g02bpf 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.
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 ${\mathit{xm}}_{j}$. 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 ${x}_{ij}={\mathit{xm}}_{j}$ for any $j$ for which an ${\mathit{xm}}_{j}$ has been assigned (see also Section 7); and ${w}_{i}=1$ otherwise, for $\mathit{i}=1,2,\dots ,n$.
The quantities calculated are:
(a)Ranks
For a given variable, $j$ say, each of the observations ${x}_{ij}$ for which ${w}_{i}=1$, for $\mathit{i}=1,2,\dots ,n$, 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 ${n}_{c}={\displaystyle \sum _{i=1}^{n}}{w}_{i}$.
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 $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
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 $j$.
where ${n}_{c}={\displaystyle \sum _{i=1}^{n}}{w}_{i}$
and ${T}_{j}^{*}=\sum {t}_{j}({t}_{j}^{2}-1)$ 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 $j$.
4References
Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations or cases.
Constraint:
${\mathbf{n}}\ge 2$.
2: $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of variables.
Constraint:
${\mathbf{m}}\ge 2$.
3: $\mathbf{x}({\mathbf{ldx}},{\mathbf{m}})$ – Real (Kind=nag_wp) arrayInput/Output
On entry: ${\mathbf{x}}(\mathit{i},\mathit{j})$ must be set to ${x}_{\mathit{i}\mathit{j}}$, the value of the $\mathit{i}$th observation on the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
On exit: ${\mathbf{x}}(\mathit{i},\mathit{j})$ contains the rank ${y}_{\mathit{i}\mathit{j}}$ of the observation ${x}_{\mathit{i}\mathit{j}}$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$. (For those observations containing missing values, and, therefore, excluded from the calculation,
${y}_{\mathit{i}\mathit{j}}=0$, for $\mathit{j}=1,2,\dots ,m$.)
4: $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g02bpf is called.
On entry: ${\mathbf{miss}}\left(j\right)$ 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.
On exit: the array miss is overwritten by the routine, and the information it contained on entry is lost.
6: $\mathbf{xmiss}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: ${\mathbf{xmiss}}\left(j\right)$ 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 Section 7).
On exit: the array xmiss is overwritten by the routine, and the information it contained on entry is lost.
7: $\mathbf{itype}$ – IntegerInput
On entry: the type of correlation coefficients which are to be calculated.
${\mathbf{itype}}=\mathrm{-1}$
Only Kendall's tau coefficients are calculated.
${\mathbf{itype}}=0$
Both Kendall's tau and Spearman's coefficients are calculated.
${\mathbf{itype}}=1$
Only Spearman's coefficients are calculated.
Constraint:
${\mathbf{itype}}=\mathrm{-1}$, $0$ or $1$.
8: $\mathbf{rr}({\mathbf{ldrr}},{\mathbf{m}})$ – Real (Kind=nag_wp) arrayOutput
On exit: the requested correlation coefficients.
If only Kendall's tau coefficients are requested (${\mathbf{itype}}=\mathrm{-1}$), ${\mathbf{rr}}(j,k)$ contains Kendall's tau for the $j$th and $k$th variables.
If only Spearman's coefficients are requested (${\mathbf{itype}}=1$), ${\mathbf{rr}}(j,k)$ 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 (${\mathbf{itype}}=0$), the upper triangle of rr contains the Spearman coefficients and the lower triangle the Kendall coefficients. That is, for the
$\mathit{j}$th and $\mathit{k}$th variables, where $\mathit{j}$ is less than $\mathit{k}$, ${\mathbf{rr}}(\mathit{j},\mathit{k})$ contains the Spearman rank correlation coefficient, and ${\mathbf{rr}}(\mathit{k},\mathit{j})$ contains Kendall's tau, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
(Diagonal terms, ${\mathbf{rr}}(j,j)$, are unity for all three values of itype.)
9: $\mathbf{ldrr}$ – IntegerInput
On entry: the first dimension of the array rr as declared in the (sub)program from which g02bpf is called.
Constraint:
${\mathbf{ldrr}}\ge {\mathbf{m}}$.
10: $\mathbf{ncases}$ – IntegerOutput
On exit: the number of cases, ${n}_{\mathrm{c}}$, actually used in the calculations (when cases involving missing values have been eliminated).
On exit: ${\mathbf{incase}}\left(\mathit{i}\right)$ holds the value $1$ if the $\mathit{i}$th case was included in the calculations, and the value $0$ if the $\mathit{i}$th case contained a missing value for at least one variable. That is, ${\mathbf{incase}}\left(\mathit{i}\right)={w}_{\mathit{i}}$ (see Section 3), for $\mathit{i}=1,2,\dots ,n$.
15: $\mathbf{work1}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
16: $\mathbf{work2}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
17: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 2$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}\ge 2$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{ldrr}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldrr}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{ldx}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{itype}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{itype}}=\mathrm{-1}$ or $1$.
${\mathbf{ifail}}=5$
After observations with missing values were omitted, fewer than two cases remained.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
You are warned of the need to exercise extreme care in your selection of missing values. g02bpf treats all values in the inclusive range $(1\pm {0.1}^{({\mathbf{x02bef}}-2)})\times {xm}_{j}$, where ${\mathit{xm}}_{j}$ is the missing value for variable $j$ specified in xmiss.
You must, therefore, ensure that the missing value chosen for each variable is sufficiently different from all valid values for that variable so that none of the valid values fall within the range indicated above.
8Parallelism and Performance
g02bpf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The time taken by g02bpf depends on $n$ and $m$, and the occurrence of missing values.
10Example
This example reads in a set of data consisting of nine observations on each of three variables. Missing values of $0.99$ and $0.0$ are declared for the first and third variables respectively; no missing value is specified for the second variable. The program then calculates and prints the rank of each observation, and both Kendall's tau and Spearman's rank correlation coefficients for all three variables, omitting completely all cases containing missing values; cases $5$, $8$ and $9$ are, therefore, eliminated, leaving only six cases in the calculations.