Maximum Likelihood Estimation (Generic models)
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.. _generic_mle_notebook:
`Link to Notebook GitHub <https://github.com/statsmodels/statsmodels/blob/master/examples/notebooks/generic_mle.ipynb>`_
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<p>This tutorial explains how to quickly implement new maximum likelihood models in <code>statsmodels</code>. We give two examples:</p>
<ol>
<li>Probit model for binary dependent variables</li>
<li>Negative binomial model for count data</li>
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<p>The <code>GenericLikelihoodModel</code> class eases the process by providing tools such as automatic numeric differentiation and a unified interface to <code>scipy</code> optimization functions. Using <code>statsmodels</code>, users can fit new MLE models simply by "plugging-in" a log-likelihood function.</p>
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<h2 id="Example-1:-Probit-model">Example 1: Probit model<a class="anchor-link" href="#Example-1:-Probit-model">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="kn">from</span> <span class="nn">__future__</span> <span class="k">import</span> <span class="n">print_function</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">scipy</span> <span class="k">import</span> <span class="n">stats</span>
<span class="kn">import</span> <span class="nn">statsmodels.api</span> <span class="k">as</span> <span class="nn">sm</span>
<span class="kn">from</span> <span class="nn">statsmodels.base.model</span> <span class="k">import</span> <span class="n">GenericLikelihoodModel</span>
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<p>The <code>Spector</code> dataset is distributed with <code>statsmodels</code>. You can access a vector of values for the dependent variable (<code>endog</code>) and a matrix of regressors (<code>exog</code>) like this:</p>
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<div class=" highlight hl-ipython3"><pre><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">spector</span><span class="o">.</span><span class="n">load_pandas</span><span class="p">()</span>
<span class="n">exog</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">exog</span>
<span class="n">endog</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">endog</span>
<span class="nb">print</span><span class="p">(</span><span class="n">sm</span><span class="o">.</span><span class="n">datasets</span><span class="o">.</span><span class="n">spector</span><span class="o">.</span><span class="n">NOTE</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">exog</span><span class="o">.</span><span class="n">head</span><span class="p">())</span>
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<p>Them, we add a constant to the matrix of regressors:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">exog</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">add_constant</span><span class="p">(</span><span class="n">exog</span><span class="p">,</span> <span class="n">prepend</span><span class="o">=</span><span class="k">True</span><span class="p">)</span>
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<pre>::
Number of Observations - 32
Number of Variables - 4
Variable name definitions::
Grade - binary variable indicating whether or not a student's grade
improved. 1 indicates an improvement.
TUCE - Test score on economics test
PSI - participation in program
GPA - Student's grade point average
GPA TUCE PSI
0 2.66 20 0
1 2.89 22 0
2 3.28 24 0
3 2.92 12 0
4 4.00 21 0
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<p>To create your own Likelihood Model, you simply need to overwrite the loglike method.</p>
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<div class=" highlight hl-ipython3"><pre><span class="k">class</span> <span class="nc">MyProbit</span><span class="p">(</span><span class="n">GenericLikelihoodModel</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">loglike</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">params</span><span class="p">):</span>
<span class="n">exog</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exog</span>
<span class="n">endog</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">endog</span>
<span class="n">q</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">endog</span> <span class="o">-</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">logcdf</span><span class="p">(</span><span class="n">q</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">exog</span><span class="p">,</span> <span class="n">params</span><span class="p">))</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
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<p>Estimate the model and print a summary:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">sm_probit_manual</span> <span class="o">=</span> <span class="n">MyProbit</span><span class="p">(</span><span class="n">endog</span><span class="p">,</span> <span class="n">exog</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="n">sm_probit_manual</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
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<p>Compare your Probit implementation to <code>statsmodels</code>' "canned" implementation:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">sm_probit_canned</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">Probit</span><span class="p">(</span><span class="n">endog</span><span class="p">,</span> <span class="n">exog</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
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<pre>Optimization terminated successfully.
Current function value: 0.400588
Iterations: 292
Function evaluations: 494
MyProbit Results
==============================================================================
Dep. Variable: GRADE Log-Likelihood: -12.819
Model: MyProbit AIC: 33.64
Method: Maximum Likelihood BIC: 39.50
Date: Mon, 20 Jul 2015
Time: 17:43:29
No. Observations: 32
Df Residuals: 28
Df Model: 3
==============================================================================
coef std err z P>|z| [95.0% Conf. Int.]
------------------------------------------------------------------------------
const -7.4523 2.542 -2.931 0.003 -12.435 -2.469
GPA 1.6258 0.694 2.343 0.019 0.266 2.986
TUCE 0.0517 0.084 0.617 0.537 -0.113 0.216
PSI 1.4263 0.595 2.397 0.017 0.260 2.593
==============================================================================
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<div class=" highlight hl-ipython3"><pre><span class="nb">print</span><span class="p">(</span><span class="n">sm_probit_canned</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">sm_probit_manual</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
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<pre>Optimization terminated successfully.
Current function value: 0.400588
Iterations 6
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<div class=" highlight hl-ipython3"><pre><span class="nb">print</span><span class="p">(</span><span class="n">sm_probit_canned</span><span class="o">.</span><span class="n">cov_params</span><span class="p">())</span>
<span class="nb">print</span><span class="p">(</span><span class="n">sm_probit_manual</span><span class="o">.</span><span class="n">cov_params</span><span class="p">())</span>
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<pre>const -7.452320
GPA 1.625810
TUCE 0.051729
PSI 1.426332
dtype: float64
[-7.45233176 1.62580888 0.05172971 1.42631954]
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<p>Notice that the <code>GenericMaximumLikelihood</code> class provides automatic differentiation, so we didn't have to provide Hessian or Score functions in order to calculate the covariance estimates.</p>
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<h2 id="Example-2:-Negative-Binomial-Regression-for-Count-Data">Example 2: Negative Binomial Regression for Count Data<a class="anchor-link" href="#Example-2:-Negative-Binomial-Regression-for-Count-Data">¶</a></h2><p>Consider a negative binomial regression model for count data with
log-likelihood (type NB-2) function expressed as:</p>
$$
\mathcal{L}(\beta_j; y, \alpha) = \sum_{i=1}^n y_i ln
\left ( \frac{\alpha exp(X_i'\beta)}{1+\alpha exp(X_i'\beta)} \right ) -
\frac{1}{\alpha} ln(1+\alpha exp(X_i'\beta)) + ln \Gamma (y_i + 1/\alpha) - ln \Gamma (y_i+1) - ln \Gamma (1/\alpha)
$$<p>with a matrix of regressors $X$, a vector of coefficients $\beta$,
and the negative binomial heterogeneity parameter $\alpha$.</p>
<p>Using the <code>nbinom</code> distribution from <code>scipy</code>, we can write this likelihood
simply as:</p>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="k">import</span> <span class="n">nbinom</span>
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<pre> const GPA TUCE PSI
const 6.464166 -1.169668 -0.101173 -0.594792
GPA -1.169668 0.481473 -0.018914 0.105439
TUCE -0.101173 -0.018914 0.007038 0.002472
PSI -0.594792 0.105439 0.002472 0.354070
[[ 6.46416770e+00 -1.16966617e+00 -1.01173180e-01 -5.94788993e-01]
[ -1.16966617e+00 4.81472116e-01 -1.89134586e-02 1.05438227e-01]
[ -1.01173180e-01 -1.89134586e-02 7.03758392e-03 2.47189191e-03]
[ -5.94788993e-01 1.05438227e-01 2.47189191e-03 3.54069512e-01]]
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<div class=" highlight hl-ipython3"><pre><span class="k">def</span> <span class="nf">_ll_nb2</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">alph</span><span class="p">):</span>
<span class="n">mu</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">beta</span><span class="p">))</span>
<span class="n">size</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">alph</span>
<span class="n">prob</span> <span class="o">=</span> <span class="n">size</span><span class="o">/</span><span class="p">(</span><span class="n">size</span><span class="o">+</span><span class="n">mu</span><span class="p">)</span>
<span class="n">ll</span> <span class="o">=</span> <span class="n">nbinom</span><span class="o">.</span><span class="n">logpmf</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">size</span><span class="p">,</span> <span class="n">prob</span><span class="p">)</span>
<span class="k">return</span> <span class="n">ll</span>
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<h3 id="New-Model-Class">New Model Class<a class="anchor-link" href="#New-Model-Class">¶</a></h3><p>We create a new model class which inherits from <code>GenericLikelihoodModel</code>:</p>
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<div class=" highlight hl-ipython3"><pre><span class="kn">from</span> <span class="nn">statsmodels.base.model</span> <span class="k">import</span> <span class="n">GenericLikelihoodModel</span>
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<div class=" highlight hl-ipython3"><pre><span class="k">class</span> <span class="nc">NBin</span><span class="p">(</span><span class="n">GenericLikelihoodModel</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">endog</span><span class="p">,</span> <span class="n">exog</span><span class="p">,</span> <span class="o">**</span><span class="n">kwds</span><span class="p">):</span>
<span class="nb">super</span><span class="p">(</span><span class="n">NBin</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="n">endog</span><span class="p">,</span> <span class="n">exog</span><span class="p">,</span> <span class="o">**</span><span class="n">kwds</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">nloglikeobs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">params</span><span class="p">):</span>
<span class="n">alph</span> <span class="o">=</span> <span class="n">params</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">beta</span> <span class="o">=</span> <span class="n">params</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">ll</span> <span class="o">=</span> <span class="n">_ll_nb2</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">endog</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">exog</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">alph</span><span class="p">)</span>
<span class="k">return</span> <span class="o">-</span><span class="n">ll</span>
<span class="k">def</span> <span class="nf">fit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">start_params</span><span class="o">=</span><span class="k">None</span><span class="p">,</span> <span class="n">maxiter</span><span class="o">=</span><span class="mi">10000</span><span class="p">,</span> <span class="n">maxfun</span><span class="o">=</span><span class="mi">5000</span><span class="p">,</span> <span class="o">**</span><span class="n">kwds</span><span class="p">):</span>
<span class="c"># we have one additional parameter and we need to add it for summary</span>
<span class="bp">self</span><span class="o">.</span><span class="n">exog_names</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">'alpha'</span><span class="p">)</span>
<span class="k">if</span> <span class="n">start_params</span> <span class="o">==</span> <span class="k">None</span><span class="p">:</span>
<span class="c"># Reasonable starting values</span>
<span class="n">start_params</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exog</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="o">.</span><span class="mi">5</span><span class="p">)</span>
<span class="c"># intercept</span>
<span class="n">start_params</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">endog</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span>
<span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">NBin</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">start_params</span><span class="o">=</span><span class="n">start_params</span><span class="p">,</span>
<span class="n">maxiter</span><span class="o">=</span><span class="n">maxiter</span><span class="p">,</span> <span class="n">maxfun</span><span class="o">=</span><span class="n">maxfun</span><span class="p">,</span>
<span class="o">**</span><span class="n">kwds</span><span class="p">)</span>
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<p>Two important things to notice:</p>
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<li><code>nloglikeobs</code>: This function should return one evaluation of the negative log-likelihood function per observation in your dataset (i.e. rows of the endog/X matrix). </li>
<li><code>start_params</code>: A one-dimensional array of starting values needs to be provided. The size of this array determines the number of parameters that will be used in optimization.</li>
</ul>
<p>That's it! You're done!</p>
<h3 id="Usage-Example">Usage Example<a class="anchor-link" href="#Usage-Example">¶</a></h3><p>The <a href="http://vincentarelbundock.github.com/Rdatasets/doc/COUNT/medpar.html">Medpar</a>
dataset is hosted in CSV format at the <a href="http://vincentarelbundock.github.com/Rdatasets">Rdatasets repository</a>. We use the <code>read_csv</code>
function from the <a href="http://pandas.pydata.org">Pandas library</a> to load the data
in memory. We then print the first few columns:</p>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">statsmodels.api</span> <span class="k">as</span> <span class="nn">sm</span>
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<div class=" highlight hl-ipython3"><pre><span class="n">medpar</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="s">"medpar"</span><span class="p">,</span> <span class="s">"COUNT"</span><span class="p">,</span> <span class="n">cache</span><span class="o">=</span><span class="k">True</span><span class="p">)</span><span class="o">.</span><span class="n">data</span>
<span class="n">medpar</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
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<p>The model we are interested in has a vector of non-negative integers as
dependent variable (<code>los</code>), and 5 regressors: <code>Intercept</code>, <code>type2</code>,
<code>type3</code>, <code>hmo</code>, <code>white</code>.</p>
<p>For estimation, we need to create two variables to hold our regressors and the outcome variable. These can be ndarrays or pandas objects.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">y</span> <span class="o">=</span> <span class="n">medpar</span><span class="o">.</span><span class="n">los</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">medpar</span><span class="p">[[</span><span class="s">"type2"</span><span class="p">,</span> <span class="s">"type3"</span><span class="p">,</span> <span class="s">"hmo"</span><span class="p">,</span> <span class="s">"white"</span><span class="p">]]</span>
<span class="n">X</span><span class="p">[</span><span class="s">"constant"</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
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<p>Then, we fit the model and extract some information:</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">mod</span> <span class="o">=</span> <span class="n">NBin</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
<span class="n">res</span> <span class="o">=</span> <span class="n">mod</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
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<pre>/Users/tom.augspurger/Envs/py3/lib/python3.4/site-packages/IPython/kernel/__main__.py:3: SettingWithCopyWarning:
A value is trying to be set on a copy of a slice from a DataFrame.
Try using .loc[row_indexer,col_indexer] = value instead
See the the caveats in the documentation: http://pandas.pydata.org/pandas-docs/stable/indexing.html#indexing-view-versus-copy
app.launch_new_instance()
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<p>Extract parameter estimates, standard errors, p-values, AIC, etc.:</p>
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<div class=" highlight hl-ipython3"><pre><span class="nb">print</span><span class="p">(</span><span class="s">'Parameters: '</span><span class="p">,</span> <span class="n">res</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s">'Standard errors: '</span><span class="p">,</span> <span class="n">res</span><span class="o">.</span><span class="n">bse</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s">'P-values: '</span><span class="p">,</span> <span class="n">res</span><span class="o">.</span><span class="n">pvalues</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s">'AIC: '</span><span class="p">,</span> <span class="n">res</span><span class="o">.</span><span class="n">aic</span><span class="p">)</span>
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<pre>Optimization terminated successfully.
Current function value: 3.209014
Iterations: 805
Function evaluations: 1238
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<p>As usual, you can obtain a full list of available information by typing
<code>dir(res)</code>.
We can also look at the summary of the estimation results.</p>
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<div class=" highlight hl-ipython3"><pre><span class="nb">print</span><span class="p">(</span><span class="n">res</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
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<pre>Parameters: [ 0.2212642 0.70613942 -0.06798155 -0.12903932 2.31026565 0.44575147]
Standard errors: [ 0.05059259 0.07613047 0.05326097 0.06854141 0.06794692 0.01981542]
P-values: [ 1.22297920e-005 1.76978024e-020 2.01819161e-001 5.97481600e-002
2.15079626e-253 4.62688439e-112]
AIC: 9604.95320583
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<h3 id="Testing">Testing<a class="anchor-link" href="#Testing">¶</a></h3>
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<p>We can check the results by using the statsmodels implementation of the Negative Binomial model, which uses the analytic score function and Hessian.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">res_nbin</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">NegativeBinomial</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">disp</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">res_nbin</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
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<pre> NBin Results
==============================================================================
Dep. Variable: los Log-Likelihood: -4797.5
Model: NBin AIC: 9605.
Method: Maximum Likelihood BIC: 9632.
Date: Mon, 20 Jul 2015
Time: 17:43:31
No. Observations: 1495
Df Residuals: 1490
Df Model: 4
==============================================================================
coef std err z P>|z| [95.0% Conf. Int.]
------------------------------------------------------------------------------
type2 0.2213 0.051 4.373 0.000 0.122 0.320
type3 0.7061 0.076 9.275 0.000 0.557 0.855
hmo -0.0680 0.053 -1.276 0.202 -0.172 0.036
white -0.1290 0.069 -1.883 0.060 -0.263 0.005
constant 2.3103 0.068 34.001 0.000 2.177 2.443
alpha 0.4458 0.020 22.495 0.000 0.407 0.485
==============================================================================
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<div class=" highlight hl-ipython3"><pre><span class="nb">print</span><span class="p">(</span><span class="n">res_nbin</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
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<pre> NegativeBinomial Regression Results
==============================================================================
Dep. Variable: los No. Observations: 1495
Model: NegativeBinomial Df Residuals: 1490
Method: MLE Df Model: 4
Date: Mon, 20 Jul 2015 Pseudo R-squ.: 0.01215
Time: 17:43:31 Log-Likelihood: -4797.5
converged: True LL-Null: -4856.5
LLR p-value: 1.404e-24
==============================================================================
coef std err z P>|z| [95.0% Conf. Int.]
------------------------------------------------------------------------------
type2 0.2212 0.051 4.373 0.000 0.122 0.320
type3 0.7062 0.076 9.276 0.000 0.557 0.855
hmo -0.0680 0.053 -1.277 0.202 -0.172 0.036
white -0.1291 0.069 -1.883 0.060 -0.263 0.005
constant 2.3103 0.068 34.001 0.000 2.177 2.443
alpha 0.4458 0.020 22.495 0.000 0.407 0.485
==============================================================================
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<pre>type2 0.221231
type3 0.706175
hmo -0.067990
white -0.129065
constant 2.310288
alpha 0.445758
dtype: float64
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<p>Or we could compare them to results obtained using the MASS implementation for R:</p>
<pre><code>url = 'http://vincentarelbundock.github.com/Rdatasets/csv/COUNT/medpar.csv'
medpar = read.csv(url)
f = los~factor(type)+hmo+white
library(MASS)
mod = glm.nb(f, medpar)
coef(summary(mod))
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.31027893 0.06744676 34.253370 3.885556e-257
factor(type)2 0.22124898 0.05045746 4.384861 1.160597e-05
factor(type)3 0.70615882 0.07599849 9.291748 1.517751e-20
hmo -0.06795522 0.05321375 -1.277024 2.015939e-01
white -0.12906544 0.06836272 -1.887951 5.903257e-02
</code></pre>
<h3 id="Numerical-precision">Numerical precision<a class="anchor-link" href="#Numerical-precision">¶</a></h3><p>The <code>statsmodels</code> generic MLE and <code>R</code> parameter estimates agree up to the fourth decimal. The standard errors, however, agree only up to the second decimal. This discrepancy is the result of imprecision in our Hessian numerical estimates. In the current context, the difference between <code>MASS</code> and <code>statsmodels</code> standard error estimates is substantively irrelevant, but it highlights the fact that users who need very precise estimates may not always want to rely on default settings when using numerical derivatives. In such cases, it is better to use analytical derivatives with the <code>LikelihoodModel</code> class.</p>
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