.. _spearman_test: Spearman correlation test ------------------------- This method deals with the modelling of a probability distribution of a random vector :math:`\vect{X} = \left( X^1,\ldots,X^{n_X} \right)`. It seeks to find a type of dependency (here a monotonous correlation) which may exist between two components :math:`X^i` and :math:`X^j`. The Spearman’s correlation coefficient :math:`\rho^S_{U,V}`, defined in :ref:`Spearman’s coefficient ` , measures the strength of a monotonous relationship between two random variables :math:`U` and :math:`V`. If we have a sample made up of :math:`N` pairs :math:`\left\{ (u_1,v_1),(u_2,v_2),(u_N,v_N) \right\}`, we denote :math:`\widehat{\rho}^S_{U,V}` to be the estimated coefficient. Even in the case where two variables :math:`U` and :math:`V` have a Spearman’s coefficient :math:`\rho^S_{U,V}` equal to zero, the estimate :math:`\widehat{\rho}^S_{U,V}` obtained from the sample may be non-zero: the limited sample size does not provide the perfect image of the real correlation. Pearson’s test nevertheless enables one to determine if the value obtained by :math:`\widehat{\rho}^S_{U,V}` is significantly different from zero. More precisely, the user first chooses a probability :math:`\alpha`. From this value the critical value :math:`d_\alpha` is calculated automatically such that: - if :math:`\left| \widehat{\rho}^S_{U,V} \right| > d_\alpha`, one can conclude that the real Spearman’s correlation coefficient :math:`\rho^S_{U,V}` is not zero; the risk of error in making this assertion is controlled and equal to :math:`\alpha`; - if :math:`\left| \widehat{\rho}^S_{U,V} \right| \leq d_\alpha`, there is insufficient evidence to reject the null hypothesis :math:`\rho^S_{U,V} = 0`. An important notion is the so-called “:math:`p`-value” of the test. This quantity is equal to the limit error probability :math:`\alpha_\textrm{lim}` under which the null correlation hypothesis is rejected. Thus, Spearman’s’s coefficient is supposed non zero if and only if :math:`\alpha_\textrm{lim}` is greater than the value :math:`\alpha` desired by the user. Note that the higher :math:`\alpha_\textrm{lim} - \alpha`, the more robust the decision. .. plot:: import openturns as ot from openturns.viewer import View N = 5 ot.RandomGenerator.SetSeed(0) x = ot.Uniform(2.0, 8.0).getSample(N) f = ot.SymbolicFunction(['x'], ['80-0.4*(x-2)^3']) y = f(x) + ot.Normal(0.0, 20.0).getSample(N) graph = f.draw(2.0, 8.0) graph.setTitle('Non significant Spearman coefficient') graph.setXTitle('u') graph.setYTitle('v') cloud = ot.Cloud(x, y) cloud.setPointStyle('circle') cloud.setColor('orange') graph.add(cloud) View(graph) .. topic:: API: - See :py:func:`~openturns.HypothesisTest_Spearman` - See :py:func:`~openturns.HypothesisTest_PartialSpearman` - See :py:func:`~openturns.HypothesisTest_FullSpearman` .. topic:: Examples: - See :doc:`/auto_data_analysis/estimate_dependency_and_copulas/plot_independence_test` .. topic:: References: - [saporta1990]_ - [dixon1983]_ - [nisthandbook]_ - [dagostino1986]_ - [bhattacharyya1997]_ - [sprent2001]_ - [burnham2002]_