Kendall and/or Spearman non-parametric rank correlation coefficients, allows variables and observations to be selectively disregarded

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NAG[g02brc] NAG[nag_ken_spe_corr_coeff] - Kendall and/or Spearman non-parametric rank correlation coefficients, allows variables and observations to be selectively disregarded

Calling Sequence

g02brc(x, svar, sobs, corr, 'n'=n, 'm'=m, 'tdx'=tdx, 'tdc'=tdc, 'fail'=fail)

nag_ken_spe_corr_coeff(. . .)

Parameters

x - Matrix(1..n, 1..tdx, datatype=float[8], order=C_order);

On entry: must contain the th observation on the th variable, for ; .

svar - Vector(1..m, datatype=integer[kernelopts('wordsize')/8]);

On entry: indicates which variables are to be included, for the th variable to be included, . If all variables are to be included then a NULL pointer (Integer *)0 may be supplied.

Constraint: , for and there is at least 1 positive element. .

sobs - Vector(1..n, datatype=integer[kernelopts('wordsize')/8]);

On entry: indicates which observations are to be included, for the th observation to be included, . If all observations are to be included then a NULL pointer (Integer *)0 may be supplied.

Constraint: , for and there are at least 2 positive elements. .

corr - Matrix(1..m, 1..tdc, datatype=float[8], order=C_order);

On exit: the upper by part of corr contains the correlation coefficients, the upper triangle contains the Spearman coefficients and the lower triangle, the Kendall coefficients. That is, for the th and th variables, where is less than , contains the Spearman rank correlation coefficient, and contains Kendall's tau, for . The diagonal will be set to 1.

'n'=n - integer; (optional)

Default value: the first dimension of the arrays x, sobs.

On entry: the number of observations in the data set.

Constraint: . .

'm'=m - integer; (optional)

Default value: the first dimension of the arrays svar, corr and the second dimension of the arrays svar, corrthe arrays x, corr.

On entry: the number of variables.

Constraint: . .

'tdx'=tdx - integer; (optional)

On entry: the second dimension of the array x as declared in the function from which nag_ken_spe_corr_coeff (g02brc) is called.

Constraint: . .

'tdc'=tdc - integer; (optional)

On entry: the second dimension of the array corr as declared in the function from which nag_ken_spe_corr_coeff (g02brc) is called.

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_ken_spe_corr_coeff (g02brc) calculates Kendall and Spearman rank correlation coefficients.

Description

nag_ken_spe_corr_coeff (g02brc) calculates both the Kendall rank correlation coefficients and the Spearman rank correlation coefficients.

The data consists of observations for each of variables:

where is the th observation on the th variable. The function eliminates any variable for where the argument , and any observation for where the argument .

The observations are first ranked as follows:

For a given variable, say, each of the observations for which , for 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 .

The smallest of these valid observations for variable is assigned the rank 1, the second smallest observation for variable the rank 2, and so on until the largest such observation is given the rank , where is the number of observations for which .

If a number of cases all have the same value for a given variable, , then they are each given an "average" rank — e.g. if in attempting to assign the rank , observations for which were found to have the same value, then instead of giving them the ranks all observations would be assigned the rank and the next value in ascending order would be assigned the rank The process is repeated for each of the variables for which .

Let be the rank assigned to the observation when the th variable is being ranked. For those observations, , for which , , for .

For variables the following are computed:

(a) Kendall's tau correlation coefficients:

R[jk]=((∑)(∑)sign(y[hj],-,y[ij])sign(y[hk],-,y[ik]))/(sqrt([n[s],(n[s],-,1),-,T[j]][n[s],(n[s],-,1),-,T[k]])) j,k=1,2,..,m

where is the number of observations for which ,

and	sign if ,
	sign if ,
	sign if ,

and where is the number of ties of a particular value of variable , and the summation is over all tied values of variable .

(b) Spearman's rank correlation coefficients:

R[jk]=(n[s](n[s]^2,-,1)-6(∑)(y[ij],-,y[ik])^2-1/2(T[j],+,T[k]))/(sqrt([n[s],(n[s]^2,-,1),-,T[j]][n[s],(n[s]^2,-,1),-,T[k]])) j,k=1,2,..,m

where is the number of observations for which ,

and where is the number of ties of a particular value of variable , and the summation is over all tied values of variable .

Error Indicators and Warnings

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_INT_ARG_LT"

On entry, n must not be less than 2: .

"NE_INT_ARRAY_1"

Value given to not valid. Correct range for elements of svar is .

"NE_INTERNAL_ERROR"

An initial error has occurred in this function. Check the function call and any array sizes.

"NE_SOBS_LOW"

Too few observations have been selected.

"NE_SVAR_LOW"

No variables have been selected.

	Accuracy
	The computations are believed to be stable.

	Further Comments
	None.

Examples

n := 7:
m := 3:
tdx := 3:
tdc := 3:
x := Matrix([[1, 2, 4], [7, 7, 3], [2, 3, 4], [4, 4, 5], [5, 6, 7], [3, 1, 3], [6, 5, 5]], datatype=float[8], order='C_order'):
svar := Vector([1, 1, 1], datatype=integer[kernelopts('wordsize')/8]):
sobs := Vector([1, 1, 1, 1, 1, 1, 1], datatype=integer[kernelopts('wordsize')/8]):
corr := Matrix(3, 3, datatype=float[8], order='C_order'):
NAG:-g02brc(x, svar, sobs, corr, 'n' = n, 'm' = m, 'tdx' = tdx, 'tdc' = tdc):

corr;

	See Also
	Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill g02 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.