Calculates probabilities for the Mann-Whitney U statistic (D.B. Baird & J.H. Klotz).
No options
Parameters
N1 = scalars |
Sizes of the first groups of observations |
|---|---|
N2 = scalars |
Sizes of the second groups of observations |
U = scalars |
Values of the U statistic |
TIES = scalars |
Number of tied observations; default 0 |
CLPROBABILITY = scalars |
Cumulative lower probability of U |
CUPROBABILITY = scalars |
Cumulative upper probability of U |
PROBABILITY = scalars |
Probability density of U |
LPROBABILITIES = variates |
Probability densities of 0…U |
EXIT = scalars |
Set to 1 if it has not been possible to calculate the probabilities when there are ties, otherwise 0 |
Description
PRMANNWHITNEYU calculates various probabilities for the Mann-Whitney U statistic. This statistic arises from the Mann-Whitney U test, which can be used to give a nonparametric assessment as to whether two samples arise from the same probability distribution. If the samples are {xi: i=1…n1} and {yj: j=1…n2}, then the Mann-Whitney U statistic is defined as the number of pairs (xi, yj) with xi < yj. In Genstat, U can be calculated by the MANNWHITNEY procedure (which calls PRMANNWHITNEYU to obtain the required probability values).
The number of samples in the two sets of observations are specified by the N1 and N2 parameters, respectively. The U parameter specifies the value of the U statistic for which the probabilities are required, and the TIES parameter supplies the number of tied observations (if any). PRMANNWHITNEY may not be able to calculate the probabilities in every Genstat implementation when there are ties, and so there is also a parameter EXIT that you can set to check whether there have been problems (if the calculation has been successful EXIT=0, otherwise EXIT=1). The CLPROBABILITY and CUPROBABILITY parameters can specify scalars to save the cumulative lower and upper probabilities, pr(u ≤ U) and pr(u > U) respectively. PROBABILITY can save the probability density at U, pr(u = U), and LPROBABILITIES can save a variate containing the densities for 0…U.
Options: none.
Parameters: N1, N2, U, TIES, CLPROBABILITY, CUPROBABILITY, PROBABILITY, LPROBABILITIES, EXIT.
Method
The procedure calculates the coefficents of the generating function for the Mann-Whitney statistic under the null hypothesis using recurrence functions. The central limit theorem is used when the smaller of N1 and N2 exceeds 50, and a Normal approximation of the CDF is returned. (See Harding 1983). A separate program, that uses the method of Klotz & Cheung (1995), is called using PASS when there are ties. This may not be feasible in every Genstat implementation.
References
Harding, E.F. (1983) An efficient, minimal-storage procedure for calculating the Mann-Whitney U, Generalised U and similar distributions. Applied Statistics, 33, 1-6.
Klotz, J.H. & Cheung, Y.K. (1995). The Mann Whitney Wilcoxon distribution using linked lists. Statistica Sinica, 7, 805-813.
See also
Procedure: MANNWHITNEY.
Commands for: Basic and nonparametric statistics.
Example
CAPTION 'PRMANNWHITNEYU example',\
!t('Calculate the first part of Table J of Seigel (1956),',\
'Nonparametric Statistics for the Behavioural Sciences.');\
STYLE=meta,plain
VARIATE [VALUES=0...5] U; DECIMALS=0
& [NVALUES=U; VALUES=6(*)] Pr_N1[1,2,3]
FOR n1=1,2,3; umax=2,3,5
CALCULATE nu = umax + 1
PRMANNWHITNEYU #nu(n1); N2=3; U=0...umax; CLPROBABILITY=clpr[0...umax]
CALCULATE ELEMENTS(#nu(Pr_N1[n1]); 1...nu) = clpr[0...umax]
ENDFOR
PRINT [MISSING=' '] Pr_N1[]; DECIMALS=3