp">In [42]: </span><span class="nb">print</span><span class="p">(</span><span class="n">row</span><span class="p">)</span>
<span class="go">right</span>
<span class="go">1    0.255049</span>
<span class="go">2    0.297178</span>
<span class="go">3    0.335295</span>
<span class="go">4    0.112478</span>
<span class="go">dtype: float64</span>

<span class="gp">In [43]: </span><span class="nb">print</span><span class="p">(</span><span class="n">col</span><span class="p">)</span>
<span class="go">right</span>
<span class="go">1    0.264277</span>
<span class="go">2    0.301725</span>
<span class="go">3    0.328474</span>
<span class="go">4    0.105524</span>
<span class="go">dtype: float64</span>
</code></pre></div>
</div>
<p>The <code class="docutils literal notranslate"><span class="pre">summary</span></code> method prints results for the symmetry and homogeneity
testing procedures.</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [44]: </span><span class="nb">print</span><span class="p">(</span><span class="n">sqtab</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
<span class="go">            Statistic P-value DF</span>
<span class="gt">--------------------------------</span>
<span class="n">Symmetry</span>       <span class="mf">19.107</span>   <span class="mf">0.004</span>  <span class="mi">6</span>
<span class="n">Homogeneity</span>    <span class="mf">11.957</span>   <span class="mf">0.008</span>  <span class="mi">3</span>
<span class="gt">--------------------------------</span>
</code></pre></div>
</div>
<p>If we had the individual case records in a dataframe called <code class="docutils literal notranslate"><span class="pre">data</span></code>,
we could also perform the same analysis by passing the raw data using
the <code class="docutils literal notranslate"><span class="pre">SquareTable.from_data</span></code> class method.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><code><span class="n">sqtab</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">SquareTable</span><span class="o">.</span><span class="n">from_data</span><span class="p">(</span><span class="n">data</span><span class="p">[[</span><span class="s1">&#39;left&#39;</span><span class="p">,</span> <span class="s1">&#39;right&#39;</span><span class="p">]])</span>
<span class="nb">print</span><span class="p">(</span><span class="n">sqtab</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
</code></pre></div>
</div>
<h2 id="a-single-2x2-table">A single 2x2 table<a class="headerlink" href="#a-single-2x2-table" title="Link to this heading">¶</a></h2>
<p>Several methods for working with individual 2x2 tables are provided in
the <code class="xref py py-class docutils literal notranslate"><span class="pre">sm.stats.Table2x2</span></code> class.  The <code class="docutils literal notranslate"><span class="pre">summary</span></code> method displays
several measures of association between the rows and columns of the
table.</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [45]: </span><span class="n">table</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([[</span><span class="mi">35</span><span class="p">,</span> <span class="mi">21</span><span class="p">],</span> <span class="p">[</span><span class="mi">25</span><span class="p">,</span> <span class="mi">58</span><span class="p">]])</span>

<span class="gp">In [46]: </span><span class="n">t22</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">Table2x2</span><span class="p">(</span><span class="n">table</span><span class="p">)</span>

<span class="gp">In [47]: </span><span class="nb">print</span><span class="p">(</span><span class="n">t22</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
<span class="go">               Estimate   SE   LCB   UCB  p-value</span>
<span class="gt">-------------------------------------------------</span>
<span class="n">Odds</span> <span class="n">ratio</span>        <span class="mf">3.867</span>       <span class="mf">1.890</span> <span class="mf">7.912</span>   <span class="mf">0.000</span>
<span class="n">Log</span> <span class="n">odds</span> <span class="n">ratio</span>    <span class="mf">1.352</span> <span class="mf">0.365</span> <span class="mf">0.636</span> <span class="mf">2.068</span>   <span class="mf">0.000</span>
<span class="n">Risk</span> <span class="n">ratio</span>        <span class="mf">2.075</span>       <span class="mf">1.411</span> <span class="mf">3.051</span>   <span class="mf">0.000</span>
<span class="n">Log</span> <span class="n">risk</span> <span class="n">ratio</span>    <span class="mf">0.730</span> <span class="mf">0.197</span> <span class="mf">0.345</span> <span class="mf">1.115</span>   <span class="mf">0.000</span>
<span class="gt">-------------------------------------------------</span>
</code></pre></div>
</div>
<p>Note that the risk ratio is not symmetric so different results will be
obtained if the transposed table is analyzed.</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [48]: </span><span class="n">table</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([[</span><span class="mi">35</span><span class="p">,</span> <span class="mi">21</span><span class="p">],</span> <span class="p">[</span><span class="mi">25</span><span class="p">,</span> <span class="mi">58</span><span class="p">]])</span>

<span class="gp">In [49]: </span><span class="n">t22</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">Table2x2</span><span class="p">(</span><span class="n">table</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>

<span class="gp">In [50]: </span><span class="nb">print</span><span class="p">(</span><span class="n">t22</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
<span class="go">               Estimate   SE   LCB   UCB  p-value</span>
<span class="gt">-------------------------------------------------</span>
<span class="n">Odds</span> <span class="n">ratio</span>        <span class="mf">3.867</span>       <span class="mf">1.890</span> <span class="mf">7.912</span>   <span class="mf">0.000</span>
<span class="n">Log</span> <span class="n">odds</span> <span class="n">ratio</span>    <span class="mf">1.352</span> <span class="mf">0.365</span> <span class="mf">0.636</span> <span class="mf">2.068</span>   <span class="mf">0.000</span>
<span class="n">Risk</span> <span class="n">ratio</span>        <span class="mf">2.194</span>       <span class="mf">1.436</span> <span class="mf">3.354</span>   <span class="mf">0.000</span>
<span class="n">Log</span> <span class="n">risk</span> <span class="n">ratio</span>    <span class="mf">0.786</span> <span class="mf">0.216</span> <span class="mf">0.362</span> <span class="mf">1.210</span>   <span class="mf">0.000</span>
<span class="gt">-------------------------------------------------</span>
</code></pre></div>
</div>
<h2 id="stratified-2x2-tables">Stratified 2x2 tables<a class="headerlink" href="#stratified-2x2-tables" title="Link to this heading">¶</a></h2>
<p>Stratification occurs when we have a collection of contingency tables
defined by the same row and column factors.  In the example below, we
have a collection of 2x2 tables reflecting the joint distribution of
smoking and lung cancer in each of several regions of China.  It is
possible that the tables all have a common odds ratio, even while the
marginal probabilities vary among the strata.  The ‘Breslow-Day’
procedure tests whether the data are consistent with a common odds
ratio.  It appears below as the <cite>Test of constant OR</cite>.  The
Mantel-Haenszel procedure tests whether this common odds ratio is
equal to one.  It appears below as the <cite>Test of OR=1</cite>.  It is also
possible to estimate the common odds and risk ratios and obtain
confidence intervals for them.  The <code class="docutils literal notranslate"><span class="pre">summary</span></code> method displays all of
these results.  Individual results can be obtained from the class
methods and attributes.</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [51]: </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">china_smoking</span><span class="o">.</span><span class="n">load_pandas</span><span class="p">()</span>

<span class="gp">In [52]: </span><span class="n">mat</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">data</span><span class="p">)</span>

<span class="gp">In [53]: </span><span class="n">tables</span> <span class="o">=</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">tolist</span><span class="p">(),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">mat</span><span class="p">]</span>

<span class="gp">In [54]: </span><span class="n">st</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">StratifiedTable</span><span class="p">(</span><span class="n">tables</span><span class="p">)</span>

<span class="gp">In [55]: </span><span class="nb">print</span><span class="p">(</span><span class="n">st</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
<span class="go">                   Estimate   LCB    UCB </span>
<span class="gt">-----------------------------------------</span>
<span class="n">Pooled</span> <span class="n">odds</span>           <span class="mf">2.174</span>   <span class="mf">1.984</span> <span class="mf">2.383</span>
<span class="n">Pooled</span> <span class="n">log</span> <span class="n">odds</span>       <span class="mf">0.777</span>   <span class="mf">0.685</span> <span class="mf">0.868</span>
<span class="n">Pooled</span> <span class="n">risk</span> <span class="n">ratio</span>     <span class="mf">1.519</span>              
                                         
                 <span class="n">Statistic</span> <span class="n">P</span><span class="o">-</span><span class="n">value</span> 
<span class="gt">-----------------------------------</span>
<span class="n">Test</span> <span class="n">of</span> <span class="n">OR</span><span class="o">=</span><span class="mi">1</span>       <span class="mf">280.138</span>   <span class="mf">0.000</span> 
<span class="n">Test</span> <span class="n">constant</span> <span class="n">OR</span>     <span class="mf">5.200</span>   <span class="mf">0.636</span> 
                       
<span class="gt">-----------------------</span>
<span class="n">Number</span> <span class="n">of</span> <span class="n">tables</span>    <span class="mi">8</span>  
<span class="n">Min</span> <span class="n">n</span>             <span class="mi">213</span>  
<span class="n">Max</span> <span class="n">n</span>            <span class="mi">2900</span>  
<span class="n">Avg</span> <span class="n">n</span>            <span class="mi">1052</span>  
<span class="n">Total</span> <span class="n">n</span>          <span class="mi">8419</span>  
<span class="gt">-----------------------</span>
</code></pre></div>
</div>
<h2 id="module-statsmodels.stats.contingency_tables"><span id="module-reference"></span>Module Reference<a class="headerlink" href="#module-statsmodels.stats.contingency_tables" title="Link to this heading">¶</a></h2>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.stats.contingency_tables.Table.html#statsmodels.stats.contingency_tables.Table" title="statsmodels.stats.contingency_tables.Table (Python class) — A two-way contingency table."><code class="xref py py-obj docutils literal notranslate"><span class="pre">Table</span></code></a>(table[, shift_zeros])</p></td>
<td><p>A two-way contingency table.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.stats.contingency_tables.Table2x2.html#statsmodels.stats.contingency_tables.Table2x2" title="statsmodels.stats.contingency_tables.Table2x2 (Python class) — Analyses that can be performed on a 2x2 contingency table."><code class="xref py py-obj docutils literal notranslate"><span class="pre">Table2x2</span></code></a>(table[, shift_zeros])</p></td>
<td><p>Analyses that can be performed on a 2x2 contingency table.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.stats.contingency_tables.SquareTable.html#statsmodels.stats.contingency_tables.SquareTable" title="statsmodels.stats.contingency_tables.SquareTable (Python class) — Methods for analyzing a square contingency table."><code class="xref py py-obj docutils literal notranslate"><span class="pre">SquareTable</span></code></a>(table[, shift_zeros])</p></td>
<td><p>Methods for analyzing a square contingency table.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.stats.contingency_tables.StratifiedTable.html#statsmodels.stats.contingency_tables.StratifiedTable" title="statsmodels.stats.contingency_tables.StratifiedTable (Python class) — Analyses for a collection of 2x2 contingency tables."><code class="xref py py-obj docutils literal notranslate"><span class="pre">StratifiedTable</span></code></a>(tables[, shift_zeros])</p></td>
<td><p>Analyses for a collection of 2x2 contingency tables.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.stats.contingency_tables.mcnemar.html#statsmodels.stats.contingency_tables.mcnemar" title="statsmodels.stats.contingency_tables.mcnemar (Python function) — McNemar test of homogeneity."><code class="xref py py-obj docutils literal notranslate"><span class="pre">mcnemar</span></code></a>(table[, exact, correction])</p></td>
<td><p>McNemar test of homogeneity.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.stats.contingency_tables.cochrans_q.html#statsmodels.stats.contingency_tables.cochrans_q" title="statsmodels.stats.contingency_tables.cochrans_q (Python function) — Cochran's Q test for identical binomial proportions."><code class="xref py py-obj docutils literal notranslate"><span class="pre">cochrans_q</span></code></a>(x[, return_object])</p></td>
<td><p>Cochran's Q test for identical binomial proportions.</p></td>
</tr>
</tbody>
</table>
<h2 id="see-also">See also<a class="headerlink" href="#see-also" title="Link to this heading">¶</a></h2>
<p><a class="reference external" href="https://docs.scipy.org/doc/scipy-0.18.0/reference/stats.html#contingency-table-functions">Scipy</a> has several functions for analyzing contingency tables,
including Fisher’s exact test which is not currently in statsmodels.</p>


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