mars.tensor.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) –

    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 [3]_.
    "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 [5]_.


  • 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.



Lowry, Richard. “Concepts and Applications of Inferential Statistics”. Chapter 8.


“Chi-squared test”,




Sokal, R. R. and Rohlf, F. J. “Biometry: the principles and practice of statistics in biological research”, New York: Freeman (1981)


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):

>>> import mars.tensor as mt
>>> from mars.tensor.stats import power_divergence
>>> power_divergence([16, 18, 16, 14, 12, 12], lambda_='log-likelihood').execute()
(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').execute()
(3.3281031458963746, 0.6495419288047497)

When f_obs is 2-D, by default the test is applied to each column.

>>> obs = mt.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T
>>> obs.shape
(6, 2)
>>> power_divergence(obs, lambda_="log-likelihood").execute()
(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).execute()
(23.31034482758621, 0.015975692534127565)
>>> power_divergence(obs.ravel()).execute()
(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).execute()
(2.0, 0.73575888234288467)

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]))