naginterfaces.library.correg.coeffs_​kspearman

naginterfaces.library.correg.coeffs_kspearman(x, itype)

coeffs_kspearman computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data; the data array is preserved, and the ranks of the observations are not available on exit from the function.

For full information please refer to the NAG Library document for g02bq

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g02/g02bqf.html

Parameters

xfloat, array-like, shape $(n,m)$

$x[\text{i}-1,\text{j}-1]$ must be set to data value $x_{\text{i}\text{j}}$, the value of the $\text{i}$th observation on the $\text{j}$th variable, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,n$.

itypeint

The type of correlation coefficients which are to be calculated.

$itype=-1$

Only Kendall’s tau coefficients are calculated.

$itype=0$

Both Kendall’s tau and Spearman’s coefficients are calculated.

$itype=1$

Only Spearman’s coefficients are calculated.

Returns

rrfloat, ndarray, shape $(m,m)$

The requested correlation coefficients.

If only Kendall’s tau coefficients are requested ($itype=-1$), $rr[j-1,k-1]$ contains Kendall’s tau for the $j$th and $k$th variables.

If only Spearman’s coefficients are requested ($itype=1$), $rr[j-1,k-1]$ contains Spearman’s rank correlation coefficient for the $j$th and $k$th variables.

If both Kendall’s tau and Spearman’s coefficients are requested ($itype=0$), the upper triangle of $rr$ contains the Spearman coefficients and the lower triangle the Kendall coefficients.

That is, for the $\text{j}$th and $\text{k}$th variables, where $\text{j}$ is less than $\text{k}$, $rr[\text{j}-1,\text{k}-1]$ contains the Spearman rank correlation coefficient, and $rr[\text{k}-1,\text{j}-1]$ contains Kendall’s tau, for $\text{k}=1,2,\dots ,m$, for $\text{j}=1,2,\dots ,m$.

(Diagonal terms, $rr[j-1,j-1]$, are unity for all three values of $itype$.)

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 2$.

(errno $2$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 2$.

(errno $4$)

On entry, $itype=⟨value⟩$.

Constraint: $itype=-1$ or $1$.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

The input data consists of $n$ observations for each of $m$ variables, given as an array

$$\left[x_{ij}\right]\text{,}i=1,2,\dots ,n(n\ge 2),j=1,2,\dots ,m(m\ge 2)\text{,}$$

where $x_{ij}$ is the $i$th observation on the $j$th variable.

The observations are first ranked, as follows.

For a given variable, $j$ say, each of the $n$ observations, $x_{1j},x_{2j},\dots ,x_{nj}$, has associated with it an additional number, the ‘rank’ of the observation, which indicates the magnitude of that observation relative to the magnitude of the other $n-1$ observations on that same variable.

The smallest observation for variable $j$ is assigned the rank $1$, the second smallest observation for variable $j$ the rank $2$, the third smallest the rank $3$, and so on until the largest observation for variable $j$ is given the rank $n$.

If a number of cases all have the same value for the given variable, $j$, then they are each given an ‘average’ rank – e.g., if in attempting to assign the rank $h+1$, $k$ observations were found to have the same value, then instead of giving them the ranks

$$h+1,h+2,\dots ,h+k\text{,}$$

all $k$ observations would be assigned the rank

$$\frac{2h+k+1}{2}$$

and the next value in ascending order would be assigned the rank

$$h+k+1\text{.}$$

The process is repeated for each of the $m$ variables.

Let $y_{ij}$ be the rank assigned to the observation $x_{ij}$ when the $j$th variable is being ranked.

The quantities calculated are:

1. Kendall’s tau rank correlation coefficients:

   $$R_{jk}=\frac{{\sum }_{h=1}^n{\sum }_{i=1}^{n}sign(y_{hj}-y_{ij})sign(y_{hk}-y_{ik})}{\sqrt{[n(n-1)-T_j][n(n-1)-T_k]}}\text{,}j,k=1,2,\dots ,m\text{,}$$

   where	$sign\left(u\right)=1$ if $u>0$
   	$sign\left(u\right)=0$ if $u=0$
   	$sign\left(u\right)=-1$ if $u<0$

   and $T_j=\sum t_j(t_j-1)$, $t_j$ being the number of ties of a particular value of variable $j$, and the summation being over all tied values of variable $j$.

2. Spearman’s rank correlation coefficients:

   $$R_{jk}^{\ast }=\frac{n(n^2-1)-6{\sum }_{i=1}^n{(y_{ij}-y_{ik})}^2-\frac{1}{2}(T_j^{\ast }+T_k^{\ast })}{\sqrt{[n(n^2-1)-T_j^{\ast }][n(n^2-1)-T_k^{\ast }]}}\text{,}j,k=1,2,\dots ,m\text{,}$$

   where $T_j^{\ast }=\sum t_j(t_j^2-1)$ where $t_j$ is the number of ties of a particular value of variable $j$, and the summation is over all tied values of variable $j$.

References

Siegel, S, 1956, Non-parametric Statistics for the Behavioral Sciences, McGraw–Hill