statsmodels.genmod.families.family.Binomial.resid_anscombe

Binomial.resid_anscombe(endog, mu, var_weights=1.0, scale=1.0) [source]

The Anscombe residuals

Parameters:	endog (array) – The endogenous response variable mu (array) – The inverse of the link function at the linear predicted values. var_weights (array-like) – 1d array of variance (analytic) weights. The default is 1. scale (float, optional) – An optional argument to divide the residuals by sqrt(scale). The default is 1.
Returns:	resid_anscombe – The Anscombe residuals as defined below.
Return type:	array

Notes

\[n^{2/3}*(cox\_snell(endog)-cox\_snell(mu)) / (mu*(1-mu/n)*scale^3)^{1/6} * \sqrt(var\_weights)\]

where cox_snell is defined as cox_snell(x) = betainc(2/3., 2/3., x)*betainc(2/3.,2/3.) where betainc is the incomplete beta function as defined in scipy, which uses a regularized version (with the unregularized version, one would just have \(cox_snell(x) = Betainc(2/3., 2/3., x)\)).

The name ‘cox_snell’ is idiosyncratic and is simply used for convenience following the approach suggested in Cox and Snell (1968). Further note that \(cox\_snell(x) = \frac{3}{2}*x^{2/3} * hyp2f1(2/3.,1/3.,5/3.,x)\) where hyp2f1 is the hypergeometric 2f1 function. The Anscombe residuals are sometimes defined in the literature using the hyp2f1 formulation. Both betainc and hyp2f1 can be found in scipy.

References

Anscombe, FJ. (1953) “Contribution to the discussion of H. Hotelling’s

paper.” Journal of the Royal Statistical Society B. 15, 229-30.

Cox, DR and Snell, EJ. (1968) “A General Definition of Residuals.”

Journal of the Royal Statistical Society B. 30, 248-75.