scipy.stats.wilcoxon#
- scipy.stats.wilcoxon(x, y=None, zero_method='wilcox', correction=False, alternative='two-sided', method='auto', *, axis=0, nan_policy='propagate', keepdims=False)[source]#
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:
- xarray_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 casey
is not to be specified.) Must be one-dimensional.- yarray_like, optional
Either the second set of measurements (if
x
is the first set of measurements), or not specified (ifx
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 ofd = x - y
. Roundoff error in the subtraction can result in elements ofd
being assigned different ranks even when they would be tied with exact arithmetic. Rather than passing x and y separately, consider computing the differencex - 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.
- correctionbool, 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 bothx
andy
are provided, ord = 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”, “approx”}, optional
Method to calculate the p-value, see Notes. Default is “auto”.
- axisint 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, aValueError
will be raised.
- keepdimsbool, 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.
- statisticarray_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.
- pvaluearray_like
The p-value for the test depending on alternative and method.
- zstatisticarray_like
When
method = 'approx'
, 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, andse
is the standard error. Whenmethod != 'approx'
, this attribute is not available.
See also
Notes
In the following, let
d
represent the difference between the paired samples:d = x - y
if bothx
andy
are provided, ord = x
otherwise. Assume that all elements ofd
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, andmethod = 'approx'
can be used to compute the p-value.When
len(d)
is small, the normal approximation may not be accurate, andmethod='exact'
is preferred (at the cost of additional execution time).The default,
method='auto'
, selects between the two: whenlen(d) <= 50
, the exact method is used; otherwise, the approximate method is used.
The presence of “ties” (i.e. not all elements of
d
are unique) and “zeros” (i.e. elements ofd
are zero) changes the null distribution of the test statistic, andmethod='exact'
no longer calculates the exact p-value. Ifmethod='approx'
, 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. There is no clear consensus among references on which method most accurately approximates the p-value for small samples in the presence of zeros and/or ties. In any case, this is the behavior ofwilcoxon
whenmethod='auto': ``method='exact'
is used whenlen(d) <= 50
and there are no zeros; otherwise,method='approx'
is used.Beginning in SciPy 1.9,
np.matrix
inputs (not recommended for new code) are converted tonp.ndarray
before the calculation is performed. In this case, the output will be a scalar ornp.ndarray
of appropriate shape rather than a 2Dnp.matrix
. Similarly, while masked elements of masked arrays are ignored, the output will be a scalar ornp.ndarray
rather than a masked array withmask=False
.References
[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] (1,2)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='approx') >>> 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 samplesx
andy
:>>> 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=0.4375)
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 magnitude0.05
, they have slightly different magnitudes in practice, and therefore are assigned different ranks in the test. Before performing the test, consider calculatingd
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=0.4375)