power_divergence(f_obs, f_exp=None, ddof=0, axis=0, lambda_=None)¶
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 - ddofdegrees 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 [R684]_. "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 [R686]_.
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
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.
[R682] Lowry, Richard. “Concepts and Applications of Inferential Statistics”. Chapter 8. http://faculty.vassar.edu/lowry/ch8pt1.html [R683] “Chi-squared test”, http://en.wikipedia.org/wiki/Chi-squared_test [R684] “G-test”, http://en.wikipedia.org/wiki/G-test [R685] Sokal, R. R. and Rohlf, F. J. “Biometry: the principles and practice of statistics in biological research”, New York: Freeman (1981) [R686] 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.
chisquarefor 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') (3.3281031458963746, 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]))
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
>>> 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]))