scipy.stats.

anderson#

scipy.stats.anderson(x, dist='norm', *, method='interpolate', axis=0, nan_policy='propagate', keepdims=False)[source]#

Anderson-Darling test for data coming from a particular distribution.

The Anderson-Darling test tests the null hypothesis that a sample is drawn from a population that follows a particular distribution. For the Anderson-Darling test, the critical values depend on which distribution is being tested against. This function works for normal, exponential, logistic, weibull_min, or Gumbel (Extreme Value Type I) distributions.

Parameters:
xarray_like

Array of sample data.

dist{‘norm’, ‘expon’, ‘logistic’, ‘gumbel’, ‘gumbel_l’, ‘gumbel_r’, ‘extreme1’, ‘weibull_min’}, optional

The type of distribution to test against. The default is ‘norm’. The names ‘extreme1’, ‘gumbel_l’ and ‘gumbel’ are synonyms for the same distribution. Only ‘norm’ and ‘expon’ are compatible with batched input (multidimensional x).

methodstr or instance of MonteCarloMethod

Defines the method used to compute the p-value. If method is "interpolate", the p-value is interpolated from pre-calculated tables (without extrapolating). This is the only method implemented for batched input (multidimensional x). If method is an instance of MonteCarloMethod, the p-value is computed using scipy.stats.monte_carlo_test with the provided configuration options and other appropriate settings.

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, a ValueError will be raised.

nan_policy='raise' and nan_policy='omit' are not supported by lazy backends, and nan_policy='omit' is not compatible with MArray input. For multidimensional input, nan_policy='omit' is supported only by the NumPy backend. Instead, mask NaN values using MArray; and use the default nan_policy. See Notes for support information.

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:
resultSignificanceResult

If method is provided, this is an object with the following attributes:

statisticfloat

The Anderson-Darling test statistic.

pvalue: float

The p-value corresponding with the test statistic, calculated according to the specified method.

See also

kstest

The Kolmogorov-Smirnov test for goodness-of-fit.

Notes

For weibull_min, maximum likelihood estimation is known to be challenging. If the test returns successfully, then the first order conditions for a maximum likelihood estimate have been verified and the critical values correspond relatively well to the significance levels, provided that the sample is sufficiently large (>10 observations [7]). However, for some data - especially data with no left tail - anderson is likely to result in an error message. In this case, consider performing a custom goodness of fit test using scipy.stats.monte_carlo_test.

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.

Deprecated since version 2.0.0: Support for NumPy masked arrays is deprecated and will be removed in SciPy 2.4.0. See Array API Standard Support table below for whether MArrays are accepted; otherwise, use regular NumPy arrays, replacing masked values with NaNs, and pass argument nan_poliy=’omit’.

Array API Standard Support

anderson has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library

CPU

GPU

NumPy

✅

n/a

CuPy

n/a

✅

PyTorch

✅

✅

JAX

✅

✅

Dask

⛔

n/a

Only dist=’norm’ and dist=’expon’ with method=’interpolate’ are implemented for non-NumPy arrays.

See Support for the array API standard for more information.

References

[2]

Stephens, M. A. (1974). EDF Statistics for Goodness of Fit and Some Comparisons, Journal of the American Statistical Association, Vol. 69, pp. 730-737.

[3]

Stephens, M. A. (1976). Asymptotic Results for Goodness-of-Fit Statistics with Unknown Parameters, Annals of Statistics, Vol. 4, pp. 357-369.

[4]

Stephens, M. A. (1977). Goodness of Fit for the Extreme Value Distribution, Biometrika, Vol. 64, pp. 583-588.

[5]

Stephens, M. A. (1977). Goodness of Fit with Special Reference to Tests for Exponentiality , Technical Report No. 262, Department of Statistics, Stanford University, Stanford, CA.

[6]

Stephens, M. A. (1979). Tests of Fit for the Logistic Distribution Based on the Empirical Distribution Function, Biometrika, Vol. 66, pp. 591-595.

[7]

Richard A. Lockhart and Michael A. Stephens “Estimation and Tests of Fit for the Three-Parameter Weibull Distribution” Journal of the Royal Statistical Society.Series B(Methodological) Vol. 56, No. 3 (1994), pp. 491-500, Table 0.

[8]

D’Agostino, Ralph B. (1986). “Tests for the Normal Distribution”. In: Goodness-of-Fit Techniques. Ed. by Ralph B. D’Agostino and Michael A. Stephens. New York: Marcel Dekker, pp. 122-141. ISBN: 0-8247-7487-6.

Examples

Test the null hypothesis that a random sample was drawn from a normal distribution (with unspecified mean and standard deviation).

>>> import numpy as np
>>> from scipy.stats import anderson
>>> rng = np.random.default_rng()
>>> data = rng.random(size=35)
>>> res = anderson(data, dist='norm', method='interpolate')
>>> res.statistic
np.float64(0.9887620209957291)
>>> res.pvalue
np.float64(0.012111200538380142)

The p-value is approximately 0.012,, so the null hypothesis may be rejected at a significance level of 2.5%, but not at a significance level of 1%.

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