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
ndarray
The endogenous response variable
- mu
ndarray
The inverse of the link function at the linear predicted values.
- var_weightsarray_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.
- endog
- Returns:
- resid_anscombe
ndarray
The Anscombe residuals as defined below.
- resid_anscombe
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.