statsmodels.genmod.families.family.Gamma.loglike_obs

Gamma.loglike_obs(endog, mu, var_weights=1.0, scale=1.0)

The log-likelihood function for each observation in terms of the fitted mean response for the Gamma distribution.

Parameters:

endogndarray

Usually the endogenous response variable.

mundarray

Usually but not always the fitted mean response variable.

var_weightsarray_like

1d array of variance (analytic) weights. The default is 1.

scalefloat

The scale parameter. The default is 1.

Returns:

ll_ifloat

The value of the loglikelihood evaluated at (endog, mu, var_weights, scale) as defined below.

Notes

\[ll_i = var\_weights_i / scale * (\ln(var\_weights_i * endog_i / (scale * \mu_i)) - (var\_weights_i * endog_i) / (scale * \mu_i)) - \ln \Gamma(var\_weights_i / scale) - \ln(\endog_i)\]

Note on weights parameterization

statsmodels follows the SPSS/SAS definition for variance weights: Var(endog_i) = scale * mu_i² / var_weights_i — i.e., the effective dispersion is scale / var_weights_i.

McCullagh & Nelder (1989) treat var_weights_i as a pure multiplier on the log-likelihood with Var(endog_i) = scale * mu_i². This leads to a log-likelihood that differs only by a constant in (endog_i, var_weights_i, scale), not mu_i. So estimates of β and deviance are identical, but absolute log-likelihood values (e.g., AIC) will differ by a constant.

References

McCullagh, P., and Nelder, J.A. (1989). Generalized Linear Models, 2nd Edition. Chapman & Hall/CRC.