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def wilcoxon(x, y=None, zero_method='wilcox', correction=False, alternative='two-sided', method='auto', *, axis=0, nan_policy='propagate', keepdims=False)
Calculate the Wilcoxon signed-rank test.
The Wilcoxon signed-rank test tests the null hypothesis that two
related paired samples come from the same distribution. In particular,
it tests whether the distribution of the differences ``x - y`` is symmetric
about zero. It is a non-parametric version of the paired T-test.
Parameters
----------
x : array_like
Either the first set of measurements (in which case ``y`` is the second
set of measurements), or the differences between two sets of
measurements (in which case ``y`` is not to be specified.) Must be
one-dimensional.
y : array_like, optional
Either the second set of measurements (if ``x`` is the first set of
measurements), or not specified (if ``x`` is the differences between
two sets of measurements.) Must be one-dimensional.
.. warning::
When `y` is provided, `wilcoxon` calculates the test statistic
based on the ranks of the absolute values of ``d = x - y``.
Roundoff error in the subtraction can result in elements of ``d``
being assigned different ranks even when they would be tied with
exact arithmetic. Rather than passing `x` and `y` separately,
consider computing the difference ``x - y``, rounding as needed to
ensure that only truly unique elements are numerically distinct,
and passing the result as `x`, leaving `y` at the default (None).
zero_method : {"wilcox", "pratt", "zsplit"}, optional
There are different conventions for handling pairs of observations
with equal values ("zero-differences", or "zeros").
* "wilcox": Discards all zero-differences (default); see [4]_.
* "pratt": Includes zero-differences in the ranking process,
but drops the ranks of the zeros (more conservative); see [3]_.
In this case, the normal approximation is adjusted as in [5]_.
* "zsplit": Includes zero-differences in the ranking process and
splits the zero rank between positive and negative ones.
correction : bool, optional
If True, apply continuity correction by adjusting the Wilcoxon rank
statistic by 0.5 towards the mean value when computing the
z-statistic if a normal approximation is used. Default is False.
alternative : {"two-sided", "greater", "less"}, optional
Defines the alternative hypothesis. Default is 'two-sided'.
In the following, let ``d`` represent the difference between the paired
samples: ``d = x - y`` if both ``x`` and ``y`` are provided, or
``d = x`` otherwise.
* 'two-sided': the distribution underlying ``d`` is not symmetric
about zero.
* 'less': the distribution underlying ``d`` is stochastically less
than a distribution symmetric about zero.
* 'greater': the distribution underlying ``d`` is stochastically
greater than a distribution symmetric about zero.
method : {"auto", "exact", "asymptotic"} or `PermutationMethod` instance, optional
Method to calculate the p-value, see Notes. Default is "auto".
axis : int or None, default: 0
If an int, the axis of the input along which to compute the statistic.
The statistic of each axis-slice (e.g. row) of the input will appear in a
corresponding element of the output.
If ``None``, the input will be raveled before computing the statistic.
nan_policy : {'propagate', 'omit', 'raise'}
Defines how to handle input NaNs.
- ``propagate``: if a NaN is present in the axis slice (e.g. row) along
which the statistic is computed, the corresponding entry of the output
will be NaN.
- ``omit``: NaNs will be omitted when performing the calculation.
If insufficient data remains in the axis slice along which the
statistic is computed, the corresponding entry of the output will be
NaN.
- ``raise``: if a NaN is present, a ``ValueError`` will be raised.
keepdims : bool, default: False
If this is set to True, the axes which are reduced are left
in the result as dimensions with size one. With this option,
the result will broadcast correctly against the input array.
Returns
-------
An object with the following attributes.
statistic : array_like
If `alternative` is "two-sided", the sum of the ranks of the
differences above or below zero, whichever is smaller.
Otherwise the sum of the ranks of the differences above zero.
pvalue : array_like
The p-value for the test depending on `alternative` and `method`.
zstatistic : array_like
When ``method = 'asymptotic'``, this is the normalized z-statistic::
z = (T - mn - d) / se
where ``T`` is `statistic` as defined above, ``mn`` is the mean of the
distribution under the null hypothesis, ``d`` is a continuity
correction, and ``se`` is the standard error.
When ``method != 'asymptotic'``, this attribute is not available.
See Also
--------
:func:`kruskal`, :func:`mannwhitneyu`
..
Notes
-----
In the following, let ``d`` represent the difference between the paired
samples: ``d = x - y`` if both ``x`` and ``y`` are provided, or ``d = x``
otherwise. Assume that all elements of ``d`` are independent and
identically distributed observations, and all are distinct and nonzero.
- When ``len(d)`` is sufficiently large, the null distribution of the
normalized test statistic (`zstatistic` above) is approximately normal,
and ``method = 'asymptotic'`` can be used to compute the p-value.
- When ``len(d)`` is small, the normal approximation may not be accurate,
and ``method='exact'`` is preferred (at the cost of additional
execution time).
- The default, ``method='auto'``, selects between the two:
``method='exact'`` is used when ``len(d) <= 50``, and
``method='asymptotic'`` is used otherwise.
The presence of "ties" (i.e. not all elements of ``d`` are unique) or
"zeros" (i.e. elements of ``d`` are zero) changes the null distribution
of the test statistic, and ``method='exact'`` no longer calculates
the exact p-value. If ``method='asymptotic'``, the z-statistic is adjusted
for more accurate comparison against the standard normal, but still,
for finite sample sizes, the standard normal is only an approximation of
the true null distribution of the z-statistic. For such situations, the
`method` parameter also accepts instances of `PermutationMethod`. In this
case, the p-value is computed using `permutation_test` with the provided
configuration options and other appropriate settings.
The presence of ties and zeros affects the resolution of ``method='auto'``
accordingly: exhasutive permutations are performed when ``len(d) <= 13``,
and the asymptotic method is used otherwise. Note that they asymptotic
method may not be very accurate even for ``len(d) > 14``; the threshold
was chosen as a compromise between execution time and accuracy under the
constraint that the results must be deterministic. Consider providing an
instance of `PermutationMethod` method manually, choosing the
``n_resamples`` parameter to balance time constraints and accuracy
requirements.
Please also note that in the edge case that all elements of ``d`` are zero,
the p-value relying on the normal approximaton cannot be computed (NaN)
if ``zero_method='wilcox'`` or ``zero_method='pratt'``.
Beginning in SciPy 1.9, ``np.matrix`` inputs (not recommended for new
code) are converted to ``np.ndarray`` before the calculation is performed. In
this case, the output will be a scalar or ``np.ndarray`` of appropriate shape
rather than a 2D ``np.matrix``. Similarly, while masked elements of masked
arrays are ignored, the output will be a scalar or ``np.ndarray`` rather than a
masked array with ``mask=False``.
References
----------
.. [1] https://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test
.. [2] Conover, W.J., Practical Nonparametric Statistics, 1971.
.. [3] Pratt, J.W., Remarks on Zeros and Ties in the Wilcoxon Signed
Rank Procedures, Journal of the American Statistical Association,
Vol. 54, 1959, pp. 655-667. :doi:`10.1080/01621459.1959.10501526`
.. [4] Wilcoxon, F., Individual Comparisons by Ranking Methods,
Biometrics Bulletin, Vol. 1, 1945, pp. 80-83. :doi:`10.2307/3001968`
.. [5] Cureton, E.E., The Normal Approximation to the Signed-Rank
Sampling Distribution When Zero Differences are Present,
Journal of the American Statistical Association, Vol. 62, 1967,
pp. 1068-1069. :doi:`10.1080/01621459.1967.10500917`
Examples
--------
In [4]_, the differences in height between cross- and self-fertilized
corn plants is given as follows:
>>> d = [6, 8, 14, 16, 23, 24, 28, 29, 41, -48, 49, 56, 60, -67, 75]
Cross-fertilized plants appear to be higher. To test the null
hypothesis that there is no height difference, we can apply the
two-sided test:
>>> from scipy.stats import wilcoxon
>>> res = wilcoxon(d)
>>> res.statistic, res.pvalue
(24.0, 0.041259765625)
Hence, we would reject the null hypothesis at a confidence level of 5%,
concluding that there is a difference in height between the groups.
To confirm that the median of the differences can be assumed to be
positive, we use:
>>> res = wilcoxon(d, alternative='greater')
>>> res.statistic, res.pvalue
(96.0, 0.0206298828125)
This shows that the null hypothesis that the median is negative can be
rejected at a confidence level of 5% in favor of the alternative that
the median is greater than zero. The p-values above are exact. Using the
normal approximation gives very similar values:
>>> res = wilcoxon(d, method='asymptotic')
>>> res.statistic, res.pvalue
(24.0, 0.04088813291185591)
Note that the statistic changed to 96 in the one-sided case (the sum
of ranks of positive differences) whereas it is 24 in the two-sided
case (the minimum of sum of ranks above and below zero).
In the example above, the differences in height between paired plants are
provided to `wilcoxon` directly. Alternatively, `wilcoxon` accepts two
samples of equal length, calculates the differences between paired
elements, then performs the test. Consider the samples ``x`` and ``y``:
>>> import numpy as np
>>> x = np.array([0.5, 0.825, 0.375, 0.5])
>>> y = np.array([0.525, 0.775, 0.325, 0.55])
>>> res = wilcoxon(x, y, alternative='greater')
>>> res
WilcoxonResult(statistic=5.0, pvalue=0.5625)
Note that had we calculated the differences by hand, the test would have
produced different results:
>>> d = [-0.025, 0.05, 0.05, -0.05]
>>> ref = wilcoxon(d, alternative='greater')
>>> ref
WilcoxonResult(statistic=6.0, pvalue=0.5)
The substantial difference is due to roundoff error in the results of
``x-y``:
>>> d - (x-y)
array([2.08166817e-17, 6.93889390e-17, 1.38777878e-17, 4.16333634e-17])
Even though we expected all the elements of ``(x-y)[1:]`` to have the same
magnitude ``0.05``, they have slightly different magnitudes in practice,
and therefore are assigned different ranks in the test. Before performing
the test, consider calculating ``d`` and adjusting it as necessary to
ensure that theoretically identically values are not numerically distinct.
For example:
>>> d2 = np.around(x - y, decimals=3)
>>> wilcoxon(d2, alternative='greater')
WilcoxonResult(statistic=6.0, pvalue=0.5)
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