scipy.stats.kstest

scipy.stats.kstest(rvs, cdf, args=(), N=20, alternative='two-sided', mode='approx')

Perform the Kolmogorov-Smirnov test for goodness of fit.

This performs a test of the distribution G(x) of an observed random variable against a given distribution F(x). Under the null hypothesis the two distributions are identical, G(x)=F(x). The alternative hypothesis can be either ‘two-sided’ (default), ‘less’ or ‘greater’. The KS test is only valid for continuous distributions.

Parameters:

rvs : str, array or callable

If a string, it should be the name of a distribution in scipy.stats. 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.

cdf : str or callable

If a string, it should be the name of a distribution in scipy.stats. If rvs is a string then cdf can be False or the same as rvs. If a callable, that callable is used to calculate the cdf.

args : tuple, sequence, optional

Distribution parameters, used if rvs or cdf are strings.

N : int, optional

Sample size if rvs is string or callable. Default is 20.

alternative : {‘two-sided’, ‘less’,’greater’}, optional

Defines the alternative hypothesis (see explanation above). Default is ‘two-sided’.

mode : ‘approx’ (default) or ‘asymp’, optional

Defines the distribution used for calculating the p-value.

• ‘approx’ : use approximation to exact distribution of test statistic
• ‘asymp’ : use asymptotic distribution of test statistic

Returns:

statistic : float

KS test statistic, either D, D+ or D-.

pvalue : float

One-tailed or two-tailed p-value.

Notes

In the one-sided test, the alternative is that the empirical cumulative distribution function of the random variable is “less” or “greater” than the cumulative distribution function F(x) of the hypothesis, G(x)<=F(x), resp. G(x)>=F(x).

Examples

>>> from scipy import stats

>>> x = np.linspace(-15, 15, 9)
>>> stats.kstest(x, 'norm')
(0.44435602715924361, 0.038850142705171065)

>>> np.random.seed(987654321) # set random seed to get the same result
>>> stats.kstest('norm', False, N=100)
(0.058352892479417884, 0.88531190944151261)

The above lines are equivalent to:

>>> np.random.seed(987654321)
>>> stats.kstest(stats.norm.rvs(size=100), 'norm')
(0.058352892479417884, 0.88531190944151261)

Test against one-sided alternative hypothesis

Shift distribution to larger values, so that cdf_dgp(x) < norm.cdf(x):

>>> np.random.seed(987654321)
>>> x = stats.norm.rvs(loc=0.2, size=100)
>>> stats.kstest(x,'norm', alternative = 'less')
(0.12464329735846891, 0.040989164077641749)

Reject equal distribution against alternative hypothesis: less

>>> stats.kstest(x,'norm', alternative = 'greater')
(0.0072115233216311081, 0.98531158590396395)

Don’t reject equal distribution against alternative hypothesis: greater

>>> stats.kstest(x,'norm', mode='asymp')
(0.12464329735846891, 0.08944488871182088)

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:

>>> np.random.seed(987654321)
>>> stats.kstest(stats.t.rvs(100,size=100),'norm')
(0.072018929165471257, 0.67630062862479168)

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:

>>> np.random.seed(987654321)
>>> stats.kstest(stats.t.rvs(3,size=100),'norm')
(0.131016895759829, 0.058826222555312224)