naginterfaces.library.nonpar.test_​wilcoxon

naginterfaces.library.nonpar.test_wilcoxon(x, xme, tail, zer)

test_wilcoxon performs the Wilcoxon signed rank test on a single sample of size $n$.

For full information please refer to the NAG Library document for g08ag

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g08/g08agf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

The sample observations, $x_1,x_2,\dots ,x_n$.

xmefloat

The median test value, $X_{med}$.

tailstr, length 1

Indicates the choice of tail probability, and hence the alternative hypothesis.

$tail=\text{'T'}$

A two tailed probability is calculated and the alternative hypothesis is $H_1$: population median $\ne X_{med}$.

$tail=\text{'U'}$

An upper tailed probability is calculated and the alternative hypothesis is $H_1$: population median $>X_{med}$.

$tail=\text{'L'}$

A lower tailed probability is calculated and the alternative hypothesis is $H_1$: population median $<X_{med}$.

zerstr, length 1

Indicates whether or not to include the cases where $d_i=0.0$ in the ranking of the $d_i$s.

$zer=\text{'Y'}$

All $d_i=0.0$ are included when ranking.

$zer=\text{'N'}$

All $d_i=0.0$, are ignored, that is all cases where $d_i=0.0$ are removed before ranking.

Returns

wfloat

The Wilcoxon rank sum statistic, $W$, being the sum of the positive ranks.

wnorfloat

The approximate Normal test statistic, $z$, as described in Notes.

pfloat

The tail probability, $p$, as specified by the argument $tail$.

n1int

The number of nonzero $d_i$’s, $n_1$.

Raises

NagValueError

(errno $1$)

On entry, $zer=⟨value⟩$.

Constraint: $zer=\text{'Y'}$ or $\text{'N'}$.

(errno $1$)

On entry, $tail=⟨value⟩$.

Constraint: $tail=\text{'T'}$, $\text{'U'}$ or $\text{'L'}$.

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

Warns

NagAlgorithmicWarning

(errno $3$)

All elements of the sample are equal to $xme$, i.e., $\text{variance}=0$.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

The Wilcoxon one-sample signed rank test may be used to test whether a particular sample came from a population with a specified median. It is assumed that the population distribution is symmetric. The data consists of a single sample of $n$ observations denoted by $x_1,x_2,\dots ,x_n$. This sample may arise from the difference between pairs of observations from two matched samples of equal size taken from two populations, in which case the test may be used to test whether the median of the first population is the same as that of the second population.

The hypothesis under test, $H_0$, often called the null hypothesis, is that the median is equal to some given value $\left(X_{med}\right)$, and this is to be tested against an alternative hypothesis $H_1$ which is

$H_1$: population median $\ne X_{med}$; or

$H_1$: population median $>X_{med}$; or

$H_1$: population median $<X_{med}$,

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 Parameters).

The Wilcoxon test differs from the Sign test (see test_sign()) in that the magnitude of the scores is taken into account, rather than simply the direction of such scores.

The test procedure is as follows

1. For each $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$, the signed difference $d_i=x_i-X_{med}$ is found, where $X_{med}$ is a given test value for the median of the sample.

2. The absolute differences $\left|d_i\right|$ are ranked with rank $r_i$ and any tied values of $\left|d_i\right|$ are assigned the average of the tied ranks. You may choose whether or not to ignore any cases where $d_i=0$ by removing them before or after ranking (see the description of the argument $zer$ in Parameters).

3. The number of nonzero $d_i$ is found.

4. To each rank is affixed the sign of the $d_i$ to which it corresponds. Let $s_i=sign\left(d_i\right)r_i$.

5. The sum of the positive-signed ranks, $W={\sum }_{s_i>0}{s}_i={\sum }_{i=1}^{n}max(s_i,0.0)$, is calculated.

test_wilcoxon returns

1. the test statistic $W$;

2. the number $n_1$ of nonzero $d_i$;

3. the approximate Normal test statistic $z$, where

   $$z=\frac{(W-\frac{n_1(n_1+1)}{4})-sign(W-\frac{n_1(n_1+1)}{4})\times \frac{1}{2}}{\sqrt{\frac{1}{4}{\sum }_{i=1}^n{s}_i^2}}\text{;}$$

4. the tail probability, $p$, corresponding to $W$, depending on the choice of the alternative hypothesis, $H_1$.

If $n_1\le 80$, $p$ is computed exactly; otherwise, an approximation to $p$ is returned based on an approximate Normal statistic corrected for continuity according to the tail specified.

The value of $p$ can be used to perform a significance test on the median against the alternative hypothesis. 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 $0.01$.

References

Conover, W J, 1980, Practical Nonparametric Statistics, Wiley

Neumann, N, 1988, Some procedures for calculating the distributions of elementary nonparametric teststatistics, Statistical Software Newsletter (14(3)), 120–126

Siegel, S, 1956, Non-parametric Statistics for the Behavioral Sciences, McGraw–Hill