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bundles / scipy latest / scipy / stats / _stats_py / power_divergence

function

scipy.stats._stats_py:power_divergence

source: /scipy/stats/_stats_py.py :6977

Signature

def   power_divergence ( f_obs ,  f_exp = None ,  ddof = 0 ,  axis = 0 ,  lambda_ = None ,  * ,  nan_policy = propagate ,  keepdims = False )

Summary

Cressie-Read power divergence statistic and goodness of fit test.

Extended Summary

This function tests the null hypothesis that the categorical data has the given frequencies, using the Cressie-Read power divergence statistic.

Parameters

f_obs : array_like

Observed frequencies in each category.

f_exp : array_like, optional

Expected frequencies in each category. By default the categories are assumed to be equally likely.

ddof : int, optional

"Delta degrees of freedom": adjustment to the degrees of freedom for the p-value. The p-value is computed using a chi-squared distribution with k - 1 - ddof degrees of freedom, where k is the number of observed frequencies. The default value of ddof is 0.

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.

lambda_ : float or str, optional

The power in the Cressie-Read power divergence statistic. The default is 1. For convenience, lambda_ may be assigned one of the following strings, in which case the corresponding numerical value is used:

  • "pearson" (value 1)

    Pearson's chi-squared statistic. In this case, the function is equivalent to chisquare.

  • "log-likelihood" (value 0)

    Log-likelihood ratio. Also known as the G-test [3].

  • "freeman-tukey" (value -1/2)

    Freeman-Tukey statistic.

  • "mod-log-likelihood" (value -1)

    Modified log-likelihood ratio.

  • "neyman" (value -2)

    Neyman's statistic.

  • "cressie-read" (value 2/3)

    The power recommended in [5].

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: Power_divergenceResult

An object containing attributes:

statistic

statistic

pvalue

pvalue

Notes

This test is invalid when the observed or expected frequencies in each category are too small. A typical rule is that all of the observed and expected frequencies should be at least 5.

Also, the sum of the observed and expected frequencies must be the same for the test to be valid; power_divergence raises an error if the sums do not agree within a relative tolerance of eps**0.5, where eps is the precision of the input dtype.

When lambda_ is less than zero, the formula for the statistic involves dividing by f_obs, so a warning or error may be generated if any value in f_obs is 0.

Similarly, a warning or error may be generated if any value in f_exp is zero when lambda_ >= 0.

The default degrees of freedom, k-1, are for the case when no parameters of the distribution are estimated. If p parameters are estimated by efficient maximum likelihood then the correct degrees of freedom are k-1-p. If the parameters are estimated in a different way, then the dof can be between k-1-p and k-1. However, it is also possible that the asymptotic distribution is not a chisquare, in which case this test is not appropriate.

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.

Array API Standard Support

power_divergence has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

====================  ====================  ====================
Library               CPU                   GPU
====================  ====================  ====================
NumPy                 ✅                     n/a                 
CuPy                  n/a                   ✅                   
PyTorch               ✅                     ✅                   
JAX                   ⚠️ no JIT             ⚠️ no JIT           
Dask                  ⚠️ computes graph     n/a                 
====================  ====================  ====================

See dev-arrayapi for more information.

Examples

(See `chisquare` for more examples.) When just `f_obs` is given, it is assumed that the expected frequencies are uniform and given by the mean of the observed frequencies. Here we perform a G-test (i.e. use the log-likelihood ratio statistic): 
import numpy as np
from scipy.stats import power_divergence
✓
power_divergence([16, 18, 16, 14, 12, 12], lambda_='log-likelihood')
✗
The expected frequencies can be given with the `f_exp` argument: 
power_divergence([16, 18, 16, 14, 12, 12],
                 f_exp=[16, 16, 16, 16, 16, 8],
                 lambda_='log-likelihood')
✗
When `f_obs` is 2-D, by default the test is applied to each column. 
obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T
obs.shape
✓
power_divergence(obs, lambda_="log-likelihood")
✗
By setting ``axis=None``, the test is applied to all data in the array, which is equivalent to applying the test to the flattened array. 
power_divergence(obs, axis=None)
power_divergence(obs.ravel())
✗
`ddof` is the change to make to the default degrees of freedom. 
power_divergence([16, 18, 16, 14, 12, 12], ddof=1)
✗
The calculation of the p-values is done by broadcasting the test statistic with `ddof`. 
power_divergence([16, 18, 16, 14, 12, 12], ddof=[0,1,2])
✗
`f_obs` and `f_exp` are also broadcast. In the following, `f_obs` has shape (6,) and `f_exp` has shape (2, 6), so the result of broadcasting `f_obs` and `f_exp` has shape (2, 6). To compute the desired chi-squared statistics, we must use ``axis=1``: 
power_divergence([16, 18, 16, 14, 12, 12],
                 f_exp=[[16, 16, 16, 16, 16, 8],
                        [8, 20, 20, 16, 12, 12]],
                 axis=1)
✗

See also

chisquare

Aliases

  • scipy.stats.power_divergence

Referenced by

This package