Repository: statsmodels/statsmodels

Issue:
ENH: add GLM Margins and supporting helper functions
GEE has margins, although incomplete, see #3704
GLM has nothing related yet.

To compute hatmatrix or, more precisely, generalized leverage for generic MLE that have predict, I need one method d mu / d params, which is in discrete_models used for the margin computations.

GEE has `_derivative_exog` which is almost the same as `_derivative_predict` which is needed for margins but not yet in GEE.
GEE has model specific methods that compute with different argument, eg, `mean_deriv`, which should extend to GLM also.

I need `_derivative_predict` and similar derivatives in a generic interface, so I don't have to access the link directly.
To go the other way as a generic cross-model pattern see #3915
(e.g. I added the family to Poisson so I have access to the inverse link deriv)

naming: I think we need more explicit names to refere to the function and wrt which variable we take the derivative, e.g.
`deriv_predict_params` or `deriv_mean_params`
`deriv_mean_exog`
`deriv_mean_linpred`
`deriv_score_endog`      (that's what I also need for generalized leverage)
or maybe another `d` in there
`deriv_mean_dlinpred`

(The new count models have many "which" in predict and we need derivatives to be able to get standard errors. It will be similar for other two part models like BetaRegression)

To get the equivalent of DFFITS, I also need` deriv_mean_params` or `derivative_predict`
e.g. currently I have it specific to the availability of `model.family.link`, i.e. GLM, and Poisson for now

```
    def d_fittedvalues(self):
        # in discrete we cannot reuse results.fittedvalues
        fittedvalues = self.results.predict()
        deriv = self.results.model.family.link.inverse_deriv(fittedvalues)
        return deriv * self.d_linpred
```




Discussion:
I just realized (again) that GLMResults still does not have `get_margeff`
We need it to e.g. compare logit and probit, Binomial with different link functions, params have different scaling and cannot be directly compared.
see also #8802 for another reason to look at margins in binomial

If we don't want to support weights in the first version #7341, then we could raise NotImplementedError for non-default weights.



Resolving patch:
diff --git a/statsmodels/genmod/generalized_linear_model.py b/statsmodels/genmod/generalized_linear_model.py
--- a/statsmodels/genmod/generalized_linear_model.py
+++ b/statsmodels/genmod/generalized_linear_model.py
@@ -54,6 +54,7 @@
 # need import in module instead of lazily to copy `__doc__`
 from . import families
 
+
 __all__ = ['GLM', 'PredictionResultsMean']
 
 
@@ -624,6 +625,90 @@ def information(self, params, scale=None):
         scale = float_like(scale, "scale", optional=True)
         return self.hessian(params, scale=scale, observed=False)
 
+    def _derivative_exog(self, params, exog=None, transform="dydx",
+                         dummy_idx=None, count_idx=None,
+                         offset=None, exposure=None):
+        """
+        Derivative of mean, expected endog with respect to the parameters
+        """
+        if exog is None:
+            exog = self.exog
+        if (offset is not None) or (exposure is not None):
+            raise NotImplementedError("offset and exposure not supported")
+
+        lin_pred = self.predict(params, exog, which="linear",
+                                offset=offset, exposure=exposure)
+
+        k_extra = getattr(self, 'k_extra', 0)
+        params_exog = params if k_extra == 0 else params[:-k_extra]
+
+        margeff = (self.family.link.inverse_deriv(lin_pred)[:, None] *
+                   params_exog)
+        if 'ex' in transform:
+            margeff *= exog
+        if 'ey' in transform:
+            mean = self.family.link.inverse(lin_pred)
+            margeff /= mean[:,None]
+
+        return self._derivative_exog_helper(margeff, params, exog,
+                                            dummy_idx, count_idx, transform)
+
+    def _derivative_exog_helper(self, margeff, params, exog, dummy_idx,
+                                count_idx, transform):
+        """
+        Helper for _derivative_exog to wrap results appropriately
+        """
+        from statsmodels.discrete.discrete_margins import (
+            _get_count_effects,
+            _get_dummy_effects,
+            )
+
+        if count_idx is not None:
+            margeff = _get_count_effects(margeff, exog, count_idx, transform,
+                                         self, params)
+        if dummy_idx is not None:
+            margeff = _get_dummy_effects(margeff, exog, dummy_idx, transform,
+                                         self, params)
+
+        return margeff
+
+    def _derivative_predict(self, params, exog=None, transform='dydx',
+                            offset=None, exposure=None):
+        """
+        Derivative of the expected endog with respect to the parameters.
+
+        Parameters
+        ----------
+        params : ndarray
+            parameter at which score is evaluated
+        exog : ndarray or None
+            Explanatory variables at which derivative are computed.
+            If None, then the estimation exog is used.
+        offset, exposure : None
+            Not yet implemented.
+
+        Returns
+        -------
+        The value of the derivative of the expected endog with respect
+        to the parameter vector.
+        """
+        # core part is same as derivative_mean_params
+        # additionally handles exog and transform
+        if exog is None:
+            exog = self.exog
+        if (offset is not None) or (exposure is not None) or (
+                getattr(self, 'offset', None) is not None):
+            raise NotImplementedError("offset and exposure not supported")
+
+        lin_pred = self.predict(params, exog=exog, which="linear")
+        idl = self.family.link.inverse_deriv(lin_pred)
+        dmat = exog * idl[:, None]
+        if 'ey' in transform:
+            mean = self.family.link.inverse(lin_pred)
+            dmat /= mean[:, None]
+
+        return dmat
+
     def _deriv_mean_dparams(self, params):
         """
         Derivative of the expected endog with respect to the parameters.
@@ -2210,6 +2295,92 @@ def get_distribution(self, exog=None, exposure=None,
             mu, scale, var_weights=var_weights, **kwds)
         return distr
 
+    def get_margeff(self, at='overall', method='dydx', atexog=None,
+            dummy=False, count=False):
+        """Get marginal effects of the fitted model.
+
+        Warning: offset, exposure and weights (var_weights and freq_weights)
+        are not supported by margeff.
+
+        Parameters
+        ----------
+        at : str, optional
+            Options are:
+
+            - 'overall', The average of the marginal effects at each
+              observation.
+            - 'mean', The marginal effects at the mean of each regressor.
+            - 'median', The marginal effects at the median of each regressor.
+            - 'zero', The marginal effects at zero for each regressor.
+            - 'all', The marginal effects at each observation. If `at` is all
+              only margeff will be available from the returned object.
+
+            Note that if `exog` is specified, then marginal effects for all
+            variables not specified by `exog` are calculated using the `at`
+            option.
+        method : str, optional
+            Options are:
+
+            - 'dydx' - dy/dx - No transformation is made and marginal effects
+              are returned.  This is the default.
+            - 'eyex' - estimate elasticities of variables in `exog` --
+              d(lny)/d(lnx)
+            - 'dyex' - estimate semi-elasticity -- dy/d(lnx)
+            - 'eydx' - estimate semi-elasticity -- d(lny)/dx
+
+            Note that tranformations are done after each observation is
+            calculated.  Semi-elasticities for binary variables are computed
+            using the midpoint method. 'dyex' and 'eyex' do not make sense
+            for discrete variables. For interpretations of these methods
+            see notes below.
+        atexog : array_like, optional
+            Optionally, you can provide the exogenous variables over which to
+            get the marginal effects.  This should be a dictionary with the key
+            as the zero-indexed column number and the value of the dictionary.
+            Default is None for all independent variables less the constant.
+        dummy : bool, optional
+            If False, treats binary variables (if present) as continuous.  This
+            is the default.  Else if True, treats binary variables as
+            changing from 0 to 1.  Note that any variable that is either 0 or 1
+            is treated as binary.  Each binary variable is treated separately
+            for now.
+        count : bool, optional
+            If False, treats count variables (if present) as continuous.  This
+            is the default.  Else if True, the marginal effect is the
+            change in probabilities when each observation is increased by one.
+
+        Returns
+        -------
+        DiscreteMargins : marginal effects instance
+            Returns an object that holds the marginal effects, standard
+            errors, confidence intervals, etc. See
+            `statsmodels.discrete.discrete_margins.DiscreteMargins` for more
+            information.
+
+        Notes
+        -----
+        Interpretations of methods:
+
+        - 'dydx' - change in `endog` for a change in `exog`.
+        - 'eyex' - proportional change in `endog` for a proportional change
+          in `exog`.
+        - 'dyex' - change in `endog` for a proportional change in `exog`.
+        - 'eydx' - proportional change in `endog` for a change in `exog`.
+
+        When using after Poisson, returns the expected number of events per
+        period, assuming that the model is loglinear.
+
+        Status : unsupported features offset, exposure and weights. Default
+        handling of freq_weights for average effect "overall" might change.
+
+        """
+        if getattr(self.model, "offset", None) is not None:
+            raise NotImplementedError("Margins with offset are not available.")
+        if (np.any(self.model.var_weights != 1) or
+                np.any(self.model.freq_weights != 1)):
+            warnings.warn("weights are not taken into account by margeff")
+        from statsmodels.discrete.discrete_margins import DiscreteMargins
+        return DiscreteMargins(self, (at, method, atexog, dummy, count))
 
     @Appender(base.LikelihoodModelResults.remove_data.__doc__)
     def remove_data(self):
diff --git a/statsmodels/stats/_delta_method.py b/statsmodels/stats/_delta_method.py
--- a/statsmodels/stats/_delta_method.py
+++ b/statsmodels/stats/_delta_method.py
@@ -151,6 +151,9 @@ def var(self):
         """
         g = self.grad()
         var = (np.dot(g, self.cov_params) * g).sum(-1)
+
+        if var.ndim == 2:
+            var = var.T
         return var
 
     def se_vectorized(self):


Test patch:
diff --git a/statsmodels/discrete/tests/test_sandwich_cov.py b/statsmodels/discrete/tests/test_sandwich_cov.py
--- a/statsmodels/discrete/tests/test_sandwich_cov.py
+++ b/statsmodels/discrete/tests/test_sandwich_cov.py
@@ -9,6 +9,8 @@
 import os
 import numpy as np
 import pandas as pd
+import pytest
+
 import statsmodels.discrete.discrete_model as smd
 from statsmodels.genmod.generalized_linear_model import GLM
 from statsmodels.genmod import families
@@ -499,6 +501,21 @@ def test_score_test(self):
         res_lm2 = res2.score_test(exog_extra, cov_type='nonrobust')
         assert_allclose(np.hstack(res_lm1), np.hstack(res_lm2), rtol=5e-7)
 
+    def test_margeff(self):
+        if (isinstance(self.res2.model, OLS) or
+                hasattr(self.res1.model, "offset")):
+            pytest.skip("not available yet")
+
+        marg1 = self.res1.get_margeff()
+        marg2 = self.res2.get_margeff()
+        assert_allclose(marg1.margeff, marg2.margeff, rtol=1e-10)
+        assert_allclose(marg1.margeff_se, marg2.margeff_se, rtol=1e-10)
+
+        marg1 = self.res1.get_margeff(count=True, dummy=True)
+        marg2 = self.res2.get_margeff(count=True, dummy=True)
+        assert_allclose(marg1.margeff, marg2.margeff, rtol=1e-10)
+        assert_allclose(marg1.margeff_se, marg2.margeff_se, rtol=1e-10)
+
 
 class TestGLMPoisson(CheckDiscreteGLM):