Spearman correlation testΒΆ

The Spearman Test checks if there exists a monotonic relationship between two random variables X and Y.

The Spearman test is based on the Spearman correlation coefficient defined in Spearman coefficient. It tests if the Spearman correlation coefficient is significantly different from zero.

The Spearman compares the null hypothesis \cH_0 = \left\{ \rho_S(X,Y) = 0 \right\} against the alternative hypothesis \cH_1 = \left\{ \rho_S(X,Y) \neq 0 \right\}.

The Spearman correlation coefficient \rho_S(X,Y) is evaluated on a sample generated by the bivariate random vector (X,Y) of size \sampleSize and denoted by \hat{\rho}_S(X,Y) as detailed in (1).

The statistics T(X,Y) used in the test is defined by:

T(X,Y) = \hat{\rho}_S(X,Y) \sqrt{\dfrac{\sampleSize-2}{1-(\hat{\rho}_S(X,Y))^2}}

Under the null hypothesis \cH_0, the statistics T follows a Student distribution with \sampleSize-2 degrees of freedom in the case of a Gaussian vector. In the other cases, the Student distribution T(\sampleSize-2) is equivalent to the asymptotic distribution of T. The library uses the Student distribution T(\sampleSize-2) in all the cases.

The p-value p_v is the probability p_v = \Prob{|T| \geq |t(X,Y)|} where t(X,Y) is the realization of T(X,Y) computed on the sample. The null hypothesis \cH_0 is rejected if p_v < p_v^\ell where p_v^\ell is specified (usually 0.1 or 0.05). The p-value limit p_v^\ell is the probability to wrongly reject the null hypothesis \cH_0, which means to commit a Type I error.

When the null hypothesis \cH_0 is rejected, it means that there is a significant monotonic relationship between both variables X and Y.