docs/source/mixed_linear.rst
.. currentmodule:: statsmodels.regression.mixed_linear_model
.. _mixedlmmod:
Linear Mixed Effects models are used for regression analyses involving dependent data. Such data arise when working with longitudinal and other study designs in which multiple observations are made on each subject. Some specific linear mixed effects models are
Random intercepts models, where all responses in a group are additively shifted by a value that is specific to the group.
Random slopes models, where the responses in a group follow a (conditional) mean trajectory that is linear in the observed covariates, with the slopes (and possibly intercepts) varying by group.
Variance components models, where the levels of one or more categorical covariates are associated with draws from distributions. These random terms additively determine the conditional mean of each observation based on its covariate values.
The statsmodels implementation of LME is primarily group-based, meaning that random effects must be independently-realized for responses in different groups. There are two types of random effects in our implementation of mixed models: (i) random coefficients (possibly vectors) that have an unknown covariance matrix, and (ii) random coefficients that are independent draws from a common univariate distribution. For both (i) and (ii), the random effects influence the conditional mean of a group through their matrix/vector product with a group-specific design matrix.
A simple example of random coefficients, as in (i) above, is:
.. math::
Y_{ij} = \beta_0 + \beta_1X_{ij} + \gamma_{0i} + \gamma_{1i}X_{ij} + \epsilon_{ij}
Here, :math:Y_{ij} is the :math:j^\rm{th} measured response for subject
:math:i, and :math:X_{ij} is a covariate for this response. The
"fixed effects parameters" :math:\beta_0 and :math:\beta_1 are
shared by all subjects, and the errors :math:\epsilon_{ij} are
independent of everything else, and identically distributed (with mean
zero). The "random effects parameters" :math:\gamma_{0i} and
:math:\gamma_{1i} follow a bivariate distribution with mean zero,
described by three parameters: :math:{\rm var}(\gamma_{0i}),
:math:{\rm var}(\gamma_{1i}), and :math:{\rm cov}(\gamma_{0i}, \gamma_{1i}). There is also a parameter for :math:{\rm var}(\epsilon_{ij}).
A simple example of variance components, as in (ii) above, is:
.. math::
Y_{ijk} = \beta_0 + \eta_{1i} + \eta_{2j} + \epsilon_{ijk}
Here, :math:Y_{ijk} is the :math:k^\rm{th} measured response under
conditions :math:i, j. The only "mean structure parameter" is
:math:\beta_0. The :math:\eta_{1i} are independent and
identically distributed with zero mean, and variance :math:\tau_1^2,
and the :math:\eta_{2j} are independent and identically distributed
with zero mean, and variance :math:\tau_2^2.
statsmodels MixedLM handles most non-crossed random effects models, and some crossed models. To include crossed random effects in a model, it is necessary to treat the entire dataset as a single group. The variance components arguments to the model can then be used to define models with various combinations of crossed and non-crossed random effects.
The statsmodels LME framework currently supports post-estimation inference via Wald tests and confidence intervals on the coefficients, profile likelihood analysis, likelihood ratio testing, and AIC.
.. ipython:: python
import statsmodels.api as sm import statsmodels.formula.api as smf
data = sm.datasets.get_rdataset("dietox", "geepack").data
md = smf.mixedlm("Weight ~ Time", data, groups=data["Pig"]) mdf = md.fit() print(mdf.summary())
Detailed examples can be found here
Mixed LM <examples/notebooks/generated/mixed_lm_example.ipynb>_There are some notebook examples on the Wiki:
Wiki notebooks for MixedLM <https://github.com/statsmodels/statsmodels/wiki/Examples#linear-mixed-models>_
The data are partitioned into disjoint groups.
The probability model for group :math:i is:
.. math::
Y = X\beta + Z\gamma + Q_1\eta_1 + \cdots + Q_k\eta_k + \epsilon
where
n_i is the number of observations in group :math:iY is a :math:n_i dimensional response vectorX is a :math:n_i * k_{fe} dimensional matrix of fixed effects
coefficients\beta is a :math:k_{fe}-dimensional vector of fixed effects slopesZ is a :math:n_i * k_{re} dimensional matrix of random effects
coefficients\gamma is a :math:k_{re}-dimensional random vector with mean 0
and covariance matrix :math:\Psi; note that each group
gets its own independent realization of gamma.Q_j is a :math:n_i \times q_j dimensional design matrix for the
:math:j^\rm{th} variance component.\eta_j is a :math:q_j-dimensional random vector containing independent
and identically distributed values with variance :math:\tau_j^2.\epsilon is a :math:n_i dimensional vector of i.i.d normal
errors with mean 0 and variance :math:\sigma^2; the :math:\epsilon
values are independent both within and between groups:math:Y, X, \{Q_j\} and :math:Z must be entirely observed. :math:\beta,
:math:\Psi, and :math:\sigma^2 are estimated using ML or REML estimation,
and :math:\gamma, :math:\{\eta_j\} and :math:\epsilon are
random so define the probability model.
The marginal mean structure is :math:E[Y|X,Z] = X*\beta. If only
the marginal mean structure is of interest, GEE is a good alternative
to mixed models.
Notation:
cov_{re} is the random effects covariance matrix (referred
to above as :math:\Psi) and :math:scale is the (scalar) error
variance. There is also a single estimated variance parameter
:math:\tau_j^2 for each variance component. For a single group,
the marginal covariance matrix of endog given exog is
:math:scale*I + Z * cov_{re} * Z, where :math:Z is the design
matrix for the random effects in one group.References ^^^^^^^^^^
The primary reference for the implementation details is:
See also this more recent document:
All the likelihood, gradient, and Hessian calculations closely follow Lindstrom and Bates.
The following two documents are written more from the perspective of users:
.. Class hierarchy: TODO
General references for this class of models are
.. module:: statsmodels.regression.mixed_linear_model :synopsis: Mixed Linear Models
The model class is:
.. autosummary:: :toctree: generated/
MixedLM
The result class is:
.. autosummary:: :toctree: generated/
MixedLMResults