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 <div class="header">
   <div class="headertitle">
 <div class="title">Logistic Regression<div class="ingroups"><a class="el" href="group__grp__glm.html">Generalized Linear Models</a></div></div>  </div>
 </div><!--header-->
 <div class="contents">
 <div class="toc"><b>Contents</b></p>
 <ul>
 <li class="level1">
 <a href="#about">About</a> </li>
 <li class="level1">
 <a href="#train">Training Function</a> </li>
 <li class="level1">
 <a href="#output">Output Table</a> </li>
 <li class="level1">
 <a href="#predict">Prediction Function</a> </li>
 <li class="level1">
 <a href="#examples">Examples</a> </li>
 <li class="level1">
 <a href="#seealso">See Also</a> </li>
 <li class="level1">
 <a href="#background">Technical Background</a> </li>
 <li class="level1">
 <a href="#literature">Literature</a> </li>
 </ul>
 </div><p><a class="anchor" id="about"></a></p>
 <dl class="section user"><dt>About</dt><dd>Binomial logistic regression models the relationship between a dichotomous dependent variable and one or more predictor variables.</dd></dl>
 <p>The dependent variable may be a Boolean value or a categorial variable that can be represented with a Boolean expression.</p>
 <p><a class="anchor" id="train"></a></p>
 <dl class="section user"><dt>Training Function</dt><dd>The logistic regression training function has the following format: <pre class="fragment">logregr_train(
     tbl_source,
     tbl_output,
     dep_col,
     ind_col,
     grouping_col,
     max_iter,
     optimizer,
     tolerance,
     verbose)
 </pre>  <dl class="arglist">
 <dt>tbl_source </dt>
 <dd><p class="startdd">TEXT. The name of the table containing the training data.</p>
 <p class="enddd"></p>
 </dd>
 <dt>tbl_output </dt>
 <dd><p class="startdd">TEXT. Name of the generated table containing the output model.</p>
 <p class="enddd"></p>
 </dd>
 <dt>dep_col </dt>
 <dd><p class="startdd">TEXT. Name of the dependent variable column (of type BOOLEAN) in the training data or an expression evaluating to a BOOLEAN.</p>
 <p class="enddd"></p>
 </dd>
 <dt>ind_col </dt>
 <dd><p class="startdd">TEXT. Expression list to evaluate for the independent variables. An intercept variable is not assumed. It is common to provide an explicit intercept term by including a single constant <code>1</code> term in the independent variable list.</p>
 <p class="enddd"></p>
 </dd>
 <dt>grouping_col (optional) </dt>
 <dd><p class="startdd">TEXT, default: NULL. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL "GROUP BY" clause. When this value is NULL, no grouping is used and a single result model is generated.</p>
 <p class="enddd"></p>
 </dd>
 <dt>max_iter (optional) </dt>
 <dd><p class="startdd">INTEGER, default: 20. The maximum number of iterations that are allowed.</p>
 <p class="enddd"></p>
 </dd>
 <dt>optimizer (optional)( </dt>
 <dd><p class="startdd">TEXT, default: 'irls'. The name of the optimizer to use:</p>
 <ul>
 <li>
 'newton' or 'irls': Iteratively reweighted least squares </li>
 <li>
 'cg': conjugate gradient </li>
 <li>
 'igd': incremental gradient descent.</li>
 </ul>
 <p class="enddd"></p>
 </dd>
 <dt>tolerance (optional) </dt>
 <dd><p class="startdd">FLOAT8, default: 0.0001. The difference between log-likelihood values in successive iterations that should indicate convergence. A zero disables the convergence criterion, so that execution stops after <code>n</code> iterations have completed.</p>
 <p class="enddd"></p>
 </dd>
 <dt>verbose (optional) </dt>
 <dd>BOOLEAN, default: FALSE. Provides verbose output of the results of training. </dd>
 </dl>
 </dd></dl>
 <p><a class="anchor" id="notes"></a></p>
 <dl class="section note"><dt>Note</dt><dd>All table names can be optionally schema qualified (current_schemas() would be searched if a schema name is not provided) and all table and column names should follow case-sensitivity and quoting rules per the database. (For instance, 'mytable' and 'MyTable' both resolve to the same entity, i.e. 'mytable'. If mixed-case or multi-byte characters are desired for entity names then the string should be double-quoted; in this case the input would be '"MyTable"').</dd></dl>
 <p><a class="anchor" id="output"></a></p>
 <dl class="section user"><dt>Output Table</dt><dd>The output table produced by the logistic regression training function contains the following columns: <dl class="arglist">
 <dt>&lt;...&gt; </dt>
 <dd><p class="startdd">Text. Grouping columns, if provided in input. This could be multiple columns depending on the <code>grouping_col</code> input.</p>
 <p class="enddd"></p>
 </dd>
 <dt>coef </dt>
 <dd><p class="startdd">FLOAT8. Vector of the coefficients of the regression.</p>
 <p class="enddd"></p>
 </dd>
 <dt>log_likelihood </dt>
 <dd><p class="startdd">FLOAT8. The log-likelihood \( l(\boldsymbol c) \).</p>
 <p class="enddd"></p>
 </dd>
 <dt>std_err </dt>
 <dd><p class="startdd">FLOAT8[]. Vector of the standard error of the coefficients.</p>
 <p class="enddd"></p>
 </dd>
 <dt>z_stats </dt>
 <dd><p class="startdd">FLOAT8[]. Vector of the z-statistics of the coefficients.</p>
 <p class="enddd"></p>
 </dd>
 <dt>p_values </dt>
 <dd><p class="startdd">FLOAT8[]. Vector of the p-values of the coefficients.</p>
 <p class="enddd"></p>
 </dd>
 <dt>odds_ratios </dt>
 <dd><p class="startdd">FLOAT8[]. The odds ratio, \( \exp(c_i) \).</p>
 <p class="enddd"></p>
 </dd>
 <dt>condition_no </dt>
 <dd><p class="startdd">FLOAT8[]. The condition number of the \(X^{*}X\) matrix. A high condition number is usually an indication that there may be some numeric instability in the result yielding a less reliable model. A high condition number often results when there is a significant amount of colinearity in the underlying design matrix, in which case other regression techniques may be more appropriate.</p>
 <p class="enddd"></p>
 </dd>
 <dt>num_iterations </dt>
 <dd>INTEGER. The number of iterations actually completed. This would be different from the <code>nIterations</code> argument if a <code>tolerance</code> parameter is provided and the algorithm converges before all iterations are completed. </dd>
 </dl>
 </dd></dl>
 <p><a class="anchor" id="examples"></a></p>
 <dl class="section user"><dt>Examples</dt><dd><ol type="1">
 <li>Create the training data table. <pre class="fragment">sql&gt; CREATE TABLE patients (id INTEGER NOT NULL, second_attack INTEGER,
     treatment INTEGER, trait_anxiety INTEGER);
 sql&gt; COPY patients FROM STDIN WITH DELIMITER '|';
   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
  19 |             0 |         0 |            50
   2 |             1 |         1 |            80
   4 |             1 |         0 |            60
   6 |             1 |         0 |            65
   8 |             1 |         0 |            80
  10 |             1 |         0 |            60
  12 |             0 |         1 |            50
  14 |             0 |         1 |            35
  16 |             0 |         1 |            50
  18 |             0 |         0 |            45
  20 |             0 |         0 |            60
 \.
 </pre></li>
 <li>Train a regression model. <pre class="fragment">SELECT madlib.logregr_train(
     'patients',
     'patients_logregr',
     'second_attack',
     'ARRAY[1, treatment, trait_anxiety]',
     NULL,
     20,
     'irls');
 </pre></li>
 <li>View the regression results: <pre class="fragment">-- Set extended display on for easier reading of output
 \x on
 SELECT * from patients_logregr;
 </pre> Query result: <pre class="fragment">coef           | {5.59049410898112,2.11077546770772,-0.237276684606453}
 log_likelihood | -467.214718489873
 std_err        | {0.318943457652178,0.101518723785383,0.294509929481773}
 z_stats        | {17.5281667482197,20.7919819024719,-0.805666162169712}
 p_values       | {8.73403463417837e-69,5.11539430631541e-96,0.420435365338518}
 odds_ratios    | {267.867942976278,8.2546400100702,0.788773016471171}
 condition_no   | 179.186118573205
 num_iterations | 9
 </pre> Alternatively, unnest the arrays in the results for easier reading of output: <pre class="fragment">\x off
 SELECT unnest(array['intercept', 'treatment', 'trait_anxiety']) as attribute,
        unnest(coef) as coefficient,
        unnest(std_err) as standard_error,
        unnest(z_stats) as z_stat,
        unnest(p_values) as pvalue,
        unnest(odds_ratios) as odds_ratio
     FROM patients_logregr;
 </pre></li>
 </ol>
 </dd></dl>
 <p><a class="anchor" id="seealso"></a></p>
 <dl class="section see"><dt>See Also</dt><dd>File <a class="el" href="logistic_8sql__in.html" title="SQL functions for logistic regression. ">logistic.sql_in</a> documenting the training function </dd>
 <dd>
 <a class="el" href="logistic_8sql__in.html#a32880a39de2e36b6c6be72691a6a4a40" title="Compute logistic-regression coefficients and diagnostic statistics. ">logregr_train()</a> </dd>
 <dd>
 <a class="el" href="elastic__net_8sql__in.html#a735038a5090c112505c740a90a203e83" title="Interface for elastic net. ">elastic_net_train()</a> </dd>
 <dd>
 <a class="el" href="group__grp__linreg.html">Linear Regression</a> </dd>
 <dd>
 <a class="el" href="group__grp__mlogreg.html">Multinomial Logistic Regression</a> </dd>
 <dd>
 <a class="el" href="group__grp__robust.html">Huber White Variance</a> </dd>
 <dd>
 <a class="el" href="group__grp__clustered__errors.html">Clustered Variance</a> </dd>
 <dd>
 <a class="el" href="group__grp__validation.html">Cross Validation</a> </dd>
 <dd>
 <a class="el" href="group__grp__marginal.html">Marginal Effects</a></dd></dl>
 <p><a class="anchor" id="background"></a></p>
 <dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
 <p>(Binomial) logistic regression refers to a stochastic model in which the conditional mean of the dependent dichotomous variable (usually denoted \( Y \in \{ 0,1 \} \)) is the logistic function of an affine function of the vector of independent variables (usually denoted \( \boldsymbol x \)). That is, </p>
 <p class="formulaDsp">
 \[ E[Y \mid \boldsymbol x] = \sigma(\boldsymbol c^T \boldsymbol x) \]
 </p>
 <p> for some unknown vector of coefficients \( \boldsymbol c \) and where \( \sigma(x) = \frac{1}{1 + \exp(-x)} \) is the logistic function. Logistic regression finds the vector of coefficients \( \boldsymbol c \) that maximizes the likelihood of the observations.</p>
 <p>Let</p>
 <ul>
 <li>\( \boldsymbol y \in \{ 0,1 \}^n \) denote the vector of observed dependent variables, with \( n \) rows, containing the observed values of the dependent variable,</li>
 <li>\( X \in \mathbf R^{n \times k} \) denote the design matrix with \( k \) columns and \( n \) rows, containing all observed vectors of independent variables \( \boldsymbol x_i \) as rows.</li>
 </ul>
 <p>By definition, </p>
 <p class="formulaDsp">
 \[ P[Y = y_i | \boldsymbol x_i] = \sigma((-1)^{y_i} \cdot \boldsymbol c^T \boldsymbol x_i) \,. \]
 </p>
 <p> Maximizing the likelihood \( \prod_{i=1}^n \Pr(Y = y_i \mid \boldsymbol x_i) \) is equivalent to maximizing the log-likelihood \( \sum_{i=1}^n \log \Pr(Y = y_i \mid \boldsymbol x_i) \), which simplifies to </p>
 <p class="formulaDsp">
 \[ l(\boldsymbol c) = -\sum_{i=1}^n \log(1 + \exp((-1)^{y_i} \cdot \boldsymbol c^T \boldsymbol x_i)) \,. \]
 </p>
 <p> The Hessian of this objective is \( H = -X^T A X \) where \( A = \text{diag}(a_1, \dots, a_n) \) is the diagonal matrix with \( a_i = \sigma(\boldsymbol c^T \boldsymbol x) \cdot \sigma(-\boldsymbol c^T \boldsymbol x) \,. \) Since \( H \) is non-positive definite, \( l(\boldsymbol c) \) is convex. There are many techniques for solving convex optimization problems. Currently, logistic regression in MADlib can use one of three algorithms:</p>
 <ul>
 <li>Iteratively Reweighted Least Squares</li>
 <li>A conjugate-gradient approach, also known as Fletcher-Reeves method in the literature, where we use the Hestenes-Stiefel rule for calculating the step size.</li>
 <li>Incremental gradient descent, also known as incremental gradient methods or stochastic gradient descent in the literature.</li>
 </ul>
 <p>We estimate the standard error for coefficient \( i \) as </p>
 <p class="formulaDsp">
 \[ \mathit{se}(c_i) = \left( (X^T A X)^{-1} \right)_{ii} \,. \]
 </p>
 <p> The Wald z-statistic is </p>
 <p class="formulaDsp">
 \[ z_i = \frac{c_i}{\mathit{se}(c_i)} \,. \]
 </p>
 <p>The Wald \( p \)-value for coefficient \( i \) gives the probability (under the assumptions inherent in the Wald test) of seeing a value at least as extreme as the one observed, provided that the null hypothesis ( \( c_i = 0 \)) is true. Letting \( F \) denote the cumulative density function of a standard normal distribution, the Wald \( p \)-value for coefficient \( i \) is therefore </p>
 <p class="formulaDsp">
 \[ p_i = \Pr(|Z| \geq |z_i|) = 2 \cdot (1 - F( |z_i| )) \]
 </p>
 <p> where \( Z \) is a standard normally distributed random variable.</p>
 <p>The odds ratio for coefficient \( i \) is estimated as \( \exp(c_i) \).</p>
 <p>The condition number is computed as \( \kappa(X^T A X) \) during the iteration immediately <em>preceding</em> convergence (i.e., \( A \) is computed using the coefficients of the previous iteration). A large condition number (say, more than 1000) indicates the presence of significant multicollinearity.</p>
 <dl class="section user"><dt>Literature</dt><dd></dd></dl>
 <p>A somewhat random selection of nice write-ups, with valuable pointers into further literature.</p>
 <p>[1] Cosma Shalizi: Statistics 36-350: Data Mining, Lecture Notes, 18 November 2009, <a href="http://www.stat.cmu.edu/~cshalizi/350/lectures/26/lecture-26.pdf">http://www.stat.cmu.edu/~cshalizi/350/lectures/26/lecture-26.pdf</a></p>
 <p>[2] Thomas P. Minka: A comparison of numerical optimizers for logistic regression, 2003 (revised Mar 26, 2007), <a href="http://research.microsoft.com/en-us/um/people/minka/papers/logreg/minka-logreg.pdf">http://research.microsoft.com/en-us/um/people/minka/papers/logreg/minka-logreg.pdf</a></p>
 <p>[3] Paul Komarek, Andrew W. Moore: Making Logistic Regression A Core Data Mining Tool With TR-IRLS, IEEE International Conference on Data Mining 2005, pp. 685-688, <a href="http://komarix.org/ac/papers/tr-irls.short.pdf">http://komarix.org/ac/papers/tr-irls.short.pdf</a></p>
 <p>[4] D. P. Bertsekas: Incremental gradient, subgradient, and proximal methods for convex optimization: a survey, Technical report, Laboratory for Information and Decision Systems, 2010, <a href="http://web.mit.edu/dimitrib/www/Incremental_Survey_LIDS.pdf">http://web.mit.edu/dimitrib/www/Incremental_Survey_LIDS.pdf</a></p>
 <p>[5] A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro: Robust stochastic approximation approach to stochastic programming, SIAM Journal on Optimization, 19(4), 2009, <a href="http://www2.isye.gatech.edu/~nemirovs/SIOPT_RSA_2009.pdf">http://www2.isye.gatech.edu/~nemirovs/SIOPT_RSA_2009.pdf</a> </p>
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	<div class="header">
	<div class="headertitle">
	<div class="title">Logistic Regression<div class="ingroups"><a class="el" href="group__grp__glm.html">Generalized Linear Models</a></div></div> </div>
	</div><!--header-->
	<div class="contents">
	<div class="toc"><b>Contents</b></p>
	<ul>
	<li class="level1">
	<a href="#about">About</a> </li>
	<li class="level1">
	<a href="#train">Training Function</a> </li>
	<li class="level1">
	<a href="#output">Output Table</a> </li>
	<li class="level1">
	<a href="#predict">Prediction Function</a> </li>
	<li class="level1">
	<a href="#examples">Examples</a> </li>
	<li class="level1">
	<a href="#seealso">See Also</a> </li>
	<li class="level1">
	<a href="#background">Technical Background</a> </li>
	<li class="level1">
	<a href="#literature">Literature</a> </li>
	</ul>
	</div><p><a class="anchor" id="about"></a></p>
	<dl class="section user"><dt>About</dt><dd>Binomial logistic regression models the relationship between a dichotomous dependent variable and one or more predictor variables.</dd></dl>
	<p>The dependent variable may be a Boolean value or a categorial variable that can be represented with a Boolean expression.</p>
	<p><a class="anchor" id="train"></a></p>
	<dl class="section user"><dt>Training Function</dt><dd>The logistic regression training function has the following format: <pre class="fragment">logregr_train(
	tbl_source,
	tbl_output,
	dep_col,
	ind_col,
	grouping_col,
	max_iter,
	optimizer,
	tolerance,
	verbose)
	</pre> <dl class="arglist">
	<dt>tbl_source </dt>
	<dd><p class="startdd">TEXT. The name of the table containing the training data.</p>
	<p class="enddd"></p>
	</dd>
	<dt>tbl_output </dt>
	<dd><p class="startdd">TEXT. Name of the generated table containing the output model.</p>
	<p class="enddd"></p>
	</dd>
	<dt>dep_col </dt>
	<dd><p class="startdd">TEXT. Name of the dependent variable column (of type BOOLEAN) in the training data or an expression evaluating to a BOOLEAN.</p>
	<p class="enddd"></p>
	</dd>
	<dt>ind_col </dt>
	<dd><p class="startdd">TEXT. Expression list to evaluate for the independent variables. An intercept variable is not assumed. It is common to provide an explicit intercept term by including a single constant <code>1</code> term in the independent variable list.</p>
	<p class="enddd"></p>
	</dd>
	<dt>grouping_col (optional) </dt>
	<dd><p class="startdd">TEXT, default: NULL. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL "GROUP BY" clause. When this value is NULL, no grouping is used and a single result model is generated.</p>
	<p class="enddd"></p>
	</dd>
	<dt>max_iter (optional) </dt>
	<dd><p class="startdd">INTEGER, default: 20. The maximum number of iterations that are allowed.</p>
	<p class="enddd"></p>
	</dd>
	<dt>optimizer (optional)( </dt>
	<dd><p class="startdd">TEXT, default: 'irls'. The name of the optimizer to use:</p>
	<ul>
	<li>
	'newton' or 'irls': Iteratively reweighted least squares </li>
	<li>
	'cg': conjugate gradient </li>
	<li>
	'igd': incremental gradient descent.</li>
	</ul>
	<p class="enddd"></p>
	</dd>
	<dt>tolerance (optional) </dt>
	<dd><p class="startdd">FLOAT8, default: 0.0001. The difference between log-likelihood values in successive iterations that should indicate convergence. A zero disables the convergence criterion, so that execution stops after <code>n</code> iterations have completed.</p>
	<p class="enddd"></p>
	</dd>
	<dt>verbose (optional) </dt>
	<dd>BOOLEAN, default: FALSE. Provides verbose output of the results of training. </dd>
	</dl>
	</dd></dl>
	<p><a class="anchor" id="notes"></a></p>
	<dl class="section note"><dt>Note</dt><dd>All table names can be optionally schema qualified (current_schemas() would be searched if a schema name is not provided) and all table and column names should follow case-sensitivity and quoting rules per the database. (For instance, 'mytable' and 'MyTable' both resolve to the same entity, i.e. 'mytable'. If mixed-case or multi-byte characters are desired for entity names then the string should be double-quoted; in this case the input would be '"MyTable"').</dd></dl>
	<p><a class="anchor" id="output"></a></p>
	<dl class="section user"><dt>Output Table</dt><dd>The output table produced by the logistic regression training function contains the following columns: <dl class="arglist">
	<dt><...> </dt>
	<dd><p class="startdd">Text. Grouping columns, if provided in input. This could be multiple columns depending on the <code>grouping_col</code> input.</p>
	<p class="enddd"></p>
	</dd>
	<dt>coef </dt>
	<dd><p class="startdd">FLOAT8. Vector of the coefficients of the regression.</p>
	<p class="enddd"></p>
	</dd>
	<dt>log_likelihood </dt>
	<dd><p class="startdd">FLOAT8. The log-likelihood \( l(\boldsymbol c) \).</p>
	<p class="enddd"></p>
	</dd>
	<dt>std_err </dt>
	<dd><p class="startdd">FLOAT8[]. Vector of the standard error of the coefficients.</p>
	<p class="enddd"></p>
	</dd>
	<dt>z_stats </dt>
	<dd><p class="startdd">FLOAT8[]. Vector of the z-statistics of the coefficients.</p>
	<p class="enddd"></p>
	</dd>
	<dt>p_values </dt>
	<dd><p class="startdd">FLOAT8[]. Vector of the p-values of the coefficients.</p>
	<p class="enddd"></p>
	</dd>
	<dt>odds_ratios </dt>
	<dd><p class="startdd">FLOAT8[]. The odds ratio, \( \exp(c_i) \).</p>
	<p class="enddd"></p>
	</dd>
	<dt>condition_no </dt>
	<dd><p class="startdd">FLOAT8[]. The condition number of the \(X^{*}X\) matrix. A high condition number is usually an indication that there may be some numeric instability in the result yielding a less reliable model. A high condition number often results when there is a significant amount of colinearity in the underlying design matrix, in which case other regression techniques may be more appropriate.</p>
	<p class="enddd"></p>
	</dd>
	<dt>num_iterations </dt>
	<dd>INTEGER. The number of iterations actually completed. This would be different from the <code>nIterations</code> argument if a <code>tolerance</code> parameter is provided and the algorithm converges before all iterations are completed. </dd>
	</dl>
	</dd></dl>
	<p><a class="anchor" id="examples"></a></p>
	<dl class="section user"><dt>Examples</dt><dd><ol type="1">
	<li>Create the training data table. <pre class="fragment">sql> CREATE TABLE patients (id INTEGER NOT NULL, second_attack INTEGER,
	treatment INTEGER, trait_anxiety INTEGER);
	sql> COPY patients FROM STDIN WITH DELIMITER '\|';
	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
	19 \| 0 \| 0 \| 50
	2 \| 1 \| 1 \| 80
	4 \| 1 \| 0 \| 60
	6 \| 1 \| 0 \| 65
	8 \| 1 \| 0 \| 80
	10 \| 1 \| 0 \| 60
	12 \| 0 \| 1 \| 50
	14 \| 0 \| 1 \| 35
	16 \| 0 \| 1 \| 50
	18 \| 0 \| 0 \| 45
	20 \| 0 \| 0 \| 60
	\.
	</pre></li>
	<li>Train a regression model. <pre class="fragment">SELECT madlib.logregr_train(
	'patients',
	'patients_logregr',
	'second_attack',
	'ARRAY[1, treatment, trait_anxiety]',
	NULL,
	20,
	'irls');
	</pre></li>
	<li>View the regression results: <pre class="fragment">-- Set extended display on for easier reading of output
	\x on
	SELECT * from patients_logregr;
	</pre> Query result: <pre class="fragment">coef \| {5.59049410898112,2.11077546770772,-0.237276684606453}
	log_likelihood \| -467.214718489873
	std_err \| {0.318943457652178,0.101518723785383,0.294509929481773}
	z_stats \| {17.5281667482197,20.7919819024719,-0.805666162169712}
	p_values \| {8.73403463417837e-69,5.11539430631541e-96,0.420435365338518}
	odds_ratios \| {267.867942976278,8.2546400100702,0.788773016471171}
	condition_no \| 179.186118573205
	num_iterations \| 9
	</pre> Alternatively, unnest the arrays in the results for easier reading of output: <pre class="fragment">\x off
	SELECT unnest(array['intercept', 'treatment', 'trait_anxiety']) as attribute,
	unnest(coef) as coefficient,
	unnest(std_err) as standard_error,
	unnest(z_stats) as z_stat,
	unnest(p_values) as pvalue,
	unnest(odds_ratios) as odds_ratio
	FROM patients_logregr;
	</pre></li>
	</ol>
	</dd></dl>
	<p><a class="anchor" id="seealso"></a></p>
	<dl class="section see"><dt>See Also</dt><dd>File <a class="el" href="logistic_8sql__in.html" title="SQL functions for logistic regression. ">logistic.sql_in</a> documenting the training function </dd>
	<dd>
	<a class="el" href="logistic_8sql__in.html#a32880a39de2e36b6c6be72691a6a4a40" title="Compute logistic-regression coefficients and diagnostic statistics. ">logregr_train()</a> </dd>
	<dd>
	<a class="el" href="elastic__net_8sql__in.html#a735038a5090c112505c740a90a203e83" title="Interface for elastic net. ">elastic_net_train()</a> </dd>
	<dd>
	<a class="el" href="group__grp__linreg.html">Linear Regression</a> </dd>
	<dd>
	<a class="el" href="group__grp__mlogreg.html">Multinomial Logistic Regression</a> </dd>
	<dd>
	<a class="el" href="group__grp__robust.html">Huber White Variance</a> </dd>
	<dd>
	<a class="el" href="group__grp__clustered__errors.html">Clustered Variance</a> </dd>
	<dd>
	<a class="el" href="group__grp__validation.html">Cross Validation</a> </dd>
	<dd>
	<a class="el" href="group__grp__marginal.html">Marginal Effects</a></dd></dl>
	<p><a class="anchor" id="background"></a></p>
	<dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
	<p>(Binomial) logistic regression refers to a stochastic model in which the conditional mean of the dependent dichotomous variable (usually denoted \( Y \in \{ 0,1 \} \)) is the logistic function of an affine function of the vector of independent variables (usually denoted \( \boldsymbol x \)). That is, </p>
	<p class="formulaDsp">
	\[ E[Y \mid \boldsymbol x] = \sigma(\boldsymbol c^T \boldsymbol x) \]
	</p>
	<p> for some unknown vector of coefficients \( \boldsymbol c \) and where \( \sigma(x) = \frac{1}{1 + \exp(-x)} \) is the logistic function. Logistic regression finds the vector of coefficients \( \boldsymbol c \) that maximizes the likelihood of the observations.</p>
	<p>Let</p>
	<ul>
	<li>\( \boldsymbol y \in \{ 0,1 \}^n \) denote the vector of observed dependent variables, with \( n \) rows, containing the observed values of the dependent variable,</li>
	<li>\( X \in \mathbf R^{n \times k} \) denote the design matrix with \( k \) columns and \( n \) rows, containing all observed vectors of independent variables \( \boldsymbol x_i \) as rows.</li>
	</ul>
	<p>By definition, </p>
	<p class="formulaDsp">
	\[ P[Y = y_i \| \boldsymbol x_i] = \sigma((-1)^{y_i} \cdot \boldsymbol c^T \boldsymbol x_i) \,. \]
	</p>
	<p> Maximizing the likelihood \( \prod_{i=1}^n \Pr(Y = y_i \mid \boldsymbol x_i) \) is equivalent to maximizing the log-likelihood \( \sum_{i=1}^n \log \Pr(Y = y_i \mid \boldsymbol x_i) \), which simplifies to </p>
	<p class="formulaDsp">
	\[ l(\boldsymbol c) = -\sum_{i=1}^n \log(1 + \exp((-1)^{y_i} \cdot \boldsymbol c^T \boldsymbol x_i)) \,. \]
	</p>
	<p> The Hessian of this objective is \( H = -X^T A X \) where \( A = \text{diag}(a_1, \dots, a_n) \) is the diagonal matrix with \( a_i = \sigma(\boldsymbol c^T \boldsymbol x) \cdot \sigma(-\boldsymbol c^T \boldsymbol x) \,. \) Since \( H \) is non-positive definite, \( l(\boldsymbol c) \) is convex. There are many techniques for solving convex optimization problems. Currently, logistic regression in MADlib can use one of three algorithms:</p>
	<ul>
	<li>Iteratively Reweighted Least Squares</li>
	<li>A conjugate-gradient approach, also known as Fletcher-Reeves method in the literature, where we use the Hestenes-Stiefel rule for calculating the step size.</li>
	<li>Incremental gradient descent, also known as incremental gradient methods or stochastic gradient descent in the literature.</li>
	</ul>
	<p>We estimate the standard error for coefficient \( i \) as </p>
	<p class="formulaDsp">
	\[ \mathit{se}(c_i) = \left( (X^T A X)^{-1} \right)_{ii} \,. \]
	</p>
	<p> The Wald z-statistic is </p>
	<p class="formulaDsp">
	\[ z_i = \frac{c_i}{\mathit{se}(c_i)} \,. \]
	</p>
	<p>The Wald \( p \)-value for coefficient \( i \) gives the probability (under the assumptions inherent in the Wald test) of seeing a value at least as extreme as the one observed, provided that the null hypothesis ( \( c_i = 0 \)) is true. Letting \( F \) denote the cumulative density function of a standard normal distribution, the Wald \( p \)-value for coefficient \( i \) is therefore </p>
	<p class="formulaDsp">
	\[ p_i = \Pr(\|Z\| \geq \|z_i\|) = 2 \cdot (1 - F( \|z_i\| )) \]
	</p>
	<p> where \( Z \) is a standard normally distributed random variable.</p>
	<p>The odds ratio for coefficient \( i \) is estimated as \( \exp(c_i) \).</p>
	<p>The condition number is computed as \( \kappa(X^T A X) \) during the iteration immediately <em>preceding</em> convergence (i.e., \( A \) is computed using the coefficients of the previous iteration). A large condition number (say, more than 1000) indicates the presence of significant multicollinearity.</p>
	<dl class="section user"><dt>Literature</dt><dd></dd></dl>
	<p>A somewhat random selection of nice write-ups, with valuable pointers into further literature.</p>
	<p>[1] Cosma Shalizi: Statistics 36-350: Data Mining, Lecture Notes, 18 November 2009, <a href="http://www.stat.cmu.edu/~cshalizi/350/lectures/26/lecture-26.pdf">http://www.stat.cmu.edu/~cshalizi/350/lectures/26/lecture-26.pdf</a></p>
	<p>[2] Thomas P. Minka: A comparison of numerical optimizers for logistic regression, 2003 (revised Mar 26, 2007), <a href="http://research.microsoft.com/en-us/um/people/minka/papers/logreg/minka-logreg.pdf">http://research.microsoft.com/en-us/um/people/minka/papers/logreg/minka-logreg.pdf</a></p>
	<p>[3] Paul Komarek, Andrew W. Moore: Making Logistic Regression A Core Data Mining Tool With TR-IRLS, IEEE International Conference on Data Mining 2005, pp. 685-688, <a href="http://komarix.org/ac/papers/tr-irls.short.pdf">http://komarix.org/ac/papers/tr-irls.short.pdf</a></p>
	<p>[4] D. P. Bertsekas: Incremental gradient, subgradient, and proximal methods for convex optimization: a survey, Technical report, Laboratory for Information and Decision Systems, 2010, <a href="http://web.mit.edu/dimitrib/www/Incremental_Survey_LIDS.pdf">http://web.mit.edu/dimitrib/www/Incremental_Survey_LIDS.pdf</a></p>
	<p>[5] A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro: Robust stochastic approximation approach to stochastic programming, SIAM Journal on Optimization, 19(4), 2009, <a href="http://www2.isye.gatech.edu/~nemirovs/SIOPT_RSA_2009.pdf">http://www2.isye.gatech.edu/~nemirovs/SIOPT_RSA_2009.pdf</a> </p>
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