.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        Click :ref:`here <sphx_glr_download_auto_data_analysis_estimate_dependency_and_copulas_plot_independence_test.py>`     to download the full example code
    .. rst-class:: sphx-glr-example-title

    .. _sphx_glr_auto_data_analysis_estimate_dependency_and_copulas_plot_independence_test.py:


Sample independence test
========================

In this example we are going to perform tests to assess whether two 1-d samples are independent or not.

The following tests are available:

- the ChiSquared test: it tests if both scalar samples (discrete ones only) are independent.
  If :math:`n_{ij}` is the number of values of the sample :math:`i=(1,2)` in the modality :math:`1 \leq j \leq m`, :math:`\displaystyle n_{i.} = \sum_{j=1}^m n_{ij}` :math:`\displaystyle n_{.j} = \sum_{i=1}^2 n_{ij}`, and the ChiSquared test evaluates the decision variable:

.. math::
   D^2 = \sum_{i=1}^2 \sum_{j=1}^m \frac{( n_{ij} - \frac{n_{i.} n_{.j}}{n} )^2}{\frac{n_{i.} n_{.j}}{n}}

which tends towards the :math:`\chi^2(m-1)` distribution. The hypothesis of independence is rejected if :math:`D^2` is too high (depending on the p-value threshold).

- the Pearson test: it tests if there exists a linear relation between two scalar samples which form a gaussian vector (which is equivalent to have a linear correlation coefficient not equal to zero).
  If both samples are :math:`\underline{x} = (x_i)_{1 \leq i \leq n}` and :math:`\underline{y} = (y_i)_{1 \leq i \leq n}`, and :math:`\bar{x} = \displaystyle \frac{1}{n}\sum_{i=1}^n x_i` and :math:`\bar{y} = \displaystyle \frac{1}{n}\sum_{i=1}^n y_i`, the Pearson test evaluates the decision variable:

 .. math::
     D = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^n (x_i - \bar{x})^2\sum_{i=1}^n (y_i - \bar{y})^2}}

The variable :math:`D` tends towards a :math:`\chi^2(n-2)`, under the hypothesis of normality of both samples. The hypothesis of a linear coefficient equal to 0 is rejected (which is equivalent to the independence of the samples) if D is too high (depending on the p-value threshold).

- the Spearman test: it tests if there exists a monotonous relation between two scalar samples.
  If both samples are :math:`\underline{x} = (x_i)_{1 \leq i \leq n}` and :math:`\underline{y}= (y_i)_{1 \leq i \leq n}`,, the Spearman test evaluates the decision variable:

.. math::
     D = 1-\frac{6\sum_{i=1}^n (r_i - s_i)^2}{n(n^2-1)}

where :math:`r_i = rank(x_i)` and  :math:`s_i = rank(y_i)`. :math:`D` is such that :math:`\sqrt{n-1}D` tends towards the standard normal distribution.



.. code-block:: default

    from __future__ import print_function
    import openturns as ot
    ot.Log.Show(ot.Log.NONE)








**continuous samples**

Create continuous samples


.. code-block:: default

    sample1 = ot.Normal().getSample(100)
    sample2 = ot.Normal().getSample(100)








Using the Pearson test


.. code-block:: default

    ot.HypothesisTest.Pearson(sample1, sample2, 0.10)






.. raw:: html

    <p>class=TestResult name=Unnamed type=Pearson binaryQualityMeasure=true p-value threshold=0.1 p-value=0.697478 statistic=0.389871 description=[]</p>
    <br />
    <br />

Using the Spearman test


.. code-block:: default

    ot.HypothesisTest.Spearman(sample1, sample2, 0.10)






.. raw:: html

    <p>class=TestResult name=Unnamed type=Spearman binaryQualityMeasure=true p-value threshold=0.1 p-value=0.485495 statistic=0.699007 description=[]</p>
    <br />
    <br />

**discrete samples**

Create discrete samples


.. code-block:: default

    sample1 = ot.Poisson(0.2).getSample(100)
    sample2 = ot.Poisson(0.2).getSample(100)








Using the Chi2 test


.. code-block:: default

    ot.HypothesisTest.ChiSquared(sample1, sample2, 0.10)





.. raw:: html

    <p>class=TestResult name=Unnamed type=ChiSquared binaryQualityMeasure=true p-value threshold=0.1 p-value=0.165616 statistic=3.59616 description=[]</p>
    <br />
    <br />


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** ( 0 minutes  0.002 seconds)


.. _sphx_glr_download_auto_data_analysis_estimate_dependency_and_copulas_plot_independence_test.py:


.. only :: html

 .. container:: sphx-glr-footer
    :class: sphx-glr-footer-example



  .. container:: sphx-glr-download sphx-glr-download-python

     :download:`Download Python source code: plot_independence_test.py <plot_independence_test.py>`



  .. container:: sphx-glr-download sphx-glr-download-jupyter

     :download:`Download Jupyter notebook: plot_independence_test.ipynb <plot_independence_test.ipynb>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_