naginterfaces.library.correg.coeffs_​kspearman_​miss_​case_​overwrite

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

coeffs_kspearman_miss_case_overwrite computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data omitting completely any cases with a missing observation for any variable; the data array is overwritten with the ranks of the observations.

For full information please refer to the NAG Library document for g02bp

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g02/g02bpf.html

Parameters

xfloat, ndarray, shape $(n,m)$, modified in place

On entry: $x[\text{i}-1,\text{j}-1]$ 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 $\text{i}=1,2,\dots ,n$.

On exit: $x[\text{i}-1,\text{j}-1]$ contains the rank $y_{\text{i}\text{j}}$ of the observation $x_{\text{i}\text{j}}$, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,n$. (For those observations containing missing values, and, therefore, excluded from the calculation, $y_{\text{i}\text{j}}=0$, for $\text{j}=1,2,\dots ,m$.)

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

$miss[j-1]$ must be set 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 number of cases, $n_c$, actually used in the calculations (when cases involving missing values have been eliminated).

incaseint, ndarray, shape $\left(n\right)$

$incase[\text{i}-1]$ holds the value $1$ if the $\text{i}$th case was included in the calculations, and the value $0$ if the $\text{i}$th case contained a missing value for at least one variable. That is, $incase[\text{i}-1]=w_{\text{i}}$ (see Notes), for $\text{i}=1,2,\dots ,n$.

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 $1$.

(errno $5$)

After observations with missing values were omitted, fewer than two cases remained.

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. 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 $w_i=0$ if observation $i$ contains a missing value for any of those variables for which missing values have been declared; i.e., if $x_{ij}={\text{xm}}_j$ for any $j$ for which an ${\text{xm}}_j$ has been assigned (see also Accuracy); and $w_i=1$ otherwise, for $\text{i}=1,2,\dots ,n$.

The quantities calculated are:

1. Ranks

   For a given variable, $j$ say, each of the observations $x_{ij}$ for which $w_i=1$, 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 magnitudes of the other observations on that same variable for which $w_i=1$.

   The smallest of these valid observations 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 such observation is given the rank $n_c$, where $n_c={\sum }_{i=1}^n{w}_i$.

   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 for which $w_i=1$ 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 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. For those observations, $i$, for which $w_i=0$, $y_{ij}=0$, for $j=1,2,\dots ,m$.

   The actual observations $x_{ij}$ are replaced by the ranks $y_{ij}$, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,n$.

2. Nonparametric rank correlation coefficients

   1. Kendall’s tau:

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

      where	$n_c={\sum }_{i=1}^n{w}_i$
      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 $T_j=\sum t_j(t_j-1)$ 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 $j$.

   2. Spearman’s:

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

      where $n_c={\sum }_{i=1}^n{w}_i$

      and $T_j^{\ast }=\sum t_j(t_j^2-1)$ 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