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 <div class="header">
   <div class="headertitle">
 <div class="title">Marginal Effects<div class="ingroups"><a class="el" href="group__grp__glm.html">Generalized Linear Models</a></div></div>  </div>
 </div><!--header-->
 <div class="contents">
 <dl class="section user"><dt>About</dt><dd></dd></dl>
 <p>A marginal effect (ME) or partial effect measures the effect on the conditional mean of \( y \) of a change in one of the regressors, say \(X_k\). In the linear regression model, the ME equals the relevant slope coefficient, greatly simplifying analysis. For nonlinear models, we require specialized algorithms for calculating ME.</p>
 <p>The standard approach to modeling dichotomous/binary variables (so \(y \in \{0, 1\} \)) is to estimate a generalized linear model under the assumption that \( y \) follows some form of Bernoulli distribution. Thus the expected value of \( y \) becomes, </p>
 <p class="formulaDsp">
 \[ y = G(X&#39; \beta), \]
 </p>
 <p>where G is the specified binomial distribution. For logistic regression, the funciton \( G \) represents the inverse logit function.</p>
 <p>In logistic regression: </p>
 <p class="formulaDsp">
 \[ P = \frac{1}{1 + e^{-(\beta_0 + \beta_1 x_1 + \dots \beta_j x_j)}} = \frac{1}{1 + e^{-z}} \implies \frac{\partial P}{\partial X_k} = \beta_k \cdot \frac{1}{1 + e^{-z}} \cdot \frac{e^{-z}}{1 + e^{-z}} \\ = \beta_k \cdot P \cdot (1-P) \]
 </p>
 <p>There are several methods for calculating the marginal effects for dichotomous dependent variables. This package uses the average of the marginal effects at every sample observation.</p>
 <p>This is calculated as follows: </p>
 <p class="formulaDsp">
 \[ \frac{\partial y}{\partial x_k} = \beta_k \frac{\sum_{i=1}^n P(y_i = 1)(1-P(y_i = 1))}{n}, \\ \text{where}, P(y_i=1) = g(X^{(i)}\beta) \]
 </p>
 <p>We use the delta method for calculating standard errors on the marginal effects.</p>
 <dl class="section user"><dt>Input</dt><dd></dd></dl>
 <p>The input parameters are expected to be of the following form: </p>
 <pre> <em>margins_logregr</em> (
     <em>sourceTable</em> VARCHAR,
     <em>outputTable</em> VARCHAR,
     <em>dependentVariable</em> VARCHAR,
     <em>independentVariable</em> VARCHAR,
     [<em>groupingCol</em> VARCHAR,
     <em>marginal_vars</em> INTEGER[],
     <em>max_iter</em> INTEGER,
     <em>optimizer</em> VARCHAR,
     <em>tolerance</em> DOUBLE PRECISION
     ]
 )</pre><p>For multinomial logistic regression the input parameters are expected to be of the following form: </p>
 <pre> <em>margins_mlogregr</em> (
     <em>sourceTable</em> VARCHAR,
     <em>outputTable</em> VARCHAR,
     <em>dependentVariable</em> VARCHAR,
     <em>independentVariable</em> VARCHAR,
     [<em>groupingCol</em> VARCHAR,
     <em>referenceCategory</em> INTEGER,
     <em>marginal_vars</em> INTEGER[],
     <em>max_iter</em> INTEGER,
     <em>optimizer</em> VARCHAR,
     <em>tolerance</em> DOUBLE PRECISION
     ]
 )</pre><dl class="section warning"><dt>Warning</dt><dd>The <b>'groupingCol'</b> input parameter for <em>margins_logregr</em> and <em>margins_mlogregr</em> is a placeholder in the Madlib V1.0. These inputs will be implemented in a future release. </dd></dl>
 <dl class="section user"><dt>Usage</dt><dd></dd></dl>
 <p><b> The Interface</b></p>
 <p>For logistic regression, one can call the following function </p>
 <pre>
 SELECT madlib.<a class="el" href="marginal_8sql__in.html#ad37e0349ac3924daba27c4600311ed71">margins_logregr</a>(
     <em>'source_table'</em>,              -- name of input table, VARCHAR
     <em>'out_table'</em>,                 -- name of output table, VARCHAR
     <em>'dependent_varname'</em>,         -- dependent variable, VARCHAR
     <em>'independent_varname'</em>,       -- independent variable, VARCHAR
     [ <em>'grouping_cols'</em>,           -- comma separated list of grouping vars, VARCHAR (Default NULL)
     <em>'marginal_effect_variables'</em>, -- Index list (base 1) with positions representing which marginal variable to calculate, INTEGER[] (Default NULL)
     <em> max_iter</em>,                   -- Maximum number of iterations to run the logistic regression INTEGER (Default 20)
     <em>'optimizer'</em>,                 -- Optimizer used for logistic regression VARCHAR (Default &amp; recommended 'irls')
     <em>tolerance</em>,                   -- Tolerance for the logistic regression DOUBLE PRECISION (default 1e-4)
     ]
 );
 </pre><p>For multinomial logistic regression, one can call the following function </p>
 <pre>
 SELECT madlib.margins_mlogregr(
     <em>'source_table'</em>,              -- name of input table, VARCHAR
     <em>'out_table'</em>,                 -- name of output table, VARCHAR
     <em>'dependent_varname'</em>,         -- dependent variable, VARCHAR
     <em>'independent_varname'</em>,       -- independent variable, VARCHAR
     [ <em>'reference_category'</em>,      -- Reference category for multinomial logistic regression INTEGER
      <em>'grouping_cols'</em>,           -- comma separated list of grouping vars, VARCHAR (Default NULL)
     <em>'marginal_effect_variables'</em>, -- Index list (base 1) with positions representing which marginal variable to calculate, INTEGER[] (Default NULL)
     <em> max_iter</em>,                   -- Maximum number of iterations to run the logistic regression INTEGER (Default 20)
     <em>'optimizer'</em>,                 -- Optimizer used for logistic regression VARCHAR (Default &amp; recommended 'irls')
     <em>tolerance</em>,                   -- Tolerance for the logistic regression DOUBLE PRECISION (default 1e-4)
     ]
 );
 </pre><p>Output is stored in the <em>out_table</em>: </p>
 <pre>
 [ margins | std_err | t_stats | p_values |
 +------+---------+-------+----------+
 </pre><p><b> Multinomial-Logistic Regression Notes</b></p>
 <ul>
 <li>The reference category ranges from [0, numCategories-1]. The default reference category is zero.</li>
 <li>The marginal effects and supporting statistics are output in the following order. For a problem with K dependent variables (1, ..., K) and J categories (0,..., J-1). The output is \( \{ m_{k_1, j_0} \ldots m_{k_K, j_{0}}, m_{k_1, j_1} \ldots m_{k_K, j_{1}}, m_{k_1, j_{J-1}} \ldots m_{k_K, j_{J-1}} \} \). The order is NOT CONSISTENT with mlogregr, robust_variance_mlogregr and clustered_variance_mlogregr. This is deliberate because the interfaces of all the other functions will be moved to match that used in marginal.</li>
 <li>Selectively picking the marginal effects of the independent variables will return the marginal effects of ALL categories associated with the independent variable.</li>
 <li>Here the <em>'independent_varname'</em> can be the name of a column, which contains array of numeric values. It can also have a format of string 'array[1, x1, x2, x3]', where <em>x1</em>, <em>x2</em> and <em>x3</em> are all column names.</li>
 <li>Here the <em>'vars'</em> is an index list (base 1) with the set of indices amongst the independent variables that must be selected. One can also chose <em>NULL</em> to pick all indepependent variables.</li>
 </ul>
 <dl class="section user"><dt>Examples</dt><dd><ol type="1">
 <li>Create the sample data set: <pre class="fragment">sql&gt; SELECT * FROM data;
  id | second_attack | treatment | trait_anxiety
 ----+---------------+-----------+---------------
   1 |             1 |         1 |            70
   3 |             1 |         1 |            50
   5 |             1 |         0 |            40
   7 |             1 |         0 |            75
   9 |             1 |         0 |            70
  11 |             0 |         1 |            65
  13 |             0 |         1 |            45
  15 |             0 |         1 |            40
  17 |             0 |         0 |            55
 ...
 </pre></li>
 <li>For function summary information. Run <pre class="fragment">sql&gt; select margins_logregr('help');
 OR
 sql&gt; select margins_logregr();
 OR
 sql&gt; select margins_logregr('');
 </pre></li>
 <li>Run the logistic regression function and then compute the marginal effects of all variables in the regression: <pre class="fragment">sql&gt; select margins_logregr('patients', 'result_table', 'second_attack', 'ARRAY[1, treatment, trait_anxiety]');
 sql&gt; select * from result_table;
 margins  | {-0.970665392796,-0.156214190168,0.0181587690137}
 coef     | {-6.36346994178179,-1.02410605239327,0.119044916668605}
 std_err  | {0.802871454422,0.292691682191,0.0137459874022}
 t_stats  | {-1.2089922832,-0.533715850748,1.32102325446}
 p_values | {0.243212810329,0.600447858606,0.204000202116}


 -# Alternate Syntax: Run the logistic regression function and then compute the marginal effects if all variables in the regression:
 sql&gt; select margins_logregr('patients', 'result_table', 'second_attack', 'ARRAY[1, treatment, trait_anxiety]', NULL, NULL);
 sql&gt; select * from result_table;
 margins  | {-0.970665392796,-0.156214190168,0.0181587690137}
 coef     | {-6.36346994178179,-1.02410605239327,0.119044916668605}
 std_err  | {0.802871454422,0.292691682191,0.0137459874022}
 t_stats  | {-1.2089922832,-0.533715850748,1.32102325446}
 p_values | {0.243212810329,0.600447858606,0.204000202116}


 -# Run the logistic regression function and then compute the marginal effects of the first variable in the regression
 sql&gt; select margins_logregr('patients', 'result_table', 'second_attack', 'ARRAY[1, treatment, trait_anxiety]', NULL, ARRAY[1]);
 sql&gt; select * from result_table;
 margins  | {-0.970665392796}
 coef     | {-6.36346994178179}
 std_err  | {0.802871454422}
 t_stats  | {-1.2089922832}
 p_values | {0.243212810329}
 </pre></li>
 </ol>
 </dd></dl>
 <dl class="section user"><dt>Examples</dt><dd><ol type="1">
 <li>Create the sample data set (the full dataset has 3 categories): <pre class="fragment">sql&gt; SELECT * FROM data;
  id |  feature_1    | feature_2 | category
 ----+---------------+-----------+---------------
   1 |             1 |        35 |            1
   3 |             2 |        33 |            0
   5 |             3 |        39 |            1
   7 |             1 |        37 |            1
   9 |             2 |        31 |            1
  11 |             3 |        36 |            1
  13 |             2 |        36 |            1
  15 |             2 |        36 |            0
  17 |             2 |        31 |            5
 ...
 </pre></li>
 <li>For function summary information. Run <pre class="fragment">sql&gt; select margins_mlogregr('help');
 OR
 sql&gt; select margins_mlogregr();
 OR
 sql&gt; select margins_mlogregr('');
 </pre></li>
 <li>Run the regression function and then compute the marginal effects of all variables in the regression (see docs for detailed order) <pre class="fragment">sql&gt; select margins_mlogregr('test_data', 'result_table', 'category', 'ARRAY[1, feature_1, feature_2]');
 sql&gt; select * from result_table;
 margins  | {0.741613239156,-0.032868883552,-0.0144502990691,-0.972055011831,0.112337273885,0.0172621628253}
 std_err  | {0.183172236055,0.044184899499,0.00332608999704,0.263532615748,0.0555196094594,0.00457999429836}
 t_stats  | {4.04872078394,-0.743894043547,-4.34453038911,-3.68855676202,2.02338011702,3.76903587663}
 p_values | {7.43784735554e-05,0.457840607871,2.24855476205e-05,0.000292799037776,0.0444060346517,0.000217384008015}
 </pre></li>
 <li>Run the regression and compute the marginals effects for the first dependent variable (all categories) <pre class="fragment">sql&gt; select margins_mlogregr('test_data', 'result_table', 'category', 'ARRAY[1, feature_1, feature_2]', 0, NULL, ARRAY[1]);
 sql&gt; select * from result_table;
 margins  | {0.741613239156,-0.972055011831}
 std_err  | {0.183172236055,0.263532615748}
 t_stats  | {4.04872078394,-3.68855676202}
 p_values | {7.43784735554e-05,0.000292799037776}
 </pre></li>
 </ol>
 </dd></dl>
 <pre class="fragment">-#  For function usage information.
 sql&gt; select margins_mlogregr('usage');
 </pre><dl class="section user"><dt>Literature</dt><dd></dd></dl>
 <p>[1] mfx function in STATA: <a href="http://www.stata.com/help.cgi?mfx_option">http://www.stata.com/help.cgi?mfx_option</a></p>
 <dl class="section see"><dt>See Also</dt><dd>File <a class="el" href="marginal_8sql__in.html" title="SQL functions for linear regression. ">marginal.sql_in</a> documenting the SQL functions. </dd></dl>
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	<div class="headertitle">
	<div class="title">Marginal Effects<div class="ingroups"><a class="el" href="group__grp__glm.html">Generalized Linear Models</a></div></div> </div>
	</div><!--header-->
	<div class="contents">
	<dl class="section user"><dt>About</dt><dd></dd></dl>
	<p>A marginal effect (ME) or partial effect measures the effect on the conditional mean of \( y \) of a change in one of the regressors, say \(X_k\). In the linear regression model, the ME equals the relevant slope coefficient, greatly simplifying analysis. For nonlinear models, we require specialized algorithms for calculating ME.</p>
	<p>The standard approach to modeling dichotomous/binary variables (so \(y \in \{0, 1\} \)) is to estimate a generalized linear model under the assumption that \( y \) follows some form of Bernoulli distribution. Thus the expected value of \( y \) becomes, </p>
	<p class="formulaDsp">
	\[ y = G(X' \beta), \]
	</p>
	<p>where G is the specified binomial distribution. For logistic regression, the funciton \( G \) represents the inverse logit function.</p>
	<p>In logistic regression: </p>
	<p class="formulaDsp">
	\[ P = \frac{1}{1 + e^{-(\beta_0 + \beta_1 x_1 + \dots \beta_j x_j)}} = \frac{1}{1 + e^{-z}} \implies \frac{\partial P}{\partial X_k} = \beta_k \cdot \frac{1}{1 + e^{-z}} \cdot \frac{e^{-z}}{1 + e^{-z}} \\ = \beta_k \cdot P \cdot (1-P) \]
	</p>
	<p>There are several methods for calculating the marginal effects for dichotomous dependent variables. This package uses the average of the marginal effects at every sample observation.</p>
	<p>This is calculated as follows: </p>
	<p class="formulaDsp">
	\[ \frac{\partial y}{\partial x_k} = \beta_k \frac{\sum_{i=1}^n P(y_i = 1)(1-P(y_i = 1))}{n}, \\ \text{where}, P(y_i=1) = g(X^{(i)}\beta) \]
	</p>
	<p>We use the delta method for calculating standard errors on the marginal effects.</p>
	<dl class="section user"><dt>Input</dt><dd></dd></dl>
	<p>The input parameters are expected to be of the following form: </p>
	<pre> <em>margins_logregr</em> (
	<em>sourceTable</em> VARCHAR,
	<em>outputTable</em> VARCHAR,
	<em>dependentVariable</em> VARCHAR,
	<em>independentVariable</em> VARCHAR,
	[<em>groupingCol</em> VARCHAR,
	<em>marginal_vars</em> INTEGER[],
	<em>max_iter</em> INTEGER,
	<em>optimizer</em> VARCHAR,
	<em>tolerance</em> DOUBLE PRECISION
	]
	)</pre><p>For multinomial logistic regression the input parameters are expected to be of the following form: </p>
	<pre> <em>margins_mlogregr</em> (
	<em>sourceTable</em> VARCHAR,
	<em>outputTable</em> VARCHAR,
	<em>dependentVariable</em> VARCHAR,
	<em>independentVariable</em> VARCHAR,
	[<em>groupingCol</em> VARCHAR,
	<em>referenceCategory</em> INTEGER,
	<em>marginal_vars</em> INTEGER[],
	<em>max_iter</em> INTEGER,
	<em>optimizer</em> VARCHAR,
	<em>tolerance</em> DOUBLE PRECISION
	]
	)</pre><dl class="section warning"><dt>Warning</dt><dd>The <b>'groupingCol'</b> input parameter for <em>margins_logregr</em> and <em>margins_mlogregr</em> is a placeholder in the Madlib V1.0. These inputs will be implemented in a future release. </dd></dl>
	<dl class="section user"><dt>Usage</dt><dd></dd></dl>
	<p><b> The Interface</b></p>
	<p>For logistic regression, one can call the following function </p>
	<pre>
	SELECT madlib.<a class="el" href="marginal_8sql__in.html#ad37e0349ac3924daba27c4600311ed71">margins_logregr</a>(
	<em>'source_table'</em>, -- name of input table, VARCHAR
	<em>'out_table'</em>, -- name of output table, VARCHAR
	<em>'dependent_varname'</em>, -- dependent variable, VARCHAR
	<em>'independent_varname'</em>, -- independent variable, VARCHAR
	[ <em>'grouping_cols'</em>, -- comma separated list of grouping vars, VARCHAR (Default NULL)
	<em>'marginal_effect_variables'</em>, -- Index list (base 1) with positions representing which marginal variable to calculate, INTEGER[] (Default NULL)
	<em> max_iter</em>, -- Maximum number of iterations to run the logistic regression INTEGER (Default 20)
	<em>'optimizer'</em>, -- Optimizer used for logistic regression VARCHAR (Default & recommended 'irls')
	<em>tolerance</em>, -- Tolerance for the logistic regression DOUBLE PRECISION (default 1e-4)
	]
	);
	</pre><p>For multinomial logistic regression, one can call the following function </p>
	<pre>
	SELECT madlib.margins_mlogregr(
	<em>'source_table'</em>, -- name of input table, VARCHAR
	<em>'out_table'</em>, -- name of output table, VARCHAR
	<em>'dependent_varname'</em>, -- dependent variable, VARCHAR
	<em>'independent_varname'</em>, -- independent variable, VARCHAR
	[ <em>'reference_category'</em>, -- Reference category for multinomial logistic regression INTEGER
	<em>'grouping_cols'</em>, -- comma separated list of grouping vars, VARCHAR (Default NULL)
	<em>'marginal_effect_variables'</em>, -- Index list (base 1) with positions representing which marginal variable to calculate, INTEGER[] (Default NULL)
	<em> max_iter</em>, -- Maximum number of iterations to run the logistic regression INTEGER (Default 20)
	<em>'optimizer'</em>, -- Optimizer used for logistic regression VARCHAR (Default & recommended 'irls')
	<em>tolerance</em>, -- Tolerance for the logistic regression DOUBLE PRECISION (default 1e-4)
	]
	);
	</pre><p>Output is stored in the <em>out_table</em>: </p>
	<pre>
	[ margins \| std_err \| t_stats \| p_values \|
	+------+---------+-------+----------+
	</pre><p><b> Multinomial-Logistic Regression Notes</b></p>
	<ul>
	<li>The reference category ranges from [0, numCategories-1]. The default reference category is zero.</li>
	<li>The marginal effects and supporting statistics are output in the following order. For a problem with K dependent variables (1, ..., K) and J categories (0,..., J-1). The output is \( \{ m_{k_1, j_0} \ldots m_{k_K, j_{0}}, m_{k_1, j_1} \ldots m_{k_K, j_{1}}, m_{k_1, j_{J-1}} \ldots m_{k_K, j_{J-1}} \} \). The order is NOT CONSISTENT with mlogregr, robust_variance_mlogregr and clustered_variance_mlogregr. This is deliberate because the interfaces of all the other functions will be moved to match that used in marginal.</li>
	<li>Selectively picking the marginal effects of the independent variables will return the marginal effects of ALL categories associated with the independent variable.</li>
	<li>Here the <em>'independent_varname'</em> can be the name of a column, which contains array of numeric values. It can also have a format of string 'array[1, x1, x2, x3]', where <em>x1</em>, <em>x2</em> and <em>x3</em> are all column names.</li>
	<li>Here the <em>'vars'</em> is an index list (base 1) with the set of indices amongst the independent variables that must be selected. One can also chose <em>NULL</em> to pick all indepependent variables.</li>
	</ul>
	<dl class="section user"><dt>Examples</dt><dd><ol type="1">
	<li>Create the sample data set: <pre class="fragment">sql> SELECT * FROM data;
	id \| second_attack \| treatment \| trait_anxiety
	----+---------------+-----------+---------------
	1 \| 1 \| 1 \| 70
	3 \| 1 \| 1 \| 50
	5 \| 1 \| 0 \| 40
	7 \| 1 \| 0 \| 75
	9 \| 1 \| 0 \| 70
	11 \| 0 \| 1 \| 65
	13 \| 0 \| 1 \| 45
	15 \| 0 \| 1 \| 40
	17 \| 0 \| 0 \| 55
	...
	</pre></li>
	<li>For function summary information. Run <pre class="fragment">sql> select margins_logregr('help');
	OR
	sql> select margins_logregr();
	OR
	sql> select margins_logregr('');
	</pre></li>
	<li>Run the logistic regression function and then compute the marginal effects of all variables in the regression: <pre class="fragment">sql> select margins_logregr('patients', 'result_table', 'second_attack', 'ARRAY[1, treatment, trait_anxiety]');
	sql> select * from result_table;
	margins \| {-0.970665392796,-0.156214190168,0.0181587690137}
	coef \| {-6.36346994178179,-1.02410605239327,0.119044916668605}
	std_err \| {0.802871454422,0.292691682191,0.0137459874022}
	t_stats \| {-1.2089922832,-0.533715850748,1.32102325446}
	p_values \| {0.243212810329,0.600447858606,0.204000202116}


	-# Alternate Syntax: Run the logistic regression function and then compute the marginal effects if all variables in the regression:
	sql> select margins_logregr('patients', 'result_table', 'second_attack', 'ARRAY[1, treatment, trait_anxiety]', NULL, NULL);
	sql> select * from result_table;
	margins \| {-0.970665392796,-0.156214190168,0.0181587690137}
	coef \| {-6.36346994178179,-1.02410605239327,0.119044916668605}
	std_err \| {0.802871454422,0.292691682191,0.0137459874022}
	t_stats \| {-1.2089922832,-0.533715850748,1.32102325446}
	p_values \| {0.243212810329,0.600447858606,0.204000202116}


	-# Run the logistic regression function and then compute the marginal effects of the first variable in the regression
	sql> select margins_logregr('patients', 'result_table', 'second_attack', 'ARRAY[1, treatment, trait_anxiety]', NULL, ARRAY[1]);
	sql> select * from result_table;
	margins \| {-0.970665392796}
	coef \| {-6.36346994178179}
	std_err \| {0.802871454422}
	t_stats \| {-1.2089922832}
	p_values \| {0.243212810329}
	</pre></li>
	</ol>
	</dd></dl>
	<dl class="section user"><dt>Examples</dt><dd><ol type="1">
	<li>Create the sample data set (the full dataset has 3 categories): <pre class="fragment">sql> SELECT * FROM data;
	id \| feature_1 \| feature_2 \| category
	----+---------------+-----------+---------------
	1 \| 1 \| 35 \| 1
	3 \| 2 \| 33 \| 0
	5 \| 3 \| 39 \| 1
	7 \| 1 \| 37 \| 1
	9 \| 2 \| 31 \| 1
	11 \| 3 \| 36 \| 1
	13 \| 2 \| 36 \| 1
	15 \| 2 \| 36 \| 0
	17 \| 2 \| 31 \| 5
	...
	</pre></li>
	<li>For function summary information. Run <pre class="fragment">sql> select margins_mlogregr('help');
	OR
	sql> select margins_mlogregr();
	OR
	sql> select margins_mlogregr('');
	</pre></li>
	<li>Run the regression function and then compute the marginal effects of all variables in the regression (see docs for detailed order) <pre class="fragment">sql> select margins_mlogregr('test_data', 'result_table', 'category', 'ARRAY[1, feature_1, feature_2]');
	sql> select * from result_table;
	margins \| {0.741613239156,-0.032868883552,-0.0144502990691,-0.972055011831,0.112337273885,0.0172621628253}
	std_err \| {0.183172236055,0.044184899499,0.00332608999704,0.263532615748,0.0555196094594,0.00457999429836}
	t_stats \| {4.04872078394,-0.743894043547,-4.34453038911,-3.68855676202,2.02338011702,3.76903587663}
	p_values \| {7.43784735554e-05,0.457840607871,2.24855476205e-05,0.000292799037776,0.0444060346517,0.000217384008015}
	</pre></li>
	<li>Run the regression and compute the marginals effects for the first dependent variable (all categories) <pre class="fragment">sql> select margins_mlogregr('test_data', 'result_table', 'category', 'ARRAY[1, feature_1, feature_2]', 0, NULL, ARRAY[1]);
	sql> select * from result_table;
	margins \| {0.741613239156,-0.972055011831}
	std_err \| {0.183172236055,0.263532615748}
	t_stats \| {4.04872078394,-3.68855676202}
	p_values \| {7.43784735554e-05,0.000292799037776}
	</pre></li>
	</ol>
	</dd></dl>
	<pre class="fragment">-# For function usage information.
	sql> select margins_mlogregr('usage');
	</pre><dl class="section user"><dt>Literature</dt><dd></dd></dl>
	<p>[1] mfx function in STATA: <a href="http://www.stata.com/help.cgi?mfx_option">http://www.stata.com/help.cgi?mfx_option</a></p>
	<dl class="section see"><dt>See Also</dt><dd>File <a class="el" href="marginal_8sql__in.html" title="SQL functions for linear regression. ">marginal.sql_in</a> documenting the SQL functions. </dd></dl>
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