"""
Linear mixed effects models for Statsmodels
The data are partitioned into disjoint groups. The probability model
for group i is:
Y = X*beta + Z*gamma + epsilon
where
* n_i is the number of observations in group i
* Y is a n_i dimensional response vector
* X is a n_i x k_fe design matrix for the fixed effects
* beta is a k_fe-dimensional vector of fixed effects slopes
* Z is a n_i x k_re design matrix for the random effects
* gamma is a k_re-dimensional random vector with mean 0
and covariance matrix Psi; note that each group
gets its own independent realization of gamma.
* epsilon is a n_i dimensional vector of iid normal
errors with mean 0 and variance sigma^2; the epsilon
values are independent both within and between groups
Y, X and Z must be entirely observed. beta, Psi, and sigma^2 are
estimated using ML or REML estimation, and gamma and epsilon are
random so define the probability model.
The mean structure is E[Y|X,Z] = X*beta. If only the mean structure
is of interest, GEE is a good alternative to mixed models.
The primary reference for the implementation details is:
MJ Lindstrom, DM Bates (1988). "Newton Raphson and EM algorithms for
linear mixed effects models for repeated measures data". Journal of
the American Statistical Association. Volume 83, Issue 404, pages
1014-1022.
See also this more recent document:
http://econ.ucsb.edu/~doug/245a/Papers/Mixed%20Effects%20Implement.pdf
All the likelihood, gradient, and Hessian calculations closely follow
Lindstrom and Bates.
The following two documents are written more from the perspective of
users:
http://lme4.r-forge.r-project.org/lMMwR/lrgprt.pdf
http://lme4.r-forge.r-project.org/slides/2009-07-07-Rennes/3Longitudinal-4.pdf
Notation:
* `cov_re` is the random effects covariance matrix (referred to above
as Psi) and `scale` is the (scalar) error variance. For a single
group, the marginal covariance matrix of endog given exog is scale*I
+ Z * cov_re * Z', where Z is the design matrix for the random
effects in one group.
Notes:
1. Three different parameterizations are used here in different
places. The regression slopes (usually called `fe_params`) are
identical in all three parameterizations, but the variance parameters
differ. The parameterizations are:
* The "natural parameterization" in which cov(endog) = scale*I + Z *
cov_re * Z', as described above. This is the main parameterization
visible to the user.
* The "profile parameterization" in which cov(endog) = I +
Z * cov_re1 * Z'. This is the parameterization of the profile
likelihood that is maximized to produce parameter estimates.
(see Lindstrom and Bates for details). The "natural" cov_re is
equal to the "profile" cov_re1 times scale.
* The "square root parameterization" in which we work with the
Cholesky factor of cov_re1 instead of cov_re1 directly.
All three parameterizations can be "packed" by concatenating fe_params
together with the lower triangle of the dependence structure. Note
that when unpacking, it is important to either square or reflect the
dependence structure depending on which parameterization is being
used.
2. The situation where the random effects covariance matrix is
singular is numerically challenging. Small changes in the covariance
parameters may lead to large changes in the likelihood and
derivatives.
3. The optimization strategy is to first use OLS to get starting
values for the mean structure. Then we optionally perform a few EM
steps, followed by optionally performing a few steepest ascent steps.
This is followed by conjugate gradient optimization using one of the
scipy gradient optimizers. The EM and steepest ascent steps are used
to get adequate starting values for the conjugate gradient
optimization, which is much faster.
"""
import numpy as np
import statsmodels.base.model as base
from scipy.optimize import fmin_ncg, fmin_cg, fmin_bfgs, fmin
from statsmodels.tools.decorators import cache_readonly
from statsmodels.tools import data as data_tools
from scipy.stats.distributions import norm
import pandas as pd
import patsy
from statsmodels.compat.collections import OrderedDict
from statsmodels.compat import range
import warnings
from statsmodels.tools.sm_exceptions import ConvergenceWarning
from statsmodels.base._penalties import Penalty
from statsmodels.compat.numpy import np_matrix_rank
from pandas import DataFrame
def _get_exog_re_names(exog_re):
"""
Passes through if given a list of names. Otherwise, gets pandas names
or creates some generic variable names as needed.
"""
if isinstance(exog_re, pd.DataFrame):
return exog_re.columns.tolist()
elif isinstance(exog_re, pd.Series) and exog_re.name is not None:
return [exog_re.name]
elif isinstance(exog_re, list):
return exog_re
return ["Z{0}".format(k + 1) for k in range(exog_re.shape[1])]
class MixedLMParams(object):
"""
This class represents a parameter state for a mixed linear model.
Parameters
----------
k_fe : integer
The number of covariates with fixed effects.
k_re : integer
The number of covariates with random effects.
use_sqrt : boolean
If True, the covariance matrix is stored using as the lower
triangle of its Cholesky square root, otherwise it is stored
as the lower triangle of the covariance matrix.
Notes
-----
This object represents the parameter state for the model in which
the scale parameter has been profiled out.
"""
def __init__(self, k_fe, k_re, use_sqrt=True):
self.k_fe = k_fe
self.k_re = k_re
self.k_re2 = k_re * (k_re + 1) / 2
self.k_tot = self.k_fe + self.k_re2
self.use_sqrt = use_sqrt
self._ix = np.tril_indices(self.k_re)
self._params = np.zeros(self.k_tot)
def from_packed(params, k_fe, use_sqrt):
"""
Create a MixedLMParams object from packed parameter vector.
Parameters
----------
params : array-like
The mode parameters packed into a single vector.
k_fe : integer
The number of covariates with fixed effects
use_sqrt : boolean
If True, the random effects covariance matrix is stored as
its Cholesky factor, otherwise the lower triangle of the
covariance matrix is stored.
Returns
-------
A MixedLMParams object.
"""
k_re2 = len(params) - k_fe
k_re = (-1 + np.sqrt(1 + 8*k_re2)) / 2
if k_re != int(k_re):
raise ValueError("Length of `packed` not compatible with value of `fe`.")
k_re = int(k_re)
pa = MixedLMParams(k_fe, k_re, use_sqrt)
pa.set_packed(params)
return pa
from_packed = staticmethod(from_packed)
def from_components(fe_params, cov_re=None, cov_re_sqrt=None,
use_sqrt=True):
"""
Create a MixedLMParams object from each parameter component.
Parameters
----------
fe_params : array-like
The fixed effects parameter (a 1-dimensional array).
cov_re : array-like
The random effects covariance matrix (a square, symmetric
2-dimensional array).
cov_re_sqrt : array-like
The Cholesky (lower triangular) square root of the random
effects covariance matrix.
use_sqrt : boolean
If True, the random effects covariance matrix is stored as
the lower triangle of its Cholesky factor, otherwise the
lower triangle of the covariance matrix is stored.
Returns
-------
A MixedLMParams object.
"""
k_fe = len(fe_params)
k_re = cov_re.shape[0]
pa = MixedLMParams(k_fe, k_re, use_sqrt)
pa.set_fe_params(fe_params)
pa.set_cov_re(cov_re)
return pa
from_components = staticmethod(from_components)
def copy(self):
"""
Returns a copy of the object.
"""
obj = MixedLMParams(self.k_fe, self.k_re, self.use_sqrt)
obj.set_packed(self.get_packed().copy())
return obj
def get_packed(self, use_sqrt=None):
"""
Returns the model parameters packed into a single vector.
Parameters
----------
use_sqrt : None or bool
If None, `use_sqrt` has the value of this instance's
`use_sqrt`. Otherwise it is set to the given value.
"""
if (use_sqrt is None) or (use_sqrt == self.use_sqrt):
return self._params
pa = self._params.copy()
cov_re = self.get_cov_re()
if use_sqrt:
L = np.linalg.cholesky(cov_re)
pa[self.k_fe:] = L[self._ix]
else:
pa[self.k_fe:] = cov_re[self._ix]
return pa
def set_packed(self, params):
"""
Sets the packed parameter vector to the given vector, without
any validity checking.
"""
self._params = params
def get_fe_params(self):
"""
Returns the fixed effects paramaters as a ndarray.
"""
return self._params[0:self.k_fe]
def set_fe_params(self, fe_params):
"""
Set the fixed effect parameters to the given vector.
"""
self._params[0:self.k_fe] = fe_params
def set_cov_re(self, cov_re=None, cov_re_sqrt=None):
"""
Set the random effects covariance matrix to the given value.
Parameters
----------
cov_re : array-like
The random effects covariance matrix.
cov_re_sqrt : array-like
The Cholesky square root of the random effects covariance
matrix. Only the lower triangle is read.
Notes
-----
The first of `cov_re` and `cov_re_sqrt` that is not None is
used.
"""
if cov_re is not None:
if self.use_sqrt:
cov_re_sqrt = np.linalg.cholesky(cov_re)
self._params[self.k_fe:] = cov_re_sqrt[self._ix]
else:
self._params[self.k_fe:] = cov_re[self._ix]
elif cov_re_sqrt is not None:
if self.use_sqrt:
self._params[self.k_fe:] = cov_re_sqrt[self._ix]
else:
cov_re = np.dot(cov_re_sqrt, cov_re_sqrt.T)
self._params[self.k_fe:] = cov_re[self._ix]
def get_cov_re(self):
"""
Returns the random effects covariance matrix.
"""
pa = self._params[self.k_fe:]
cov_re = np.zeros((self.k_re, self.k_re))
cov_re[self._ix] = pa
if self.use_sqrt:
cov_re = np.dot(cov_re, cov_re.T)
else:
cov_re = (cov_re + cov_re.T) - np.diag(np.diag(cov_re))
return cov_re
# This is a global switch to use direct linear algebra calculations
# for solving factor-structured linear systems and calculating
# factor-structured determinants. If False, use the
# Sherman-Morrison-Woodbury update which is more efficient for
# factor-structured matrices. Should be False except when testing.
_no_smw = False
def _smw_solve(s, A, AtA, B, BI, rhs):
"""
Solves the system (s*I + A*B*A') * x = rhs for x and returns x.
Parameters
----------
s : scalar
See above for usage
A : square symmetric ndarray
See above for usage
AtA : square ndarray
A.T * A
B : square symmetric ndarray
See above for usage
BI : square symmetric ndarray
The inverse of `B`. Can be None if B is singular
rhs : ndarray
See above for usage
Returns
-------
x : ndarray
See above
If the global variable `_no_smw` is True, this routine uses direct
linear algebra calculations. Otherwise it uses the
Sherman-Morrison-Woodbury identity to speed up the calculation.
"""
# Direct calculation
if _no_smw or BI is None:
mat = np.dot(A, np.dot(B, A.T))
# Add constant to diagonal
mat.flat[::mat.shape[0]+1] += s
return np.linalg.solve(mat, rhs)
# Use SMW identity
qmat = BI + AtA / s
u = np.dot(A.T, rhs)
qmat = np.linalg.solve(qmat, u)
qmat = np.dot(A, qmat)
rslt = rhs / s - qmat / s**2
return rslt
def _smw_logdet(s, A, AtA, B, BI, B_logdet):
"""
Use the matrix determinant lemma to accelerate the calculation of
the log determinant of s*I + A*B*A'.
Parameters
----------
s : scalar
See above for usage
A : square symmetric ndarray
See above for usage
AtA : square matrix
A.T * A
B : square symmetric ndarray
See above for usage
BI : square symmetric ndarray
The inverse of `B`; can be None if B is singular.
B_logdet : real
The log determinant of B
Returns
-------
The log determinant of s*I + A*B*A'.
"""
p = A.shape[0]
if _no_smw or BI is None:
mat = np.dot(A, np.dot(B, A.T))
# Add constant to diagonal
mat.flat[::p+1] += s
_, ld = np.linalg.slogdet(mat)
return ld
ld = p * np.log(s)
qmat = BI + AtA / s
_, ld1 = np.linalg.slogdet(qmat)
return B_logdet + ld + ld1
[docs]class MixedLM(base.LikelihoodModel):
"""
An object specifying a linear mixed effects model. Use the `fit`
method to fit the model and obtain a results object.
Parameters
----------
endog : 1d array-like
The dependent variable
exog : 2d array-like
A matrix of covariates used to determine the
mean structure (the "fixed effects" covariates).
groups : 1d array-like
A vector of labels determining the groups -- data from
different groups are independent
exog_re : 2d array-like
A matrix of covariates used to determine the variance and
covariance structure (the "random effects" covariates). If
None, defaults to a random intercept for each group.
use_sqrt : bool
If True, optimization is carried out using the lower
triangle of the square root of the random effects
covariance matrix, otherwise it is carried out using the
lower triangle of the random effects covariance matrix.
missing : string
The approach to missing data handling
Notes
-----
The covariates in `exog` and `exog_re` may (but need not)
partially or wholly overlap.
`use_sqrt` should almost always be set to True. The main use case
for use_sqrt=False is when complicated patterns of fixed values in
the covariance structure are set (using the `free` argument to
`fit`) that cannot be expressed in terms of the Cholesky factor L.
"""
def __init__(self, endog, exog, groups, exog_re=None,
use_sqrt=True, missing='none', **kwargs):
self.use_sqrt = use_sqrt
# Some defaults
self.reml = True
self.fe_pen = None
self.re_pen = None
# If there is one covariate, it may be passed in as a column
# vector, convert these to 2d arrays.
# TODO: Can this be moved up in the class hierarchy?
# yes, it should be done up the hierarchy
if (exog is not None and
data_tools._is_using_ndarray_type(exog, None) and
exog.ndim == 1):
exog = exog[:, None]
if (exog_re is not None and
data_tools._is_using_ndarray_type(exog_re, None) and
exog_re.ndim == 1):
exog_re = exog_re[:, None]
# Calling super creates self.endog, etc. as ndarrays and the
# original exog, endog, etc. are self.data.endog, etc.
super(MixedLM, self).__init__(endog, exog, groups=groups,
exog_re=exog_re, missing=missing,
**kwargs)
self._init_keys.extend(["use_sqrt"])
self.k_fe = exog.shape[1] # Number of fixed effects parameters
if exog_re is None:
# Default random effects structure (random intercepts).
self.k_re = 1
self.k_re2 = 1
self.exog_re = np.ones((len(endog), 1), dtype=np.float64)
self.data.exog_re = self.exog_re
self.data.param_names = self.exog_names + ['Intercept RE']
else:
# Process exog_re the same way that exog is handled
# upstream
# TODO: this is wrong and should be handled upstream wholly
self.data.exog_re = exog_re
self.exog_re = np.asarray(exog_re)
if self.exog_re.ndim == 1:
self.exog_re = self.exog_re[:, None]
if not self.data._param_names:
# HACK: could've been set in from_formula already
# needs refactor
(param_names,
exog_re_names,
exog_re_names_full) = self._make_param_names(exog_re)
self.data.param_names = param_names
self.data.exog_re_names = exog_re_names
self.data.exog_re_names_full = exog_re_names_full
# Model dimensions
# Number of random effect covariates
self.k_re = self.exog_re.shape[1]
# Number of covariance parameters
self.k_re2 = self.k_re * (self.k_re + 1) // 2
self.k_params = self.k_fe + self.k_re2
# Convert the data to the internal representation, which is a
# list of arrays, corresponding to the groups.
group_labels = list(set(groups))
group_labels.sort()
row_indices = dict((s, []) for s in group_labels)
for i,g in enumerate(groups):
row_indices[g].append(i)
self.row_indices = row_indices
self.group_labels = group_labels
self.n_groups = len(self.group_labels)
# Split the data by groups
self.endog_li = self.group_list(self.endog)
self.exog_li = self.group_list(self.exog)
self.exog_re_li = self.group_list(self.exog_re)
# Precompute this.
self.exog_re2_li = [np.dot(x.T, x) for x in self.exog_re_li]
# The total number of observations, summed over all groups
self.n_totobs = sum([len(y) for y in self.endog_li])
# why do it like the above?
self.nobs = len(self.endog)
# Set the fixed effects parameter names
if self.exog_names is None:
self.exog_names = ["FE%d" % (k + 1) for k in
range(self.exog.shape[1])]
def _make_param_names(self, exog_re):
"""
Returns the full parameter names list, just the exogenous random
effects variables, and the exogenous random effects variables with
the interaction terms.
"""
exog_names = list(self.exog_names)
exog_re_names = _get_exog_re_names(exog_re)
param_names = []
jj = self.k_fe
for i in range(len(exog_re_names)):
for j in range(i + 1):
if i == j:
param_names.append(exog_re_names[i] + " RE")
else:
param_names.append(exog_re_names[j] + " RE x " +
exog_re_names[i] + " RE")
jj += 1
return exog_names + param_names, exog_re_names, param_names
@classmethod
[docs] def group_list(self, array):
"""
Returns `array` split into subarrays corresponding to the
grouping structure.
"""
if array.ndim == 1:
return [np.array(array[self.row_indices[k]])
for k in self.group_labels]
else:
return [np.array(array[self.row_indices[k], :])
for k in self.group_labels]
[docs] def fit_regularized(self, start_params=None, method='l1', alpha=0,
ceps=1e-4, ptol=1e-6, maxit=200, **fit_kwargs):
"""
Fit a model in which the fixed effects parameters are
penalized. The dependence parameters are held fixed at their
estimated values in the unpenalized model.
Parameters
----------
method : string of Penalty object
Method for regularization. If a string, must be 'l1'.
alpha : array-like
Scalar or vector of penalty weights. If a scalar, the
same weight is applied to all coefficients; if a vector,
it contains a weight for each coefficient. If method is a
Penalty object, the weights are scaled by alpha. For L1
regularization, the weights are used directly.
ceps : positive real scalar
Fixed effects parameters smaller than this value
in magnitude are treaded as being zero.
ptol : positive real scalar
Convergence occurs when the sup norm difference
between successive values of `fe_params` is less than
`ptol`.
maxit : integer
The maximum number of iterations.
fit_kwargs : keywords
Additional keyword arguments passed to fit.
Returns
-------
A MixedLMResults instance containing the results.
Notes
-----
The covariance structure is not updated as the fixed effects
parameters are varied.
The algorithm used here for L1 regularization is a"shooting"
or cyclic coordinate descent algorithm.
If method is 'l1', then `fe_pen` and `cov_pen` are used to
obtain the covariance structure, but are ignored during the
L1-penalized fitting.
References
----------
Friedman, J. H., Hastie, T. and Tibshirani, R. Regularized
Paths for Generalized Linear Models via Coordinate
Descent. Journal of Statistical Software, 33(1) (2008)
http://www.jstatsoft.org/v33/i01/paper
http://statweb.stanford.edu/~tibs/stat315a/Supplements/fuse.pdf
"""
if type(method) == str and (method.lower() != 'l1'):
raise ValueError("Invalid regularization method")
# If method is a smooth penalty just optimize directly.
if isinstance(method, Penalty):
# Scale the penalty weights by alpha
method.alpha = alpha
fit_kwargs.update({"fe_pen": method})
return self.fit(**fit_kwargs)
if np.isscalar(alpha):
alpha = alpha * np.ones(self.k_fe, dtype=np.float64)
# Fit the unpenalized model to get the dependence structure.
mdf = self.fit(**fit_kwargs)
fe_params = mdf.fe_params
cov_re = mdf.cov_re
scale = mdf.scale
try:
cov_re_inv = np.linalg.inv(cov_re)
except np.linalg.LinAlgError:
cov_re_inv = None
for itr in range(maxit):
fe_params_s = fe_params.copy()
for j in range(self.k_fe):
if abs(fe_params[j]) < ceps:
continue
# The residuals
fe_params[j] = 0.
expval = np.dot(self.exog, fe_params)
resid_all = self.endog - expval
# The loss function has the form
# a*x^2 + b*x + pwt*|x|
a, b = 0., 0.
for k, lab in enumerate(self.group_labels):
exog = self.exog_li[k]
ex_r = self.exog_re_li[k]
ex2_r = self.exog_re2_li[k]
resid = resid_all[self.row_indices[lab]]
x = exog[:,j]
u = _smw_solve(scale, ex_r, ex2_r, cov_re,
cov_re_inv, x)
a += np.dot(u, x)
b -= 2 * np.dot(u, resid)
pwt1 = alpha[j]
if b > pwt1:
fe_params[j] = -(b - pwt1) / (2 * a)
elif b < -pwt1:
fe_params[j] = -(b + pwt1) / (2 * a)
if np.abs(fe_params_s - fe_params).max() < ptol:
break
# Replace the fixed effects estimates with their penalized
# values, leave the dependence parameters in their unpenalized
# state.
params_prof = mdf.params.copy()
params_prof[0:self.k_fe] = fe_params
scale = self.get_scale(fe_params, mdf.cov_re_unscaled)
# Get the Hessian including only the nonzero fixed effects,
# then blow back up to the full size after inverting.
hess = self.hessian_full(params_prof)
pcov = np.nan * np.ones_like(hess)
ii = np.abs(params_prof) > ceps
ii[self.k_fe:] = True
ii = np.flatnonzero(ii)
hess1 = hess[ii, :][:, ii]
pcov[np.ix_(ii,ii)] = np.linalg.inv(-hess1)
results = MixedLMResults(self, params_prof, pcov / scale)
results.fe_params = fe_params
results.cov_re = cov_re
results.scale = scale
results.cov_re_unscaled = mdf.cov_re_unscaled
results.method = mdf.method
results.converged = True
results.cov_pen = self.cov_pen
results.k_fe = self.k_fe
results.k_re = self.k_re
results.k_re2 = self.k_re2
return MixedLMResultsWrapper(results)
def _reparam(self):
"""
Returns parameters of the map converting parameters from the
form used in optimization to the form returned to the user.
Returns
-------
lin : list-like
Linear terms of the map
quad : list-like
Quadratic terms of the map
Notes
-----
If P are the standard form parameters and R are the
modified parameters (i.e. with square root covariance),
then P[i] = lin[i] * R + R' * quad[i] * R
"""
k_fe, k_re, k_re2 = self.k_fe, self.k_re, self.k_re2
k_tot = k_fe + k_re2
ix = np.tril_indices(self.k_re)
lin = []
for k in range(k_fe):
e = np.zeros(k_tot)
e[k] = 1
lin.append(e)
for k in range(k_re2):
lin.append(np.zeros(k_tot))
quad = []
for k in range(k_tot):
quad.append(np.zeros((k_tot, k_tot)))
ii = np.tril_indices(k_re)
ix = [(a,b) for a,b in zip(ii[0], ii[1])]
for i1 in range(k_re2):
for i2 in range(k_re2):
ix1 = ix[i1]
ix2 = ix[i2]
if (ix1[1] == ix2[1]) and (ix1[0] <= ix2[0]):
ii = (ix2[0], ix1[0])
k = ix.index(ii)
quad[k_fe+k][k_fe+i2, k_fe+i1] += 1
for k in range(k_tot):
quad[k] = 0.5*(quad[k] + quad[k].T)
return lin, quad
[docs] def hessian_sqrt(self, params):
"""
Returns the Hessian matrix of the log-likelihood evaluated at
a given point, calculated with respect to the parameterization
in which the random effects covariance matrix is represented
through its Cholesky square root.
Parameters
----------
params : MixedLMParams or array-like
The model parameters. If array-like, must contain packed
parameters that are compatible with this model.
Returns
-------
The Hessian matrix of the profile log likelihood function,
evaluated at `params`.
Notes
-----
If `params` is provided as a MixedLMParams object it may be of
any parameterization.
"""
if type(params) is not MixedLMParams:
params = MixedLMParams.from_packed(params, self.k_fe,
self.use_sqrt)
score0 = self.score_full(params)
hess0 = self.hessian_full(params)
params_vec = params.get_packed(use_sqrt=True)
lin, quad = self._reparam()
k_tot = self.k_fe + self.k_re2
# Convert Hessian to new coordinates
hess = 0.
for i in range(k_tot):
hess += 2 * score0[i] * quad[i]
for i in range(k_tot):
vi = lin[i] + 2*np.dot(quad[i], params_vec)
for j in range(k_tot):
vj = lin[j] + 2*np.dot(quad[j], params_vec)
hess += hess0[i, j] * np.outer(vi, vj)
return hess
[docs] def loglike(self, params):
"""
Evaluate the (profile) log-likelihood of the linear mixed
effects model.
Parameters
----------
params : MixedLMParams, or array-like.
The parameter value. If array-like, must be a packed
parameter vector compatible with this model.
Returns
-------
The log-likelihood value at `params`.
Notes
-----
This is the profile likelihood in which the scale parameter
`scale` has been profiled out.
The input parameter state, if provided as a MixedLMParams
object, can be with respect to any parameterization.
"""
if type(params) is not MixedLMParams:
params = MixedLMParams.from_packed(params, self.k_fe,
self.use_sqrt)
fe_params = params.get_fe_params()
cov_re = params.get_cov_re()
try:
cov_re_inv = np.linalg.inv(cov_re)
except np.linalg.LinAlgError:
cov_re_inv = None
_, cov_re_logdet = np.linalg.slogdet(cov_re)
# The residuals
expval = np.dot(self.exog, fe_params)
resid_all = self.endog - expval
likeval = 0.
# Handle the covariance penalty
if self.cov_pen is not None:
likeval -= self.cov_pen.func(cov_re, cov_re_inv)
# Handle the fixed effects penalty
if self.fe_pen is not None:
likeval -= self.fe_pen.func(fe_params)
xvx, qf = 0., 0.
for k, lab in enumerate(self.group_labels):
exog = self.exog_li[k]
ex_r = self.exog_re_li[k]
ex2_r = self.exog_re2_li[k]
resid = resid_all[self.row_indices[lab]]
# Part 1 of the log likelihood (for both ML and REML)
ld = _smw_logdet(1., ex_r, ex2_r, cov_re, cov_re_inv,
cov_re_logdet)
likeval -= ld / 2.
# Part 2 of the log likelihood (for both ML and REML)
u = _smw_solve(1., ex_r, ex2_r, cov_re, cov_re_inv, resid)
qf += np.dot(resid, u)
# Adjustment for REML
if self.reml:
mat = _smw_solve(1., ex_r, ex2_r, cov_re, cov_re_inv,
exog)
xvx += np.dot(exog.T, mat)
if self.reml:
likeval -= (self.n_totobs - self.k_fe) * np.log(qf) / 2.
_,ld = np.linalg.slogdet(xvx)
likeval -= ld / 2.
likeval -= (self.n_totobs - self.k_fe) * np.log(2 * np.pi) / 2.
likeval += ((self.n_totobs - self.k_fe) *
np.log(self.n_totobs - self.k_fe) / 2.)
likeval -= (self.n_totobs - self.k_fe) / 2.
else:
likeval -= self.n_totobs * np.log(qf) / 2.
likeval -= self.n_totobs * np.log(2 * np.pi) / 2.
likeval += self.n_totobs * np.log(self.n_totobs) / 2.
likeval -= self.n_totobs / 2.
return likeval
def _gen_dV_dPsi(self, ex_r, max_ix=None):
"""
A generator that yields the derivative of the covariance
matrix V (=I + Z*Psi*Z') with respect to the free elements of
Psi. Each call to the generator yields the index of Psi with
respect to which the derivative is taken, and the derivative
matrix with respect to that element of Psi. Psi is a
symmetric matrix, so the free elements are the lower triangle.
If max_ix is not None, the iterations terminate after max_ix
values are yielded.
"""
jj = 0
for j1 in range(self.k_re):
for j2 in range(j1 + 1):
if max_ix is not None and jj > max_ix:
return
mat = np.outer(ex_r[:,j1], ex_r[:,j2])
if j1 != j2:
mat += mat.T
yield jj,mat
jj += 1
[docs] def score(self, params):
"""
Returns the score vector of the profile log-likelihood.
Notes
-----
The score vector that is returned is computed with respect to
the parameterization defined by this model instance's
`use_sqrt` attribute. The input value `params` can be with
respect to any parameterization.
"""
if self.use_sqrt:
return self.score_sqrt(params)
else:
return self.score_full(params)
[docs] def hessian(self, params):
"""
Returns the Hessian matrix of the profile log-likelihood.
Notes
-----
The Hessian matrix that is returned is computed with respect
to the parameterization defined by this model's `use_sqrt`
attribute. The input value `params` can be with respect to
any parameterization.
"""
if self.use_sqrt:
return self.hessian_sqrt(params)
else:
return self.hessian_full(params)
[docs] def score_full(self, params):
"""
Calculates the score vector for the profiled log-likelihood of
the mixed effects model with respect to the parameterization
in which the random effects covariance matrix is represented
in its full form (not using the Cholesky factor).
Parameters
----------
params : MixedLMParams or array-like
The parameter at which the score function is evaluated.
If array-like, must contain packed parameter values that
are compatible with the current model.
Returns
-------
The score vector, calculated at `params`.
Notes
-----
The score vector that is returned is taken with respect to the
parameterization in which `cov_re` is represented through its
lower triangle (without taking the Cholesky square root).
The input, if provided as a MixedLMParams object, can be of
any parameterization.
"""
if type(params) is not MixedLMParams:
params = MixedLMParams.from_packed(params, self.k_fe,
self.use_sqrt)
fe_params = params.get_fe_params()
cov_re = params.get_cov_re()
try:
cov_re_inv = np.linalg.inv(cov_re)
except np.linalg.LinAlgError:
cov_re_inv = None
score_fe = np.zeros(self.k_fe, dtype=np.float64)
score_re = np.zeros(self.k_re2, dtype=np.float64)
# Handle the covariance penalty.
if self.cov_pen is not None:
score_re -= self.cov_pen.grad(cov_re, cov_re_inv)
# Handle the fixed effects penalty.
if self.fe_pen is not None:
score_fe -= self.fe_pen.grad(fe_params)
# resid' V^{-1} resid, summed over the groups (a scalar)
rvir = 0.
# exog' V^{-1} resid, summed over the groups (a k_fe
# dimensional vector)
xtvir = 0.
# exog' V^{_1} exog, summed over the groups (a k_fe x k_fe
# matrix)
xtvix = 0.
# V^{-1} exog' dV/dQ_jj exog V^{-1}, where Q_jj is the jj^th
# covariance parameter.
xtax = [0.,] * self.k_re2
# Temporary related to the gradient of log |V|
dlv = np.zeros(self.k_re2, dtype=np.float64)
# resid' V^{-1} dV/dQ_jj V^{-1} resid (a scalar)
rvavr = np.zeros(self.k_re2, dtype=np.float64)
for k in range(self.n_groups):
exog = self.exog_li[k]
ex_r = self.exog_re_li[k]
ex2_r = self.exog_re2_li[k]
# The residuals
expval = np.dot(exog, fe_params)
resid = self.endog_li[k] - expval
if self.reml:
viexog = _smw_solve(1., ex_r, ex2_r, cov_re,
cov_re_inv, exog)
xtvix += np.dot(exog.T, viexog)
# Contributions to the covariance parameter gradient
jj = 0
vex = _smw_solve(1., ex_r, ex2_r, cov_re, cov_re_inv,
ex_r)
vir = _smw_solve(1., ex_r, ex2_r, cov_re, cov_re_inv,
resid)
for jj,mat in self._gen_dV_dPsi(ex_r):
dlv[jj] = np.trace(_smw_solve(1., ex_r, ex2_r, cov_re,
cov_re_inv, mat))
rvavr[jj] += np.dot(vir, np.dot(mat, vir))
if self.reml:
xtax[jj] += np.dot(viexog.T, np.dot(mat, viexog))
# Contribution of log|V| to the covariance parameter
# gradient.
score_re -= 0.5 * dlv
# Needed for the fixed effects params gradient
rvir += np.dot(resid, vir)
xtvir += np.dot(exog.T, vir)
fac = self.n_totobs
if self.reml:
fac -= self.exog.shape[1]
score_fe += fac * xtvir / rvir
score_re += 0.5 * fac * rvavr / rvir
if self.reml:
for j in range(self.k_re2):
score_re[j] += 0.5 * np.trace(np.linalg.solve(
xtvix, xtax[j]))
score_vec = np.concatenate((score_fe, score_re))
if self._freepat is not None:
return self._freepat.get_packed() * score_vec
else:
return score_vec
[docs] def score_sqrt(self, params):
"""
Returns the score vector with respect to the parameterization
in which the random effects covariance matrix is represented
through its Cholesky square root.
Parameters
----------
params : MixedLMParams or array-like
The model parameters. If array-like must contain packed
parameters that are compatible with this model instance.
Returns
-------
The score vector.
Notes
-----
The input, if provided as a MixedLMParams object, can be of
any parameterization.
"""
if type(params) is not MixedLMParams:
params = MixedLMParams.from_packed(params, self.k_fe,
self.use_sqrt)
score_full = self.score_full(params)
params_vec = params.get_packed(use_sqrt=True)
lin, quad = self._reparam()
scr = 0.
for i in range(len(params_vec)):
v = lin[i] + 2 * np.dot(quad[i], params_vec)
scr += score_full[i] * v
if self._freepat is not None:
return self._freepat.get_packed() * scr
else:
return scr
[docs] def hessian_full(self, params):
"""
Calculates the Hessian matrix for the mixed effects model with
respect to the parameterization in which the covariance matrix
is represented directly (without square-root transformation).
Parameters
----------
params : MixedLMParams or array-like
The model parameters at which the Hessian is calculated.
If array-like, must contain the packed parameters in a
form that is compatible with this model instance.
Returns
-------
hess : 2d ndarray
The Hessian matrix, evaluated at `params`.
Notes
-----
Tf provided as a MixedLMParams object, the input may be of
any parameterization.
"""
if type(params) is not MixedLMParams:
params = MixedLMParams.from_packed(params, self.k_fe,
self.use_sqrt)
fe_params = params.get_fe_params()
cov_re = params.get_cov_re()
try:
cov_re_inv = np.linalg.inv(cov_re)
except np.linalg.LinAlgError:
cov_re_inv = None
# Blocks for the fixed and random effects parameters.
hess_fe = 0.
hess_re = np.zeros((self.k_re2, self.k_re2), dtype=np.float64)
hess_fere = np.zeros((self.k_re2, self.k_fe),
dtype=np.float64)
fac = self.n_totobs
if self.reml:
fac -= self.exog.shape[1]
rvir = 0.
xtvix = 0.
xtax = [0.,] * self.k_re2
B = np.zeros(self.k_re2, dtype=np.float64)
D = np.zeros((self.k_re2, self.k_re2), dtype=np.float64)
F = [[0.,]*self.k_re2 for k in range(self.k_re2)]
for k in range(self.n_groups):
exog = self.exog_li[k]
ex_r = self.exog_re_li[k]
ex2_r = self.exog_re2_li[k]
# The residuals
expval = np.dot(exog, fe_params)
resid = self.endog_li[k] - expval
viexog = _smw_solve(1., ex_r, ex2_r, cov_re, cov_re_inv,
exog)
xtvix += np.dot(exog.T, viexog)
vir = _smw_solve(1., ex_r, ex2_r, cov_re, cov_re_inv,
resid)
rvir += np.dot(resid, vir)
for jj1,mat1 in self._gen_dV_dPsi(ex_r):
hess_fere[jj1,:] += np.dot(viexog.T,
np.dot(mat1, vir))
if self.reml:
xtax[jj1] += np.dot(viexog.T, np.dot(mat1, viexog))
B[jj1] += np.dot(vir, np.dot(mat1, vir))
E = _smw_solve(1., ex_r, ex2_r, cov_re, cov_re_inv,
mat1)
for jj2,mat2 in self._gen_dV_dPsi(ex_r, jj1):
Q = np.dot(mat2, E)
Q1 = Q + Q.T
vt = np.dot(vir, np.dot(Q1, vir))
D[jj1, jj2] += vt
if jj1 != jj2:
D[jj2, jj1] += vt
R = _smw_solve(1., ex_r, ex2_r, cov_re,
cov_re_inv, Q)
rt = np.trace(R) / 2
hess_re[jj1, jj2] += rt
if jj1 != jj2:
hess_re[jj2, jj1] += rt
if self.reml:
F[jj1][jj2] += np.dot(viexog.T,
np.dot(Q, viexog))
hess_fe -= fac * xtvix / rvir
hess_re = hess_re - 0.5 * fac * (D/rvir - np.outer(B, B) / rvir**2)
hess_fere = -fac * hess_fere / rvir
if self.reml:
for j1 in range(self.k_re2):
Q1 = np.linalg.solve(xtvix, xtax[j1])
for j2 in range(j1 + 1):
Q2 = np.linalg.solve(xtvix, xtax[j2])
a = np.trace(np.dot(Q1, Q2))
a -= np.trace(np.linalg.solve(xtvix, F[j1][j2]))
a *= 0.5
hess_re[j1, j2] += a
if j1 > j2:
hess_re[j2, j1] += a
# Put the blocks together to get the Hessian.
m = self.k_fe + self.k_re2
hess = np.zeros((m, m), dtype=np.float64)
hess[0:self.k_fe, 0:self.k_fe] = hess_fe
hess[0:self.k_fe, self.k_fe:] = hess_fere.T
hess[self.k_fe:, 0:self.k_fe] = hess_fere
hess[self.k_fe:, self.k_fe:] = hess_re
return hess
[docs] def steepest_ascent(self, params, n_iter):
"""
Take steepest ascent steps to increase the log-likelihood
function.
Parameters
----------
params : array-like
The starting point of the optimization.
n_iter: non-negative integer
Return once this number of iterations have occured.
Returns
-------
A MixedLMParameters object containing the final value of the
optimization.
"""
fval = self.loglike(params)
params = params.copy()
params1 = params.copy()
for itr in range(n_iter):
gro = self.score(params)
gr = gro / np.max(np.abs(gro))
sl = 0.5
while sl > 1e-20:
params1._params = params._params + sl*gr
fval1 = self.loglike(params1)
if fval1 > fval:
cov_re = params1.get_cov_re()
if np.min(np.diag(cov_re)) > 1e-2:
params, params1 = params1, params
fval = fval1
break
sl /= 2
return params
[docs] def Estep(self, fe_params, cov_re, scale):
"""
The E-step of the EM algorithm.
This is for ML (not REML), but it seems to be good enough to use for
REML starting values.
Parameters
----------
fe_params : 1d ndarray
The current value of the fixed effect coefficients
cov_re : 2d ndarray
The current value of the covariance matrix of random
effects
scale : positive scalar
The current value of the error variance
Returns
-------
m1x : 1d ndarray
sum_groups :math:`X'*Z*E[gamma | Y]`, where X and Z are the fixed
and random effects covariates, gamma is the random
effects, and Y is the observed data
m1y : scalar
sum_groups :math:`Y'*E[gamma | Y]`
m2 : 2d ndarray
sum_groups :math:`E[gamma * gamma' | Y]`
m2xx : 2d ndarray
sum_groups :math:`Z'*Z * E[gamma * gamma' | Y]`
"""
m1x, m1y, m2, m2xx = 0., 0., 0., 0.
try:
cov_re_inv = np.linalg.inv(cov_re)
except np.linalg.LinAlgError:
cov_re_inv = None
for k in range(self.n_groups):
# Get the residuals
expval = np.dot(self.exog_li[k], fe_params)
resid = self.endog_li[k] - expval
# Contruct the marginal covariance matrix for this group
ex_r = self.exog_re_li[k]
ex2_r = self.exog_re2_li[k]
vr1 = _smw_solve(scale, ex_r, ex2_r, cov_re, cov_re_inv,
resid)
vr1 = np.dot(ex_r.T, vr1)
vr1 = np.dot(cov_re, vr1)
vr2 = _smw_solve(scale, ex_r, ex2_r, cov_re, cov_re_inv,
self.exog_re_li[k])
vr2 = np.dot(vr2, cov_re)
vr2 = np.dot(ex_r.T, vr2)
vr2 = np.dot(cov_re, vr2)
rg = np.dot(ex_r, vr1)
m1x += np.dot(self.exog_li[k].T, rg)
m1y += np.dot(self.endog_li[k].T, rg)
egg = cov_re - vr2 + np.outer(vr1, vr1)
m2 += egg
m2xx += np.dot(np.dot(ex_r.T, ex_r), egg)
return m1x, m1y, m2, m2xx
[docs] def EM(self, fe_params, cov_re, scale, niter_em=10,
hist=None):
"""
Run the EM algorithm from a given starting point. This is for
ML (not REML), but it seems to be good enough to use for REML
starting values.
Returns
-------
fe_params : 1d ndarray
The final value of the fixed effects coefficients
cov_re : 2d ndarray
The final value of the random effects covariance
matrix
scale : float
The final value of the error variance
Notes
-----
This uses the parameterization of the likelihood
:math:`scale*I + Z'*V*Z`, note that this differs from the profile
likelihood used in the gradient calculations.
"""
xxtot = 0.
for x in self.exog_li:
xxtot += np.dot(x.T, x)
xytot = 0.
for x,y in zip(self.exog_li, self.endog_li):
xytot += np.dot(x.T, y)
pp = []
for itr in range(niter_em):
m1x, m1y, m2, m2xx = self.Estep(fe_params, cov_re, scale)
fe_params = np.linalg.solve(xxtot, xytot - m1x)
cov_re = m2 / self.n_groups
scale = 0.
for x,y in zip(self.exog_li, self.endog_li):
scale += np.sum((y - np.dot(x, fe_params))**2)
scale -= 2 * m1y
scale += 2 * np.dot(fe_params, m1x)
scale += np.trace(m2xx)
scale /= self.n_totobs
if hist is not None:
hist.append(["EM", fe_params, cov_re, scale])
return fe_params, cov_re, scale
[docs] def get_scale(self, fe_params, cov_re):
"""
Returns the estimated error variance based on given estimates
of the slopes and random effects covariance matrix.
Parameters
----------
fe_params : array-like
The regression slope estimates
cov_re : 2d array
Estimate of the random effects covariance matrix (Psi).
Returns
-------
scale : float
The estimated error variance.
"""
try:
cov_re_inv = np.linalg.inv(cov_re)
except np.linalg.LinAlgError:
cov_re_inv = None
qf = 0.
for k in range(self.n_groups):
exog = self.exog_li[k]
ex_r = self.exog_re_li[k]
ex2_r = self.exog_re2_li[k]
# The residuals
expval = np.dot(exog, fe_params)
resid = self.endog_li[k] - expval
mat = _smw_solve(1., ex_r, ex2_r, cov_re, cov_re_inv,
resid)
qf += np.dot(resid, mat)
if self.reml:
qf /= (self.n_totobs - self.k_fe)
else:
qf /= self.n_totobs
return qf
[docs] def starting_values(self, start_params):
# It's already in the desired form.
if type(start_params) is MixedLMParams:
return start_params
# Given a packed parameter vector, assume it has the same
# dimensions as the current model.
if type(start_params) in [np.ndarray, pd.Series]:
return MixedLMParams.from_packed(start_params,
self.k_fe, self.use_sqrt)
# Very crude starting values.
from statsmodels.regression.linear_model import OLS
fe_params = OLS(self.endog, self.exog).fit().params
cov_re = np.eye(self.k_re)
pa = MixedLMParams.from_components(fe_params, cov_re=cov_re,
use_sqrt=self.use_sqrt)
return pa
[docs] def fit(self, start_params=None, reml=True, niter_em=0,
niter_sa=0, do_cg=True, fe_pen=None, cov_pen=None,
free=None, full_output=False, **kwargs):
"""
Fit a linear mixed model to the data.
Parameters
----------
start_params: array-like or MixedLMParams
If a `MixedLMParams` the state provides the starting
value. If array-like, this is the packed parameter
vector, assumed to be in the same state as this model.
reml : bool
If true, fit according to the REML likelihood, else
fit the standard likelihood using ML.
niter_sa : integer
The number of steepest ascent iterations
niter_em : non-negative integer
The number of EM steps. The EM steps always
preceed steepest descent and conjugate gradient
optimization. The EM algorithm implemented here
is for ML estimation.
do_cg : bool
If True, a conjugate gradient algorithm is
used for optimization (following any steepest
descent or EM steps).
cov_pen : CovariancePenalty object
A penalty for the random effects covariance matrix
fe_pen : Penalty object
A penalty on the fixed effects
free : MixedLMParams object
If not `None`, this is a mask that allows parameters to be
held fixed at specified values. A 1 indicates that the
correspondinig parameter is estimated, a 0 indicates that
it is fixed at its starting value. Setting the `cov_re`
component to the identity matrix fits a model with
independent random effects. The state of `use_sqrt` for
`free` must agree with that of the parent model.
full_output : bool
If true, attach iteration history to results
Returns
-------
A MixedLMResults instance.
Notes
-----
If `start` is provided as an array, it must have the same
`use_sqrt` state as the parent model.
The value of `free` must have the same `use_sqrt` state as the
parent model.
"""
self.reml = reml
self.cov_pen = cov_pen
self.fe_pen = fe_pen
self._freepat = free
if full_output:
hist = []
else:
hist = None
params = self.starting_values(start_params)
success = False
# EM iterations
if niter_em > 0:
fe_params = params.get_fe_params()
cov_re = params.get_cov_re()
scale = 1.
fe_params, cov_re, scale = self.EM(fe_params, cov_re,
scale, niter_em, hist)
params.set_fe_params(fe_params)
# Switch to profile parameterization.
params.set_cov_re(cov_re / scale)
# Try up to 10 times to make the optimization work. Usually
# only one cycle is used.
if do_cg:
for cycle in range(10):
params = self.steepest_ascent(params, niter_sa)
try:
kwargs["retall"] = hist is not None
if "disp" not in kwargs:
kwargs["disp"] = False
# Only bfgs and lbfgs seem to work
kwargs["method"] = "bfgs"
pa = params.get_packed()
rslt = super(MixedLM, self).fit(start_params=pa,
skip_hessian=True,
**kwargs)
except np.linalg.LinAlgError:
continue
# The optimization succeeded
params.set_packed(rslt.params)
success = True
if hist is not None:
hist.append(rslt.mle_retvals)
break
if not success:
msg = "Gradient optimization failed."
warnings.warn(msg, ConvergenceWarning)
# Convert to the final parameterization (i.e. undo the square
# root transform of the covariance matrix, and the profiling
# over the error variance).
fe_params = params.get_fe_params()
cov_re_unscaled = params.get_cov_re()
scale = self.get_scale(fe_params, cov_re_unscaled)
cov_re = scale * cov_re_unscaled
if np.min(np.abs(np.diag(cov_re))) < 0.01:
msg = "The MLE may be on the boundary of the parameter space."
warnings.warn(msg, ConvergenceWarning)
# Compute the Hessian at the MLE. Note that this is the
# Hessian with respect to the random effects covariance matrix
# (not its square root). It is used for obtaining standard
# errors, not for optimization.
hess = self.hessian_full(params)
if free is not None:
pcov = np.zeros_like(hess)
pat = self._freepat.get_packed()
ii = np.flatnonzero(pat)
if len(ii) > 0:
hess1 = hess[np.ix_(ii, ii)]
pcov[np.ix_(ii, ii)] = np.linalg.inv(-hess1)
else:
pcov = np.linalg.inv(-hess)
# Prepare a results class instance
params_packed = params.get_packed()
results = MixedLMResults(self, params_packed, pcov / scale)
results.params_object = params
results.fe_params = fe_params
results.cov_re = cov_re
results.scale = scale
results.cov_re_unscaled = cov_re_unscaled
results.method = "REML" if self.reml else "ML"
results.converged = success
results.hist = hist
results.reml = self.reml
results.cov_pen = self.cov_pen
results.k_fe = self.k_fe
results.k_re = self.k_re
results.k_re2 = self.k_re2
results.use_sqrt = self.use_sqrt
results.freepat = self._freepat
return MixedLMResultsWrapper(results)
[docs]class MixedLMResults(base.LikelihoodModelResults):
'''
Class to contain results of fitting a linear mixed effects model.
MixedLMResults inherits from statsmodels.LikelihoodModelResults
Parameters
----------
See statsmodels.LikelihoodModelResults
Returns
-------
**Attributes**
model : class instance
Pointer to PHreg model instance that called fit.
normalized_cov_params : array
The sampling covariance matrix of the estimates
fe_params : array
The fitted fixed-effects coefficients
re_params : array
The fitted random-effects covariance matrix
bse_fe : array
The standard errors of the fitted fixed effects coefficients
bse_re : array
The standard errors of the fitted random effects covariance
matrix
See Also
--------
statsmodels.LikelihoodModelResults
'''
def __init__(self, model, params, cov_params):
super(MixedLMResults, self).__init__(model, params,
normalized_cov_params=cov_params)
self.nobs = self.model.nobs
self.df_resid = self.nobs - np_matrix_rank(self.model.exog)
@cache_readonly
[docs] def bse_fe(self):
"""
Returns the standard errors of the fixed effect regression
coefficients.
"""
p = self.model.exog.shape[1]
return np.sqrt(np.diag(self.cov_params())[0:p])
@cache_readonly
[docs] def bse_re(self):
"""
Returns the standard errors of the variance parameters. Note
that the sampling distribution of variance parameters is
strongly skewed unless the sample size is large, so these
standard errors may not give meaningful confidence intervals
of p-values if used in the usual way.
"""
p = self.model.exog.shape[1]
return np.sqrt(self.scale * np.diag(self.cov_params())[p:])
@cache_readonly
[docs] def random_effects(self):
"""
Returns the conditional means of all random effects given the
data.
Returns
-------
random_effects : DataFrame
A DataFrame with the distinct `group` values as the index
and the conditional means of the random effects
in the columns.
"""
try:
cov_re_inv = np.linalg.inv(self.cov_re)
except np.linalg.LinAlgError:
cov_re_inv = None
ranef_dict = {}
for k in range(self.model.n_groups):
endog = self.model.endog_li[k]
exog = self.model.exog_li[k]
ex_r = self.model.exog_re_li[k]
ex2_r = self.model.exog_re2_li[k]
label = self.model.group_labels[k]
# Get the residuals
expval = np.dot(exog, self.fe_params)
resid = endog - expval
vresid = _smw_solve(self.scale, ex_r, ex2_r, self.cov_re,
cov_re_inv, resid)
ranef_dict[label] = np.dot(self.cov_re,
np.dot(ex_r.T, vresid))
column_names = dict(zip(range(self.k_re),
self.model.data.exog_re_names))
df = DataFrame.from_dict(ranef_dict, orient='index')
return df.rename(columns=column_names).ix[self.model.group_labels]
@cache_readonly
[docs] def random_effects_cov(self):
"""
Returns the conditional covariance matrix of the random
effects for each group given the data.
Returns
-------
random_effects_cov : dict
A dictionary mapping the distinct values of the `group`
variable to the conditional covariance matrix of the
random effects given the data.
"""
try:
cov_re_inv = np.linalg.inv(self.cov_re)
except np.linalg.LinAlgError:
cov_re_inv = None
ranef_dict = {}
#columns = self.model.data.exog_re_names
for k in range(self.model.n_groups):
ex_r = self.model.exog_re_li[k]
ex2_r = self.model.exog_re2_li[k]
label = self.model.group_labels[k]
mat1 = np.dot(ex_r, self.cov_re)
mat2 = _smw_solve(self.scale, ex_r, ex2_r, self.cov_re,
cov_re_inv, mat1)
mat2 = np.dot(mat1.T, mat2)
ranef_dict[label] = self.cov_re - mat2
#ranef_dict[label] = DataFrame(self.cov_re - mat2,
# index=columns, columns=columns)
return ranef_dict
[docs] def summary(self, yname=None, xname_fe=None, xname_re=None,
title=None, alpha=.05):
"""
Summarize the mixed model regression results.
Parameters
-----------
yname : string, optional
Default is `y`
xname_fe : list of strings, optional
Fixed effects covariate names
xname_re : list of strings, optional
Random effects covariate names
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
"""
from statsmodels.iolib import summary2
smry = summary2.Summary()
info = OrderedDict()
info["Model:"] = "MixedLM"
if yname is None:
yname = self.model.endog_names
info["No. Observations:"] = str(self.model.n_totobs)
info["No. Groups:"] = str(self.model.n_groups)
gs = np.array([len(x) for x in self.model.endog_li])
info["Min. group size:"] = "%.0f" % min(gs)
info["Max. group size:"] = "%.0f" % max(gs)
info["Mean group size:"] = "%.1f" % np.mean(gs)
info["Dependent Variable:"] = yname
info["Method:"] = self.method
info["Scale:"] = self.scale
info["Likelihood:"] = self.llf
info["Converged:"] = "Yes" if self.converged else "No"
smry.add_dict(info)
smry.add_title("Mixed Linear Model Regression Results")
float_fmt = "%.3f"
sdf = np.nan * np.ones((self.k_fe + self.k_re2, 6),
dtype=np.float64)
# Coefficient estimates
sdf[0:self.k_fe, 0] = self.fe_params
# Standard errors
sdf[0:self.k_fe, 1] =\
np.sqrt(np.diag(self.cov_params()[0:self.k_fe]))
# Z-scores
sdf[0:self.k_fe, 2] = sdf[0:self.k_fe, 0] / sdf[0:self.k_fe, 1]
# p-values
sdf[0:self.k_fe, 3] = 2 * norm.cdf(-np.abs(sdf[0:self.k_fe, 2]))
# Confidence intervals
qm = -norm.ppf(alpha / 2)
sdf[0:self.k_fe, 4] = sdf[0:self.k_fe, 0] - qm * sdf[0:self.k_fe, 1]
sdf[0:self.k_fe, 5] = sdf[0:self.k_fe, 0] + qm * sdf[0:self.k_fe, 1]
# Names for all pairs of random effects
jj = self.k_fe
for i in range(self.k_re):
for j in range(i + 1):
sdf[jj, 0] = self.cov_re[i, j]
sdf[jj, 1] = np.sqrt(self.scale) * self.bse[jj]
jj += 1
sdf = pd.DataFrame(index=self.model.data.param_names, data=sdf)
sdf.columns = ['Coef.', 'Std.Err.', 'z', 'P>|z|',
'[' + str(alpha/2), str(1-alpha/2) + ']']
for col in sdf.columns:
sdf[col] = [float_fmt % x if np.isfinite(x) else ""
for x in sdf[col]]
smry.add_df(sdf, align='r')
return smry
[docs] def profile_re(self, re_ix, num_low=5, dist_low=1., num_high=5,
dist_high=1.):
"""
Calculate a series of values along a 1-dimensional profile
likelihood.
Parameters
----------
re_ix : integer
The index of the variance parameter for which to construct
a profile likelihood.
num_low : integer
The number of points at which to calculate the likelihood
below the MLE of the parameter of interest.
dist_low : float
The distance below the MLE of the parameter of interest to
begin calculating points on the profile likelihood.
num_high : integer
The number of points at which to calculate the likelihood
abov the MLE of the parameter of interest.
dist_high : float
The distance above the MLE of the parameter of interest to
begin calculating points on the profile likelihood.
Returns
-------
An array with two columns. The first column contains the
values to which the parameter of interest is constrained. The
second column contains the corresponding likelihood values.
Notes
-----
Only variance parameters can be profiled. `re_ix` is the index
of the random effect that is profiled.
"""
pmodel = self.model
k_fe = pmodel.exog.shape[1]
k_re = pmodel.exog_re.shape[1]
endog, exog, groups = pmodel.endog, pmodel.exog, pmodel.groups
# Need to permute the columns of the random effects design
# matrix so that the profiled variable is in the first column.
ix = np.arange(k_re)
ix[0] = re_ix
ix[re_ix] = 0
exog_re = pmodel.exog_re.copy()[:, ix]
# Permute the covariance structure to match the permuted
# design matrix.
params = self.params_object.copy()
cov_re_unscaled = params.get_cov_re()
cov_re_unscaled = cov_re_unscaled[np.ix_(ix, ix)]
params.set_cov_re(cov_re_unscaled)
# Convert dist_low and dist_high to the profile
# parameterization
cov_re = self.scale * cov_re_unscaled
low = (cov_re[0, 0] - dist_low) / self.scale
high = (cov_re[0, 0] + dist_high) / self.scale
# Define the sequence of values to which the parameter of
# interest will be constrained.
ru0 = cov_re_unscaled[0, 0]
if low <= 0:
raise ValueError("dist_low is too large and would result in a "
"negative variance. Try a smaller value.")
left = np.linspace(low, ru0, num_low + 1)
right = np.linspace(ru0, high, num_high+1)[1:]
rvalues = np.concatenate((left, right))
# Indicators of which parameters are free and fixed.
free = MixedLMParams(k_fe, k_re, self.use_sqrt)
if self.freepat is None:
free.set_fe_params(np.ones(k_fe))
mat = np.ones((k_re, k_re))
else:
free.set_fe_params(self.freepat.get_fe_params())
mat = self.freepat.get_cov_re()
mat = mat[np.ix_(ix, ix)]
mat[0, 0] = 0
if self.use_sqrt:
free.set_cov_re(cov_re_sqrt=mat)
else:
free.set_cov_re(cov_re=mat)
klass = self.model.__class__
init_kwargs = pmodel._get_init_kwds()
init_kwargs['exog_re'] = exog_re
likev = []
for x in rvalues:
model = klass(endog, exog, **init_kwargs)
cov_re = params.get_cov_re()
cov_re[0, 0] = x
# Shrink the covariance parameters until a PSD covariance
# matrix is obtained.
dg = np.diag(cov_re)
success = False
for ks in range(50):
try:
np.linalg.cholesky(cov_re)
success = True
break
except np.linalg.LinAlgError:
cov_re /= 2
np.fill_diagonal(cov_re, dg)
if not success:
raise ValueError("unable to find PSD covariance matrix along likelihood profile")
params.set_cov_re(cov_re)
# TODO should use fit_kwargs
rslt = model.fit(start_params=params, free=free,
reml=self.reml, cov_pen=self.cov_pen)._results
likev.append([rslt.cov_re[0, 0], rslt.llf])
likev = np.asarray(likev)
return likev
class MixedLMResultsWrapper(base.LikelihoodResultsWrapper):
_attrs = {'bse_re': ('generic_columns', 'exog_re_names_full'),
'fe_params': ('generic_columns', 'xnames'),
'bse_fe': ('generic_columns', 'xnames'),
'cov_re': ('generic_columns_2d', 'exog_re_names'),
'cov_re_unscaled': ('generic_columns_2d', 'exog_re_names'),
}
_upstream_attrs = base.LikelihoodResultsWrapper._wrap_attrs
_wrap_attrs = base.wrap.union_dicts(_attrs, _upstream_attrs)
_methods = {}
_upstream_methods = base.LikelihoodResultsWrapper._wrap_methods
_wrap_methods = base.wrap.union_dicts(_methods, _upstream_methods)
