# Source code for statsmodels.genmod.generalized_estimating_equations

"""
Procedures for fitting marginal regression models to dependent data
using Generalized Estimating Equations.

References
----------
KY Liang and S Zeger. "Longitudinal data analysis using
generalized linear models". Biometrika (1986) 73 (1): 13-22.

S Zeger and KY Liang. "Longitudinal Data Analysis for Discrete and
Continuous Outcomes". Biometrics Vol. 42, No. 1 (Mar., 1986),
pp. 121-130

A Rotnitzky and NP Jewell (1990). "Hypothesis testing of regression
parameters in semiparametric generalized linear models for cluster
correlated data", Biometrika, 77, 485-497.

Xu Guo and Wei Pan (2002). "Small sample performance of the score
test in GEE".

LA Mancl LA, TA DeRouen (2001). A covariance estimator for GEE with
improved small-sample properties.  Biometrics. 2001 Mar;57(1):126-34.
"""
from statsmodels.compat.python import lzip
from statsmodels.compat.pandas import Appender

import numpy as np
from scipy import stats
import pandas as pd
import patsy
from collections import defaultdict
import statsmodels.base.model as base
# used for wrapper:
import statsmodels.regression.linear_model as lm
import statsmodels.base.wrapper as wrap

from statsmodels.genmod import families
from statsmodels.genmod.generalized_linear_model import GLM
from statsmodels.genmod import cov_struct as cov_structs

import statsmodels.genmod.families.varfuncs as varfuncs

from statsmodels.tools.sm_exceptions import (ConvergenceWarning,
DomainWarning,
IterationLimitWarning,
ValueWarning)
import warnings

from statsmodels.graphics._regressionplots_doc import (
_plot_partial_residuals_doc,
_plot_ceres_residuals_doc)
from statsmodels.discrete.discrete_margins import (
_get_margeff_exog, _check_margeff_args, _effects_at, margeff_cov_with_se,
_check_at_is_all, _transform_names, _check_discrete_args,
_get_dummy_index, _get_count_index)

class ParameterConstraint(object):
"""
A class for managing linear equality constraints for a parameter
vector.
"""

def __init__(self, lhs, rhs, exog):
"""
Parameters
----------
lhs : ndarray
A q x p matrix which is the left hand side of the
constraint lhs * param = rhs.  The number of constraints is
q >= 1 and p is the dimension of the parameter vector.
rhs : ndarray
A 1-dimensional vector of length q which is the right hand
side of the constraint equation.
exog : ndarray
The n x p exognenous data for the full model.
"""

# In case a row or column vector is passed (patsy linear
# constraints passes a column vector).
rhs = np.atleast_1d(rhs.squeeze())

if rhs.ndim > 1:
raise ValueError("The right hand side of the constraint "
"must be a vector.")

if len(rhs) != lhs.shape[0]:
raise ValueError("The number of rows of the left hand "
"side constraint matrix L must equal "
"the length of the right hand side "
"constraint vector R.")

self.lhs = lhs
self.rhs = rhs

# The columns of lhs0 are an orthogonal basis for the
# orthogonal complement to row(lhs), the columns of lhs1 are
# an orthogonal basis for row(lhs).  The columns of lhsf =
# [lhs0, lhs1] are mutually orthogonal.
lhs_u, lhs_s, lhs_vt = np.linalg.svd(lhs.T, full_matrices=1)
self.lhs0 = lhs_u[:, len(lhs_s):]
self.lhs1 = lhs_u[:, 0:len(lhs_s)]
self.lhsf = np.hstack((self.lhs0, self.lhs1))

# param0 is one solution to the underdetermined system
# L * param = R.
self.param0 = np.dot(self.lhs1, np.dot(lhs_vt, self.rhs) /
lhs_s)

self._offset_increment = np.dot(exog, self.param0)

self.orig_exog = exog
self.exog_fulltrans = np.dot(exog, self.lhsf)

def offset_increment(self):
"""
Returns a vector that should be added to the offset vector to
accommodate the constraint.

Parameters
----------
exog : array_like
The exogeneous data for the model.
"""

return self._offset_increment

def reduced_exog(self):
"""
Returns a linearly transformed exog matrix whose columns span
the constrained model space.

Parameters
----------
exog : array_like
The exogeneous data for the model.
"""
return self.exog_fulltrans[:, 0:self.lhs0.shape[1]]

def restore_exog(self):
"""
Returns the full exog matrix before it was reduced to
satisfy the constraint.
"""
return self.orig_exog

def unpack_param(self, params):
"""
Converts the parameter vector params from reduced to full
coordinates.
"""

return self.param0 + np.dot(self.lhs0, params)

def unpack_cov(self, bcov):
"""
Converts the covariance matrix bcov from reduced to full
coordinates.
"""

return np.dot(self.lhs0, np.dot(bcov, self.lhs0.T))

_gee_init_doc = """
Marginal regression model fit using Generalized Estimating Equations.

GEE can be used to fit Generalized Linear Models (GLMs) when the
data have a grouped structure, and the observations are possibly
correlated within groups but not between groups.

Parameters
----------
endog : array_like
1d array of endogenous values (i.e. responses, outcomes,
dependent variables, or 'Y' values).
exog : array_like
2d array of exogeneous values (i.e. covariates, predictors,
independent variables, regressors, or 'X' values). A nobs x
k array where nobs is the number of observations and k is
the number of regressors. An intercept is not included by
default and should be added by the user. See
statsmodels.tools.add_constant.
groups : array_like
A 1d array of length nobs containing the group labels.
time : array_like
A 2d array of time (or other index) values, used by some
dependence structures to define similarity relationships among
observations within a cluster.
family : family class instance
%(family_doc)s
cov_struct : CovStruct class instance
The default is Independence.  To specify an exchangeable
structure use cov_struct = Exchangeable().  See
statsmodels.genmod.cov_struct.CovStruct for more
information.
offset : array_like
An offset to be included in the fit.  If provided, must be
an array whose length is the number of rows in exog.
dep_data : array_like
Additional data passed to the dependence structure.
constraint : (ndarray, ndarray)
If provided, the constraint is a tuple (L, R) such that the
model parameters are estimated under the constraint L *
param = R, where L is a q x p matrix and R is a
q-dimensional vector.  If constraint is provided, a score
test is performed to compare the constrained model to the
unconstrained model.
update_dep : bool
If true, the dependence parameters are optimized, otherwise
they are held fixed at their starting values.
weights : array_like
An array of weights to use in the analysis.  The weights must
be constant within each group.  These correspond to
probability weights (pweights) in Stata.
%(extra_params)s

--------
statsmodels.genmod.families.family
:ref:families
:ref:links

Notes
-----
Only the following combinations make sense for family and link ::

+ ident log logit probit cloglog pow opow nbinom loglog logc
Gaussian     |   x    x                        x
inv Gaussian |   x    x                        x
binomial     |   x    x    x     x       x     x    x           x      x
Poisson     |   x    x                        x
neg binomial |   x    x                        x          x
gamma        |   x    x                        x

Not all of these link functions are currently available.

Endog and exog are references so that if the data they refer
to are already arrays and these arrays are changed, endog and
exog will change.

The "robust" covariance type is the standard "sandwich estimator"
(e.g. Liang and Zeger (1986)).  It is the default here and in most
other packages.  The "naive" estimator gives smaller standard
errors, but is only correct if the working correlation structure
is correctly specified.  The "bias reduced" estimator of Mancl and
DeRouen (Biometrics, 2001) reduces the downward bias of the robust
estimator.

The robust covariance provided here follows Liang and Zeger (1986)
and agrees with R's gee implementation.  To obtain the robust
standard errors reported in Stata, multiply by sqrt(N / (N - g)),
where N is the total sample size, and g is the average group size.

Examples
--------
%(example)s
"""

_gee_family_doc = """\
The default is Gaussian.  To specify the binomial
distribution use family=sm.families.Binomial(). Each family
can take a link instance as an argument.  See

_gee_ordinal_family_doc = """\
The only family supported is Binomial.  The default Logit
link may be replaced with probit if desired."""

_gee_nominal_family_doc = """\
The default value None uses a multinomial logit family
specifically designed for use with GEE.  Setting this
argument to a non-default value is not currently supported."""

_gee_fit_doc = """
Fits a marginal regression model using generalized estimating
equations (GEE).

Parameters
----------
maxiter : int
The maximum number of iterations
ctol : float
The convergence criterion for stopping the Gauss-Seidel
iterations
start_params : array_like
A vector of starting values for the regression
coefficients.  If None, a default is chosen.
params_niter : int
The number of Gauss-Seidel updates of the mean structure
parameters that take place prior to each update of the
dependence structure.
first_dep_update : int
No dependence structure updates occur before this
iteration number.
cov_type : str
One of "robust", "naive", or "bias_reduced".
ddof_scale : scalar or None
The scale parameter is estimated as the sum of squared
Pearson residuals divided by N - ddof_scale, where N
is the total sample size.  If ddof_scale is None, the
number of covariates (including an intercept if present)
is used.
scaling_factor : scalar
The estimated covariance of the parameter estimates is
scaled by this value.  Default is 1, Stata uses N / (N - g),
where N is the total sample size and g is the average group
size.
scale : str or float, optional
scale can be None, 'X2', or a float
If a float, its value is used as the scale parameter.
The default value is None, which uses X2 (Pearson's
chi-square) for Gamma, Gaussian, and Inverse Gaussian.
The default is 1 for the Binomial and Poisson families.

Returns
-------
An instance of the GEEResults class or subclass

Notes
-----
If convergence difficulties occur, increase the values of
first_dep_update and/or params_niter.  Setting
first_dep_update to a greater value (e.g. ~10-20) causes the
algorithm to move close to the GLM solution before attempting
to identify the dependence structure.

For the Gaussian family, there is no benefit to setting
params_niter to a value greater than 1, since the mean
structure parameters converge in one step.
"""

_gee_results_doc = """
Attributes
----------

cov_params_default : ndarray
default covariance of the parameter estimates. Is chosen among one
of the following three based on cov_type
cov_robust : ndarray
covariance of the parameter estimates that is robust
cov_naive : ndarray
covariance of the parameter estimates that is not robust to
correlation or variance misspecification
cov_robust_bc : ndarray
covariance of the parameter estimates that is robust and bias
reduced
converged : bool
indicator for convergence of the optimization.
True if the norm of the score is smaller than a threshold
cov_type : str
string indicating whether a "robust", "naive" or "bias_reduced"
covariance is used as default
fit_history : dict
fittedvalues : ndarray
Linear predicted values for the fitted model.
dot(exog, params)
model : class instance
Pointer to GEE model instance that called fit.
normalized_cov_params : ndarray
See GEE docstring
params : ndarray
The coefficients of the fitted model.  Note that
interpretation of the coefficients often depends on the
distribution family and the data.
scale : float
The estimate of the scale / dispersion for the model fit.
score_norm : float
norm of the score at the end of the iterative estimation.
bse : ndarray
The standard errors of the fitted GEE parameters.
"""

_gee_example = """
Logistic regression with autoregressive working dependence:

>>> import statsmodels.api as sm
>>> family = sm.families.Binomial()
>>> va = sm.cov_struct.Autoregressive()
>>> model = sm.GEE(endog, exog, group, family=family, cov_struct=va)
>>> result = model.fit()
>>> print(result.summary())

Use formulas to fit a Poisson GLM with independent working
dependence:

>>> import statsmodels.api as sm
>>> fam = sm.families.Poisson()
>>> ind = sm.cov_struct.Independence()
>>> model = sm.GEE.from_formula("y ~ age + trt + base", "subject", \
data, cov_struct=ind, family=fam)
>>> result = model.fit()
>>> print(result.summary())

Equivalent, using the formula API:

>>> import statsmodels.api as sm
>>> import statsmodels.formula.api as smf
>>> fam = sm.families.Poisson()
>>> ind = sm.cov_struct.Independence()
>>> model = smf.gee("y ~ age + trt + base", "subject", \
data, cov_struct=ind, family=fam)
>>> result = model.fit()
>>> print(result.summary())
"""

_gee_ordinal_example = """
Fit an ordinal regression model using GEE, with "global
odds ratio" dependence:

>>> import statsmodels.api as sm
>>> gor = sm.cov_struct.GlobalOddsRatio("ordinal")
>>> model = sm.OrdinalGEE(endog, exog, groups, cov_struct=gor)
>>> result = model.fit()
>>> print(result.summary())

Using formulas:

>>> import statsmodels.formula.api as smf
>>> model = smf.ordinal_gee("y ~ x1 + x2", groups, data,
cov_struct=gor)
>>> result = model.fit()
>>> print(result.summary())
"""

_gee_nominal_example = """
Fit a nominal regression model using GEE:

>>> import statsmodels.api as sm
>>> import statsmodels.formula.api as smf
>>> gor = sm.cov_struct.GlobalOddsRatio("nominal")
>>> model = sm.NominalGEE(endog, exog, groups, cov_struct=gor)
>>> result = model.fit()
>>> print(result.summary())

Using formulas:

>>> import statsmodels.api as sm
>>> model = sm.NominalGEE.from_formula("y ~ x1 + x2", groups,
data, cov_struct=gor)
>>> result = model.fit()
>>> print(result.summary())

Using the formula API:

>>> import statsmodels.formula.api as smf
>>> model = smf.nominal_gee("y ~ x1 + x2", groups, data,
cov_struct=gor)
>>> result = model.fit()
>>> print(result.summary())
"""

def _check_args(endog, exog, groups, time, offset, exposure):

if endog.size != exog.shape[0]:
raise ValueError("Leading dimension of 'exog' should match "
"length of 'endog'")

if groups.size != endog.size:
raise ValueError("'groups' and 'endog' should have the same size")

if time is not None and (time.size != endog.size):
raise ValueError("'time' and 'endog' should have the same size")

if offset is not None and (offset.size != endog.size):
raise ValueError("'offset and 'endog' should have the same size")

if exposure is not None and (exposure.size != endog.size):
raise ValueError("'exposure' and 'endog' should have the same size")

[docs]class GEE(base.Model):

__doc__ = (
"    Marginal Regression Model using Generalized Estimating "
"Equations.\n" + _gee_init_doc %
{'extra_params': base._missing_param_doc,
'family_doc': _gee_family_doc,
'example': _gee_example})

cached_means = None

def __init__(self, endog, exog, groups, time=None, family=None,
cov_struct=None, missing='none', offset=None,
exposure=None, dep_data=None, constraint=None,
update_dep=True, weights=None, **kwargs):

if family is not None:
import warnings
msg = ("The {0} link function does not respect the "
"domain of the {1} family.")
family.__class__.__name__),
DomainWarning)

groups = np.asarray(groups)  # in case groups is pandas

if "missing_idx" in kwargs and kwargs["missing_idx"] is not None:
# If here, we are entering from super.from_formula; missing
# has already been dropped from endog and exog, but not from
# the other variables.
ii = ~kwargs["missing_idx"]
groups = groups[ii]
if time is not None:
time = time[ii]
if offset is not None:
offset = offset[ii]
if exposure is not None:
exposure = exposure[ii]
del kwargs["missing_idx"]

_check_args(endog, exog, groups, time, offset, exposure)

self.missing = missing
self.dep_data = dep_data
self.constraint = constraint
self.update_dep = update_dep

self._fit_history = defaultdict(list)

# Pass groups, time, offset, and dep_data so they are
# processed for missing data along with endog and exog.
# Calling super creates self.exog, self.endog, etc. as
# ndarrays and the original exog, endog, etc. are
# self.data.endog, etc.
super(GEE, self).__init__(endog, exog, groups=groups,
time=time, offset=offset,
exposure=exposure, weights=weights,
dep_data=dep_data, missing=missing,
**kwargs)

self._init_keys.extend(["update_dep", "constraint", "family",
"cov_struct"])

# Handle the family argument
if family is None:
family = families.Gaussian()
else:
if not issubclass(family.__class__, families.Family):
raise ValueError("GEE: family must be a genmod "
"family instance")
self.family = family

# Handle the cov_struct argument
if cov_struct is None:
cov_struct = cov_structs.Independence()
else:
if not issubclass(cov_struct.__class__, cov_structs.CovStruct):
raise ValueError("GEE: cov_struct must be a genmod "
"cov_struct instance")

self.cov_struct = cov_struct

# Handle the offset and exposure
self._offset_exposure = None
if offset is not None:
self._offset_exposure = self.offset.copy()
self.offset = offset
if exposure is not None:
raise ValueError(
"exposure can only be used with the log link function")
if self._offset_exposure is not None:
self._offset_exposure += np.log(exposure)
else:
self._offset_exposure = np.log(exposure)
self.exposure = exposure

# Handle the constraint
self.constraint = None
if constraint is not None:
if len(constraint) != 2:
raise ValueError("GEE: constraint must be a 2-tuple.")
if constraint[0].shape[1] != self.exog.shape[1]:
raise ValueError(
"GEE: the left hand side of the constraint must have "
"the same number of columns as the exog matrix.")
self.constraint = ParameterConstraint(constraint[0],
constraint[1],
self.exog)

if self._offset_exposure is not None:
self._offset_exposure += self.constraint.offset_increment()
else:
self._offset_exposure = (
self.constraint.offset_increment().copy())
self.exog = self.constraint.reduced_exog()

# Create list of row indices for each group
group_labels, ix = np.unique(self.groups, return_inverse=True)
se = pd.Series(index=np.arange(len(ix)), dtype="int")
gb = se.groupby(ix).groups
dk = [(lb, np.asarray(gb[k])) for k, lb in enumerate(group_labels)]
self.group_indices = dict(dk)
self.group_labels = group_labels

# Convert the data to the internal representation, which is a
# list of arrays, corresponding to the groups.
self.endog_li = self.cluster_list(self.endog)
self.exog_li = self.cluster_list(self.exog)

if self.weights is not None:
self.weights_li = self.cluster_list(self.weights)
self.weights_li = [x[0] for x in self.weights_li]
self.weights_li = np.asarray(self.weights_li)

self.num_group = len(self.endog_li)

# Time defaults to a 1d grid with equal spacing
if self.time is not None:
self.time = np.asarray(self.time, np.float64)
if self.time.ndim == 1:
self.time = self.time[:, None]
self.time_li = self.cluster_list(self.time)
else:
self.time_li = \
[np.arange(len(y), dtype=np.float64)[:, None]
for y in self.endog_li]
self.time = np.concatenate(self.time_li)

if self._offset_exposure is not None:
self.offset_li = self.cluster_list(self._offset_exposure)
else:
self.offset_li = None
if constraint is not None:
self.constraint.exog_fulltrans_li = \
self.cluster_list(self.constraint.exog_fulltrans)

self.family = family

self.cov_struct.initialize(self)

# Total sample size
group_ns = [len(y) for y in self.endog_li]
self.nobs = sum(group_ns)
# The following are column based, not on rank see #1928
self.df_model = self.exog.shape[1] - 1  # assumes constant
self.df_resid = self.nobs - self.exog.shape[1]

# Skip the covariance updates if all groups have a single
# observation (reduces to fitting a GLM).
maxgroup = max([len(x) for x in self.endog_li])
if maxgroup == 1:
self.update_dep = False

# Override to allow groups and time to be passed as variable
# names.
[docs]    @classmethod
def from_formula(cls, formula, groups, data, subset=None,
time=None, offset=None, exposure=None,
*args, **kwargs):
"""
Create a GEE model instance from a formula and dataframe.

Parameters
----------
formula : str or generic Formula object
The formula specifying the model
groups : array_like or string
Array of grouping labels.  If a string, this is the name
of a variable in data that contains the grouping labels.
data : array_like
The data for the model.
subset : array_like
An array-like object of booleans, integers, or index
values that indicate the subset of the data to used when
fitting the model.
time : array_like or string
The time values, used for dependence structures involving
distances between observations.  If a string, this is the
name of a variable in data that contains the time
values.
offset : array_like or string
The offset values, added to the linear predictor.  If a
string, this is the name of a variable in data that
contains the offset values.
exposure : array_like or string
The exposure values, only used if the link function is the
logarithm function, in which case the log of exposure
is added to the offset (if any).  If a string, this is the
name of a variable in data that contains the offset
values.
%(missing_param_doc)s
args : extra arguments
These are passed to the model
kwargs : extra keyword arguments
These are passed to the model with two exceptions. dep_data
is processed as described below.  The eval_env keyword is
passed to patsy. It can be either a
:class:patsy:patsy.EvalEnvironment object or an integer
indicating the depth of the namespace to use. For example, the
default eval_env=0 uses the calling namespace.
If you wish to use a "clean" environment set eval_env=-1.

Optional arguments
------------------
dep_data : str or array_like
Data used for estimating the dependence structure.  See
specific dependence structure classes (e.g. Nested) for
details.  If dep_data is a string, it is interpreted as
a formula that is applied to data. If it is an array, it
must be an array of strings corresponding to column names in
data.  Otherwise it must be an array-like with the same
number of rows as data.

Returns
-------
model : GEE model instance

Notes
-----
data must define __getitem__ with the keys in the formula
terms args and kwargs are passed on to the model
instantiation. E.g., a numpy structured or rec array, a
dictionary, or a pandas DataFrame.
""" % {'missing_param_doc': base._missing_param_doc}

groups_name = "Groups"
if isinstance(groups, str):
groups_name = groups
groups = data[groups]

if isinstance(time, str):
time = data[time]

if isinstance(offset, str):
offset = data[offset]

if isinstance(exposure, str):
exposure = data[exposure]

dep_data = kwargs.get("dep_data")
dep_data_names = None
if dep_data is not None:
if isinstance(dep_data, str):
dep_data = patsy.dmatrix(dep_data, data,
return_type='dataframe')
dep_data_names = dep_data.columns.tolist()
else:
dep_data_names = list(dep_data)
dep_data = data[dep_data]
kwargs["dep_data"] = np.asarray(dep_data)

model = super(GEE, cls).from_formula(formula, data=data, subset=subset,
groups=groups, time=time,
offset=offset,
exposure=exposure,
*args, **kwargs)

if dep_data_names is not None:
model._dep_data_names = dep_data_names
model._groups_name = groups_name

return model

[docs]    def cluster_list(self, array):
"""
Returns array split into subarrays corresponding to the
cluster structure.
"""

if array.ndim == 1:
return [np.array(array[self.group_indices[k]])
for k in self.group_labels]
else:
return [np.array(array[self.group_indices[k], :])
for k in self.group_labels]

[docs]    def compare_score_test(self, submodel):
"""
Perform a score test for the given submodel against this model.

Parameters
----------
submodel : GEEResults instance
A fitted GEE model that is a submodel of this model.

Returns
-------
A dictionary with keys "statistic", "p-value", and "df",
containing the score test statistic, its chi^2 p-value,
and the degrees of freedom used to compute the p-value.

Notes
-----
The score test can be performed without calling 'fit' on the
larger model.  The provided submodel must be obtained from a
fitted GEE.

This method performs the same score test as can be obtained by
fitting the GEE with a linear constraint and calling score_test
on the results.

References
----------
Xu Guo and Wei Pan (2002). "Small sample performance of the score
test in GEE".
"""

# Since the model has not been fit, its scaletype has not been
# set.  So give it the scaletype of the submodel.
self.scaletype = submodel.model.scaletype

# Check consistency between model and submodel (not a comprehensive
# check)
submod = submodel.model
if self.exog.shape[0] != submod.exog.shape[0]:
msg = "Model and submodel have different numbers of cases."
raise ValueError(msg)
if self.exog.shape[1] == submod.exog.shape[1]:
msg = "Model and submodel have the same number of variables"
warnings.warn(msg)
if not isinstance(self.family, type(submod.family)):
msg = "Model and submodel have different GLM families."
warnings.warn(msg)
if not isinstance(self.cov_struct, type(submod.cov_struct)):
warnings.warn("Model and submodel have different GEE covariance "
"structures.")
if not np.equal(self.weights, submod.weights).all():
msg = "Model and submodel should have the same weights."
warnings.warn(msg)

# Get the positions of the submodel variables in the
# parent model
qm, qc = _score_test_submodel(self, submodel.model)
if qm is None:
msg = "The provided model is not a submodel."
raise ValueError(msg)

# Embed the submodel params into a params vector for the
# parent model
params_ex = np.dot(qm, submodel.params)

# Attempt to preserve the state of the parent model
cov_struct_save = self.cov_struct
import copy
cached_means_save = copy.deepcopy(self.cached_means)

# Get the score vector of the submodel params in
# the parent model
self.cov_struct = submodel.cov_struct
self.update_cached_means(params_ex)
_, score = self._update_mean_params()
if score is None:
msg = "Singular matrix encountered in GEE score test"
warnings.warn(msg, ConvergenceWarning)
return None

if not hasattr(self, "ddof_scale"):
self.ddof_scale = self.exog.shape[1]

if not hasattr(self, "scaling_factor"):
self.scaling_factor = 1

_, ncov1, cmat = self._covmat()
scale = self.estimate_scale()
cmat = cmat / scale ** 2
score2 = np.dot(qc.T, score) / scale

amat = np.linalg.inv(ncov1)

bmat_11 = np.dot(qm.T, np.dot(cmat, qm))
bmat_22 = np.dot(qc.T, np.dot(cmat, qc))
bmat_12 = np.dot(qm.T, np.dot(cmat, qc))

amat_11 = np.dot(qm.T, np.dot(amat, qm))
amat_12 = np.dot(qm.T, np.dot(amat, qc))

score_cov = bmat_22 - np.dot(amat_12.T,
np.linalg.solve(amat_11, bmat_12))
score_cov -= np.dot(bmat_12.T,
np.linalg.solve(amat_11, amat_12))
score_cov += np.dot(amat_12.T,
np.dot(np.linalg.solve(amat_11, bmat_11),
np.linalg.solve(amat_11, amat_12)))

# Attempt to restore state
self.cov_struct = cov_struct_save
self.cached_means = cached_means_save

from scipy.stats.distributions import chi2
score_statistic = np.dot(score2,
np.linalg.solve(score_cov, score2))
score_df = len(score2)
score_pvalue = 1 - chi2.cdf(score_statistic, score_df)
return {"statistic": score_statistic,
"df": score_df,
"p-value": score_pvalue}

[docs]    def estimate_scale(self):
"""
Estimate the dispersion/scale.
"""

if self.scaletype is None:
if isinstance(self.family, (families.Binomial, families.Poisson,
families.NegativeBinomial,
_Multinomial)):
return 1.
elif isinstance(self.scaletype, float):
return np.array(self.scaletype)

endog = self.endog_li
cached_means = self.cached_means
nobs = self.nobs
varfunc = self.family.variance

scale = 0.
fsum = 0.
for i in range(self.num_group):

if len(endog[i]) == 0:
continue

expval, _ = cached_means[i]

f = self.weights_li[i] if self.weights is not None else 1.

sdev = np.sqrt(varfunc(expval))
resid = (endog[i] - expval) / sdev

scale += f * np.sum(resid ** 2)
fsum += f * len(endog[i])

scale /= (fsum * (nobs - self.ddof_scale) / float(nobs))

return scale

[docs]    def mean_deriv(self, exog, lin_pred):
"""
Derivative of the expected endog with respect to the parameters.

Parameters
----------
exog : array_like
The exogeneous data at which the derivative is computed.
lin_pred : array_like
The values of the linear predictor.

Returns
-------
The value of the derivative of the expected endog with respect
to the parameter vector.

Notes
-----
If there is an offset or exposure, it should be added to
lin_pred prior to calling this function.
"""

dmat = exog * idl[:, None]
return dmat

[docs]    def mean_deriv_exog(self, exog, params, offset_exposure=None):
"""
Derivative of the expected endog with respect to exog.

Parameters
----------
exog : array_like
Values of the independent variables at which the derivative
is calculated.
params : array_like
Parameter values at which the derivative is calculated.
offset_exposure : array_like, optional
Combined offset and exposure.

Returns
-------
The derivative of the expected endog with respect to exog.
"""

lin_pred = np.dot(exog, params)
if offset_exposure is not None:
lin_pred += offset_exposure

dmat = np.outer(idl, params)
return dmat

def _update_mean_params(self):
"""
Returns
-------
update : array_like
The update vector such that params + update is the next
iterate when solving the score equations.
score : array_like
The current value of the score equations, not
incorporating the scale parameter.  If desired,
multiply this vector by the scale parameter to
incorporate the scale.
"""

endog = self.endog_li
exog = self.exog_li

cached_means = self.cached_means

varfunc = self.family.variance

bmat, score = 0, 0
for i in range(self.num_group):

expval, lpr = cached_means[i]
resid = endog[i] - expval
dmat = self.mean_deriv(exog[i], lpr)
sdev = np.sqrt(varfunc(expval))

rslt = self.cov_struct.covariance_matrix_solve(expval, i,
sdev, (dmat, resid))
if rslt is None:
return None, None
vinv_d, vinv_resid = tuple(rslt)

f = self.weights_li[i] if self.weights is not None else 1.

bmat += f * np.dot(dmat.T, vinv_d)
score += f * np.dot(dmat.T, vinv_resid)

update = np.linalg.solve(bmat, score)

return update, score

[docs]    def update_cached_means(self, mean_params):
"""
cached_means should always contain the most recent calculation
of the group-wise mean vectors.  This function should be
called every time the regression parameters are changed, to
keep the cached means up to date.
"""

endog = self.endog_li
exog = self.exog_li
offset = self.offset_li

self.cached_means = []

for i in range(self.num_group):

if len(endog[i]) == 0:
continue

lpr = np.dot(exog[i], mean_params)
if offset is not None:
lpr += offset[i]

self.cached_means.append((expval, lpr))

def _covmat(self):
"""
Returns the sampling covariance matrix of the regression
parameters and related quantities.

Returns
-------
cov_robust : array_like
The robust, or sandwich estimate of the covariance, which
is meaningful even if the working covariance structure is
incorrectly specified.
cov_naive : array_like
The model-based estimate of the covariance, which is
meaningful if the covariance structure is correctly
specified.
cmat : array_like
The center matrix of the sandwich expression, used in
obtaining score test results.
"""

endog = self.endog_li
exog = self.exog_li
varfunc = self.family.variance
cached_means = self.cached_means

# Calculate the naive (model-based) and robust (sandwich)
# covariances.
bmat, cmat = 0, 0
for i in range(self.num_group):

expval, lpr = cached_means[i]
resid = endog[i] - expval
dmat = self.mean_deriv(exog[i], lpr)
sdev = np.sqrt(varfunc(expval))

rslt = self.cov_struct.covariance_matrix_solve(
expval, i, sdev, (dmat, resid))
if rslt is None:
return None, None, None, None
vinv_d, vinv_resid = tuple(rslt)

f = self.weights_li[i] if self.weights is not None else 1.

bmat += f * np.dot(dmat.T, vinv_d)
dvinv_resid = f * np.dot(dmat.T, vinv_resid)
cmat += np.outer(dvinv_resid, dvinv_resid)

scale = self.estimate_scale()

bmati = np.linalg.inv(bmat)
cov_naive = bmati * scale
cov_robust = np.dot(bmati, np.dot(cmat, bmati))

cov_naive *= self.scaling_factor
cov_robust *= self.scaling_factor
return cov_robust, cov_naive, cmat

# Calculate the bias-corrected sandwich estimate of Mancl and
# DeRouen.
def _bc_covmat(self, cov_naive):

cov_naive = cov_naive / self.scaling_factor
endog = self.endog_li
exog = self.exog_li
varfunc = self.family.variance
cached_means = self.cached_means
scale = self.estimate_scale()

bcm = 0
for i in range(self.num_group):

expval, lpr = cached_means[i]
resid = endog[i] - expval
dmat = self.mean_deriv(exog[i], lpr)
sdev = np.sqrt(varfunc(expval))

rslt = self.cov_struct.covariance_matrix_solve(
expval, i, sdev, (dmat,))
if rslt is None:
return None
vinv_d = rslt[0]
vinv_d /= scale

hmat = np.dot(vinv_d, cov_naive)
hmat = np.dot(hmat, dmat.T).T

f = self.weights_li[i] if self.weights is not None else 1.

aresid = np.linalg.solve(np.eye(len(resid)) - hmat, resid)
rslt = self.cov_struct.covariance_matrix_solve(
expval, i, sdev, (aresid,))
if rslt is None:
return None
srt = rslt[0]
srt = f * np.dot(dmat.T, srt) / scale
bcm += np.outer(srt, srt)

cov_robust_bc = np.dot(cov_naive, np.dot(bcm, cov_naive))
cov_robust_bc *= self.scaling_factor

return cov_robust_bc

[docs]    def predict(self, params, exog=None, offset=None,
exposure=None, linear=False):
"""
Return predicted values for a marginal regression model fit
using GEE.

Parameters
----------
params : array_like
Parameters / coefficients of a marginal regression model.
exog : array_like, optional
Design / exogenous data. If exog is None, model exog is
used.
offset : array_like, optional
Offset for exog if provided.  If offset is None, model
offset is used.
exposure : array_like, optional
Exposure for exog, if exposure is None, model exposure is
used.  Only allowed if link function is the logarithm.
linear : bool
If True, returns the linear predicted values.  If False,
returns the value of the inverse of the model's link
function at the linear predicted values.

Returns
-------
An array of fitted values

Notes
-----
Using log(V) as the offset is equivalent to using V as the
exposure.  If exposure U and offset V are both provided, then
log(U) + V is added to the linear predictor.
"""

# TODO: many paths through this, not well covered in tests

if exposure is not None:
raise ValueError(
"exposure can only be used with the log link function")

# This is the combined offset and exposure
_offset = 0.

# Using model exog
if exog is None:
exog = self.exog

# Do not need to worry about exposure
if offset is None:
if self._offset_exposure is not None:
_offset = self._offset_exposure.copy()
else:
_offset = offset

else:
if offset is None and exposure is None:
if self._offset_exposure is not None:
_offset = self._offset_exposure
elif offset is None and exposure is not None:
_offset = np.log(exposure)
if hasattr(self, "offset"):
_offset = _offset + self.offset
elif offset is not None and exposure is None:
_offset = offset
if hasattr(self, "exposure"):
_offset = offset + np.log(self.exposure)
else:
_offset = offset + np.log(exposure)

# exog is provided: this is simpler than above because we
# never use model exog or exposure if exog is provided.
else:
if offset is not None:
_offset = _offset + offset
if exposure is not None:
_offset += np.log(exposure)

lin_pred = _offset + np.dot(exog, params)

if not linear:

return lin_pred

def _starting_params(self):

model = GLM(self.endog, self.exog, family=self.family,
offset=self._offset_exposure,
freq_weights=self.weights)
result = model.fit()
return result.params

[docs]    @Appender(_gee_fit_doc)
def fit(self, maxiter=60, ctol=1e-6, start_params=None,
params_niter=1, first_dep_update=0,
cov_type='robust', ddof_scale=None, scaling_factor=1.,
scale=None):

self.scaletype = scale

# Subtract this number from the total sample size when
# normalizing the scale parameter estimate.
if ddof_scale is None:
self.ddof_scale = self.exog.shape[1]
else:
if not ddof_scale >= 0:
raise ValueError(
"ddof_scale must be a non-negative number or None")
self.ddof_scale = ddof_scale

self.scaling_factor = scaling_factor

self._fit_history = defaultdict(list)

if self.weights is not None and cov_type == 'naive':
raise ValueError("when using weights, cov_type may not be naive")

if start_params is None:
mean_params = self._starting_params()
else:
start_params = np.asarray(start_params)
mean_params = start_params.copy()

self.update_cached_means(mean_params)

del_params = -1.
for itr in range(maxiter):

update, score = self._update_mean_params()
if update is None:
warnings.warn("Singular matrix encountered in GEE update",
ConvergenceWarning)
break
mean_params += update
self.update_cached_means(mean_params)

# L2 norm of the change in mean structure parameters at
# this iteration.
del_params = np.sqrt(np.sum(score ** 2))

self._fit_history['params'].append(mean_params.copy())
self._fit_history['score'].append(score)
self._fit_history['dep_params'].append(
self.cov_struct.dep_params)

# Do not exit until the association parameters have been
# updated at least once.
if (del_params < ctol and
(num_assoc_updates > 0 or self.update_dep is False)):
break

# Update the dependence structure
if (self.update_dep and (itr % params_niter) == 0
and (itr >= first_dep_update)):
self._update_assoc(mean_params)

if del_params >= ctol:
warnings.warn("Iteration limit reached prior to convergence",
IterationLimitWarning)

if mean_params is None:
warnings.warn("Unable to estimate GEE parameters.",
ConvergenceWarning)
return None

bcov, ncov, _ = self._covmat()
if bcov is None:
warnings.warn("Estimated covariance structure for GEE "
"estimates is singular", ConvergenceWarning)
return None
bc_cov = None
if cov_type == "bias_reduced":
bc_cov = self._bc_covmat(ncov)

if self.constraint is not None:
x = mean_params.copy()
mean_params, bcov = self._handle_constraint(mean_params, bcov)
if mean_params is None:
warnings.warn("Unable to estimate constrained GEE "
"parameters.", ConvergenceWarning)
return None

y, ncov = self._handle_constraint(x, ncov)
if y is None:
warnings.warn("Unable to estimate constrained GEE "
"parameters.", ConvergenceWarning)
return None

if bc_cov is not None:
y, bc_cov = self._handle_constraint(x, bc_cov)
if x is None:
warnings.warn("Unable to estimate constrained GEE "
"parameters.", ConvergenceWarning)
return None

scale = self.estimate_scale()

# kwargs to add to results instance, need to be available in __init__
res_kwds = dict(cov_type=cov_type,
cov_robust=bcov,
cov_naive=ncov,
cov_robust_bc=bc_cov)

# The superclass constructor will multiply the covariance
# matrix argument bcov by scale, which we do not want, so we
# divide bcov by the scale parameter here
results = GEEResults(self, mean_params, bcov / scale, scale,
cov_type=cov_type, use_t=False,
attr_kwds=res_kwds)

# attributes not needed during results__init__
results.fit_history = self._fit_history
self.fit_history = defaultdict(list)
results.score_norm = del_params
results.converged = (del_params < ctol)
results.cov_struct = self.cov_struct
results.params_niter = params_niter
results.first_dep_update = first_dep_update
results.ctol = ctol
results.maxiter = maxiter

# These will be copied over to subclasses when upgrading.
results._props = ["cov_type", "use_t",
"cov_params_default", "cov_robust",
"cov_naive", "cov_robust_bc",
"fit_history",
"score_norm", "converged", "cov_struct",
"params_niter", "first_dep_update", "ctol",
"maxiter"]

return GEEResultsWrapper(results)

def _update_regularized(self, params, pen_wt, scad_param, eps):

sn, hm = 0, 0

for i in range(self.num_group):

expval, _ = self.cached_means[i]
resid = self.endog_li[i] - expval
sdev = np.sqrt(self.family.variance(expval))

ex = self.exog_li[i] * sdev[:, None]**2
rslt = self.cov_struct.covariance_matrix_solve(
expval, i, sdev, (resid, ex))
sn0 = rslt[0]
sn += np.dot(ex.T, sn0)
hm0 = rslt[1]
hm += np.dot(ex.T, hm0)

# Wang et al. divide sn here by num_group, but that
# seems to be incorrect

ap = np.abs(params)
clipped = np.clip(scad_param * pen_wt - ap, 0, np.inf)
en = pen_wt * clipped * (ap > pen_wt)
en /= (scad_param - 1) * pen_wt
en += pen_wt * (ap <= pen_wt)
en /= eps + ap

hm.flat[::hm.shape[0] + 1] += self.num_group * en
sn -= self.num_group * en * params
update = np.linalg.solve(hm, sn)
hm *= self.estimate_scale()

return update, hm

def _regularized_covmat(self, mean_params):

self.update_cached_means(mean_params)

ma = 0

for i in range(self.num_group):

expval, _ = self.cached_means[i]
resid = self.endog_li[i] - expval
sdev = np.sqrt(self.family.variance(expval))

ex = self.exog_li[i] * sdev[:, None]**2
rslt = self.cov_struct.covariance_matrix_solve(
expval, i, sdev, (resid,))
ma0 = np.dot(ex.T, rslt[0])
ma += np.outer(ma0, ma0)

return ma

[docs]    def fit_regularized(self, pen_wt, scad_param=3.7, maxiter=100,
ddof_scale=None, update_assoc=5,
ctol=1e-5, ztol=1e-3, eps=1e-6, scale=None):
"""
Regularized estimation for GEE.

Parameters
----------
pen_wt : float
The penalty weight (a non-negative scalar).
Non-negative scalar determining the shape of the Scad
penalty.
maxiter : int
The maximum number of iterations.
ddof_scale : int
Value to subtract from nobs when calculating the
denominator degrees of freedom for t-statistics, defaults
to the number of columns in exog.
update_assoc : int
The dependence parameters are updated every update_assoc
iterations of the mean structure parameter updates.
ctol : float
Convergence criterion, default is one order of magnitude
smaller than proposed in section 3.1 of Wang et al.
ztol : float
Coefficients smaller than this value are treated as
being zero, default is based on section 5 of Wang et al.
eps : non-negative scalar
Numerical constant, see section 3.2 of Wang et al.
scale : float or string
If a float, this value is used as the scale parameter.
If "X2", the scale parameter is always estimated using
Pearson's chi-square method (e.g. as in a quasi-Poisson
analysis).  If None, the default approach for the family
is used to estimate the scale parameter.

Returns
-------
GEEResults instance.  Note that not all methods of the results
class make sense when the model has been fit with regularization.

Notes
-----
This implementation assumes that the link is canonical.

References
----------
Wang L, Zhou J, Qu A. (2012). Penalized generalized estimating
equations for high-dimensional longitudinal data analysis.
Biometrics. 2012 Jun;68(2):353-60.
doi: 10.1111/j.1541-0420.2011.01678.x.
https://www.ncbi.nlm.nih.gov/pubmed/21955051
http://users.stat.umn.edu/~wangx346/research/GEE_selection.pdf
"""

self.scaletype = scale

mean_params = np.zeros(self.exog.shape[1])
self.update_cached_means(mean_params)
converged = False
fit_history = defaultdict(list)

# Subtract this number from the total sample size when
# normalizing the scale parameter estimate.
if ddof_scale is None:
self.ddof_scale = self.exog.shape[1]
else:
if not ddof_scale >= 0:
raise ValueError(
"ddof_scale must be a non-negative number or None")
self.ddof_scale = ddof_scale

# Keep this private for now.  In some cases the early steps are
# very small so it seems necessary to ensure a certain minimum
# number of iterations before testing for convergence.
miniter = 20

for itr in range(maxiter):

update, hm = self._update_regularized(
if update is None:
msg = "Singular matrix encountered in regularized GEE update",
warnings.warn(msg, ConvergenceWarning)
break
if itr > miniter and np.sqrt(np.sum(update**2)) < ctol:
converged = True
break
mean_params += update
fit_history['params'].append(mean_params.copy())
self.update_cached_means(mean_params)

if itr != 0 and (itr % update_assoc == 0):
self._update_assoc(mean_params)

if not converged:
msg = "GEE.fit_regularized did not converge"
warnings.warn(msg)

mean_params[np.abs(mean_params) < ztol] = 0

self._update_assoc(mean_params)
ma = self._regularized_covmat(mean_params)
cov = np.linalg.solve(hm, ma)
cov = np.linalg.solve(hm, cov.T)

# kwargs to add to results instance, need to be available in __init__
res_kwds = dict(cov_type="robust", cov_robust=cov)

scale = self.estimate_scale()
rslt = GEEResults(self, mean_params, cov, scale,
regularized=True, attr_kwds=res_kwds)
rslt.fit_history = fit_history

return GEEResultsWrapper(rslt)

def _handle_constraint(self, mean_params, bcov):
"""
Expand the parameter estimate mean_params and covariance matrix
bcov to the coordinate system of the unconstrained model.

Parameters
----------
mean_params : array_like
A parameter vector estimate for the reduced model.
bcov : array_like
The covariance matrix of mean_params.

Returns
-------
mean_params : array_like
The input parameter vector mean_params, expanded to the
coordinate system of the full model
bcov : array_like
The input covariance matrix bcov, expanded to the
coordinate system of the full model
"""

# The number of variables in the full model
red_p = len(mean_params)
full_p = self.constraint.lhs.shape[1]
mean_params0 = np.r_[mean_params, np.zeros(full_p - red_p)]

# Get the score vector under the full model.
save_exog_li = self.exog_li
self.exog_li = self.constraint.exog_fulltrans_li
import copy
save_cached_means = copy.deepcopy(self.cached_means)
self.update_cached_means(mean_params0)
_, score = self._update_mean_params()

if score is None:
warnings.warn("Singular matrix encountered in GEE score test",
ConvergenceWarning)
return None, None

_, ncov1, cmat = self._covmat()
scale = self.estimate_scale()
cmat = cmat / scale ** 2
score2 = score[red_p:] / scale
amat = np.linalg.inv(ncov1)

bmat_11 = cmat[0:red_p, 0:red_p]
bmat_22 = cmat[red_p:, red_p:]
bmat_12 = cmat[0:red_p, red_p:]
amat_11 = amat[0:red_p, 0:red_p]
amat_12 = amat[0:red_p, red_p:]

score_cov = bmat_22 - np.dot(amat_12.T,
np.linalg.solve(amat_11, bmat_12))
score_cov -= np.dot(bmat_12.T,
np.linalg.solve(amat_11, amat_12))
score_cov += np.dot(amat_12.T,
np.dot(np.linalg.solve(amat_11, bmat_11),
np.linalg.solve(amat_11, amat_12)))

from scipy.stats.distributions import chi2
score_statistic = np.dot(score2,
np.linalg.solve(score_cov, score2))
score_df = len(score2)
score_pvalue = 1 - chi2.cdf(score_statistic, score_df)
self.score_test_results = {"statistic": score_statistic,
"df": score_df,
"p-value": score_pvalue}

mean_params = self.constraint.unpack_param(mean_params)
bcov = self.constraint.unpack_cov(bcov)

self.exog_li = save_exog_li
self.cached_means = save_cached_means
self.exog = self.constraint.restore_exog()

return mean_params, bcov

def _update_assoc(self, params):
"""
Update the association parameters
"""

self.cov_struct.update(params)

def _derivative_exog(self, params, exog=None, transform='dydx',
dummy_idx=None, count_idx=None):
"""
For computing marginal effects, returns dF(XB) / dX where F(.)
is the fitted mean.

transform can be 'dydx', 'dyex', 'eydx', or 'eyex'.

Not all of these make sense in the presence of discrete regressors,
but checks are done in the results in get_margeff.
"""
# This form should be appropriate for group 1 probit, logit,
# logistic, cloglog, heckprob, xtprobit.
offset_exposure = None
if exog is None:
exog = self.exog
offset_exposure = self._offset_exposure

margeff = self.mean_deriv_exog(exog, params, offset_exposure)

if 'ex' in transform:
margeff *= exog
if 'ey' in transform:
margeff /= self.predict(params, exog)[:, None]
if count_idx is not None:
from statsmodels.discrete.discrete_margins import (
_get_count_effects)
margeff = _get_count_effects(margeff, exog, count_idx, transform,
self, params)
if dummy_idx is not None:
from statsmodels.discrete.discrete_margins import (
_get_dummy_effects)
margeff = _get_dummy_effects(margeff, exog, dummy_idx, transform,
self, params)
return margeff

[docs]    def qic(self, params, scale, cov_params):
"""
Returns quasi-information criteria and quasi-likelihood values.

Parameters
----------
params : array_like
The GEE estimates of the regression parameters.
scale : scalar
Estimated scale parameter
cov_params : array_like
An estimate of the covariance matrix for the
model parameters.  Conventionally this is the robust
covariance matrix.

Returns
-------
ql : scalar
The quasi-likelihood value
qic : scalar
A QIC that can be used to compare the mean and covariance
structures of the model.
qicu : scalar
A simplified QIC that can be used to compare mean structures
but not covariance structures

Notes
-----
The quasi-likelihood used here is obtained by numerically evaluating
Wedderburn's integral representation of the quasi-likelihood function.
This approach is valid for all families and  links.  Many other
packages use analytical expressions for quasi-likelihoods that are
valid in special cases where the link function is canonical.  These
analytical expressions may omit additive constants that only depend
on the data.  Therefore, the numerical values of our QL and QIC values
will differ from the values reported by other packages.  However only
the differences between two QIC values calculated for different models
using the same data are meaningful.  Our QIC should produce the same
QIC differences as other software.

When using the QIC for models with unknown scale parameter, use a
common estimate of the scale parameter for all models being compared.

References
----------
.. [*] W. Pan (2001).  Akaike's information criterion in generalized
estimating equations.  Biometrics (57) 1.
"""

varfunc = self.family.variance

means = []
omega = 0.0
# omega^-1 is the model-based covariance assuming independence

for i in range(self.num_group):
expval, lpr = self.cached_means[i]
means.append(expval)
dmat = self.mean_deriv(self.exog_li[i], lpr)
omega += np.dot(dmat.T, dmat) / scale

means = np.concatenate(means)

# The quasi-likelihood, use change of variables so the integration is
# from -1 to 1.
du = means - self.endog
nstep = 10000
qv = np.empty(nstep)
xv = np.linspace(-0.99999, 1, nstep)
for i, g in enumerate(xv):
u = self.endog + (g + 1) * du / 2.0
vu = varfunc(u)
qv[i] = -np.sum(du**2 * (g + 1) / vu)
qv /= (4 * scale)

from scipy.integrate import trapz
ql = trapz(qv, dx=xv[1] - xv[0])

qicu = -2 * ql + 2 * self.exog.shape[1]
qic = -2 * ql + 2 * np.trace(np.dot(omega, cov_params))

return ql, qic, qicu

[docs]class GEEResults(base.LikelihoodModelResults):

__doc__ = (
"This class summarizes the fit of a marginal regression model "
"using GEE.\n" + _gee_results_doc)

def __init__(self, model, params, cov_params, scale,
cov_type='robust', use_t=False, regularized=False,
**kwds):

super(GEEResults, self).__init__(
model, params, normalized_cov_params=cov_params,
scale=scale)

self.df_resid = model.df_resid
self.df_model = model.df_model
self.family = model.family

attr_kwds = kwds.pop('attr_kwds', {})
self.__dict__.update(attr_kwds)

# we do not do this if the cov_type has already been set
# subclasses can set it through attr_kwds
if not (hasattr(self, 'cov_type') and
hasattr(self, 'cov_params_default')):
self.cov_type = cov_type  # keep alias
covariance_type = self.cov_type.lower()
allowed_covariances = ["robust", "naive", "bias_reduced"]
if covariance_type not in allowed_covariances:
msg = ("GEE: cov_type must be one of " +
", ".join(allowed_covariances))
raise ValueError(msg)

if cov_type == "robust":
cov = self.cov_robust
elif cov_type == "naive":
cov = self.cov_naive
elif cov_type == "bias_reduced":
cov = self.cov_robust_bc

self.cov_params_default = cov
else:
if self.cov_type != cov_type:
raise ValueError('cov_type in argument is different from '

[docs]    def standard_errors(self, cov_type="robust"):
"""
This is a convenience function that returns the standard
errors for any covariance type.  The value of bse is the
standard errors for whichever covariance type is specified as
an argument to fit (defaults to "robust").

Parameters
----------
cov_type : str
One of "robust", "naive", or "bias_reduced".  Determines
the covariance used to compute standard errors.  Defaults
to "robust".
"""

# Check covariance_type
covariance_type = cov_type.lower()
allowed_covariances = ["robust", "naive", "bias_reduced"]
if covariance_type not in allowed_covariances:
msg = ("GEE: covariance_type must be one of " +
", ".join(allowed_covariances))
raise ValueError(msg)

if covariance_type == "robust":
return np.sqrt(np.diag(self.cov_robust))
elif covariance_type == "naive":
return np.sqrt(np.diag(self.cov_naive))
elif covariance_type == "bias_reduced":
if self.cov_robust_bc is None:
raise ValueError(
"GEE: bias_reduced covariance not available")
return np.sqrt(np.diag(self.cov_robust_bc))

# Need to override to allow for different covariance types.
def bse(self):
return self.standard_errors(self.cov_type)

def resid(self):
"""
Returns the residuals, the endogeneous data minus the fitted
values from the model.
"""
return self.model.endog - self.fittedvalues

[docs]    def score_test(self):
"""
Return the results of a score test for a linear constraint.

Returns
-------
Adictionary containing the p-value, the test statistic,
and the degrees of freedom for the score test.

Notes
-----
a score test.  GEEResults.score_test is more general, in that it
supports testing arbitrary linear equality constraints.   However
GEE.compare_score_test might be easier to use when comparing
two explicit models.

References
----------
Xu Guo and Wei Pan (2002). "Small sample performance of the score
test in GEE".
"""

if not hasattr(self.model, "score_test_results"):
msg = "score_test on results instance only available when "
msg += " model was fit with constraints"
raise ValueError(msg)

return self.model.score_test_results

def resid_split(self):
"""
Returns the residuals, the endogeneous data minus the fitted
values from the model.  The residuals are returned as a list
of arrays containing the residuals for each cluster.
"""
sresid = []
for v in self.model.group_labels:
ii = self.model.group_indices[v]
sresid.append(self.resid[ii])
return sresid

def resid_centered(self):
"""
Returns the residuals centered within each group.
"""
cresid = self.resid.copy()
for v in self.model.group_labels:
ii = self.model.group_indices[v]
cresid[ii] -= cresid[ii].mean()
return cresid

def resid_centered_split(self):
"""
Returns the residuals centered within each group.  The
residuals are returned as a list of arrays containing the
centered residuals for each cluster.
"""
sresid = []
for v in self.model.group_labels:
ii = self.model.group_indices[v]
sresid.append(self.centered_resid[ii])
return sresid

[docs]    def qic(self, scale=None):
"""
Returns the QIC and QICu information criteria.

For families with a scale parameter (e.g. Gaussian), provide
as the scale argument the estimated scale from the largest
model under consideration.

If the scale parameter is not provided, the estimated scale
parameter is used.  Doing this does not allow comparisons of
QIC values between models.
"""

# It is easy to forget to set the scale parameter.  Sometimes
# this is intentional, so we warn.
if scale is None:
warnings.warn("QIC values obtained using scale=None are not "
"appropriate for comparing models")

if scale is None:
scale = self.scale

_, qic, qicu = self.model.qic(self.params, scale,
self.cov_params())

return qic, qicu

# FIXME: alias to be removed, temporary backwards compatibility
split_resid = resid_split
centered_resid = resid_centered
split_centered_resid = resid_centered_split

def resid_response(self):
return self.model.endog - self.fittedvalues

def resid_pearson(self):
val = self.model.endog - self.fittedvalues
val = val / np.sqrt(self.family.variance(self.fittedvalues))
return val

def resid_working(self):
val = self.resid_response
return val

def resid_anscombe(self):
return self.family.resid_anscombe(self.model.endog, self.fittedvalues)

def resid_deviance(self):
return self.family.resid_dev(self.model.endog, self.fittedvalues)

def fittedvalues(self):
"""
Returns the fitted values from the model.
"""
self.params))

use_glm_weights=True, fit_kwargs=None,
ax=None):

resid_type=resid_type,
use_glm_weights=use_glm_weights,
fit_kwargs=fit_kwargs, ax=ax)

return fig

[docs]    @Appender(_plot_partial_residuals_doc % {'extra_params_doc': ''})
def plot_partial_residuals(self, focus_exog, ax=None):

from statsmodels.graphics.regressionplots import plot_partial_residuals

return plot_partial_residuals(self, focus_exog, ax=ax)

[docs]    @Appender(_plot_ceres_residuals_doc % {'extra_params_doc': ''})
def plot_ceres_residuals(self, focus_exog, frac=0.66, cond_means=None,
ax=None):

from statsmodels.graphics.regressionplots import plot_ceres_residuals

return plot_ceres_residuals(self, focus_exog, frac,
cond_means=cond_means, ax=ax)

[docs]    def conf_int(self, alpha=.05, cols=None, cov_type=None):
"""
Returns confidence intervals for the fitted parameters.

Parameters
----------
alpha : float, optional
The alpha level for the confidence interval.  i.e., The
default alpha = .05 returns a 95% confidence interval.
cols : array_like, optional
cols specifies which confidence intervals to return
cov_type : str
The covariance type used for computing standard errors;
must be one of 'robust', 'naive', and 'bias reduced'.
See GEE for details.

Notes
-----
The confidence interval is based on the Gaussian distribution.
"""
# super does not allow to specify cov_type and method is not
# implemented,
# FIXME: remove this method here
if cov_type is None:
bse = self.bse
else:
bse = self.standard_errors(cov_type=cov_type)
params = self.params
dist = stats.norm
q = dist.ppf(1 - alpha / 2)

if cols is None:
lower = self.params - q * bse
upper = self.params + q * bse
else:
cols = np.asarray(cols)
lower = params[cols] - q * bse[cols]
upper = params[cols] + q * bse[cols]
return np.asarray(lzip(lower, upper))

[docs]    def summary(self, yname=None, xname=None, title=None, alpha=.05):
"""
Summarize the GEE regression results

Parameters
----------
yname : str, optional
Default is y
xname : list[str], optional
Names for the exogenous variables, default is var_# for ## in
the number of regressors. Must match the number of parameters in
the model
title : str, optional
Title for the top table. If not None, then this replaces
the default title
alpha : float
significance level for the confidence intervals
cov_type : str
The covariance type used to compute the standard errors;
one of 'robust' (the usual robust sandwich-type covariance
estimate), 'naive' (ignores dependence), and 'bias
reduced' (the Mancl/DeRouen estimate).

Returns
-------
smry : Summary instance
this holds the summary tables and text, which can be
printed or converted to various output formats.

--------
statsmodels.iolib.summary.Summary : class to hold summary results
"""

top_left = [('Dep. Variable:', None),
('Model:', None),
('Method:', ['Generalized']),
('', ['Estimating Equations']),
('Family:', [self.model.family.__class__.__name__]),
('Dependence structure:',
[self.model.cov_struct.__class__.__name__]),
('Date:', None),
('Covariance type: ', [self.cov_type, ])
]

NY = [len(y) for y in self.model.endog_li]

top_right = [('No. Observations:', [sum(NY)]),
('No. clusters:', [len(self.model.endog_li)]),
('Min. cluster size:', [min(NY)]),
('Max. cluster size:', [max(NY)]),
('Mean cluster size:', ["%.1f" % np.mean(NY)]),
('Num. iterations:', ['%d' %
len(self.fit_history['params'])]),
('Scale:', ["%.3f" % self.scale]),
('Time:', None),
]

# The skew of the residuals
skew1 = stats.skew(self.resid)
kurt1 = stats.kurtosis(self.resid)
skew2 = stats.skew(self.centered_resid)
kurt2 = stats.kurtosis(self.centered_resid)

diagn_left = [('Skew:', ["%12.4f" % skew1]),
('Centered skew:', ["%12.4f" % skew2])]

diagn_right = [('Kurtosis:', ["%12.4f" % kurt1]),
('Centered kurtosis:', ["%12.4f" % kurt2])
]

if title is None:
title = self.model.__class__.__name__ + ' ' +\
"Regression Results"

# Override the exog variable names if xname is provided as an
# argument.
if xname is None:
xname = self.model.exog_names

if yname is None:
yname = self.model.endog_names

# Create summary table instance
from statsmodels.iolib.summary import Summary
smry = Summary()
yname=yname, xname=xname,
title=title)
alpha=alpha, use_t=False)
gright=diagn_right, yname=yname,
xname=xname, title="")

return smry

[docs]    def get_margeff(self, at='overall', method='dydx', atexog=None,
dummy=False, count=False):
"""Get marginal effects of the fitted model.

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.

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.
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
-------
effects : ndarray
the marginal effect corresponding to the input options

Notes
-----
When using after Poisson, returns the expected number of events
per period, assuming that the model is loglinear.
"""

if self.model.constraint is not None:
warnings.warn("marginal effects ignore constraints",
ValueWarning)

return GEEMargins(self, (at, method, atexog, dummy, count))

[docs]    def plot_isotropic_dependence(self, ax=None, xpoints=10,
min_n=50):
"""
Create a plot of the pairwise products of within-group
residuals against the corresponding time differences.  This
plot can be used to assess the possible form of an isotropic
covariance structure.

Parameters
----------
ax : AxesSubplot
An axes on which to draw the graph.  If None, new
figure and axes objects are created
xpoints : scalar or array_like
If scalar, the number of points equally spaced points on
the time difference axis used to define bins for
calculating local means.  If an array, the specific points
that define the bins.
min_n : int
The minimum sample size in a bin for the mean residual
product to be included on the plot.
"""

from statsmodels.graphics import utils as gutils

resid = self.model.cluster_list(self.resid)
time = self.model.cluster_list(self.model.time)

# All within-group pairwise time distances (xdt) and the
# corresponding products of scaled residuals (xre).
xre, xdt = [], []
for re, ti in zip(resid, time):
ix = np.tril_indices(re.shape[0], 0)
re = re[ix[0]] * re[ix[1]] / self.scale ** 2
xre.append(re)
dists = np.sqrt(((ti[ix[0], :] - ti[ix[1], :]) ** 2).sum(1))
xdt.append(dists)

xre = np.concatenate(xre)
xdt = np.concatenate(xdt)

if ax is None:
fig, ax = gutils.create_mpl_ax(ax)
else:
fig = ax.get_figure()

# Convert to a correlation
ii = np.flatnonzero(xdt == 0)
v0 = np.mean(xre[ii])
xre /= v0

# Use the simple average to smooth, since fancier smoothers
# that trim and downweight outliers give biased results (we
# need the actual mean of a skewed distribution).
if np.isscalar(xpoints):
xpoints = np.linspace(0, max(xdt), xpoints)
dg = np.digitize(xdt, xpoints)
dgu = np.unique(dg)
hist = np.asarray([np.sum(dg == k) for k in dgu])
ii = np.flatnonzero(hist >= min_n)
dgu = dgu[ii]
dgy = np.asarray([np.mean(xre[dg == k]) for k in dgu])
dgx = np.asarray([np.mean(xdt[dg == k]) for k in dgu])

ax.plot(dgx, dgy, '-', color='orange', lw=5)
ax.set_xlabel("Time difference")
ax.set_ylabel("Product of scaled residuals")

return fig

[docs]    def sensitivity_params(self, dep_params_first,
dep_params_last, num_steps):
"""
Refits the GEE model using a sequence of values for the
dependence parameters.

Parameters
----------
dep_params_first : array_like
The first dep_params in the sequence
dep_params_last : array_like
The last dep_params in the sequence
num_steps : int
The number of dep_params in the sequence

Returns
-------
results : array_like
The GEEResults objects resulting from the fits.
"""

model = self.model

import copy
cov_struct = copy.deepcopy(self.model.cov_struct)

# We are fixing the dependence structure in each run.
update_dep = model.update_dep
model.update_dep = False

dep_params = []
results = []
for x in np.linspace(0, 1, num_steps):

dp = x * dep_params_last + (1 - x) * dep_params_first
dep_params.append(dp)

model.cov_struct = copy.deepcopy(cov_struct)
model.cov_struct.dep_params = dp
rslt = model.fit(start_params=self.params,
ctol=self.ctol,
params_niter=self.params_niter,
first_dep_update=self.first_dep_update,
cov_type=self.cov_type)
results.append(rslt)

model.update_dep = update_dep

return results

# FIXME: alias to be removed, temporary backwards compatibility
params_sensitivity = sensitivity_params

class GEEResultsWrapper(lm.RegressionResultsWrapper):
_attrs = {
'centered_resid': 'rows',
}
_wrap_attrs = wrap.union_dicts(lm.RegressionResultsWrapper._wrap_attrs,
_attrs)
wrap.populate_wrapper(GEEResultsWrapper, GEEResults)  # noqa:E305

[docs]class OrdinalGEE(GEE):

__doc__ = (
"    Ordinal Response Marginal Regression Model using GEE\n" +
_gee_init_doc % {'extra_params': base._missing_param_doc,
'family_doc': _gee_ordinal_family_doc,
'example': _gee_ordinal_example})

def __init__(self, endog, exog, groups, time=None, family=None,
cov_struct=None, missing='none', offset=None,
dep_data=None, constraint=None, **kwargs):

if family is None:
family = families.Binomial()
else:
if not isinstance(family, families.Binomial):
raise ValueError("ordinal GEE must use a Binomial family")

if cov_struct is None:
cov_struct = cov_structs.OrdinalIndependence()

endog, exog, groups, time, offset = self.setup_ordinal(
endog, exog, groups, time, offset)

super(OrdinalGEE, self).__init__(endog, exog, groups, time,
family, cov_struct, missing,
offset, dep_data, constraint)

[docs]    def setup_ordinal(self, endog, exog, groups, time, offset):
"""
Restructure ordinal data as binary indicators so that they can
be analyzed using Generalized Estimating Equations.
"""

self.endog_orig = endog.copy()
self.exog_orig = exog.copy()
self.groups_orig = groups.copy()
if offset is not None:
self.offset_orig = offset.copy()
else:
self.offset_orig = None
offset = np.zeros(len(endog))
if time is not None:
self.time_orig = time.copy()
else:
self.time_orig = None
time = np.zeros((len(endog), 1))

exog = np.asarray(exog)
endog = np.asarray(endog)
groups = np.asarray(groups)
time = np.asarray(time)
offset = np.asarray(offset)

# The unique outcomes, except the greatest one.
self.endog_values = np.unique(endog)
endog_cuts = self.endog_values[0:-1]
ncut = len(endog_cuts)

nrows = ncut * len(endog)
exog_out = np.zeros((nrows, exog.shape[1]),
dtype=np.float64)
endog_out = np.zeros(nrows, dtype=np.float64)
intercepts = np.zeros((nrows, ncut), dtype=np.float64)
groups_out = np.zeros(nrows, dtype=groups.dtype)
time_out = np.zeros((nrows, time.shape[1]),
dtype=np.float64)
offset_out = np.zeros(nrows, dtype=np.float64)

jrow = 0
zipper = zip(exog, endog, groups, time, offset)
for (exog_row, endog_value, group_value, time_value,
offset_value) in zipper:

# Loop over thresholds for the indicators
for thresh_ix, thresh in enumerate(endog_cuts):

exog_out[jrow, :] = exog_row
endog_out[jrow] = (int(endog_value > thresh))
intercepts[jrow, thresh_ix] = 1
groups_out[jrow] = group_value
time_out[jrow] = time_value
offset_out[jrow] = offset_value
jrow += 1

exog_out = np.concatenate((intercepts, exog_out), axis=1)

# exog column names, including intercepts
xnames = ["I(y>%.1f)" % v for v in endog_cuts]
if type(self.exog_orig) == pd.DataFrame:
xnames.extend(self.exog_orig.columns)
else:
xnames.extend(["x%d" % k for k in range(1, exog.shape[1] + 1)])
exog_out = pd.DataFrame(exog_out, columns=xnames)

# Preserve the endog name if there is one
if type(self.endog_orig) == pd.Series:
endog_out = pd.Series(endog_out, name=self.endog_orig.name)

return endog_out, exog_out, groups_out, time_out, offset_out

def _starting_params(self):
model = GEE(self.endog, self.exog, self.groups,
time=self.time, family=families.Binomial(),
offset=self.offset, exposure=self.exposure)
result = model.fit()
return result.params

[docs]    @Appender(_gee_fit_doc)
def fit(self, maxiter=60, ctol=1e-6, start_params=None,
params_niter=1, first_dep_update=0,
cov_type='robust'):

rslt = super(OrdinalGEE, self).fit(maxiter, ctol, start_params,
params_niter, first_dep_update,
cov_type=cov_type)

rslt = rslt._results   # use unwrapped instance
res_kwds = dict(((k, getattr(rslt, k)) for k in rslt._props))
# Convert the GEEResults to an OrdinalGEEResults
ord_rslt = OrdinalGEEResults(self, rslt.params,
rslt.cov_params() / rslt.scale,
rslt.scale,
cov_type=cov_type,
attr_kwds=res_kwds)
# for k in rslt._props:
#    setattr(ord_rslt, k, getattr(rslt, k))
# TODO: document or delete

return OrdinalGEEResultsWrapper(ord_rslt)

class OrdinalGEEResults(GEEResults):

__doc__ = (
"This class summarizes the fit of a marginal regression model"
"for an ordinal response using GEE.\n"
+ _gee_results_doc)

def plot_distribution(self, ax=None, exog_values=None):
"""
Plot the fitted probabilities of endog in an ordinal model,
for specified values of the predictors.

Parameters
----------
ax : AxesSubplot
An axes on which to draw the graph.  If None, new
figure and axes objects are created
exog_values : array_like
A list of dictionaries, with each dictionary mapping
variable names to values at which the variable is held
fixed.  The values P(endog=y | exog) are plotted for all
possible values of y, at the given exog value.  Variables
not included in a dictionary are held fixed at the mean
value.

Example:
--------
We have a model with covariates 'age' and 'sex', and wish to
plot the probabilities P(endog=y | exog) for males (sex=0) and
for females (sex=1), as separate paths on the plot.  Since
'age' is not included below in the map, it is held fixed at
its mean value.

>>> ev = [{"sex": 1}, {"sex": 0}]
>>> rslt.distribution_plot(exog_values=ev)
"""

from statsmodels.graphics import utils as gutils

if ax is None:
fig, ax = gutils.create_mpl_ax(ax)
else:
fig = ax.get_figure()

# If no covariate patterns are specified, create one with all
# variables set to their mean values.
if exog_values is None:
exog_values = [{}, ]

exog_means = self.model.exog.mean(0)
ix_icept = [i for i, x in enumerate(self.model.exog_names) if
x.startswith("I(")]

for ev in exog_values:

for k in ev.keys():
if k not in self.model.exog_names:
raise ValueError("%s is not a variable in the model"
% k)

# Get the fitted probability for each level, at the given
# covariate values.
pr = []
for j in ix_icept:

xp = np.zeros_like(self.params)
xp[j] = 1.
for i, vn in enumerate(self.model.exog_names):
if i in ix_icept:
continue
# User-specified value
if vn in ev:
xp[i] = ev[vn]
# Mean value
else:
xp[i] = exog_means[i]

p = 1 / (1 + np.exp(-np.dot(xp, self.params)))
pr.append(p)

pr.insert(0, 1)
pr.append(0)
pr = np.asarray(pr)
prd = -np.diff(pr)

ax.plot(self.model.endog_values, prd, 'o-')

ax.set_xlabel("Response value")
ax.set_ylabel("Probability")
ax.set_ylim(0, 1)

return fig

def _score_test_submodel(par, sub):
"""
Return transformation matrices for design matrices.

Parameters
----------
par : instance
The parent model
sub : instance
The sub-model

Returns
-------
qm : array_like
Matrix mapping the design matrix of the parent to the design matrix
for the sub-model.
qc : array_like
Matrix mapping the design matrix of the parent to the orthogonal
complement of the columnspace of the submodel in the columnspace
of the parent.

Notes
-----
Returns None, None if the provided submodel is not actually a submodel.
"""

x1 = par.exog
x2 = sub.exog

u, s, vt = np.linalg.svd(x1, 0)

# Get the orthogonal complement of col(x2) in col(x1).
a, _, _ = np.linalg.svd(x2, 0)
a = u - np.dot(a, np.dot(a.T, u))
x2c, sb, _ = np.linalg.svd(a, 0)
x2c = x2c[:, sb > 1e-12]

# x1 * qm = x2
qm = np.dot(vt.T, np.dot(u.T, x2) / s[:, None])

e = np.max(np.abs(x2 - np.dot(x1, qm)))
if e > 1e-8:
return None, None

# x1 * qc = x2c
qc = np.dot(vt.T, np.dot(u.T, x2c) / s[:, None])

return qm, qc

class OrdinalGEEResultsWrapper(GEEResultsWrapper):
pass
wrap.populate_wrapper(OrdinalGEEResultsWrapper, OrdinalGEEResults)  # noqa:E305

[docs]class NominalGEE(GEE):

__doc__ = (
"    Nominal Response Marginal Regression Model using GEE.\n" +
_gee_init_doc % {'extra_params': base._missing_param_doc,
'family_doc': _gee_nominal_family_doc,
'example': _gee_nominal_example})

def __init__(self, endog, exog, groups, time=None, family=None,
cov_struct=None, missing='none', offset=None,
dep_data=None, constraint=None, **kwargs):

endog, exog, groups, time, offset = self.setup_nominal(
endog, exog, groups, time, offset)

if family is None:
family = _Multinomial(self.ncut + 1)

if cov_struct is None:
cov_struct = cov_structs.NominalIndependence()

super(NominalGEE, self).__init__(
endog, exog, groups, time, family, cov_struct, missing,
offset, dep_data, constraint)

def _starting_params(self):
model = GEE(self.endog, self.exog, self.groups,
time=self.time, family=families.Binomial(),
offset=self.offset, exposure=self.exposure)
result = model.fit()
return result.params

[docs]    def setup_nominal(self, endog, exog, groups, time, offset):
"""
Restructure nominal data as binary indicators so that they can
be analyzed using Generalized Estimating Equations.
"""

self.endog_orig = endog.copy()
self.exog_orig = exog.copy()
self.groups_orig = groups.copy()
if offset is not None:
self.offset_orig = offset.copy()
else:
self.offset_orig = None
offset = np.zeros(len(endog))
if time is not None:
self.time_orig = time.copy()
else:
self.time_orig = None
time = np.zeros((len(endog), 1))

exog = np.asarray(exog)
endog = np.asarray(endog)
groups = np.asarray(groups)
time = np.asarray(time)
offset = np.asarray(offset)

# The unique outcomes, except the greatest one.
self.endog_values = np.unique(endog)
endog_cuts = self.endog_values[0:-1]
ncut = len(endog_cuts)
self.ncut = ncut

nrows = len(endog_cuts) * exog.shape[0]
ncols = len(endog_cuts) * exog.shape[1]
exog_out = np.zeros((nrows, ncols), dtype=np.float64)
endog_out = np.zeros(nrows, dtype=np.float64)
groups_out = np.zeros(nrows, dtype=np.float64)
time_out = np.zeros((nrows, time.shape[1]),
dtype=np.float64)
offset_out = np.zeros(nrows, dtype=np.float64)

jrow = 0
zipper = zip(exog, endog, groups, time, offset)
for (exog_row, endog_value, group_value, time_value,
offset_value) in zipper:

# Loop over thresholds for the indicators
for thresh_ix, thresh in enumerate(endog_cuts):

u = np.zeros(len(endog_cuts), dtype=np.float64)
u[thresh_ix] = 1
exog_out[jrow, :] = np.kron(u, exog_row)
endog_out[jrow] = (int(endog_value == thresh))
groups_out[jrow] = group_value
time_out[jrow] = time_value
offset_out[jrow] = offset_value
jrow += 1

# exog names
if isinstance(self.exog_orig, pd.DataFrame):
xnames_in = self.exog_orig.columns
else:
xnames_in = ["x%d" % k for k in range(1, exog.shape[1] + 1)]
xnames = []
for tr in endog_cuts:
xnames.extend(["%s[%.1f]" % (v, tr) for v in xnames_in])
exog_out = pd.DataFrame(exog_out, columns=xnames)
exog_out = pd.DataFrame(exog_out, columns=xnames)

# Preserve endog name if there is one
if isinstance(self.endog_orig, pd.Series):
endog_out = pd.Series(endog_out, name=self.endog_orig.name)

return endog_out, exog_out, groups_out, time_out, offset_out

[docs]    def mean_deriv(self, exog, lin_pred):
"""
Derivative of the expected endog with respect to the parameters.

Parameters
----------
exog : array_like
The exogeneous data at which the derivative is computed,
number of rows must be a multiple of ncut.
lin_pred : array_like
The values of the linear predictor, length must be multiple
of ncut.

Returns
-------
The derivative of the expected endog with respect to the
parameters.
"""

expval = np.exp(lin_pred)

# Reshape so that each row contains all the indicators
# corresponding to one multinomial observation.
expval_m = np.reshape(expval, (len(expval) // self.ncut,
self.ncut))

# The normalizing constant for the multinomial probabilities.
denom = 1 + expval_m.sum(1)
denom = np.kron(denom, np.ones(self.ncut, dtype=np.float64))

# The multinomial probabilities
mprob = expval / denom

# First term of the derivative: denom * expval' / denom^2 =
# expval' / denom.
dmat = mprob[:, None] * exog

# Second term of the derivative: -expval * denom' / denom^2
ddenom = expval[:, None] * exog
dmat -= mprob[:, None] * ddenom / denom[:, None]

return dmat

[docs]    def mean_deriv_exog(self, exog, params, offset_exposure=None):
"""
Derivative of the expected endog with respect to exog for the
multinomial model, used in analyzing marginal effects.

Parameters
----------
exog : array_like
The exogeneous data at which the derivative is computed,
number of rows must be a multiple of ncut.
lpr : array_like
The linear predictor values, length must be multiple of
ncut.

Returns
-------
The value of the derivative of the expected endog with respect
to exog.

Notes
-----
offset_exposure must be set at None for the multinomial family.
"""

if offset_exposure is not None:
warnings.warn("Offset/exposure ignored for the multinomial family",
ValueWarning)

lpr = np.dot(exog, params)
expval = np.exp(lpr)

expval_m = np.reshape(expval, (len(expval) // self.ncut,
self.ncut))

denom = 1 + expval_m.sum(1)
denom = np.kron(denom, np.ones(self.ncut, dtype=np.float64))

bmat0 = np.outer(np.ones(exog.shape[0]), params)

qmat = []
for j in range(self.ncut):
ee = np.zeros(self.ncut, dtype=np.float64)
ee[j] = 1
qmat.append(np.kron(ee, np.ones(len(params) // self.ncut)))
qmat = np.array(qmat)
qmat = np.kron(np.ones((exog.shape[0] // self.ncut, 1)), qmat)
bmat = bmat0 * qmat

dmat = expval[:, None] * bmat / denom[:, None]

expval_mb = np.kron(expval_m, np.ones((self.ncut, 1)))
expval_mb = np.kron(expval_mb, np.ones((1, self.ncut)))

dmat -= expval[:, None] * (bmat * expval_mb) / denom[:, None] ** 2

return dmat

[docs]    @Appender(_gee_fit_doc)
def fit(self, maxiter=60, ctol=1e-6, start_params=None,
params_niter=1, first_dep_update=0,
cov_type='robust'):

rslt = super(NominalGEE, self).fit(maxiter, ctol, start_params,
params_niter, first_dep_update,
cov_type=cov_type)
if rslt is None:
ConvergenceWarning)
return None

rslt = rslt._results   # use unwrapped instance
res_kwds = dict(((k, getattr(rslt, k)) for k in rslt._props))
# Convert the GEEResults to a NominalGEEResults
nom_rslt = NominalGEEResults(self, rslt.params,
rslt.cov_params() / rslt.scale,
rslt.scale,
cov_type=cov_type,
attr_kwds=res_kwds)
# TODO: document or delete
# for k in rslt._props:
#    setattr(nom_rslt, k, getattr(rslt, k))

return NominalGEEResultsWrapper(nom_rslt)

class NominalGEEResults(GEEResults):

__doc__ = (
"This class summarizes the fit of a marginal regression model"
"for a nominal response using GEE.\n"
+ _gee_results_doc)

def plot_distribution(self, ax=None, exog_values=None):
"""
Plot the fitted probabilities of endog in an nominal model,
for specified values of the predictors.

Parameters
----------
ax : AxesSubplot
An axes on which to draw the graph.  If None, new
figure and axes objects are created
exog_values : array_like
A list of dictionaries, with each dictionary mapping
variable names to values at which the variable is held
fixed.  The values P(endog=y | exog) are plotted for all
possible values of y, at the given exog value.  Variables
not included in a dictionary are held fixed at the mean
value.

Example:
--------
We have a model with covariates 'age' and 'sex', and wish to
plot the probabilities P(endog=y | exog) for males (sex=0) and
for females (sex=1), as separate paths on the plot.  Since
'age' is not included below in the map, it is held fixed at
its mean value.

>>> ex = [{"sex": 1}, {"sex": 0}]
>>> rslt.distribution_plot(exog_values=ex)
"""

from statsmodels.graphics import utils as gutils

if ax is None:
fig, ax = gutils.create_mpl_ax(ax)
else:
fig = ax.get_figure()

# If no covariate patterns are specified, create one with all
# variables set to their mean values.
if exog_values is None:
exog_values = [{}, ]

ncut = self.model.family.ncut

k = int(self.model.exog.shape[1] / ncut)
exog_means = self.model.exog.mean(0)[0:k]
exog_names = self.model.exog_names[0:k]
exog_names = [x.split("[")[0] for x in exog_names]

params = np.reshape(self.params,
(ncut, len(self.params) // ncut))

for ev in exog_values:

exog = exog_means.copy()

for k in ev.keys():
if k not in exog_names:
raise ValueError("%s is not a variable in the model"
% k)

ii = exog_names.index(k)
exog[ii] = ev[k]

lpr = np.dot(params, exog)
pr = np.r_[pr, 1 - pr.sum()]

ax.plot(self.model.endog_values, pr, 'o-')

ax.set_xlabel("Response value")
ax.set_ylabel("Probability")
ax.set_xticks(self.model.endog_values)
ax.set_xticklabels(self.model.endog_values)
ax.set_ylim(0, 1)

return fig

class NominalGEEResultsWrapper(GEEResultsWrapper):
pass
wrap.populate_wrapper(NominalGEEResultsWrapper, NominalGEEResults)  # noqa:E305

"""
The multinomial logit transform, only for use with GEE.

Notes
-----
The data are assumed coded as binary indicators, where each
observed multinomial value y is coded as I(y == S[0]), ..., I(y ==
S[-1]), where S is the set of possible response labels, excluding
the largest one.  Thererefore functions in this class should only
be called using vector argument whose length is a multiple of |S|
= ncut, which is an argument to be provided when initializing the
class.

call and derivative use a private method _clean to trim p by 1e-10
so that p is in (0, 1)
"""

def __init__(self, ncut):
self.ncut = ncut

def inverse(self, lpr):
"""
Inverse of the multinomial logit transform, which gives the
expected values of the data as a function of the linear
predictors.

Parameters
----------
lpr : array_like (length must be divisible by ncut)
The linear predictors

Returns
-------
prob : ndarray
Probabilities, or expected values
"""

expval = np.exp(lpr)

denom = 1 + np.reshape(expval, (len(expval) // self.ncut,
self.ncut)).sum(1)
denom = np.kron(denom, np.ones(self.ncut, dtype=np.float64))

prob = expval / denom

return prob

class _Multinomial(families.Family):
"""
Pseudo-link function for fitting nominal multinomial models with
GEE.  Not for use outside the GEE class.
"""

variance = varfuncs.binary

def __init__(self, nlevels):
"""
Parameters
----------
nlevels : int
The number of distinct categories for the multinomial
distribution.
"""
self.initialize(nlevels)

def initialize(self, nlevels):
self.ncut = nlevels - 1

[docs]class GEEMargins(object):
"""
Estimated marginal effects for a regression model fit with GEE.

Parameters
----------
results : GEEResults instance
The results instance of a fitted discrete choice model
args : tuple
Args are passed to get_margeff. This is the same as
kwargs : dict
Keyword args are passed to get_margeff. This is the same as
"""

def __init__(self, results, args, kwargs={}):
self._cache = {}
self.results = results
self.get_margeff(*args, **kwargs)

def _reset(self):
self._cache = {}

def tvalues(self):
_check_at_is_all(self.margeff_options)
return self.margeff / self.margeff_se

[docs]    def summary_frame(self, alpha=.05):
"""
Returns a DataFrame summarizing the marginal effects.

Parameters
----------
alpha : float
Number between 0 and 1. The confidence intervals have the
probability 1-alpha.

Returns
-------
frame : DataFrames
A DataFrame summarizing the marginal effects.
"""
_check_at_is_all(self.margeff_options)
from pandas import DataFrame
names = [_transform_names[self.margeff_options['method']],
'Std. Err.', 'z', 'Pr(>|z|)',
'Conf. Int. Low', 'Cont. Int. Hi.']
ind = self.results.model.exog.var(0) != 0  # True if not a constant
exog_names = self.results.model.exog_names
var_names = [name for i, name in enumerate(exog_names) if ind[i]]
table = np.column_stack((self.margeff, self.margeff_se, self.tvalues,
self.pvalues, self.conf_int(alpha)))
return DataFrame(table, columns=names, index=var_names)

def pvalues(self):
_check_at_is_all(self.margeff_options)
return stats.norm.sf(np.abs(self.tvalues)) * 2

[docs]    def conf_int(self, alpha=.05):
"""
Returns the confidence intervals of the marginal effects

Parameters
----------
alpha : float
Number between 0 and 1. The confidence intervals have the
probability 1-alpha.

Returns
-------
conf_int : ndarray
An array with lower, upper confidence intervals for the marginal
effects.
"""
_check_at_is_all(self.margeff_options)
me_se = self.margeff_se
q = stats.norm.ppf(1 - alpha / 2)
lower = self.margeff - q * me_se
upper = self.margeff + q * me_se
return np.asarray(lzip(lower, upper))

[docs]    def summary(self, alpha=.05):
"""
Returns a summary table for marginal effects

Parameters
----------
alpha : float
Number between 0 and 1. The confidence intervals have the
probability 1-alpha.

Returns
-------
Summary : SummaryTable
A SummaryTable instance
"""
_check_at_is_all(self.margeff_options)
results = self.results
model = results.model
title = model.__class__.__name__ + " Marginal Effects"
method = self.margeff_options['method']
top_left = [('Dep. Variable:', [model.endog_names]),
('Method:', [method]),
('At:', [self.margeff_options['at']]), ]

from statsmodels.iolib.summary import (Summary, summary_params,
table_extend)
exog_names = model.exog_names[:]  # copy
smry = Summary()

const_idx = model.data.const_idx
if const_idx is not None:
exog_names.pop(const_idx)

J = int(getattr(model, "J", 1))
if J > 1:
yname, yname_list = results._get_endog_name(model.endog_names,
None, all=True)
else:
yname = model.endog_names
yname_list = [yname]

yname=yname, xname=exog_names, title=title)

# NOTE: add_table_params is not general enough yet for margeff
# could use a refactor with getattr instead of hard-coded params
# tvalues etc.
table = []
conf_int = self.conf_int(alpha)
margeff = self.margeff
margeff_se = self.margeff_se
tvalues = self.tvalues
pvalues = self.pvalues
if J > 1:
for eq in range(J):
restup = (results, margeff[:, eq], margeff_se[:, eq],
tvalues[:, eq], pvalues[:, eq], conf_int[:, :, eq])
tble = summary_params(restup, yname=yname_list[eq],
xname=exog_names, alpha=alpha,
use_t=False,
tble.title = yname_list[eq]
# overwrite coef with method name
header = ['', _transform_names[method], 'std err', 'z',
'P>|z|',
'[%3.1f%% Conf. Int.]' % (100 - alpha * 100)]
# from IPython.core.debugger import Pdb; Pdb().set_trace()
table.append(tble)

else:
restup = (results, margeff, margeff_se, tvalues, pvalues, conf_int)
table = summary_params(restup, yname=yname, xname=exog_names,
header = ['', _transform_names[method], 'std err', 'z',
'P>|z|', '[%3.1f%% Conf. Int.]' % (100 - alpha * 100)]

smry.tables.append(table)
return smry

[docs]    def get_margeff(self, at='overall', method='dydx', atexog=None,
dummy=False, count=False):

self._reset()  # always reset the cache when this is called
# TODO: if at is not all or overall, we can also put atexog values
method = method.lower()
at = at.lower()
_check_margeff_args(at, method)
self.margeff_options = dict(method=method, at=at)
results = self.results
model = results.model
params = results.params
exog = model.exog.copy()  # copy because values are changed
effects_idx = exog.var(0) != 0
const_idx = model.data.const_idx

if dummy:
_check_discrete_args(at, method)
dummy_idx, dummy = _get_dummy_index(exog, const_idx)
else:
dummy_idx = None

if count:
_check_discrete_args(at, method)
count_idx, count = _get_count_index(exog, const_idx)
else:
count_idx = None

# get the exogenous variables
exog = _get_margeff_exog(exog, at, atexog, effects_idx)

# get base marginal effects, handled by sub-classes
effects = model._derivative_exog(params, exog, method,
dummy_idx, count_idx)
effects = _effects_at(effects, at)

if at == 'all':
self.margeff = effects[:, effects_idx]
else:
# Set standard error of the marginal effects by Delta method.
margeff_cov, margeff_se = margeff_cov_with_se(
model, params, exog, results.cov_params(), at,
model._derivative_exog, dummy_idx, count_idx,
method, 1)

# do not care about at constant
self.margeff_cov = margeff_cov[effects_idx][:, effects_idx]
self.margeff_se = margeff_se[effects_idx]
self.margeff = effects[effects_idx]