"""
Covariance models and estimators for GEE.
Some details for the covariance calculations can be found in the Stata
docs:
http://www.stata.com/manuals13/xtxtgee.pdf
"""
from statsmodels.compat.python import iterkeys, itervalues, zip, range
from statsmodels.stats.correlation_tools import cov_nearest
import numpy as np
import pandas as pd
from scipy import linalg as spl
from collections import defaultdict
from statsmodels.tools.sm_exceptions import (ConvergenceWarning, OutputWarning,
NotImplementedWarning)
import warnings
[docs]class CovStruct(object):
"""
Base class for correlation and covariance structures.
An implementation of this class takes the residuals from a
regression model that has been fit to grouped data, and uses
them to estimate the within-group dependence structure of the
random errors in the model.
The current state of the covariance structure is represented
through the value of the `dep_params` attribute.
The default state of a newly-created instance should always be
the identity correlation matrix.
"""
def __init__(self, cov_nearest_method="clipped"):
# Parameters describing the dependency structure
self.dep_params = None
# Keep track of the number of times that the covariance was
# adjusted.
self.cov_adjust = []
# Method for projecting the covariance matrix if it is not
# PSD.
self.cov_nearest_method = cov_nearest_method
[docs] def initialize(self, model):
"""
Called by GEE, used by implementations that need additional
setup prior to running `fit`.
Parameters
----------
model : GEE class
A reference to the parent GEE class instance.
"""
self.model = model
[docs] def update(self, params):
"""
Update the association parameter values based on the current
regression coefficients.
Parameters
----------
params : array-like
Working values for the regression parameters.
"""
raise NotImplementedError
[docs] def covariance_matrix(self, endog_expval, index):
"""
Returns the working covariance or correlation matrix for a
given cluster of data.
Parameters
----------
endog_expval: array-like
The expected values of endog for the cluster for which the
covariance or correlation matrix will be returned
index: integer
The index of the cluster for which the covariane or
correlation matrix will be returned
Returns
-------
M: matrix
The covariance or correlation matrix of endog
is_cor: bool
True if M is a correlation matrix, False if M is a
covariance matrix
"""
raise NotImplementedError
[docs] def covariance_matrix_solve(self, expval, index, stdev, rhs):
"""
Solves matrix equations of the form `covmat * soln = rhs` and
returns the values of `soln`, where `covmat` is the covariance
matrix represented by this class.
Parameters
----------
expval: array-like
The expected value of endog for each observed value in the
group.
index: integer
The group index.
stdev : array-like
The standard deviation of endog for each observation in
the group.
rhs : list/tuple of array-like
A set of right-hand sides; each defines a matrix equation
to be solved.
Returns
-------
soln : list/tuple of array-like
The solutions to the matrix equations.
Notes
-----
Returns None if the solver fails.
Some dependence structures do not use `expval` and/or `index`
to determine the correlation matrix. Some families
(e.g. binomial) do not use the `stdev` parameter when forming
the covariance matrix.
If the covariance matrix is singular or not SPD, it is
projected to the nearest such matrix. These projection events
are recorded in the fit_history attribute of the GEE model.
Systems of linear equations with the covariance matrix as the
left hand side (LHS) are solved for different right hand sides
(RHS); the LHS is only factorized once to save time.
This is a default implementation, it can be reimplemented in
subclasses to optimize the linear algebra according to the
struture of the covariance matrix.
"""
vmat, is_cor = self.covariance_matrix(expval, index)
if is_cor:
vmat *= np.outer(stdev, stdev)
# Factor the covariance matrix. If the factorization fails,
# attempt to condition it into a factorizable matrix.
threshold = 1e-2
success = False
cov_adjust = 0
for itr in range(20):
try:
vco = spl.cho_factor(vmat)
success = True
break
except np.linalg.LinAlgError:
vmat = cov_nearest(vmat, method=self.cov_nearest_method,
threshold=threshold)
threshold *= 2
cov_adjust += 1
msg = "At least one covariance matrix was not PSD "
msg += "and required projection."
warnings.warn(msg)
self.cov_adjust.append(cov_adjust)
# Last resort if we still can't factor the covariance matrix.
if not success:
warnings.warn(
"Unable to condition covariance matrix to an SPD "
"matrix using cov_nearest", ConvergenceWarning)
vmat = np.diag(np.diag(vmat))
vco = spl.cho_factor(vmat)
soln = [spl.cho_solve(vco, x) for x in rhs]
return soln
[docs] def summary(self):
"""
Returns a text summary of the current estimate of the
dependence structure.
"""
raise NotImplementedError
[docs]class Independence(CovStruct):
"""
An independence working dependence structure.
"""
# Nothing to update
[docs] def update(self, params):
return
[docs] def covariance_matrix(self, expval, index):
dim = len(expval)
return np.eye(dim, dtype=np.float64), True
[docs] def covariance_matrix_solve(self, expval, index, stdev, rhs):
v = stdev ** 2
rslt = []
for x in rhs:
if x.ndim == 1:
rslt.append(x / v)
else:
rslt.append(x / v[:, None])
return rslt
update.__doc__ = CovStruct.update.__doc__
covariance_matrix.__doc__ = CovStruct.covariance_matrix.__doc__
covariance_matrix_solve.__doc__ = CovStruct.covariance_matrix_solve.__doc__
[docs] def summary(self):
return ("Observations within a cluster are modeled "
"as being independent.")
[docs]class Exchangeable(CovStruct):
"""
An exchangeable working dependence structure.
"""
def __init__(self):
super(Exchangeable, self).__init__()
# The correlation between any two values in the same cluster
self.dep_params = 0.
[docs] def update(self, params):
endog = self.model.endog_li
nobs = self.model.nobs
varfunc = self.model.family.variance
cached_means = self.model.cached_means
has_weights = self.model.weights is not None
weights_li = self.model.weights
residsq_sum, scale = 0, 0
fsum1, fsum2, n_pairs = 0., 0., 0.
for i in range(self.model.num_group):
expval, _ = cached_means[i]
stdev = np.sqrt(varfunc(expval))
resid = (endog[i] - expval) / stdev
f = weights_li[i] if has_weights else 1.
ssr = np.sum(resid * resid)
scale += f * ssr
fsum1 += f * len(endog[i])
residsq_sum += f * (resid.sum() ** 2 - ssr) / 2
ngrp = len(resid)
npr = 0.5 * ngrp * (ngrp - 1)
fsum2 += f * npr
n_pairs += npr
ddof = self.model.ddof_scale
scale /= (fsum1 * (nobs - ddof) / float(nobs))
residsq_sum /= scale
self.dep_params = residsq_sum / \
(fsum2 * (n_pairs - ddof) / float(n_pairs))
[docs] def covariance_matrix(self, expval, index):
dim = len(expval)
dp = self.dep_params * np.ones((dim, dim), dtype=np.float64)
np.fill_diagonal(dp, 1)
return dp, True
[docs] def covariance_matrix_solve(self, expval, index, stdev, rhs):
k = len(expval)
c = self.dep_params / (1. - self.dep_params)
c /= 1. + self.dep_params * (k - 1)
rslt = []
for x in rhs:
if x.ndim == 1:
x1 = x / stdev
y = x1 / (1. - self.dep_params)
y -= c * sum(x1)
y /= stdev
else:
x1 = x / stdev[:, None]
y = x1 / (1. - self.dep_params)
y -= c * x1.sum(0)
y /= stdev[:, None]
rslt.append(y)
return rslt
update.__doc__ = CovStruct.update.__doc__
covariance_matrix.__doc__ = CovStruct.covariance_matrix.__doc__
covariance_matrix_solve.__doc__ = CovStruct.covariance_matrix_solve.__doc__
[docs] def summary(self):
return ("The correlation between two observations in the " +
"same cluster is %.3f" % self.dep_params)
[docs]class Nested(CovStruct):
"""
A nested working dependence structure.
A working dependence structure that captures a nested hierarchy of
groups. Each level of grouping contributes to the random error
structure of the model.
When using this working covariance structure, `dep_data` of the
GEE instance should contain a n_obs x k matrix of 0/1 indicators,
corresponding to the k subgroups nested under the top-level
`groups` of the GEE instance. These subgroups should be nested
from left to right, so that two observations with the same value
for column j of `dep_data` should also have the same value for all
columns j' < j (this only applies to observations in the same
top-level cluster given by the `groups` argument to GEE).
Examples
--------
Suppose our data are student test scores, and the students are in
classrooms, nested in schools, nested in school districts. The
school district is the highest level of grouping, so the school
district id would be provided to GEE as `groups`, and the school
and classroom id's would be provided to the Nested class as the
`dep_data` argument, e.g.
0 0 # School 0, classroom 0, student 0
0 0 # School 0, classroom 0, student 1
0 1 # School 0, classroom 1, student 0
0 1 # School 0, classroom 1, student 1
1 0 # School 1, classroom 0, student 0
1 0 # School 1, classroom 0, student 1
1 1 # School 1, classroom 1, student 0
1 1 # School 1, classroom 1, student 1
Labels lower in the hierarchy are recycled, so that student 0 in
classroom 0 is different fro student 0 in classroom 1, etc.
Notes
-----
The calculations for this dependence structure involve all pairs
of observations within a group (that is, within the top level
`group` structure passed to GEE). Large group sizes will result
in slow iterations.
The variance components are estimated using least squares
regression of the products r*r', for standardized residuals r and
r' in the same group, on a matrix of indicators defining which
variance components are shared by r and r'.
"""
[docs] def initialize(self, model):
"""
Called on the first call to update
`ilabels` is a list of n_i x n_i matrices containing integer
labels that correspond to specific correlation parameters.
Two elements of ilabels[i] with the same label share identical
variance components.
`designx` is a matrix, with each row containing dummy
variables indicating which variance components are associated
with the corresponding element of QY.
"""
super(Nested, self).initialize(model)
if self.model.weights is not None:
warnings.warn("weights not implemented for nested cov_struct, "
"using unweighted covariance estimate",
NotImplementedWarning)
# A bit of processing of the nest data
id_matrix = np.asarray(self.model.dep_data)
if id_matrix.ndim == 1:
id_matrix = id_matrix[:, None]
self.id_matrix = id_matrix
endog = self.model.endog_li
designx, ilabels = [], []
# The number of layers of nesting
n_nest = self.id_matrix.shape[1]
for i in range(self.model.num_group):
ngrp = len(endog[i])
glab = self.model.group_labels[i]
rix = self.model.group_indices[glab]
# Determine the number of common variance components
# shared by each pair of observations.
ix1, ix2 = np.tril_indices(ngrp, -1)
ncm = (self.id_matrix[rix[ix1], :] ==
self.id_matrix[rix[ix2], :]).sum(1)
# This is used to construct the working correlation
# matrix.
ilabel = np.zeros((ngrp, ngrp), dtype=np.int32)
ilabel[(ix1, ix2)] = ncm + 1
ilabel[(ix2, ix1)] = ncm + 1
ilabels.append(ilabel)
# This is used to estimate the variance components.
dsx = np.zeros((len(ix1), n_nest + 1), dtype=np.float64)
dsx[:, 0] = 1
for k in np.unique(ncm):
ii = np.flatnonzero(ncm == k)
dsx[ii, 1:k + 1] = 1
designx.append(dsx)
self.designx = np.concatenate(designx, axis=0)
self.ilabels = ilabels
svd = np.linalg.svd(self.designx, 0)
self.designx_u = svd[0]
self.designx_s = svd[1]
self.designx_v = svd[2].T
[docs] def update(self, params):
endog = self.model.endog_li
nobs = self.model.nobs
dim = len(params)
if self.designx is None:
self._compute_design(self.model)
cached_means = self.model.cached_means
varfunc = self.model.family.variance
dvmat = []
scale = 0.
for i in range(self.model.num_group):
expval, _ = cached_means[i]
stdev = np.sqrt(varfunc(expval))
resid = (endog[i] - expval) / stdev
ix1, ix2 = np.tril_indices(len(resid), -1)
dvmat.append(resid[ix1] * resid[ix2])
scale += np.sum(resid ** 2)
dvmat = np.concatenate(dvmat)
scale /= (nobs - dim)
# Use least squares regression to estimate the variance
# components
vcomp_coeff = np.dot(self.designx_v, np.dot(self.designx_u.T,
dvmat) / self.designx_s)
self.vcomp_coeff = np.clip(vcomp_coeff, 0, np.inf)
self.scale = scale
self.dep_params = self.vcomp_coeff.copy()
[docs] def covariance_matrix(self, expval, index):
dim = len(expval)
# First iteration
if self.dep_params is None:
return np.eye(dim, dtype=np.float64), True
ilabel = self.ilabels[index]
c = np.r_[self.scale, np.cumsum(self.vcomp_coeff)]
vmat = c[ilabel]
vmat /= self.scale
return vmat, True
update.__doc__ = CovStruct.update.__doc__
covariance_matrix.__doc__ = CovStruct.covariance_matrix.__doc__
[docs] def summary(self):
"""
Returns a summary string describing the state of the
dependence structure.
"""
dep_names = ["Groups"]
if hasattr(self.model, "_dep_data_names"):
dep_names.extend(self.model._dep_data_names)
else:
dep_names.extend(["Component %d:" % (k + 1) for k in range(len(self.vcomp_coeff) - 1)])
if hasattr(self.model, "_groups_name"):
dep_names[0] = self.model._groups_name
dep_names.append("Residual")
vc = self.vcomp_coeff.tolist()
vc.append(self.scale - np.sum(vc))
smry = pd.DataFrame({"Variance": vc}, index=dep_names)
return smry
class Stationary(CovStruct):
"""
A stationary covariance structure.
The correlation between two observations is an arbitrary function
of the distance between them. Distances up to a given maximum
value are included in the covariance model.
Parameters
----------
max_lag : float
The largest distance that is included in the covariance model.
grid : bool
If True, the index positions in the data (after dropping missing
values) are used to define distances, and the `time` variable is
ignored.
"""
def __init__(self, max_lag=1, grid=False):
super(Stationary, self).__init__()
self.max_lag = max_lag
self.grid = grid
self.dep_params = np.zeros(max_lag + 1)
def initialize(self, model):
super(Stationary, self).initialize(model)
# Time used as an index needs to be integer type.
if not self.grid:
time = self.model.time[:, 0].astype(np.int32)
self.time = self.model.cluster_list(time)
def update(self, params):
if self.grid:
self.update_grid(params)
else:
self.update_nogrid(params)
def update_grid(self, params):
endog = self.model.endog_li
cached_means = self.model.cached_means
varfunc = self.model.family.variance
dep_params = np.zeros(self.max_lag + 1)
for i in range(self.model.num_group):
expval, _ = cached_means[i]
stdev = np.sqrt(varfunc(expval))
resid = (endog[i] - expval) / stdev
dep_params[0] += np.sum(resid * resid) / len(resid)
for j in range(1, self.max_lag + 1):
v = resid[j:]
dep_params[j] += np.sum(resid[0:-j] * v) / len(v)
dep_params /= dep_params[0]
self.dep_params = dep_params
def update_nogrid(self, params):
endog = self.model.endog_li
cached_means = self.model.cached_means
varfunc = self.model.family.variance
dep_params = np.zeros(self.max_lag + 1)
dn = np.zeros(self.max_lag + 1)
resid_ssq = 0
resid_ssq_n = 0
for i in range(self.model.num_group):
expval, _ = cached_means[i]
stdev = np.sqrt(varfunc(expval))
resid = (endog[i] - expval) / stdev
j1, j2 = np.tril_indices(len(expval), -1)
dx = np.abs(self.time[i][j1] - self.time[i][j2])
ii = np.flatnonzero(dx <= self.max_lag)
j1 = j1[ii]
j2 = j2[ii]
dx = dx[ii]
vs = np.bincount(dx, weights=resid[j1] * resid[j2],
minlength=self.max_lag + 1)
vd = np.bincount(dx, minlength=self.max_lag + 1)
resid_ssq += np.sum(resid**2)
resid_ssq_n += len(resid)
ii = np.flatnonzero(vd > 0)
if len(ii) > 0:
dn[ii] += 1
dep_params[ii] += vs[ii] / vd[ii]
i0 = np.flatnonzero(dn > 0)
dep_params[i0] /= dn[i0]
resid_msq = resid_ssq / resid_ssq_n
dep_params /= resid_msq
self.dep_params = dep_params
def covariance_matrix(self, endog_expval, index):
if self.grid:
return self.covariance_matrix_grid(endog_expval, index)
j1, j2 = np.tril_indices(len(endog_expval), -1)
dx = np.abs(self.time[index][j1] - self.time[index][j2])
ii = np.flatnonzero(dx <= self.max_lag)
j1 = j1[ii]
j2 = j2[ii]
dx = dx[ii]
cmat = np.eye(len(endog_expval))
cmat[j1, j2] = self.dep_params[dx]
cmat[j2, j1] = self.dep_params[dx]
return cmat, True
def covariance_matrix_grid(self, endog_expval, index):
from scipy.linalg import toeplitz
r = np.zeros(len(endog_expval))
r[0] = 1
r[1:self.max_lag + 1] = self.dep_params[1:]
return toeplitz(r), True
def covariance_matrix_solve(self, expval, index, stdev, rhs):
if not self.grid:
return super(Stationary, self).covariance_matrix_solve(
expval, index, stdev, rhs)
from statsmodels.tools.linalg import stationary_solve
r = np.zeros(len(expval))
r[0:self.max_lag] = self.dep_params[1:]
return [stationary_solve(r, x) for x in rhs]
update.__doc__ = CovStruct.update.__doc__
covariance_matrix.__doc__ = CovStruct.covariance_matrix.__doc__
covariance_matrix_solve.__doc__ = CovStruct.covariance_matrix_solve.__doc__
def summary(self):
lag = np.arange(self.max_lag + 1)
return pd.DataFrame({"Lag": lag, "Cov": self.dep_params})
[docs]class Autoregressive(CovStruct):
"""
A first-order autoregressive working dependence structure.
The dependence is defined in terms of the `time` component of the
parent GEE class, which defaults to the index position of each
value within its cluster, based on the order of values in the
input data set. Time represents a potentially multidimensional
index from which distances between pairs of observations can be
determined.
The correlation between two observations in the same cluster is
dep_params^distance, where `dep_params` contains the (scalar)
autocorrelation parameter to be estimated, and `distance` is the
distance between the two observations, calculated from their
corresponding time values. `time` is stored as an n_obs x k
matrix, where `k` represents the number of dimensions in the time
index.
The autocorrelation parameter is estimated using weighted
nonlinear least squares, regressing each value within a cluster on
each preceeding value in the same cluster.
Parameters
----------
dist_func: function from R^k x R^k to R^+, optional
A function that computes the distance between the two
observations based on their `time` values.
References
----------
B Rosner, A Munoz. Autoregressive modeling for the analysis of
longitudinal data with unequally spaced examinations. Statistics
in medicine. Vol 7, 59-71, 1988.
"""
def __init__(self, dist_func=None):
super(Autoregressive, self).__init__()
# The function for determining distances based on time
if dist_func is None:
self.dist_func = lambda x, y: np.abs(x - y).sum()
else:
self.dist_func = dist_func
self.designx = None
# The autocorrelation parameter
self.dep_params = 0.
[docs] def update(self, params):
if self.model.weights is not None:
warnings.warn("weights not implemented for autoregressive "
"cov_struct, using unweighted covariance estimate",
NotImplementedWarning)
endog = self.model.endog_li
time = self.model.time_li
# Only need to compute this once
if self.designx is not None:
designx = self.designx
else:
designx = []
for i in range(self.model.num_group):
ngrp = len(endog[i])
if ngrp == 0:
continue
# Loop over pairs of observations within a cluster
for j1 in range(ngrp):
for j2 in range(j1):
designx.append(self.dist_func(time[i][j1, :],
time[i][j2, :]))
designx = np.array(designx)
self.designx = designx
scale = self.model.estimate_scale()
varfunc = self.model.family.variance
cached_means = self.model.cached_means
# Weights
var = 1. - self.dep_params ** (2 * designx)
var /= 1. - self.dep_params ** 2
wts = 1. / var
wts /= wts.sum()
residmat = []
for i in range(self.model.num_group):
expval, _ = cached_means[i]
stdev = np.sqrt(scale * varfunc(expval))
resid = (endog[i] - expval) / stdev
ngrp = len(resid)
for j1 in range(ngrp):
for j2 in range(j1):
residmat.append([resid[j1], resid[j2]])
residmat = np.array(residmat)
# Need to minimize this
def fitfunc(a):
dif = residmat[:, 0] - (a ** designx) * residmat[:, 1]
return np.dot(dif ** 2, wts)
# Left bracket point
b_lft, f_lft = 0., fitfunc(0.)
# Center bracket point
b_ctr, f_ctr = 0.5, fitfunc(0.5)
while f_ctr > f_lft:
b_ctr /= 2
f_ctr = fitfunc(b_ctr)
if b_ctr < 1e-8:
self.dep_params = 0
return
# Right bracket point
b_rgt, f_rgt = 0.75, fitfunc(0.75)
while f_rgt < f_ctr:
b_rgt = b_rgt + (1. - b_rgt) / 2
f_rgt = fitfunc(b_rgt)
if b_rgt > 1. - 1e-6:
raise ValueError(
"Autoregressive: unable to find right bracket")
from scipy.optimize import brent
self.dep_params = brent(fitfunc, brack=[b_lft, b_ctr, b_rgt])
[docs] def covariance_matrix(self, endog_expval, index):
ngrp = len(endog_expval)
if self.dep_params == 0:
return np.eye(ngrp, dtype=np.float64), True
idx = np.arange(ngrp)
cmat = self.dep_params ** np.abs(idx[:, None] - idx[None, :])
return cmat, True
[docs] def covariance_matrix_solve(self, expval, index, stdev, rhs):
# The inverse of an AR(1) covariance matrix is tri-diagonal.
k = len(expval)
soln = []
# LHS has 1 column
if k == 1:
return [x / stdev ** 2 for x in rhs]
# LHS has 2 columns
if k == 2:
mat = np.array([[1, -self.dep_params], [-self.dep_params, 1]])
mat /= (1. - self.dep_params ** 2)
for x in rhs:
if x.ndim == 1:
x1 = x / stdev
else:
x1 = x / stdev[:, None]
x1 = np.dot(mat, x1)
if x.ndim == 1:
x1 /= stdev
else:
x1 /= stdev[:, None]
soln.append(x1)
return soln
# LHS has >= 3 columns: values c0, c1, c2 defined below give
# the inverse. c0 is on the diagonal, except for the first
# and last position. c1 is on the first and last position of
# the diagonal. c2 is on the sub/super diagonal.
c0 = (1. + self.dep_params ** 2) / (1. - self.dep_params ** 2)
c1 = 1. / (1. - self.dep_params ** 2)
c2 = -self.dep_params / (1. - self.dep_params ** 2)
soln = []
for x in rhs:
flatten = False
if x.ndim == 1:
x = x[:, None]
flatten = True
x1 = x / stdev[:, None]
z0 = np.zeros((1, x.shape[1]))
rhs1 = np.concatenate((x[1:, :], z0), axis=0)
rhs2 = np.concatenate((z0, x[0:-1, :]), axis=0)
y = c0 * x + c2 * rhs1 + c2 * rhs2
y[0, :] = c1 * x[0, :] + c2 * x[1, :]
y[-1, :] = c1 * x[-1, :] + c2 * x[-2, :]
y /= stdev[:, None]
if flatten:
y = np.squeeze(y)
soln.append(y)
return soln
update.__doc__ = CovStruct.update.__doc__
covariance_matrix.__doc__ = CovStruct.covariance_matrix.__doc__
covariance_matrix_solve.__doc__ = CovStruct.covariance_matrix_solve.__doc__
[docs] def summary(self):
return ("Autoregressive(1) dependence parameter: %.3f\n" %
self.dep_params)
class CategoricalCovStruct(CovStruct):
"""
Parent class for covariance structure for categorical data models.
Attributes
----------
nlevel : int
The number of distinct levels for the outcome variable.
ibd : list
A list whose i^th element ibd[i] is an array whose rows
contain integer pairs (a,b), where endog_li[i][a:b] is the
subvector of binary indicators derived from the same ordinal
value.
"""
def initialize(self, model):
super(CategoricalCovStruct, self).initialize(model)
self.nlevel = len(model.endog_values)
self._ncut = self.nlevel - 1
from numpy.lib.stride_tricks import as_strided
b = np.dtype(np.int64).itemsize
ibd = []
for v in model.endog_li:
jj = np.arange(0, len(v) + 1, self._ncut, dtype=np.int64)
jj = as_strided(jj, shape=(len(jj) - 1, 2), strides=(b, b))
ibd.append(jj)
self.ibd = ibd
[docs]class GlobalOddsRatio(CategoricalCovStruct):
"""
Estimate the global odds ratio for a GEE with ordinal or nominal
data.
References
----------
PJ Heagerty and S Zeger. "Marginal Regression Models for Clustered
Ordinal Measurements". Journal of the American Statistical
Association Vol. 91, Issue 435 (1996).
Thomas Lumley. Generalized Estimating Equations for Ordinal Data:
A Note on Working Correlation Structures. Biometrics Vol. 52,
No. 1 (Mar., 1996), pp. 354-361
http://www.jstor.org/stable/2533173
Notes
-----
The following data structures are calculated in the class:
'ibd' is a list whose i^th element ibd[i] is a sequence of integer
pairs (a,b), where endog_li[i][a:b] is the subvector of binary
indicators derived from the same ordinal value.
`cpp` is a dictionary where cpp[group] is a map from cut-point
pairs (c,c') to the indices of all between-subject pairs derived
from the given cut points.
"""
def __init__(self, endog_type):
super(GlobalOddsRatio, self).__init__()
self.endog_type = endog_type
self.dep_params = 0.
[docs] def initialize(self, model):
super(GlobalOddsRatio, self).initialize(model)
if self.model.weights is not None:
warnings.warn("weights not implemented for GlobalOddsRatio "
"cov_struct, using unweighted covariance estimate",
NotImplementedWarning)
# Need to restrict to between-subject pairs
cpp = []
for v in model.endog_li:
# Number of subjects in this group
m = int(len(v) / self._ncut)
i1, i2 = np.tril_indices(m, -1)
cpp1 = {}
for k1 in range(self._ncut):
for k2 in range(k1 + 1):
jj = np.zeros((len(i1), 2), dtype=np.int64)
jj[:, 0] = i1 * self._ncut + k1
jj[:, 1] = i2 * self._ncut + k2
cpp1[(k2, k1)] = jj
cpp.append(cpp1)
self.cpp = cpp
# Initialize the dependence parameters
self.crude_or = self.observed_crude_oddsratio()
if self.model.update_dep:
self.dep_params = self.crude_or
[docs] def pooled_odds_ratio(self, tables):
"""
Returns the pooled odds ratio for a list of 2x2 tables.
The pooled odds ratio is the inverse variance weighted average
of the sample odds ratios of the tables.
"""
if len(tables) == 0:
return 1.
# Get the sampled odds ratios and variances
log_oddsratio, var = [], []
for table in tables:
lor = np.log(table[1, 1]) + np.log(table[0, 0]) -\
np.log(table[0, 1]) - np.log(table[1, 0])
log_oddsratio.append(lor)
var.append((1 / table.astype(np.float64)).sum())
# Calculate the inverse variance weighted average
wts = [1 / v for v in var]
wtsum = sum(wts)
wts = [w / wtsum for w in wts]
log_pooled_or = sum([w * e for w, e in zip(wts, log_oddsratio)])
return np.exp(log_pooled_or)
[docs] def covariance_matrix(self, expected_value, index):
vmat = self.get_eyy(expected_value, index)
vmat -= np.outer(expected_value, expected_value)
return vmat, False
[docs] def observed_crude_oddsratio(self):
"""
To obtain the crude (global) odds ratio, first pool all binary
indicators corresponding to a given pair of cut points (c,c'),
then calculate the odds ratio for this 2x2 table. The crude
odds ratio is the inverse variance weighted average of these
odds ratios. Since the covariate effects are ignored, this OR
will generally be greater than the stratified OR.
"""
cpp = self.cpp
endog = self.model.endog_li
# Storage for the contingency tables for each (c,c')
tables = {}
for ii in iterkeys(cpp[0]):
tables[ii] = np.zeros((2, 2), dtype=np.float64)
# Get the observed crude OR
for i in range(len(endog)):
# The observed joint values for the current cluster
yvec = endog[i]
endog_11 = np.outer(yvec, yvec)
endog_10 = np.outer(yvec, 1. - yvec)
endog_01 = np.outer(1. - yvec, yvec)
endog_00 = np.outer(1. - yvec, 1. - yvec)
cpp1 = cpp[i]
for ky in iterkeys(cpp1):
ix = cpp1[ky]
tables[ky][1, 1] += endog_11[ix[:, 0], ix[:, 1]].sum()
tables[ky][1, 0] += endog_10[ix[:, 0], ix[:, 1]].sum()
tables[ky][0, 1] += endog_01[ix[:, 0], ix[:, 1]].sum()
tables[ky][0, 0] += endog_00[ix[:, 0], ix[:, 1]].sum()
return self.pooled_odds_ratio(list(itervalues(tables)))
[docs] def get_eyy(self, endog_expval, index):
"""
Returns a matrix V such that V[i,j] is the joint probability
that endog[i] = 1 and endog[j] = 1, based on the marginal
probabilities of endog and the global odds ratio `current_or`.
"""
current_or = self.dep_params
ibd = self.ibd[index]
# The between-observation joint probabilities
if current_or == 1.0:
vmat = np.outer(endog_expval, endog_expval)
else:
psum = endog_expval[:, None] + endog_expval[None, :]
pprod = endog_expval[:, None] * endog_expval[None, :]
pfac = np.sqrt((1. + psum * (current_or - 1.)) ** 2 +
4 * current_or * (1. - current_or) * pprod)
vmat = 1. + psum * (current_or - 1.) - pfac
vmat /= 2. * (current_or - 1)
# Fix E[YY'] for elements that belong to same observation
for bdl in ibd:
evy = endog_expval[bdl[0]:bdl[1]]
if self.endog_type == "ordinal":
vmat[bdl[0]:bdl[1], bdl[0]:bdl[1]] =\
np.minimum.outer(evy, evy)
else:
vmat[bdl[0]:bdl[1], bdl[0]:bdl[1]] = np.diag(evy)
return vmat
[docs] def update(self, params):
"""
Update the global odds ratio based on the current value of
params.
"""
cpp = self.cpp
cached_means = self.model.cached_means
# This will happen if all the clusters have only
# one observation
if len(cpp[0]) == 0:
return
tables = {}
for ii in cpp[0]:
tables[ii] = np.zeros((2, 2), dtype=np.float64)
for i in range(self.model.num_group):
endog_expval, _ = cached_means[i]
emat_11 = self.get_eyy(endog_expval, i)
emat_10 = endog_expval[:, None] - emat_11
emat_01 = -emat_11 + endog_expval
emat_00 = 1. - (emat_11 + emat_10 + emat_01)
cpp1 = cpp[i]
for ky in iterkeys(cpp1):
ix = cpp1[ky]
tables[ky][1, 1] += emat_11[ix[:, 0], ix[:, 1]].sum()
tables[ky][1, 0] += emat_10[ix[:, 0], ix[:, 1]].sum()
tables[ky][0, 1] += emat_01[ix[:, 0], ix[:, 1]].sum()
tables[ky][0, 0] += emat_00[ix[:, 0], ix[:, 1]].sum()
cor_expval = self.pooled_odds_ratio(list(itervalues(tables)))
self.dep_params *= self.crude_or / cor_expval
if not np.isfinite(self.dep_params):
self.dep_params = 1.
warnings.warn("dep_params became inf, resetting to 1",
ConvergenceWarning)
update.__doc__ = CovStruct.update.__doc__
covariance_matrix.__doc__ = CovStruct.covariance_matrix.__doc__
[docs] def summary(self):
return "Global odds ratio: %.3f\n" % self.dep_params
class OrdinalIndependence(CategoricalCovStruct):
"""
An independence covariance structure for ordinal models.
The working covariance between indicators derived from different
observations is zero. The working covariance between indicators
derived form a common observation is determined from their current
mean values.
There are no parameters to estimate in this covariance structure.
"""
def covariance_matrix(self, expected_value, index):
ibd = self.ibd[index]
n = len(expected_value)
vmat = np.zeros((n, n))
for bdl in ibd:
ev = expected_value[bdl[0]:bdl[1]]
vmat[bdl[0]:bdl[1], bdl[0]:bdl[1]] =\
np.minimum.outer(ev, ev) - np.outer(ev, ev)
return vmat, False
# Nothing to update
def update(self, params):
pass
class NominalIndependence(CategoricalCovStruct):
"""
An independence covariance structure for nominal models.
The working covariance between indicators derived from different
observations is zero. The working covariance between indicators
derived form a common observation is determined from their current
mean values.
There are no parameters to estimate in this covariance structure.
"""
def covariance_matrix(self, expected_value, index):
ibd = self.ibd[index]
n = len(expected_value)
vmat = np.zeros((n, n))
for bdl in ibd:
ev = expected_value[bdl[0]:bdl[1]]
vmat[bdl[0]:bdl[1], bdl[0]:bdl[1]] =\
np.diag(ev) - np.outer(ev, ev)
return vmat, False
# Nothing to update
def update(self, params):
pass
class Equivalence(CovStruct):
"""
A covariance structure defined in terms of equivalence classes.
An 'equivalence class' is a set of pairs of observations such that
the covariance of every pair within the equivalence class has a
common value.
Parameters
----------
pairs : dict-like
A dictionary of dictionaries, where `pairs[group][label]`
provides the indices of all pairs of observations in the group
that have the same covariance value. Specifically,
`pairs[group][label]` is a tuple `(j1, j2)`, where `j1` and `j2`
are integer arrays of the same length. `j1[i], j2[i]` is one
index pair that belongs to the `label` equivalence class. Only
one triangle of each covariance matrix should be included.
Positions where j1 and j2 have the same value are variance
parameters.
labels : array-like
An array of labels such that every distinct pair of labels
defines an equivalence class. Either `labels` or `pairs` must
be provided. When the two labels in a pair are equal two
equivalence classes are defined: one for the diagonal elements
(corresponding to variances) and one for the off-diagonal
elements (corresponding to covariances).
return_cov : boolean
If True, `covariance_matrix` returns an estimate of the
covariance matrix, otherwise returns an estimate of the
correlation matrix.
Notes
-----
Using `labels` to define the class is much easier than using
`pairs`, but is less general.
Any pair of values not contained in `pairs` will be assigned zero
covariance.
The index values in `pairs` are row indices into the `exog`
matrix. They are not updated if missing data are present. When
using this covariance structure, missing data should be removed
before constructing the model.
If using `labels`, after a model is defined using the covariance
structure it is possible to remove a label pair from the second
level of the `pairs` dictionary to force the corresponding
covariance to be zero.
Examples
--------
The following sets up the `pairs` dictionary for a model with two
groups, equal variance for all observations, and constant
covariance for all pairs of observations within each group.
>> pairs = {0: {}, 1: {}}
>> pairs[0][0] = (np.r_[0, 1, 2], np.r_[0, 1, 2])
>> pairs[0][1] = np.tril_indices(3, -1)
>> pairs[1][0] = (np.r_[3, 4, 5], np.r_[3, 4, 5])
>> pairs[1][2] = 3 + np.tril_indices(3, -1)
"""
def __init__(self, pairs=None, labels=None, return_cov=False):
super(Equivalence, self).__init__()
if (pairs is None) and (labels is None):
raise ValueError(
"Equivalence cov_struct requires either `pairs` or `labels`")
if (pairs is not None) and (labels is not None):
raise ValueError(
"Equivalence cov_struct accepts only one of `pairs` "
"and `labels`")
if pairs is not None:
import copy
self.pairs = copy.deepcopy(pairs)
if labels is not None:
self.labels = np.asarray(labels)
self.return_cov = return_cov
def _make_pairs(self, i, j):
"""
Create arrays containing all unique ordered pairs of i, j.
The arrays i and j must be one-dimensional containing non-negative
integers.
"""
mat = np.zeros((len(i) * len(j), 2), dtype=np.int32)
# Create the pairs and order them
f = np.ones(len(j))
mat[:, 0] = np.kron(f, i).astype(np.int32)
f = np.ones(len(i))
mat[:, 1] = np.kron(j, f).astype(np.int32)
mat.sort(1)
# Remove repeated rows
try:
dtype = np.dtype((np.void, mat.dtype.itemsize * mat.shape[1]))
bmat = np.ascontiguousarray(mat).view(dtype)
_, idx = np.unique(bmat, return_index=True)
except TypeError:
# workaround for old numpy that can't call unique with complex
# dtypes
rs = np.random.RandomState(4234)
bmat = np.dot(mat, rs.uniform(size=mat.shape[1]))
_, idx = np.unique(bmat, return_index=True)
mat = mat[idx, :]
return mat[:, 0], mat[:, 1]
def _pairs_from_labels(self):
from collections import defaultdict
pairs = defaultdict(lambda: defaultdict(lambda: None))
model = self.model
df = pd.DataFrame({"labels": self.labels, "groups": model.groups})
gb = df.groupby(["groups", "labels"])
ulabels = np.unique(self.labels)
for g_ix, g_lb in enumerate(model.group_labels):
# Loop over label pairs
for lx1 in range(len(ulabels)):
for lx2 in range(lx1 + 1):
lb1 = ulabels[lx1]
lb2 = ulabels[lx2]
try:
i1 = gb.groups[(g_lb, lb1)]
i2 = gb.groups[(g_lb, lb2)]
except KeyError:
continue
i1, i2 = self._make_pairs(i1, i2)
clabel = str(lb1) + "/" + str(lb2)
# Variance parameters belong in their own equiv class.
jj = np.flatnonzero(i1 == i2)
if len(jj) > 0:
clabelv = clabel + "/v"
pairs[g_lb][clabelv] = (i1[jj], i2[jj])
# Covariance parameters
jj = np.flatnonzero(i1 != i2)
if len(jj) > 0:
i1 = i1[jj]
i2 = i2[jj]
pairs[g_lb][clabel] = (i1, i2)
self.pairs = pairs
def initialize(self, model):
super(Equivalence, self).initialize(model)
if self.model.weights is not None:
warnings.warn("weights not implemented for equalence cov_struct, "
"using unweighted covariance estimate",
NotImplementedWarning)
if not hasattr(self, 'pairs'):
self._pairs_from_labels()
# Initialize so that any equivalence class containing a
# variance parameter has value 1.
self.dep_params = defaultdict(lambda: 0.)
self._var_classes = set([])
for gp in self.model.group_labels:
for lb in self.pairs[gp]:
j1, j2 = self.pairs[gp][lb]
if np.any(j1 == j2):
if not np.all(j1 == j2):
warnings.warn(
"equivalence class contains both variance "
"and covariance parameters", OutputWarning)
self._var_classes.add(lb)
self.dep_params[lb] = 1
# Need to start indexing at 0 within each group.
# rx maps olds indices to new indices
rx = -1 * np.ones(len(self.model.endog), dtype=np.int32)
for g_ix, g_lb in enumerate(self.model.group_labels):
ii = self.model.group_indices[g_lb]
rx[ii] = np.arange(len(ii), dtype=np.int32)
# Reindex
for gp in self.model.group_labels:
for lb in self.pairs[gp].keys():
a, b = self.pairs[gp][lb]
self.pairs[gp][lb] = (rx[a], rx[b])
def update(self, params):
endog = self.model.endog_li
varfunc = self.model.family.variance
cached_means = self.model.cached_means
dep_params = defaultdict(lambda: [0., 0., 0.])
n_pairs = defaultdict(lambda: 0)
dim = len(params)
for k, gp in enumerate(self.model.group_labels):
expval, _ = cached_means[k]
stdev = np.sqrt(varfunc(expval))
resid = (endog[k] - expval) / stdev
for lb in self.pairs[gp].keys():
if (not self.return_cov) and lb in self._var_classes:
continue
jj = self.pairs[gp][lb]
dep_params[lb][0] += np.sum(resid[jj[0]] * resid[jj[1]])
if not self.return_cov:
dep_params[lb][1] += np.sum(resid[jj[0]] ** 2)
dep_params[lb][2] += np.sum(resid[jj[1]] ** 2)
n_pairs[lb] += len(jj[0])
if self.return_cov:
for lb in dep_params.keys():
dep_params[lb] = dep_params[lb][0] / (n_pairs[lb] - dim)
else:
for lb in dep_params.keys():
den = np.sqrt(dep_params[lb][1] * dep_params[lb][2])
dep_params[lb] = dep_params[lb][0] / den
for lb in self._var_classes:
dep_params[lb] = 1.
self.dep_params = dep_params
self.n_pairs = n_pairs
def covariance_matrix(self, expval, index):
dim = len(expval)
cmat = np.zeros((dim, dim))
g_lb = self.model.group_labels[index]
for lb in self.pairs[g_lb].keys():
j1, j2 = self.pairs[g_lb][lb]
cmat[j1, j2] = self.dep_params[lb]
cmat = cmat + cmat.T
np.fill_diagonal(cmat, cmat.diagonal() / 2)
return cmat, not self.return_cov
update.__doc__ = CovStruct.update.__doc__
covariance_matrix.__doc__ = CovStruct.covariance_matrix.__doc__
