scipy.stats.kstest¶
- scipy.stats.kstest(rvs, cdf, args=(), N=20, alternative='two-sided', mode='auto')[source]¶
Performs the (one-sample or two-sample) Kolmogorov-Smirnov test for goodness of fit.
The one-sample test compares the underlying distribution F(x) of a sample against a given distribution G(x). The two-sample test compares the underlying distributions of two independent samples. Both tests are valid only for continuous distributions.
- Parameters
- rvsstr, array_like, or callable
If an array, it should be a 1-D array of observations of random variables. If a callable, it should be a function to generate random variables; it is required to have a keyword argument size. If a string, it should be the name of a distribution in
scipy.stats
, which will be used to generate random variables.- cdfstr, array_like or callable
If array_like, it should be a 1-D array of observations of random variables, and the two-sample test is performed (and rvs must be array_like). If a callable, that callable is used to calculate the cdf. If a string, it should be the name of a distribution in
scipy.stats
, which will be used as the cdf function.- argstuple, sequence, optional
Distribution parameters, used if rvs or cdf are strings or callables.
- Nint, optional
Sample size if rvs is string or callable. Default is 20.
- alternative{‘two-sided’, ‘less’, ‘greater’}, optional
Defines the null and alternative hypotheses. Default is ‘two-sided’. Please see explanations in the Notes below.
- mode{‘auto’, ‘exact’, ‘approx’, ‘asymp’}, optional
Defines the distribution used for calculating the p-value. The following options are available (default is ‘auto’):
‘auto’ : selects one of the other options.
‘exact’ : uses the exact distribution of test statistic.
‘approx’ : approximates the two-sided probability with twice the one-sided probability
‘asymp’: uses asymptotic distribution of test statistic
- Returns
- statisticfloat
KS test statistic, either D, D+ or D-.
- pvaluefloat
One-tailed or two-tailed p-value.
See also
Notes
There are three options for the null and corresponding alternative hypothesis that can be selected using the alternative parameter.
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.
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.
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.
Note that the alternative hypotheses describe the CDFs of the underlying distributions, not the observed values. 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.
Examples
>>> from scipy import stats >>> rng = np.random.default_rng()
>>> x = np.linspace(-15, 15, 9) >>> stats.kstest(x, 'norm') KstestResult(statistic=0.444356027159..., pvalue=0.038850140086...)
>>> stats.kstest(stats.norm.rvs(size=100, random_state=rng), stats.norm.cdf) KstestResult(statistic=0.165471391799..., pvalue=0.007331283245...)
The above lines are equivalent to:
>>> stats.kstest(stats.norm.rvs, 'norm', N=100) KstestResult(statistic=0.113810164200..., pvalue=0.138690052319...) # may vary
Test against one-sided alternative hypothesis
Shift distribution to larger values, so that
CDF(x) < norm.cdf(x)
:>>> x = stats.norm.rvs(loc=0.2, size=100, random_state=rng) >>> stats.kstest(x, 'norm', alternative='less') KstestResult(statistic=0.1002033514..., pvalue=0.1255446444...)
Reject null hypothesis in favor of alternative hypothesis: less
>>> stats.kstest(x, 'norm', alternative='greater') KstestResult(statistic=0.018749806388..., pvalue=0.920581859791...)
Don’t reject null hypothesis in favor of alternative hypothesis: greater
>>> stats.kstest(x, 'norm') KstestResult(statistic=0.100203351482..., pvalue=0.250616879765...)
Testing t distributed random variables against normal distribution
With 100 degrees of freedom the t distribution looks close to the normal distribution, and the K-S test does not reject the hypothesis that the sample came from the normal distribution:
>>> stats.kstest(stats.t.rvs(100, size=100, random_state=rng), 'norm') KstestResult(statistic=0.064273776544..., pvalue=0.778737758305...)
With 3 degrees of freedom the t distribution looks sufficiently different from the normal distribution, that we can reject the hypothesis that the sample came from the normal distribution at the 10% level:
>>> stats.kstest(stats.t.rvs(3, size=100, random_state=rng), 'norm') KstestResult(statistic=0.128678487493..., pvalue=0.066569081515...)