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def ks_2samp(data1, data2, alternative='two-sided', method='auto', *, axis=0, nan_policy='propagate', keepdims=False)
Performs the two-sample Kolmogorov-Smirnov test for goodness of fit.
This test compares the underlying continuous distributions F(x) and G(x)
of two independent samples. See Notes for a description of the available
null and alternative hypotheses.
Parameters
----------
data1, data2 : array_like, 1-Dimensional
Two arrays of sample observations assumed to be drawn from a continuous
distribution, sample sizes can be different.
alternative : {'two-sided', 'less', 'greater'}, optional
Defines the null and alternative hypotheses. Default is 'two-sided'.
Please see explanations in the Notes below.
method : {'auto', 'exact', 'asymp'}, optional
Defines the method used for calculating the p-value.
The following options are available (default is 'auto'):
* 'auto' : use 'exact' for small size arrays, 'asymp' for large
* 'exact' : use exact distribution of test statistic
* 'asymp' : use asymptotic distribution of test statistic
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
-------
res: KstestResult
An object containing attributes:
statistic : float
KS test statistic.
pvalue : float
One-tailed or two-tailed p-value.
statistic_location : float
Value from `data1` or `data2` corresponding with the KS statistic;
i.e., the distance between the empirical distribution functions is
measured at this observation.
statistic_sign : int
+1 if the empirical distribution function of `data1` exceeds
the empirical distribution function of `data2` at
`statistic_location`, otherwise -1.
See Also
--------
:func:`kstest`, :func:`ks_1samp`, :func:`epps_singleton_2samp`, :func:`anderson_ksamp`
..
Notes
-----
There are three options for the null and corresponding alternative
hypothesis that can be selected using the `alternative` parameter.
- `less`: The null hypothesis is that F(x) >= G(x) for all x; the
alternative is that F(x) < G(x) for at least one x. The statistic
is the magnitude of the minimum (most negative) difference between the
empirical distribution functions of the samples.
- `greater`: The null hypothesis is that F(x) <= G(x) for all x; the
alternative is that F(x) > G(x) for at least one x. The statistic
is the maximum (most positive) difference between the empirical
distribution functions of the samples.
- `two-sided`: The null hypothesis is that the two distributions are
identical, F(x)=G(x) for all x; the alternative is that they are not
identical. The statistic is the maximum absolute difference between the
empirical distribution functions of the samples.
Note that the alternative hypotheses describe the *CDFs* of the
underlying distributions, not the observed values of the data. For example,
suppose x1 ~ F and x2 ~ G. If F(x) > G(x) for all x, the values in
x1 tend to be less than those in x2.
If the KS statistic is large, then the p-value will be small, and this may
be taken as evidence against the null hypothesis in favor of the
alternative.
If ``method='exact'``, `ks_2samp` attempts to compute an exact p-value,
that is, the probability under the null hypothesis of obtaining a test
statistic value as extreme as the value computed from the data.
If ``method='asymp'``, the asymptotic Kolmogorov-Smirnov distribution is
used to compute an approximate p-value.
If ``method='auto'``, an exact p-value computation is attempted if both
sample sizes are less than 10000; otherwise, the asymptotic method is used.
In any case, if an exact p-value calculation is attempted and fails, a
warning will be emitted, and the asymptotic p-value will be returned.
The 'two-sided' 'exact' computation computes the complementary probability
and then subtracts from 1. As such, the minimum probability it can return
is about 1e-16. While the algorithm itself is exact, numerical
errors may accumulate for large sample sizes. It is most suited to
situations in which one of the sample sizes is only a few thousand.
We generally follow Hodges' treatment of Drion/Gnedenko/Korolyuk [1]_.
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] Hodges, J.L. Jr., "The Significance Probability of the Smirnov
Two-Sample Test," Arkiv fiur Matematik, 3, No. 43 (1958), 469-486.
Examples
--------
Suppose we wish to test the null hypothesis that two samples were drawn
from the same distribution.
We choose a confidence level of 95%; that is, we will reject the null
hypothesis in favor of the alternative if the p-value is less than 0.05.
If the first sample were drawn from a uniform distribution and the second
were drawn from the standard normal, we would expect the null hypothesis
to be rejected.
>>> import numpy as np
>>> from scipy import stats
>>> rng = np.random.default_rng()
>>> sample1 = stats.uniform.rvs(size=100, random_state=rng)
>>> sample2 = stats.norm.rvs(size=110, random_state=rng)
>>> stats.ks_2samp(sample1, sample2)
KstestResult(statistic=0.5454545454545454,
pvalue=7.37417839555191e-15,
statistic_location=-0.014071496412861274,
statistic_sign=-1)
Indeed, the p-value is lower than our threshold of 0.05, so we reject the
null hypothesis in favor of the default "two-sided" alternative: the data
were *not* drawn from the same distribution.
When both samples are drawn from the same distribution, we expect the data
to be consistent with the null hypothesis most of the time.
>>> sample1 = stats.norm.rvs(size=105, random_state=rng)
>>> sample2 = stats.norm.rvs(size=95, random_state=rng)
>>> stats.ks_2samp(sample1, sample2)
KstestResult(statistic=0.10927318295739348,
pvalue=0.5438289009927495,
statistic_location=-0.1670157701848795,
statistic_sign=-1)
As expected, the p-value of 0.54 is not below our threshold of 0.05, so
we cannot reject the null hypothesis.
Suppose, however, that the first sample were drawn from
a normal distribution shifted toward greater values. In this case,
the cumulative density function (CDF) of the underlying distribution tends
to be *less* than the CDF underlying the second sample. Therefore, we would
expect the null hypothesis to be rejected with ``alternative='less'``:
>>> sample1 = stats.norm.rvs(size=105, loc=0.5, random_state=rng)
>>> stats.ks_2samp(sample1, sample2, alternative='less')
KstestResult(statistic=0.4055137844611529,
pvalue=3.5474563068855554e-08,
statistic_location=-0.13249370614972575,
statistic_sign=-1)
and indeed, with p-value smaller than our threshold, we reject the null
hypothesis in favor of the alternative.
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