import numpy as np
from collections import defaultdict
import statsmodels.base.model as base
from statsmodels.genmod import families
from statsmodels.genmod.families import links
from statsmodels.genmod.families import varfuncs
import statsmodels.regression.linear_model as lm
import statsmodels.base.wrapper as wrap
from statsmodels.tools.decorators import cache_readonly
class QIFCovariance(object):
"""
A covariance model for quadratic inference function regression.
The mat method returns a basis matrix B such that the inverse
of the working covariance lies in the linear span of the
basis matrices.
Subclasses should set the number of basis matrices `num_terms`,
so that `mat(d, j)` for j=0, ..., num_terms-1 gives the basis
of dimension d.`
"""
def mat(self, dim, term):
"""
Returns the term'th basis matrix, which is a dim x dim
matrix.
"""
raise NotImplementedError
class QIFIndependence(QIFCovariance):
"""
Independent working covariance for QIF regression. This covariance
model gives identical results to GEE with the independence working
covariance. When using QIFIndependence as the working covariance,
the QIF value will be zero, and cannot be used for chi^2 testing, or
for model selection using AIC, BIC, etc.
"""
def __init__(self):
self.num_terms = 1
def mat(self, dim, term):
if term == 0:
return np.eye(dim)
else:
return None
class QIFExchangeable(QIFCovariance):
"""
Exchangeable working covariance for QIF regression.
"""
def __init__(self):
self.num_terms = 2
def mat(self, dim, term):
if term == 0:
return np.eye(dim)
elif term == 1:
return np.ones((dim, dim))
else:
return None
class QIFAutoregressive(QIFCovariance):
"""
Autoregressive working covariance for QIF regression.
"""
def __init__(self):
self.num_terms = 3
def mat(self, dim, term):
if dim < 3:
msg = ("Groups must have size at least 3 for " +
"autoregressive covariance.")
raise ValueError(msg)
if term == 0:
return np.eye(dim)
elif term == 1:
mat = np.zeros((dim, dim))
mat.flat[1::(dim+1)] = 1
mat += mat.T
return mat
elif term == 2:
mat = np.zeros((dim, dim))
mat[0, 0] = 1
mat[dim-1, dim-1] = 1
return mat
else:
return None
[docs]class QIF(base.Model):
"""
Fit a regression model using quadratic inference functions (QIF).
QIF is an alternative to GEE that can be more efficient, and that
offers different approaches for model selection and inference.
Parameters
----------
endog : array-like
The dependent variables of the regression.
exog : array-like
The independent variables of the regression.
groups : array-like
Labels indicating which group each observation belongs to.
Observations in different groups should be independent.
family : genmod family
An instance of a GLM family.
cov_struct : QIFCovariance instance
An instance of a QIFCovariance.
References
----------
A. Qu, B. Lindsay, B. Li (2000). Improving Generalized Estimating
Equations using Quadratic Inference Functions, Biometrika 87:4.
www.jstor.org/stable/2673612
"""
def __init__(self, endog, exog, groups, family=None,
cov_struct=None, missing='none', **kwargs):
# Handle the family argument
if family is None:
family = families.Gaussian()
else:
if not issubclass(family.__class__, families.Family):
raise ValueError("QIF: `family` must be a genmod "
"family instance")
self.family = family
self._fit_history = defaultdict(list)
# Handle the cov_struct argument
if cov_struct is None:
cov_struct = QIFIndependence()
else:
if not isinstance(cov_struct, QIFCovariance):
raise ValueError(
"QIF: `cov_struct` must be a QIFCovariance instance")
self.cov_struct = cov_struct
groups = np.asarray(groups)
super(QIF, self).__init__(endog, exog, groups=groups,
missing=missing, **kwargs)
self.group_names = list(set(groups))
self.nobs = len(self.endog)
groups_ix = defaultdict(list)
for i, g in enumerate(groups):
groups_ix[g].append(i)
self.groups_ix = [groups_ix[na] for na in self.group_names]
self._check_args(groups)
def _check_args(self, groups):
if len(groups) != len(self.endog):
msg = "QIF: groups and endog should have the same length"
raise ValueError(msg)
if len(self.endog) != self.exog.shape[0]:
msg = ("QIF: the length of endog should be equal to the "
"number of rows of exog.")
raise ValueError(msg)
[docs] def objective(self, params):
"""
Calculate the gradient of the QIF objective function.
Parameters
----------
params : array-like
The model parameters at which the gradient is evaluated.
Returns
-------
grad : array-like
The gradient vector of the QIF objective function.
gn_deriv : array-like
The gradients of each estimating equation with
respect to the parameter.
"""
endog = self.endog
exog = self.exog
lpr = np.dot(exog, params)
mean = self.family.link.inverse(lpr)
va = self.family.variance(mean)
# Mean derivative
idl = self.family.link.inverse_deriv(lpr)
idl2 = self.family.link.inverse_deriv2(lpr)
vd = self.family.variance.deriv(mean)
m = self.cov_struct.num_terms
p = exog.shape[1]
d = p * m
gn = np.zeros(d)
gi = np.zeros(d)
gi_deriv = np.zeros((d, p))
gn_deriv = np.zeros((d, p))
cn_deriv = [0] * p
cmat = np.zeros((d, d))
fastvar = self.family.variance is varfuncs.constant
fastlink = isinstance(self.family.link, links.identity)
for ix in self.groups_ix:
sd = np.sqrt(va[ix])
resid = endog[ix] - mean[ix]
sresid = resid / sd
deriv = exog[ix, :] * idl[ix, None]
jj = 0
for j in range(m):
# The derivative of each term in (5) of Qu et al.
# There are four terms involving beta in a product.
# Iterated application of the product rule gives
# the gradient as a sum of four terms.
c = self.cov_struct.mat(len(ix), j)
crs1 = np.dot(c, sresid) / sd
gi[jj:jj+p] = np.dot(deriv.T, crs1)
crs2 = np.dot(c, -deriv / sd[:, None]) / sd[:, None]
gi_deriv[jj:jj+p, :] = np.dot(deriv.T, crs2)
if not (fastlink and fastvar):
for k in range(p):
m1 = np.dot(exog[ix, :].T,
idl2[ix] * exog[ix, k] * crs1)
if not fastvar:
vx = -0.5 * vd[ix] * deriv[:, k] / va[ix]**1.5
m2 = np.dot(deriv.T, vx * np.dot(c, sresid))
m3 = np.dot(deriv.T, np.dot(c, vx * resid) / sd)
else:
m2, m3 = 0, 0
gi_deriv[jj:jj+p, k] += m1 + m2 + m3
jj += p
for j in range(p):
u = np.outer(gi, gi_deriv[:, j])
cn_deriv[j] += u + u.T
gn += gi
gn_deriv += gi_deriv
cmat += np.outer(gi, gi)
ngrp = len(self.groups_ix)
gn /= ngrp
gn_deriv /= ngrp
cmat /= ngrp**2
qif = np.dot(gn, np.linalg.solve(cmat, gn))
gcg = np.zeros(p)
for j in range(p):
cn_deriv[j] /= len(self.groups_ix)**2
u = np.linalg.solve(cmat, cn_deriv[j]).T
u = np.linalg.solve(cmat, u)
gcg[j] = np.dot(gn, np.dot(u, gn))
grad = 2 * np.dot(gn_deriv.T, np.linalg.solve(cmat, gn)) - gcg
return qif, grad, cmat, gn, gn_deriv
[docs] def estimate_scale(self, params):
"""
Estimate the dispersion/scale.
The scale parameter for binomial and Poisson families is
fixed at 1, otherwise it is estimated from the data.
"""
if isinstance(self.family, (families.Binomial, families.Poisson)):
return 1.
if hasattr(self, "ddof_scale"):
ddof_scale = self.ddof_scale
else:
ddof_scale = self.exog[1]
lpr = np.dot(self.exog, params)
mean = self.family.link.inverse(lpr)
resid = self.endog - mean
scale = np.sum(resid**2) / (self.nobs - ddof_scale)
return scale
[docs] def fit(self, maxiter=100, start_params=None, tol=1e-6, gtol=1e-4,
ddof_scale=None):
"""
Fit a GLM to correlated data using QIF.
Parameters
----------
maxiter : integer
Maximum number of iterations.
start_params : array-like, optional
Starting values
tol : float
Convergence threshold for difference of successive
estimates.
gtol : float
Convergence threshold for gradient.
ddof_scale : int, optional
Degrees of freedom for the scale parameter
Returns
-------
QIFResults object
"""
if ddof_scale is None:
self.ddof_scale = self.exog.shape[1]
else:
self.ddof_scale = ddof_scale
if start_params is None:
params = np.zeros(self.exog.shape[1])
else:
params = start_params
for _ in range(maxiter):
qif, grad, cmat, _, gn_deriv = self.objective(params)
gnorm = np.sqrt(np.sum(grad * grad))
self._fit_history["qif"].append(qif)
self._fit_history["gradnorm"].append(gnorm)
if gnorm < gtol:
break
cjac = 2 * np.dot(gn_deriv.T, np.linalg.solve(cmat, gn_deriv))
step = np.linalg.solve(cjac, grad)
snorm = np.sqrt(np.sum(step * step))
self._fit_history["stepnorm"].append(snorm)
if snorm < tol:
break
params -= step
vcov = np.dot(gn_deriv.T, np.linalg.solve(cmat, gn_deriv))
vcov = np.linalg.inv(vcov)
scale = self.estimate_scale(params)
rslt = QIFResults(self, params, vcov / scale, scale)
rslt.fit_history = self._fit_history
self._fit_history = defaultdict(list)
return QIFResultsWrapper(rslt)
[docs]class QIFResults(base.LikelihoodModelResults):
"""Results class for QIF Regression"""
def __init__(self, model, params, cov_params, scale,
use_t=False, **kwds):
super(QIFResults, self).__init__(
model, params, normalized_cov_params=cov_params,
scale=scale)
self.qif, _, _, _, _ = self.model.objective(params)
[docs] @cache_readonly
def aic(self):
"""
An AIC-like statistic for models fit using QIF.
"""
if isinstance(self.model.cov_struct, QIFIndependence):
msg = "AIC not available with QIFIndependence covariance"
raise ValueError(msg)
df = self.model.exog.shape[1]
return self.qif + 2*df
[docs] @cache_readonly
def bic(self):
"""
A BIC-like statistic for models fit using QIF.
"""
if isinstance(self.model.cov_struct, QIFIndependence):
msg = "BIC not available with QIFIndependence covariance"
raise ValueError(msg)
df = self.model.exog.shape[1]
return self.qif + np.log(self.model.nobs)*df
[docs] @cache_readonly
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 summary(self, yname=None, xname=None, title=None, alpha=.05):
"""
Summarize the QIF 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
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),
('Method:', ['QIF']),
('Family:', [self.model.family.__class__.__name__]),
('Covariance structure:',
[self.model.cov_struct.__class__.__name__]),
('Date:', None),
('Time:', None),
]
NY = [len(y) for y in self.model.groups_ix]
top_right = [('No. Observations:', [sum(NY)]),
('No. clusters:', [len(NY)]),
('Min. cluster size:', [min(NY)]),
('Max. cluster size:', [max(NY)]),
('Mean cluster size:', ["%.1f" % np.mean(NY)]),
('Scale:', ["%.3f" % self.scale]),
]
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()
smry.add_table_2cols(self, gleft=top_left, gright=top_right,
yname=yname, xname=xname,
title=title)
smry.add_table_params(self, yname=yname, xname=xname,
alpha=alpha, use_t=False)
return smry
class QIFResultsWrapper(lm.RegressionResultsWrapper):
pass
wrap.populate_wrapper(QIFResultsWrapper, QIFResults)
