g08akf calculates the exact tail probability for the Mann–Whitney rank sum test statistic for the case where there are ties in the two samples pooled together.
The routine may be called by the names g08akf or nagf_nonpar_prob_mwu_ties.
3Description
g08akf computes the exact tail probability for the Mann–Whitney $U$ test statistic (calculated by g08ahf and returned through the argument u) using a method based on an algorithm developed by Neumann (1988), for the case where there are ties in the pooled sample.
The Mann–Whitney $U$ test investigates the difference between two populations defined by the distribution functions $F\left(x\right)$ and $G\left(y\right)$ respectively. The data consist of two independent samples of size ${\mathit{n}}_{1}$ and ${\mathit{n}}_{2}$, denoted by ${x}_{1},{x}_{2},\dots ,{x}_{{\mathit{n}}_{1}}$ and ${y}_{1},{y}_{2},\dots ,{y}_{{\mathit{n}}_{2}}$, taken from the two populations.
The hypothesis under test, ${H}_{0}$, often called the null hypothesis, is that the two distributions are the same, that is $F\left(x\right)=G\left(x\right)$, and this is to be tested against an alternative hypothesis ${H}_{1}$ which is
${H}_{1}$: $F\left(x\right)\ne G\left(y\right)$; or
${H}_{1}$: $F\left(x\right)<G\left(y\right)$, i.e., the $x$'s tend to be greater than the $y$'s; or
${H}_{1}$: $F\left(x\right)>G\left(y\right)$, i.e., the $x$'s tend to be less than the $y$'s,
using a two tailed, upper tailed or lower tailed probability respectively. You select the alternative hypothesis by choosing the appropriate tail probability to be computed (see the description of argument tail in Section 5).
Note that when using this test to test for differences in the distributions one is primarily detecting differences in the location of the two distributions. That is to say, if we reject the null hypothesis ${H}_{0}$ in favour of the alternative hypothesis ${H}_{1}$: $F\left(x\right)>G\left(y\right)$ we have evidence to suggest that the location, of the distribution defined by $F\left(x\right)$, is less than the location of the distribution defined by $G\left(y\right)$.
g08akf returns the exact tail probability, $p$, corresponding to $U$, depending on the choice of alternative hypothesis, ${H}_{1}$.
The value of $p$ can be used to perform a significance test on the null hypothesis ${H}_{0}$ against the alternative hypothesis ${H}_{1}$. Let $\alpha $ be the size of the significance test (that is $\alpha $ is the probability of rejecting ${H}_{0}$ when ${H}_{0}$ is true). If $p<\alpha $ then the null hypothesis is rejected. Typically $\alpha $ might be $0.05$ or $0.01$.
4References
Conover W J (1980) Practical Nonparametric Statistics Wiley
Neumann N (1988) Some procedures for calculating the distributions of elementary nonparametric teststatistics Statistical Software Newsletter14(3) 120–126
Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill
5Arguments
1: $\mathbf{n1}$ – IntegerInput
On entry: the number of non-tied pairs, ${\mathit{n}}_{1}$.
Constraint:
${\mathbf{n1}}\ge 1$.
2: $\mathbf{n2}$ – IntegerInput
On entry: the size of the second sample, ${\mathit{n}}_{2}$.
Constraint:
${\mathbf{n2}}\ge 1$.
3: $\mathbf{tail}$ – Character(1)Input
On entry: indicates the choice of tail probability, and hence the alternative hypothesis.
${\mathbf{tail}}=\text{'T'}$
A two tailed probability is calculated and the alternative hypothesis is ${H}_{1}:F\left(x\right)\ne G\left(y\right)$.
${\mathbf{tail}}=\text{'U'}$
An upper tailed probability is calculated and the alternative hypothesis ${H}_{1}:F\left(x\right)<G\left(y\right)$, i.e., the $x$'s tend to be greater than the $y$'s.
${\mathbf{tail}}=\text{'L'}$
A lower tailed probability is calculated and the alternative hypothesis ${H}_{1}:F\left(x\right)>G\left(y\right)$, i.e., the $x$'s tend to be less than the $y$'s.
Constraint:
${\mathbf{tail}}=\text{'T'}$, $\text{'U'}$ or $\text{'L'}$.
4: $\mathbf{ranks}\left({\mathbf{n1}}+{\mathbf{n2}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the ranks of the pooled sample. These ranks are output in the array ranks by g08ahf and should not be altered in any way if you are using the same ${\mathit{n}}_{1}$, ${\mathit{n}}_{2}$ and ${\mathbf{u}}$ as used in g08ahf.
5: $\mathbf{u}$ – Real (Kind=nag_wp)Input
On entry: $U$, the value of the Mann–Whitney rank sum test statistic. This is the statistic returned through the argument u by g08ahf.
Constraint:
${\mathbf{u}}\ge 0.0$.
6: $\mathbf{p}$ – Real (Kind=nag_wp)Output
On exit: the tail probability, $p$, as specified by the argument tail.
7: $\mathbf{wrk}\left({\mathbf{lwrk}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
8: $\mathbf{lwrk}$ – IntegerInput
On entry: the dimension of the array wrk as declared in the (sub)program from which g08akf is called.
Constraint:
${\mathbf{lwrk}}\ge \mathit{n}+\mathit{n}(\mathit{n}+1)(\mathit{n}+\mathit{m})-\frac{\mathit{n}(\mathit{n}+1)(2\times \mathit{n}+1)}{3}+1$, where $\mathit{n}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}({\mathbf{n1}},{\mathbf{n2}})$ and $\mathit{m}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}({\mathbf{n1}},{\mathbf{n2}})$.
On entry: ifail must be set to $0$, $-1$ 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 $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ 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 $-1$, 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, ${\mathbf{n1}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{n1}}\ge 1$.
On entry, ${\mathbf{n2}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{n2}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{tail}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{tail}}=\text{'T'}$, $\text{'U'}$ or $\text{'L'}$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{u}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{u}}\ge 0.0$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{lwrk}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{lwrk}}\ge \u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=5$
On entry, at least one rank, given in ranks, was outside the expected range.
${\mathbf{ifail}}=-99$
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
The exact tail probability, $p$, is computed to an accuracy of at least $4$ significant figures.
8Parallelism and Performance
g08akf 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 g08akf increases with ${\mathit{n}}_{1}$ and ${\mathit{n}}_{2}$ and the product ${\mathit{n}}_{1}{\mathit{n}}_{2}$. Note that the amount of workspace required becomes very large for even moderate sizes of ${\mathit{n}}_{1}$ and ${\mathit{n}}_{2}$.
10Example
This example finds the Mann–Whitney test statistic, using g08ahf for two independent samples of size $16$ and $23$ respectively. This is used to test the null hypothesis that the distributions of the two populations from which the samples were taken are the same against the alternative hypothesis that the distributions are different. The test statistic, the approximate Normal statistic and the approximate two tail probability are printed. g08akf is then called to obtain the exact two tailed probability. The exact probability is also printed.