"""
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".
http://www.sph.umn.edu/faculty1/wp-content/uploads/2012/11/rr2002-013.pdf
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 iterkeys, range, lrange, lzip, zip
import numpy as np
from scipy import stats
import pandas as pd
from statsmodels.tools.decorators import (cache_readonly,
resettable_cache)
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.cov_struct import (Independence,
GlobalOddsRatio,
CovStruct)
import statsmodels.genmod.families.varfuncs as varfuncs
from statsmodels.genmod.families.links import Link
from statsmodels.tools.sm_exceptions import (ConvergenceWarning,
IterationLimitWarning)
import warnings
# Workaround for block_diag, not available until scipy version
# 0.11. When the statsmodels scipy dependency moves to version 0.11,
# we can remove this function and use:
# from scipy.sparse import block_diag
[docs]def block_diag(dblocks, format=None):
from scipy.sparse import bmat
n = len(dblocks)
blocks = []
for i in range(n):
b = [None,] * n
b[i] = dblocks[i]
blocks.append(b)
return bmat(blocks, format)
[docs]class ParameterConstraint(object):
"""
A class for managing linear equality constraints for a parameter
vector.
"""
[docs] 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)
[docs] 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
[docs] 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]]
[docs] def restore_exog(self):
"""
Returns the full exog matrix before it was reduced to
satisfy the constraint.
"""
return self.orig_exog
[docs] def unpack_param(self, params):
"""
Converts the parameter vector `params` from reduced to full
coordinates.
"""
return self.param0 + np.dot(self.lhs0, params)
[docs] 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 = """
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.
%(extra_params)s
See Also
--------
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
Poission | 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 downard bias of the robust
estimator.
Examples
--------
%(example)s
"""
_gee_family_doc = """\
The default is Gaussian. To specify the binomial
distribution use `family=sm.family.Binomial()`. Each family
can take a link instance as an argument. See
statsmodels.family.family for more information."""
_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 : integer
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 : integer
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 : integer
No dependence structure updates occur before this
iteration number.
cov_type : string
One of "robust", "naive", or "bias_reduced".
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 = """
Returns
-------
**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 : string
string indicating whether a "robust", "naive" or "bias_reduced"
covariance is used as default
fit_history : dict
Contains information about the iterations.
fittedvalues : array
Linear predicted values for the fitted model.
dot(exog, params)
model : class instance
Pointer to GEE model instance that called `fit`.
normalized_cov_params : array
See GEE docstring
params : array
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.
See GEE.fit for more information.
score_norm : float
norm of the score at the end of the iterative estimation.
bse : array
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()
"""
[docs]class GEE(base.Model):
__doc__ = (
" Estimation of marginal regression models using Generalized\n"
" Estimating Equations (GEE).\n" + _gee_init_doc %
{'extra_params': base._missing_param_doc,
'family_doc': _gee_family_doc,
'example': _gee_example})
cached_means = None
[docs] 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, **kwargs):
self.missing = missing
self.dep_data = dep_data
self.constraint = constraint
self.update_dep = update_dep
groups = np.array(groups) # in case groups is pandas
# 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,
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 = Independence()
else:
if not issubclass(cov_struct.__class__, 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 = np.zeros(len(self.endog))
if offset is not None:
self._offset_exposure += self.offset
self.offset = offset
if exposure is not None:
if not isinstance(self.family.link, families.links.Log):
raise ValueError("exposure can only be used with the log link function")
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)
self._offset_exposure += self.constraint.offset_increment()
self.exog = self.constraint.reduced_exog()
# Convert the data to the internal representation, which is a
# list of arrays, corresponding to the clusters.
group_labels = sorted(set(self.groups))
group_indices = dict((s, []) for s in group_labels)
for i in range(len(self.endog)):
group_indices[self.groups[i]].append(i)
for k in iterkeys(group_indices):
group_indices[k] = np.asarray(group_indices[k])
self.group_indices = group_indices
self.group_labels = group_labels
self.endog_li = self.cluster_list(self.endog)
self.exog_li = self.cluster_list(self.exog)
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)
self.offset_li = self.cluster_list(self._offset_exposure)
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]
# mean_deriv is the derivative of E[endog|exog] with respect
# to params
try:
# This custom mean_deriv is currently only used for the
# multinomial logit model
self.mean_deriv = self.family.link.mean_deriv
except AttributeError:
# Otherwise it can be obtained easily from inverse_deriv
mean_deriv_lpr = self.family.link.inverse_deriv
def mean_deriv(exog, lpr):
dmat = exog * mean_deriv_lpr(lpr)[:, None]
return dmat
self.mean_deriv = mean_deriv
# mean_deriv_exog is the derivative of E[endog|exog] with
# respect to exog
try:
# This custom mean_deriv_exog is currently only used for
# the multinomial logit model
self.mean_deriv_exog = self.family.link.mean_deriv_exog
except AttributeError:
# Otherwise it can be obtained easily from inverse_deriv
mean_deriv_lpr = self.family.link.inverse_deriv
def mean_deriv_exog(exog, params):
lpr = np.dot(exog, params)
dmat = np.outer(mean_deriv_lpr(lpr), params)
return dmat
self.mean_deriv_exog = mean_deriv_exog
# 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.
@classmethod
[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 estimate_scale(self):
"""
Returns an estimate of the scale parameter `phi` at the
current parameter value.
"""
endog = self.endog_li
exog = self.exog_li
offset = self.offset_li
cached_means = self.cached_means
nobs = self.nobs
exog_dim = exog[0].shape[1]
varfunc = self.family.variance
scale = 0.
for i in range(self.num_group):
if len(endog[i]) == 0:
continue
expval, _ = cached_means[i]
sdev = np.sqrt(varfunc(expval))
resid = (endog[i] - offset[i] - expval) / sdev
scale += np.sum(resid**2)
scale /= (nobs - exog_dim)
return scale
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)
bmat += np.dot(dmat.T, vinv_d)
score += np.dot(dmat.T, vinv_resid)
update = np.linalg.solve(bmat, score)
self._fit_history["cov_adjust"].append(
self.cov_struct.cov_adjust)
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
linkinv = self.family.link.inverse
self.cached_means = []
for i in range(self.num_group):
if len(endog[i]) == 0:
continue
lpr = offset[i] + np.dot(exog[i], mean_params)
expval = linkinv(lpr)
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.
cov_robust_bc : array-like
The "bias corrected" robust covariance of Mancl and
DeRouen.
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)
bmat += np.dot(dmat.T, vinv_d)
dvinv_resid = 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))
# Calculate the bias-corrected sandwich estimate of Mancl and
# DeRouen (requires cov_naive so cannot be calculated
# in the previous loop).
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, None, None, None
vinv_d = rslt[0]
vinv_d /= scale
hmat = np.dot(vinv_d, cov_naive)
hmat = np.dot(hmat, dmat.T).T
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, None, None, None
srt = rslt[0]
srt = np.dot(dmat.T, srt) / scale
bcm += np.outer(srt, srt)
cov_robust_bc = np.dot(cov_naive, np.dot(bcm, cov_naive))
return (cov_robust, cov_naive, cov_robust_bc, cmat)
[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 and not isinstance(self.family.link, families.links.Log):
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
if not isinstance(self.family.link, families.links.Log):
# Don't need to worry about exposure
if offset is None:
_offset = self._offset_exposure
else:
_offset = offset
else:
if offset is None and exposure is 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
if exposure is not None:
_offset += np.log(exposure)
lin_pred = _offset + np.dot(exog, params)
if not linear:
return self.family.link.inverse(lin_pred)
return lin_pred
def _starting_params(self):
"""
Returns a starting value for the mean parameters and a list of
variable names.
"""
dm = self.exog.shape[1]
# For categorical models, use independence cov_struct to get
# starting values.
if isinstance(self.cov_struct, GlobalOddsRatio):
ind = Independence()
md = GEE(self.endog, self.exog, self.groups,
time=self.time, family=self.family,
offset=self.offset, exposure=self.exposure)
mdf = md.fit()
return mdf.params
# TODO: use GLM to get Poisson starting values
else:
return np.zeros(dm, dtype=np.float64)
[docs] def fit(self, maxiter=60, ctol=1e-6, start_params=None,
params_niter=1, first_dep_update=0,
cov_type='robust'):
self._fit_history = {'params': [],
'score': [],
'dep_params': [],
'cov_adjust': []}
if start_params is None:
mean_params = self._starting_params()
else:
mean_params = start_params.copy()
self.update_cached_means(mean_params)
del_params = -1.
num_assoc_updates = 0
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)
# Don't exit until the association parameters have been
# updated at least once.
if (del_params < ctol and
(num_assoc_updates > 0 or self.update_dep == 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)
num_assoc_updates += 1
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, bc_cov, _ = self._covmat()
if bcov is None:
warnings.warn("Estimated covariance structure for GEE "
"estimates is singular", ConvergenceWarning)
return None
if self.constraint is not None:
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
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 don't 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
delattr(self, "_fit_history")
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)
fit.__doc__ = _gee_fit_doc
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 predicted probabilities
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.
"""
#note, this form should be appropriate for
## group 1 probit, logit, logistic, cloglog, heckprob, xtprobit
if exog is None:
exog = self.exog
margeff = self.mean_deriv_exog(exog, params)
# lpr = np.dot(exog, params)
# margeff = (self.mean_deriv(exog, lpr) / exog) * params
# margeff = np.dot(self.pdf(np.dot(exog, params))[:, None],
# params[None,:])
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]class GEEResults(base.LikelihoodModelResults):
__doc__ = (
"This class summarizes the fit of a marginal regression model using GEE.\n"
+ _gee_results_doc)
[docs] def __init__(self, model, params, cov_params, scale,
cov_type='robust', use_t=False, **kwds):
super(GEEResults, self).__init__(model, params,
normalized_cov_params=cov_params, scale=scale)
# not added by super
self.df_resid = model.df_resid
self.df_model = model.df_model
attr_kwds = kwds.pop('attr_kwds', {})
self.__dict__.update(attr_kwds)
# we don't 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 '
'already attached cov_type')
[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 : string
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":
return np.sqrt(np.diag(self.cov_robust_bc))
# Need to override to allow for different covariance types.
@cache_readonly
[docs] def bse(self):
return self.standard_errors(self.cov_type)
@cache_readonly
[docs] def resid(self):
"""
Returns the residuals, the endogeneous data minus the fitted
values from the model.
"""
return self.model.endog - self.fittedvalues
@cache_readonly
[docs] 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
@cache_readonly
[docs] 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
@cache_readonly
[docs] 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
# FIXME: alias to be removed, temporary backwards compatibility
split_resid = resid_split
centered_resid = resid_centered
split_centered_resid = resid_centered_split
@cache_readonly
[docs] def fittedvalues(self):
"""
Returns the fitted values from the model.
"""
return self.model.family.link.inverse(np.dot(self.model.exog,
self.params))
[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 : string
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 doesn't 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 : string, optional
Default is `y`
xname : list of strings, optional
Default is `var_##` for ## in p the number of regressors
title : string, 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 : string
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.
See Also
--------
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 dataframe names if xname is provided as an
# argument.
if xname is not None:
xna = xname
else:
xna = self.model.exog_names
# Create summary table instance
from statsmodels.iolib.summary import Summary
smry = Summary()
smry.add_table_2cols(self, gleft=top_left, gright=top_right,
yname=self.model.endog_names, xname=xna,
title=title)
smry.add_table_params(self, yname=yname, xname=xna,
alpha=alpha, use_t=False)
smry.add_table_2cols(self, gleft=diagn_left,
gright=diagn_right, yname=yname,
xname=xna, title="")
return smry
[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 : Matplotlib axes instance
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 : integer
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
[docs]class GEEResultsWrapper(lm.RegressionResultsWrapper):
_attrs = {
'centered_resid' : 'rows',
}
_wrap_attrs = wrap.union_dicts(lm.RegressionResultsWrapper._wrap_attrs,
_attrs)
wrap.populate_wrapper(GEEResultsWrapper, GEEResults)
[docs]class OrdinalGEE(GEE):
__doc__ = (
" Estimation of ordinal response marginal regression models\n"
" using Generalized Estimating Equations (GEE).\n" +
_gee_init_doc % {'extra_params': base._missing_param_doc,
'family_doc': _gee_ordinal_family_doc,
'example': _gee_ordinal_example})
[docs] def __init__(self, endog, exog, groups, time=None, family=None,
cov_struct=None, missing='none', offset=None,
dep_data=None, constraint=None):
if family is None:
family = families.Binomial()
else:
if not isinstance(family, families.Binomial):
raise ValueError("ordinal GEE must use a Binomial family")
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 analysed 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
[docs] 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))
return OrdinalGEEResultsWrapper(ord_rslt)
fit.__doc__ = _gee_fit_doc
[docs]class OrdinalGEEResults(GEEResults):
__doc__ = (
"This class summarizes the fit of a marginal regression model"
"for an ordinal response using GEE.\n"
+ _gee_results_doc)
[docs] def plot_distribution(self, ax=None, exog_values=None):
"""
Plot the fitted probabilities of endog in an ordinal model,
for specifed values of the predictors.
Parameters
----------
ax : Matplotlib axes instance
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
[docs]class OrdinalGEEResultsWrapper(GEEResultsWrapper):
pass
wrap.populate_wrapper(OrdinalGEEResultsWrapper, OrdinalGEEResults)
[docs]class NominalGEE(GEE):
__doc__ = (
" Estimation of nominal response marginal regression models\n"
" using Generalized Estimating Equations (GEE).\n" +
_gee_init_doc % {'extra_params': base._missing_param_doc,
'family_doc': _gee_nominal_family_doc,
'example': _gee_nominal_example})
[docs] def __init__(self, endog, exog, groups, time=None, family=None,
cov_struct=None, missing='none', offset=None,
dep_data=None, constraint=None):
endog, exog, groups, time, offset = self.setup_nominal(endog,
exog, groups, time, offset)
if family is None:
family = _Multinomial(self.ncut+1)
super(NominalGEE, self).__init__(endog, exog, groups,
time, family, cov_struct, missing, offset, dep_data,
constraint)
[docs] def setup_nominal(self, endog, exog, groups, time, offset):
"""
Restructure nominal data as binary indicators so that they can
be analysed 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 type(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 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
[docs] 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:
warnings.warn("GEE updates did not converge",
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)
#for k in rslt._props:
# setattr(nom_rslt, k, getattr(rslt, k))
return NominalGEEResultsWrapper(nom_rslt)
fit.__doc__ = _gee_fit_doc
[docs]class NominalGEEResults(GEEResults):
__doc__ = (
"This class summarizes the fit of a marginal regression model"
"for a nominal response using GEE.\n"
+ _gee_results_doc)
[docs] def plot_distribution(self, ax=None, exog_values=None):
"""
Plot the fitted probabilities of endog in an nominal model,
for specifed values of the predictors.
Parameters
----------
ax : Matplotlib axes instance
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 = [{},]
link = self.model.family.link.inverse
ncut = self.model.family.ncut
k = 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 = link(lpr)
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
[docs]class NominalGEEResultsWrapper(GEEResultsWrapper):
pass
wrap.populate_wrapper(NominalGEEResultsWrapper, NominalGEEResults)
class _MultinomialLogit(Link):
"""
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 : array
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
def mean_deriv(self, exog, lpr):
"""
Derivative of the expected endog with respect to param.
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 param
"""
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))
dmat = expval[:, None] * exog / denom[:, None]
ones = np.ones(self.ncut, dtype=np.float64)
cmat = block_diag([np.outer(ones, x) for x in expval_m], "csr")
rmat = cmat.dot(exog)
dmat -= expval[:, None] * rmat / denom[:, None]**2
return dmat
# Minimally tested
def mean_deriv_exog(self, exog, params):
"""
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.
"""
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)
# Masking matrix
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
class _Multinomial(families.Family):
"""
Pseudo-link function for fitting nominal multinomial models with
GEE. Not for use outside the GEE class.
"""
links = [_MultinomialLogit,]
variance = varfuncs.binary
def __init__(self, nlevels):
"""
Parameters
----------
nlevels : integer
The number of distinct categories for the multinomial
distribution.
"""
self.initialize(nlevels)
def initialize(self, nlevels):
self.ncut = nlevels - 1
self.link = _MultinomialLogit(self.ncut)
from statsmodels.discrete.discrete_margins import \
_get_margeff_exog, _get_const_index, _check_margeff_args, \
_effects_at, margeff_cov_with_se, _check_at_is_all, \
_transform_names, \
_check_discrete_args, _get_dummy_index, _get_count_index
[docs]class GEEMargins(object):
"""Estimate the marginal effects of a model fit using generalized
estimating equations.
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
results.get_margeff. See there for more information.
kwargs : dict
Keyword args are passed to `get_margeff`. This is the same as
results.get_margeff. See there for more information.
"""
[docs] def __init__(self, results, args, kwargs={}):
self._cache = resettable_cache()
self.results = results
self.get_margeff(*args, **kwargs)
def _reset(self):
self._cache = resettable_cache()
@cache_readonly
[docs] 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)
@cache_readonly
[docs] 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()
# sigh, we really need to hold on to this in _data...
_, const_idx = _get_const_index(model.exog)
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]
smry.add_table_2cols(self, gleft=top_left, gright=[],
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,
skip_header=True)
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)]
tble.insert_header_row(0, header)
#from IPython.core.debugger import Pdb; Pdb().set_trace()
table.append(tble)
table = table_extend(table, keep_headers=True)
else:
restup = (results, margeff, margeff_se, tvalues, pvalues, conf_int)
table = summary_params(restup, yname=yname, xname=exog_names,
alpha=alpha, use_t=False, skip_header=True)
header = ['', _transform_names[method], 'std err', 'z',
'P>|z|', '[%3.1f%% Conf. Int.]' % (100-alpha*100)]
table.insert_header_row(0, header)
smry.tables.append(table)
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 semielasticity -- dy/d(lnx)
- 'eydx' - estimate semeilasticity -- 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.
"""
self._reset() # always reset the cache when this is called
#TODO: if at is not all or overall, we can also put atexog values
# in summary table head
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, const_idx = _get_const_index(exog)
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)
J = getattr(model, 'J', 1)
effects_idx = np.tile(effects_idx, J) # adjust for multi-equation.
effects = _effects_at(effects, at)
if at == 'all':
if J > 1:
K = model.K - np.any(~effects_idx) # subtract constant
self.margeff = effects[:, effects_idx].reshape(-1, K, J,
order='F')
else:
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, J)
# reshape for multi-equation
if J > 1:
K = model.K - np.any(~effects_idx) # subtract constant
self.margeff = effects[effects_idx].reshape(K, J, order='F')
self.margeff_se = margeff_se[effects_idx].reshape(K, J,
order='F')
self.margeff_cov = margeff_cov[effects_idx][:, effects_idx]
else:
# don't 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]