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 {*x _{i}*:

*i*=1…

*n*

_{1}} and {

*y*:

_{j}*j*=1…

*n*

_{2}}, then the Mann-Whitney U statistic is defined as the number of pairs (

*x*,

_{i}*y*) with

_{j}*x*<

_{i}*y*. In Genstat, U can be calculated by the

_{j}`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