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Linear Mixed Effects Models
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<h1 id="linear-mixed-effects-models"><span id="mixedlmmod"></span>Linear Mixed Effects Models<a class="headerlink" href="#linear-mixed-effects-models" title="Link to this heading">¶</a></h1>
<p>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</p>
<ul class="simple">
<li><p><em>Random intercepts models</em>, where all responses in a group are
additively shifted by a value that is specific to the group.</p></li>
<li><p><em>Random slopes models</em>, 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.</p></li>
<li><p><em>Variance components models</em>, 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.</p></li>
</ul>
<p>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.</p>
<p>A simple example of random coefficients, as in (i) above, is:</p>
<div class="math notranslate nohighlight">
\[Y_{ij} = \beta_0 + \beta_1X_{ij} + \gamma_{0i} + \gamma_{1i}X_{ij} + \epsilon_{ij}\]</div>
<p>Here, <span class="math notranslate nohighlight">\(Y_{ij}\)</span> is the <span class="math notranslate nohighlight">\(j^\rm{th}\)</span> measured response for subject
<span class="math notranslate nohighlight">\(i\)</span>, and <span class="math notranslate nohighlight">\(X_{ij}\)</span> is a covariate for this response. The
“fixed effects parameters” <span class="math notranslate nohighlight">\(\beta_0\)</span> and <span class="math notranslate nohighlight">\(\beta_1\)</span> are
shared by all subjects, and the errors <span class="math notranslate nohighlight">\(\epsilon_{ij}\)</span> are
independent of everything else, and identically distributed (with mean
zero). The “random effects parameters” <span class="math notranslate nohighlight">\(\gamma_{0i}\)</span> and
<span class="math notranslate nohighlight">\(\gamma_{1i}\)</span> follow a bivariate distribution with mean zero,
described by three parameters: <span class="math notranslate nohighlight">\({\rm var}(\gamma_{0i})\)</span>,
<span class="math notranslate nohighlight">\({\rm var}(\gamma_{1i})\)</span>, and <span class="math notranslate nohighlight">\({\rm cov}(\gamma_{0i},
\gamma_{1i})\)</span>. There is also a parameter for <span class="math notranslate nohighlight">\({\rm
var}(\epsilon_{ij})\)</span>.</p>
<p>A simple example of variance components, as in (ii) above, is:</p>
<div class="math notranslate nohighlight">
\[Y_{ijk} = \beta_0 + \eta_{1i} + \eta_{2j} + \epsilon_{ijk}\]</div>
<p>Here, <span class="math notranslate nohighlight">\(Y_{ijk}\)</span> is the <span class="math notranslate nohighlight">\(k^\rm{th}\)</span> measured response under
conditions <span class="math notranslate nohighlight">\(i, j\)</span>. The only “mean structure parameter” is
<span class="math notranslate nohighlight">\(\beta_0\)</span>. The <span class="math notranslate nohighlight">\(\eta_{1i}\)</span> are independent and
identically distributed with zero mean, and variance <span class="math notranslate nohighlight">\(\tau_1^2\)</span>,
and the <span class="math notranslate nohighlight">\(\eta_{2j}\)</span> are independent and identically distributed
with zero mean, and variance <span class="math notranslate nohighlight">\(\tau_2^2\)</span>.</p>
<p>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.</p>
<p>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.</p>
<h2 id="examples">Examples<a class="headerlink" href="#examples" title="Link to this heading">¶</a></h2>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [1]: </span><span class="kn">import</span><span class="w"> </span><span class="nn">statsmodels.api</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">sm</span>
<span class="gp">In [2]: </span><span class="kn">import</span><span class="w"> </span><span class="nn">statsmodels.formula.api</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">smf</span>
<span class="gp">In [3]: </span><span class="n">data</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">datasets</span><span class="o">.</span><span class="n">get_rdataset</span><span class="p">(</span><span class="s2">"dietox"</span><span class="p">,</span> <span class="s2">"geepack"</span><span class="p">)</span><span class="o">.</span><span class="n">data</span>
<span class="gp">In [4]: </span><span class="n">md</span> <span class="o">=</span> <span class="n">smf</span><span class="o">.</span><span class="n">mixedlm</span><span class="p">(</span><span class="s2">"Weight ~ Time"</span><span class="p">,</span> <span class="n">data</span><span class="p">,</span> <span class="n">groups</span><span class="o">=</span><span class="n">data</span><span class="p">[</span><span class="s2">"Pig"</span><span class="p">])</span>
<span class="gp">In [5]: </span><span class="n">mdf</span> <span class="o">=</span> <span class="n">md</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="gp">In [6]: </span><span class="nb">print</span><span class="p">(</span><span class="n">mdf</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
<span class="go"> Mixed Linear Model Regression Results</span>
<span class="go">========================================================</span>
<span class="go">Model: MixedLM Dependent Variable: Weight </span>
<span class="go">No. Observations: 861 Method: REML </span>
<span class="go">No. Groups: 72 Scale: 11.3669 </span>
<span class="go">Min. group size: 11 Log-Likelihood: -2404.7753</span>
<span class="go">Max. group size: 12 Converged: Yes </span>
<span class="go">Mean group size: 12.0 </span>
<span class="gt">--------------------------------------------------------</span>
<span class="n">Coef</span><span class="o">.</span> <span class="n">Std</span><span class="o">.</span><span class="n">Err</span><span class="o">.</span> <span class="n">z</span> <span class="n">P</span><span class="o">>|</span><span class="n">z</span><span class="o">|</span> <span class="p">[</span><span class="mf">0.025</span> <span class="mf">0.975</span><span class="p">]</span>
<span class="gt">--------------------------------------------------------</span>
<span class="n">Intercept</span> <span class="mf">15.724</span> <span class="mf">0.788</span> <span class="mf">19.952</span> <span class="mf">0.000</span> <span class="mf">14.179</span> <span class="mf">17.268</span>
<span class="n">Time</span> <span class="mf">6.943</span> <span class="mf">0.033</span> <span class="mf">207.939</span> <span class="mf">0.000</span> <span class="mf">6.877</span> <span class="mf">7.008</span>
<span class="n">Group</span> <span class="n">Var</span> <span class="mf">40.394</span> <span class="mf">2.149</span>
<span class="o">========================================================</span>
</code></pre></div>
</div>
<p>Detailed examples can be found here</p>
<ul class="simple">
<li><p><a class="reference internal" href="examples/notebooks/generated/mixed_lm_example.html"><span class="doc">Mixed LM</span></a></p></li>
</ul>
<p>There are some notebook examples on the Wiki:
<a class="reference external" href="https://github.com/statsmodels/statsmodels/wiki/Examples#linear-mixed-models">Wiki notebooks for MixedLM</a></p>
<h2 id="technical-documentation">Technical Documentation<a class="headerlink" href="#technical-documentation" title="Link to this heading">¶</a></h2>
<p>The data are partitioned into disjoint groups.
The probability model for group <span class="math notranslate nohighlight">\(i\)</span> is:</p>
<div class="math notranslate nohighlight">
\[Y = X\beta + Z\gamma + Q_1\eta_1 + \cdots + Q_k\eta_k + \epsilon\]</div>