scipy.stats.power_divergence(f_obs, f_exp=None, ddof=0, axis=0, lambda_=None)[source]

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

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


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, optional

The axis of the broadcast result of f_obs and f_exp along which to apply the test. If axis is None, all values in f_obs are treated as a single data set. Default is 0.

lambda_ : float or str, optional

lambda_ gives 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:

String              Value   Description
"pearson"             1     Pearson's chi-squared statistic.
                            In this case, the function is
                            equivalent to `stats.chisquare`.
"log-likelihood"      0     Log-likelihood ratio. Also known as
                            the G-test [R354]_.
"freeman-tukey"      -1/2   Freeman-Tukey statistic.
"mod-log-likelihood" -1     Modified log-likelihood ratio.
"neyman"             -2     Neyman's statistic.
"cressie-read"        2/3   The power recommended in [R356]_.

statistic : float or ndarray

The Cressie-Read power divergence test statistic. The value is a float if axis is None or if` f_obs and f_exp are 1-D.

pvalue : float or ndarray

The p-value of the test. The value is a float if ddof and the return value stat are scalars.

See also



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.

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.

This function handles masked arrays. If an element of f_obs or f_exp is masked, then data at that position is ignored, and does not count towards the size of the data set.

New in version 0.13.0.


[R352]Lowry, Richard. “Concepts and Applications of Inferential Statistics”. Chapter 8.
[R353]“Chi-squared test”,
[R355]Sokal, R. R. and Rohlf, F. J. “Biometry: the principles and practice of statistics in biological research”, New York: Freeman (1981)
[R356]Cressie, N. and Read, T. R. C., “Multinomial Goodness-of-Fit Tests”, J. Royal Stat. Soc. Series B, Vol. 46, No. 3 (1984), pp. 440-464.


(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):

>>> from scipy.stats import power_divergence
>>> power_divergence([16, 18, 16, 14, 12, 12], lambda_='log-likelihood')
(2.006573162632538, 0.84823476779463769)

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')
(3.5, 0.62338762774958223)

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
(6, 2)
>>> power_divergence(obs, lambda_="log-likelihood")
(array([ 2.00657316,  6.77634498]), array([ 0.84823477,  0.23781225]))

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)
(23.31034482758621, 0.015975692534127565)
>>> power_divergence(obs.ravel())
(23.31034482758621, 0.015975692534127565)

ddof is the change to make to the default degrees of freedom.

>>> power_divergence([16, 18, 16, 14, 12, 12], ddof=1)
(2.0, 0.73575888234288467)

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])
(2.0, array([ 0.84914504,  0.73575888,  0.5724067 ]))

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)
(array([ 3.5 ,  9.25]), array([ 0.62338763,  0.09949846]))

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