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scipy.stats.

# power_divergence

scipy.stats.power_divergence(f_obs, f_exp=None, ddof=0, axis=0, lambda_=None, *, nan_policy='propagate', keepdims=False)

  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.

  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:

      statisticfloat 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.

      pvaluefloat or ndarray

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

  See also

  `chisquare`

  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 | ✅ | ✅
  Dask | ✅ | n/a

  `power_divergence` also accepts MArrays backed by the backends indicated above; masked values will be treated as though they were not present.

  See Support for the array API standard for more information.

  References

  [1]

  Lowry, Richard. “Concepts and Applications of Inferential Statistics”. Chapter 8. https://web.archive.org/web/20171015035606/http://faculty.vassar.edu/lowry/ch8pt1.html

  [2]

  “Chi-squared test”, https://en.wikipedia.org/wiki/Chi-squared_test

  [3]

  “G-test”, https://en.wikipedia.org/wiki/G-test

  [4]

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

  [5]

  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.

  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')
      (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')
      (3.3281031458963746, 0.6495419288047497)

  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]))