naginterfaces.library.correg.coeffs_​kspearman_​miss_​pair

naginterfaces.library.correg.coeffs_kspearman_miss_pair(x, miss, xmiss, itype)

coeffs_kspearman_miss_pair computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data omitting cases with missing values from only those calculations involving the variables for which the values are missing; 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 g02bs

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g02/g02bsf.html

Parameters

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

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

missint, array-like, shape $\left(m\right)$

$miss[j-1]$ must be set equal to $1$ if a missing value, $x{m}_j$, is to be specified for the $j$th variable in the array $x$, or set equal to $0$ otherwise. Values of $miss$ must be given for all $m$ variables in the array $x$.

xmissfloat, array-like, shape $\left(m\right)$

$xmiss[j-1]$ must be set to the missing value, $x{m}_j$, to be associated with the $j$th variable in the array $x$, for those variables for which missing values are specified by means of the array $miss$ (see Accuracy).

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 $itype$.)

ncasesint

The minimum number of cases used in the calculation of any of the correlation coefficients (when cases involving missing values have been eliminated).

cntfloat, ndarray, shape $(m,m)$

The number of cases, $n_{\text{j}\text{k}}$, actually used in the calculation of the rank correlation coefficient for the $\text{j}$th and $\text{k}$th variables, for $\text{k}=1,2,\dots ,m$, for $\text{j}=1,2,\dots ,m$.

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 $1$.

Warns

NagAlgorithmicWarning

(errno $5$)

After observations with missing values were omitted, fewer than two cases remained for at least one pair of variables. (The pairs of variables involved can be determined by examination of the contents of the array $cnt$). All means, standard deviations, sums of squares and cross-products, and correlation-like coefficients based on two or more cases are returned by the function even if $errno$ = 5.

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

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

where $x_{ij}$ is the $i$th observation on the $j$th variable. In addition each of the $m$ variables may optionally have associated with it a value which is to be considered as representing a missing observation for that variable; the missing value for the $j$th variable is denoted by ${\text{xm}}_j$. Missing values need not be specified for all variables.

Let $w_{\text{i}\text{j}}=0$ if the $\text{i}$th observation for the $\text{j}$th variable is a missing value, i.e., if a missing value, ${\text{xm}}_{\text{j}}$, has been declared for the $\text{j}$th variable, and $x_{\text{i}\text{j}}={\text{xm}}_{\text{j}}$ (see also Accuracy); and $w_{\text{i}\text{j}}=1$ otherwise, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,n$.

The observations are first ranked, a pair of variables at a time as follows:

For a given pair of variables, $j$ and $l$ say, each of the observations $x_{\text{i}j}$ for which the product $w_{\text{i}j}{w}_{\text{i}l}=1$, for $\text{i}=1,2,\dots ,n$, 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 observations on variable $j$ for which $w_{ij}{w}_{il}=1$.

The smallest of these valid observations for variable $j$ is assigned to rank $1$, the second smallest valid observation for variable $j$ the rank $2$, the third smallest rank $3$, and so on until the largest such observation is given the rank $n_{jl}$, where

$$n_{jl}=\sum _{i=1}^n{w}_{ij}{w}_{il}\text{.}$$

If a number of cases all have the same value for the variable $j$, then they are each given an ‘average’ rank, e.g., if in attempting to assign the rank $h+1$, $k$ observations for which $w_{ij}{w}_{il}=1$ 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 variable $\text{l}$ is then ranked in a similar way. The process is then repeated for all pairs of variables $\text{j}$ and $\text{l}$, for $\text{l}=\text{j},\dots ,m$, for $\text{j}=1,2,\dots ,m$. Let $y_{\text{i}\text{j}\left(\text{l}\right)}$ be the rank assigned to the observation $x_{\text{i}\text{j}}$ when the $\text{j}$th and $\text{l}$th variables are being ranked, and $y_{\text{i}\text{l}\left(\text{j}\right)}$ be the rank assigned to the observation $x_{\text{i}\text{l}}$ during the same process, for $\text{l}=j,\dots ,m$, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,n$.

The quantities calculated are:

1. Kendall’s tau rank correlation coefficients:

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

   where	$n_{jk}={\sum }_{i=1}^n{w}_{ij}{w}_{ik}$
   and	$sign\left(u\right)=1$ if $u>0$
   	$sign\left(u\right)=0$ if $u=0$
   	$sign\left(u\right)=-1$ if $u<0$

   and $T_{j\left(k\right)}=\sum t_j(t_j-1)$ where $t_j$ is the number of ties of a particular value of variable $j$ when the $j$th and $k$th variables are being ranked, and the summation is over all tied values of variable $j$.

2. Spearman’s rank correlation coefficients:

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

   where $n_{jk}={\sum }_{i=1}^n{w}_{ij}{w}_{ik}$

   and $T_{j\left(k\right)}^{\ast }=\sum t_j(t_j^2-1)$, where $t_j$ is the number of ties of a particular value of variable $j$ when the $j$th and $k$th variables are being ranked, and the summation is over all tied values of variable $j$.

References

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