statsmodels.stats.gof.powerdiscrepancy

statsmodels.stats.gof.powerdiscrepancy(observed, expected, lambd=0.0, axis=0, ddof=0)

Calculates power discrepancy, a class of goodness-of-fit tests as a measure of discrepancy between observed and expected data.

This contains several goodness-of-fit tests as special cases, see the describtion of lambd, the exponent of the power discrepancy. The pvalue is based on the asymptotic chi-square distribution of the test statistic.

freeman_tukey: D(x| heta) = sum_j (sqrt{x_j} - sqrt{e_j})^2

Parameters:

• o (Iterable) – Observed values
• e (Iterable) – Expected values
• lambd (float or string) –
  • float : exponent a for power discrepancy
  • ’loglikeratio’: a = 0
  • ’freeman_tukey’: a = -0.5
  • ’pearson’: a = 1 (standard chisquare test statistic)
  • ’modified_loglikeratio’: a = -1
  • ’cressie_read’: a = 2/3
  • ’neyman’ : a = -2 (Neyman-modified chisquare, reference from a book?)
• axis (int) – axis for observations of one series
• ddof (int) – degrees of freedom correction,

Returns:

• D_obs (Discrepancy of observed values)
• pvalue (pvalue)

References

Cressie, Noel and Timothy R. C. Read, Multinomial Goodness-of-Fit Tests,

Journal of the Royal Statistical Society. Series B (Methodological), Vol. 46, No. 3 (1984), pp. 440-464

Campbell B. Read: Freeman-Tukey chi-squared goodness-of-fit statistics,

Statistics & Probability Letters 18 (1993) 271-278

Nobuhiro Taneichi, Yuri Sekiya, Akio Suzukawa, Asymptotic Approximations

for the Distributions of the Multinomial Goodness-of-Fit Statistics under Local Alternatives, Journal of Multivariate Analysis 81, 335?359 (2002)

Steele, M. 1,2, C. Hurst 3 and J. Chaseling, Simulated Power of Discrete

Goodness-of-Fit Tests for Likert Type Data

Examples

>>> observed = np.array([ 2.,  4.,  2.,  1.,  1.])
>>> expected = np.array([ 0.2,  0.2,  0.2,  0.2,  0.2])

for checking correct dimension with multiple series

>>> powerdiscrepancy(np.column_stack((observed,observed)).T, 10*expected, lambd='freeman_tukey',axis=1)
(array([[ 2.745166,  2.745166]]), array([[ 0.6013346,  0.6013346]]))
>>> powerdiscrepancy(np.column_stack((observed,observed)).T, 10*expected,axis=1)
(array([[ 2.77258872,  2.77258872]]), array([[ 0.59657359,  0.59657359]]))
>>> powerdiscrepancy(np.column_stack((observed,observed)).T, 10*expected, lambd=0,axis=1)
(array([[ 2.77258872,  2.77258872]]), array([[ 0.59657359,  0.59657359]]))
>>> powerdiscrepancy(np.column_stack((observed,observed)).T, 10*expected, lambd=1,axis=1)
(array([[ 3.,  3.]]), array([[ 0.5578254,  0.5578254]]))
>>> powerdiscrepancy(np.column_stack((observed,observed)).T, 10*expected, lambd=2/3.0,axis=1)
(array([[ 2.89714546,  2.89714546]]), array([[ 0.57518277,  0.57518277]]))
>>> powerdiscrepancy(np.column_stack((observed,observed)).T, expected, lambd=2/3.0,axis=1)
(array([[ 2.89714546,  2.89714546]]), array([[ 0.57518277,  0.57518277]]))
>>> powerdiscrepancy(np.column_stack((observed,observed)), expected, lambd=2/3.0, axis=0)
(array([[ 2.89714546,  2.89714546]]), array([[ 0.57518277,  0.57518277]]))

each random variable can have different total count/sum

>>> powerdiscrepancy(np.column_stack((observed,2*observed)), expected, lambd=2/3.0, axis=0)
(array([[ 2.89714546,  5.79429093]]), array([[ 0.57518277,  0.21504648]]))
>>> powerdiscrepancy(np.column_stack((observed,2*observed)), expected, lambd=2/3.0, axis=0)
(array([[ 2.89714546,  5.79429093]]), array([[ 0.57518277,  0.21504648]]))
>>> powerdiscrepancy(np.column_stack((2*observed,2*observed)), expected, lambd=2/3.0, axis=0)
(array([[ 5.79429093,  5.79429093]]), array([[ 0.21504648,  0.21504648]]))
>>> powerdiscrepancy(np.column_stack((2*observed,2*observed)), 20*expected, lambd=2/3.0, axis=0)
(array([[ 5.79429093,  5.79429093]]), array([[ 0.21504648,  0.21504648]]))
>>> powerdiscrepancy(np.column_stack((observed,2*observed)), np.column_stack((10*expected,20*expected)), lambd=2/3.0, axis=0)
(array([[ 2.89714546,  5.79429093]]), array([[ 0.57518277,  0.21504648]]))
>>> powerdiscrepancy(np.column_stack((observed,2*observed)), np.column_stack((10*expected,20*expected)), lambd=-1, axis=0)
(array([[ 2.77258872,  5.54517744]]), array([[ 0.59657359,  0.2357868 ]]))