BHAKV

Performs a Bhapkar V test.

Required Arguments

NI — Vector of length NGROUP containing the number of responses for each of the NGROUP groups. (Input)

Y — Vector of length NI(1) + NI(2) + … + NI(NGROUP) containing the responses for each of the NGROUP groups. (Input)
Y must be sorted by group with the NI(1) observations for group 1 coming first.

V — Bhapkar V statistic. (Output)

PROB — Asymptotic probability of exceeding V under the null hypothesis that the populations are equal. (Output)
Asymptotically, V follows a chi‑squared distribution with NGROUP ‑ 1 degrees of freedom.

Optional Arguments

NGROUP — Number of groups. (Input)
Default: NGROUP = size (NI,1).

FORTRAN 90 Interface

Generic: CALL BHAKV (NI, Y, V, PROB [, …])

Specific: The specific interface names are S_BHAKV and D_BHAKV.

FORTRAN 77 Interface

Single: CALL BHAKV (NGROUP, NI, Y, V, PROB)

Double: The double precision name is DBHAKV.

Description

Routine BHAKV tests the hypothesis that several samples are from the same population using the Bhapkar V statistic. Let the number of samples be denoted by K = NGROUP. To compute the Bhapkar V statistic, one first computes, for each group i, the statistic ti = the number of K‑tuples that can be formed with one observation from each sample such that the element from population i is the smallest. The sample variance of the ratio of ti to the total number of such k‑tuples is then computed. The Bhapkar V statistic is then a constant c multiplied by this variance, where c = n(2m ‑ 1), m = NGROUP, and n is the sum of the sample sizes (after missing values are eliminated).

Comments

Workspace may be explicitly provided, if desired, by use of B2AKV/DB2AKV. The reference is

CALL B2AKV (NGROUP, NI, Y, V, PROB, IWK, WK, YWK)

The additional arguments are as follows:

IWK — Integer work vector of length NI(1) + … + NI(NGROUP) + NGROUP

WK — Work vector of length NGROUP

YWK — Work vector of length NI(1) + … + NI(NGROUP). If Y is not needed, Y and YWK can share the same storage locations.

Example

We want to test the null hypothesis that three samples of size 3, 2, and 4, respectively, are from the same population using the Bhapkar V statistic.

 

USE BHAKV_INT

USE UMACH_INT

 

IMPLICIT NONE

INTEGER NGROUP

PARAMETER (NGROUP=3)

!

INTEGER NI(NGROUP), NOUT

REAL PROB, V, Y(9)

!

DATA NI/3, 2, 4/

DATA Y/1, 3, 2, -1, 5, 4, 7, 2, 9/

! Perform Bhapkar V test

CALL BHAKV (NI, Y, V, PROB)

! Print results

CALL UMACH (2, NOUT)

WRITE (NOUT,99998) V

WRITE (NOUT,99999) PROB

!

99998 FORMAT (' V = ', F12.5)

99999 FORMAT (' Prob = ', F12.5)

!

END

Output

 

V = 1.89429

Prob = 0.38785