docs/v1.2/group__grp__cox__prop__hazards.html - madlib-site - Git at Google

 <!-- HTML header for doxygen 1.8.4-->
 <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
 <html xmlns="http://www.w3.org/1999/xhtml">
 <head>
 <meta http-equiv="Content-Type" content="text/xhtml;charset=UTF-8"/>
 <meta http-equiv="X-UA-Compatible" content="IE=9"/>
 <meta name="generator" content="Doxygen 1.8.4"/>
 <meta name="keywords" content="madlib,postgres,greenplum,machine learning,data mining,deep learning,ensemble methods,data science,market basket analysis,affinity analysis,pca,lda,regression,elastic net,huber white,proportional hazards,k-means,latent dirichlet allocation,bayes,support vector machines,svm"/>
 <title>MADlib: Cox-Proportional Hazards Regression</title>
 <link href="tabs.css" rel="stylesheet" type="text/css"/>
 <script type="text/javascript" src="jquery.js"></script>
 <script type="text/javascript" src="dynsections.js"></script>
 <link href="navtree.css" rel="stylesheet" type="text/css"/>
 <script type="text/javascript" src="resize.js"></script>
 <script type="text/javascript" src="navtree.js"></script>
 <script type="text/javascript">
   $(document).ready(initResizable);
   $(window).load(resizeHeight);
 </script>
 <link href="search/search.css" rel="stylesheet" type="text/css"/>
 <script type="text/javascript" src="search/search.js"></script>
 <script type="text/javascript">
   $(document).ready(function() { searchBox.OnSelectItem(0); });
 </script>
 <script type="text/x-mathjax-config">
   MathJax.Hub.Config({
     extensions: ["tex2jax.js", "TeX/AMSmath.js", "TeX/AMSsymbols.js"],
     jax: ["input/TeX","output/HTML-CSS"],
 });
 </script><script src="../mathjax/MathJax.js"></script>
 <link href="doxygen.css" rel="stylesheet" type="text/css" />
 <link href="madlib_extra.css" rel="stylesheet" type="text/css"/>
 <!-- google analytics -->
 <script>
   (function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){
   (i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
   m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
   })(window,document,'script','//www.google-analytics.com/analytics.js','ga');
   ga('create', 'UA-45382226-1', 'auto');
   ga('send', 'pageview');
 </script>
 </head>
 <body>
 <div id="top"><!-- do not remove this div, it is closed by doxygen! -->
 <div id="titlearea">
 <table cellspacing="0" cellpadding="0">
  <tbody>
  <tr style="height: 56px;">
   <td style="padding-left: 0.5em;">
    <div id="projectname">MADlib
    &#160;<span id="projectnumber">1.2</span> <span style="font-size:10pt; font-style:italic"><a href="../latest/./group__grp__cox__prop__hazards.html"> A newer version is available</a></span>
    </div>
    <div id="projectbrief">User Documentation</div>
   </td>
   <!--BEGIN VERSIONS LINKS-->
     <td style="padding-left: 0.5em;">
       <div class="versionlist"><ul>
         <li class="head">More versions:</li>
         <li><a href="../v1.1/index.html">v1.1</li>
         <li><a href="../v1.0/index.html">v1.0</li>
         <li><a href="../v0.7/index.html">v0.7</li>
         <li><a href="../v0.5/index.html">v0.5</li></ul>
       </div>
     </td>
    <td>        <div id="MSearchBox" class="MSearchBoxInactive">
         <span class="left">
           <img id="MSearchSelect" src="search/mag_sel.png"
                onmouseover="return searchBox.OnSearchSelectShow()"
                onmouseout="return searchBox.OnSearchSelectHide()"
                alt=""/>
           <input type="text" id="MSearchField" value="Search" accesskey="S"
                onfocus="searchBox.OnSearchFieldFocus(true)"
                onblur="searchBox.OnSearchFieldFocus(false)"
                onkeyup="searchBox.OnSearchFieldChange(event)"/>
           </span><span class="right">
             <a id="MSearchClose" href="javascript:searchBox.CloseResultsWindow()"><img id="MSearchCloseImg" border="0" src="search/close.png" alt=""/></a>
           </span>
         </div>
 </td>
  </tr>
  </tbody>
 </table>
 </div>
 <!-- end header part -->
 <!-- Generated by Doxygen 1.8.4 -->
 <script type="text/javascript">
 var searchBox = new SearchBox("searchBox", "search",false,'Search');
 </script>
 </div><!-- top -->
 <div id="side-nav" class="ui-resizable side-nav-resizable">
   <div id="nav-tree">
     <div id="nav-tree-contents">
       <div id="nav-sync" class="sync"></div>
     </div>
   </div>
   <div id="splitbar" style="-moz-user-select:none;"
        class="ui-resizable-handle">
   </div>
 </div>
 <script type="text/javascript">
 $(document).ready(function(){initNavTree('group__grp__cox__prop__hazards.html','');});
 </script>
 <div id="doc-content">
 <!-- window showing the filter options -->
 <div id="MSearchSelectWindow"
      onmouseover="return searchBox.OnSearchSelectShow()"
      onmouseout="return searchBox.OnSearchSelectHide()"
      onkeydown="return searchBox.OnSearchSelectKey(event)">
 <a class="SelectItem" href="javascript:void(0)" onclick="searchBox.OnSelectItem(0)"><span class="SelectionMark">&#160;</span>All</a><a class="SelectItem" href="javascript:void(0)" onclick="searchBox.OnSelectItem(1)"><span class="SelectionMark">&#160;</span>Files</a><a class="SelectItem" href="javascript:void(0)" onclick="searchBox.OnSelectItem(2)"><span class="SelectionMark">&#160;</span>Functions</a><a class="SelectItem" href="javascript:void(0)" onclick="searchBox.OnSelectItem(3)"><span class="SelectionMark">&#160;</span>Variables</a><a class="SelectItem" href="javascript:void(0)" onclick="searchBox.OnSelectItem(4)"><span class="SelectionMark">&#160;</span>Groups</a></div>

 <!-- iframe showing the search results (closed by default) -->
 <div id="MSearchResultsWindow">
 <iframe src="javascript:void(0)" frameborder="0"
         name="MSearchResults" id="MSearchResults">
 </iframe>
 </div>

 <div class="header">
   <div class="headertitle">
 <div class="title">Cox-Proportional Hazards 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"> <ul>
 <li class="level1">
 <a href="#about">About</a> </li>
 <li class="level1">
 <a href="#help">Online Help</a> </li>
 <li class="level1">
 <a href="#training">Training Function</a> </li>
 <li class="level1">
 <a href="#output">Output Table</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>
 </ul>
 </div><p><a class="anchor" id="about"></a></p>
 <dl class="section user"><dt>About</dt><dd>Proportional-Hazard models enable the comparison of various survival models. These survival models are functions describing the probability of a one-item event (prototypically, this event is death) with respect to time. The interval of time before death occurs is the survival time. Let T be a random variable representing the survival time, with a cumulative probability function P(t). Informally, P(t) is the probability that death has happened before time t.</dd></dl>
 <p><a class="anchor" id="help"></a></p>
 <dl class="section user"><dt>Online Help</dt><dd></dd></dl>
 <p>To display a brief summary of the Cox Proportional Hazards module, call the cox_prop_hazards() function with no argument: </p>
 <pre class="fragment">SELECT madlib.cox_prop_hazards();
 </pre><p><a class="anchor" id="train"></a></p>
 <dl class="section user"><dt>Training Function</dt><dd></dd></dl>
 <p>Following is the syntax for the cox_prop_hazards() training function: </p>
 <pre class="fragment">cox_prop_hazards(
     source_table,
     out_table,
     dependent_variable,
     independent_variable,
     right_censoring_status)
 )
 </pre><p> <b>Arguments</b> </p>
 <dl class="arglist">
 <dt>source_table </dt>
 <dd>TEXT. The name of the table containing input data. </dd>
 <dt>out_table </dt>
 <dd>TEXT. The name of the table where the output model is saved. </dd>
 <dt>dependent_variable </dt>
 <dd>TEXT. A string containing the name of a column that contains an array of numeric values, or a string expression in the format 'array[1, x1, x2, x3]', where <em>x1</em>, <em>x2</em> and <em>x3</em> are column names. Dependent variables refer to the time of death. There is no need to pre-sort the data. </dd>
 <dt>independent_variable </dt>
 <dd>TEXT. The name of the independent variable. </dd>
 <dt>right_censoring_status (optional) </dt>
 <dd>TEXT, default: TRUE for all observations. A string containing an expression that evaluates to the right-censoring status for the observation&mdash;TRUE if the observation is not censored and FALSE if the observation is censored. The string could contain the name of the column containing the right-censoring status, a fixed Boolean expression (i.e., 'true', 'false', '0', '1') that applies to all observations, or a Boolean expression such as 'column_name &lt; 10' that can be evaluated for each observation. </dd>
 </dl>
 <p><a class="anchor" id="notes"></a></p>
 <dl class="section user"><dt>Notes</dt><dd><ul>
 <li>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"').</li>
 <li>The <a class="el" href="cox__prop__hazards_8sql__in.html#a3310cf98478b7c1e400e8fb1b3965d30" title="Compute cox-regression coefficients and diagnostic statistics. ">cox_prop_hazards_regr()</a> function has been deprecated, and cox_prop_hazards() should be used instead.</li>
 </ul>
 </dd></dl>
 <p><a class="anchor" id="output"></a></p>
 <dl class="section user"><dt>Output Table</dt><dd></dd></dl>
 <p>The output is saved in the table named by the <em>out_table</em> argument. It has the following columns: </p>
 <dl class="arglist">
 <dt>coef </dt>
 <dd>FLOAT8[]. Vector of the coefficients. </dd>
 <dt>std_err </dt>
 <dd>FLOAT8[]. Vector of the standard error of the coefficients. </dd>
 <dt>stats </dt>
 <dd>FLOAT8[]. Vector of the statistics of the coefficients. </dd>
 <dt>p_values </dt>
 <dd>FLOAT8[]. Vector of the p-values of the coefficients. </dd>
 </dl>
 <p><a class="anchor" id="examples"></a></p>
 <dl class="section user"><dt>Examples</dt><dd><ol type="1">
 <li>Create an input data set: <pre class="fragment">DROP TABLE IF EXISTS sample_data;
 CREATE TABLE sample_data (
     id INTEGER NOT NULL,
     grp DOUBLE PRECISION,
     wbc DOUBLE PRECISION,
     timedeath INTEGER,
     status BOOLEAN
 );
 COPY sample_data FROM STDIN DELIMITED BY '|';
   0 |   0 | 1.45 |        35 | t
   1 |   0 | 1.47 |        34 | t
   3 |   0 |  2.2 |        32 | t
   4 |   0 | 1.78 |        25 | t
   5 |   0 | 2.57 |        23 | t
   6 |   0 | 2.32 |        22 | t
   7 |   0 | 2.01 |        20 | t
   8 |   0 | 2.05 |        19 | t
   9 |   0 | 2.16 |        17 | t
  10 |   0 |  3.6 |        16 | t
  11 |   1 |  2.3 |        15 | t
  12 |   0 | 2.88 |        13 | t
  13 |   1 |  1.5 |        12 | t
  14 |   0 |  2.6 |        11 | t
  15 |   0 |  2.7 |        10 | t
  16 |   0 |  2.8 |         9 | t
  17 |   1 | 2.32 |         8 | t
  18 |   0 | 4.43 |         7 | t
  19 |   0 | 2.31 |         6 | t
  20 |   1 | 3.49 |         5 | t
  21 |   1 | 2.42 |         4 | t
  22 |   1 | 4.01 |         3 | t
  23 |   1 | 4.91 |         2 | t
  24 |   1 |    5 |         1 | t
 \.
 </pre></li>
 <li>Run the cox regression function: <pre class="fragment">SELECT * FROM madlib.cox_prop_hazards(
     'sample_data',
     'sample_cox',
     'timedeath',
     'ARRAY[grp,wbc]',
     'status');
 </pre></li>
 <li>View the results of the regression: <pre class="fragment">\x on
 SELECT * FROM sample_cox;
 </pre> Results: <pre class="fragment">-[ RECORD 1 ]-----------------------------------------
 coef     | {2.54449137803027,1.67183255057665}
 std_err  | {0.677308807341768,0.387308633304678}
 z_stats  | {3.75676700265663,4.31653830257251}
 p_values | {0.000172122613528057,1.58495189046891e-05}
 </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="cox__prop__hazards_8sql__in.html" title="SQL functions for cox proportional hazards. ">cox_prop_hazards.sql_in</a> documenting the functions</dd></dl>
 <p><a class="anchor" id="Literature"></a></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] John Fox: Cox Proportional-Hazards Regression for Survival Data, Appendix to An R and S-PLUS companion to Applied Regression Feb 2012, <a href="http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-cox-regression.pdf">http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-cox-regression.pdf</a></p>
 <p>[2] Stephen J Walters: What is a Cox model? <a href="http://www.medicine.ox.ac.uk/bandolier/painres/download/whatis/cox_model.pdf">http://www.medicine.ox.ac.uk/bandolier/painres/download/whatis/cox_model.pdf</a></p>
 <p><a class="anchor" id="background"></a></p>
 <dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
 <p>Generally, proportional-hazard models start with a list of \( \boldsymbol n \) observations, each with \( \boldsymbol m \) covariates and a time of death. From this \( \boldsymbol n \times m \) matrix, we would like to derive the correlation between the covariates and the hazard function. This amounts to finding the parameters \( \boldsymbol \beta \) that best fit the model described below.</p>
 <p>Let us define:</p>
 <ul>
 <li>\( \boldsymbol t \in \mathbf R^{m} \) denote the vector of observed dependent variables, with \( n \) rows.</li>
 <li>\( X \in \mathbf R^{m} \) denote the design matrix with \( m \) columns and \( n \) rows, containing all observed vectors of independent variables \( \boldsymbol x_i \) as rows.</li>
 <li>\( R(t_i) \) denote the set of observations still alive at time \( t_i \)</li>
 </ul>
 <p>Note that this model <b>does not</b> include a <b>constant term</b>, and the data cannot contain a column of 1s.</p>
 <p>By definition, </p>
 <p class="formulaDsp">
 \[ P[T_k = t_i | \boldsymbol R(t_i)] = \frac{e^{\beta^T x_k} }{ \sum_{j \in R(t_i)} e^{\beta^T x_j}}. \,. \]
 </p>
 <p>The <b>partial likelihood </b>function can now be generated as the product of conditional probabilities: </p>
 <p class="formulaDsp">
 \[ \mathcal L = \prod_{i = 1}^n \left( \frac{e^{\beta^T x_i}}{ \sum_{j \in R(t_i)} e^{\beta^T x_j}} \right). \]
 </p>
 <p>The log-likelihood form of this equation is </p>
 <p class="formulaDsp">
 \[ L = \sum_{i = 1}^n \left[ \beta^T x_i - \log\left(\sum_{j \in R(t_i)} e^{\beta^T x_j }\right) \right]. \]
 </p>
 <p>Using this score function and Hessian matrix, the partial likelihood can be maximized using the <b> Newton-Raphson algorithm</b>. <b>Breslow's method</b> is used to resolved tied times of deaths. The time of death for two records are considered "equal" if they differ by less than 1.0e-6</p>
 <p>The inverse of the Hessian matrix, evaluated at the estimate of \( \boldsymbol \beta \), can be used as an <b>approximate variance-covariance matrix </b> for the estimate, and used to produce approximate <b>standard errors</b> for the regression coefficients.</p>
 <p class="formulaDsp">
 \[ \mathit{se}(c_i) = \left( (H)^{-1} \right)_{ii} \,. \]
 </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 condition number is computed as \( \kappa(H) \) 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>
 </div><!-- contents -->
 </div><!-- doc-content -->
 <!-- start footer part -->
 <div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
   <ul>
     <li class="footer">Generated on Thu Jan 9 2014 20:35:40 for MADlib by
     <a href="http://www.doxygen.org/index.html">
     <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.4 </li>
   </ul>
 </div>
 </body>
 </html>
	<!-- HTML header for doxygen 1.8.4-->
	<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
	<html xmlns="http://www.w3.org/1999/xhtml">
	<head>
	<meta http-equiv="Content-Type" content="text/xhtml;charset=UTF-8"/>
	<meta http-equiv="X-UA-Compatible" content="IE=9"/>
	<meta name="generator" content="Doxygen 1.8.4"/>
	<meta name="keywords" content="madlib,postgres,greenplum,machine learning,data mining,deep learning,ensemble methods,data science,market basket analysis,affinity analysis,pca,lda,regression,elastic net,huber white,proportional hazards,k-means,latent dirichlet allocation,bayes,support vector machines,svm"/>
	<title>MADlib: Cox-Proportional Hazards Regression</title>
	<link href="tabs.css" rel="stylesheet" type="text/css"/>
	<script type="text/javascript" src="jquery.js"></script>
	<script type="text/javascript" src="dynsections.js"></script>
	<link href="navtree.css" rel="stylesheet" type="text/css"/>
	<script type="text/javascript" src="resize.js"></script>
	<script type="text/javascript" src="navtree.js"></script>
	<script type="text/javascript">
	$(document).ready(initResizable);
	$(window).load(resizeHeight);
	</script>
	<link href="search/search.css" rel="stylesheet" type="text/css"/>
	<script type="text/javascript" src="search/search.js"></script>
	<script type="text/javascript">
	$(document).ready(function() { searchBox.OnSelectItem(0); });
	</script>
	<script type="text/x-mathjax-config">
	MathJax.Hub.Config({
	extensions: ["tex2jax.js", "TeX/AMSmath.js", "TeX/AMSsymbols.js"],
	jax: ["input/TeX","output/HTML-CSS"],
	});
	</script><script src="../mathjax/MathJax.js"></script>
	<link href="doxygen.css" rel="stylesheet" type="text/css" />
	<link href="madlib_extra.css" rel="stylesheet" type="text/css"/>
	<!-- google analytics -->
	<script>
	(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]\|\|function(){
	(i[r].q=i[r].q\|\|[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
	m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
	})(window,document,'script','//www.google-analytics.com/analytics.js','ga');
	ga('create', 'UA-45382226-1', 'auto');
	ga('send', 'pageview');
	</script>
	</head>
	<body>
	<div id="top"><!-- do not remove this div, it is closed by doxygen! -->
	<div id="titlearea">
	<table cellspacing="0" cellpadding="0">
	<tbody>
	<tr style="height: 56px;">
	<td style="padding-left: 0.5em;">
	<div id="projectname">MADlib
	<span id="projectnumber">1.2</span> <span style="font-size:10pt; font-style:italic"><a href="../latest/./group__grp__cox__prop__hazards.html"> A newer version is available</a></span>
	</div>
	<div id="projectbrief">User Documentation</div>
	</td>
	<!--BEGIN VERSIONS LINKS-->
	<td style="padding-left: 0.5em;">
	<div class="versionlist"><ul>
	<li class="head">More versions:</li>
	<li><a href="../v1.1/index.html">v1.1</li>
	<li><a href="../v1.0/index.html">v1.0</li>
	<li><a href="../v0.7/index.html">v0.7</li>
	<li><a href="../v0.5/index.html">v0.5</li></ul>
	</div>
	</td>
	<td> <div id="MSearchBox" class="MSearchBoxInactive">
	<span class="left">
	<img id="MSearchSelect" src="search/mag_sel.png"
	onmouseover="return searchBox.OnSearchSelectShow()"
	onmouseout="return searchBox.OnSearchSelectHide()"
	alt=""/>
	<input type="text" id="MSearchField" value="Search" accesskey="S"
	onfocus="searchBox.OnSearchFieldFocus(true)"
	onblur="searchBox.OnSearchFieldFocus(false)"
	onkeyup="searchBox.OnSearchFieldChange(event)"/>
	</span><span class="right">
	<a id="MSearchClose" href="javascript:searchBox.CloseResultsWindow()"><img id="MSearchCloseImg" border="0" src="search/close.png" alt=""/></a>
	</span>
	</div>
	</td>
	</tr>
	</tbody>
	</table>
	</div>
	<!-- end header part -->
	<!-- Generated by Doxygen 1.8.4 -->
	<script type="text/javascript">
	var searchBox = new SearchBox("searchBox", "search",false,'Search');
	</script>
	</div><!-- top -->
	<div id="side-nav" class="ui-resizable side-nav-resizable">
	<div id="nav-tree">
	<div id="nav-tree-contents">
	<div id="nav-sync" class="sync"></div>
	</div>
	</div>
	<div id="splitbar" style="-moz-user-select:none;"
	class="ui-resizable-handle">
	</div>
	</div>
	<script type="text/javascript">
	$(document).ready(function(){initNavTree('group__grp__cox__prop__hazards.html','');});
	</script>
	<div id="doc-content">
	<!-- window showing the filter options -->
	<div id="MSearchSelectWindow"
	onmouseover="return searchBox.OnSearchSelectShow()"
	onmouseout="return searchBox.OnSearchSelectHide()"
	onkeydown="return searchBox.OnSearchSelectKey(event)">
	<a class="SelectItem" href="javascript:void(0)" onclick="searchBox.OnSelectItem(0)"><span class="SelectionMark"> </span>All</a><a class="SelectItem" href="javascript:void(0)" onclick="searchBox.OnSelectItem(1)"><span class="SelectionMark"> </span>Files</a><a class="SelectItem" href="javascript:void(0)" onclick="searchBox.OnSelectItem(2)"><span class="SelectionMark"> </span>Functions</a><a class="SelectItem" href="javascript:void(0)" onclick="searchBox.OnSelectItem(3)"><span class="SelectionMark"> </span>Variables</a><a class="SelectItem" href="javascript:void(0)" onclick="searchBox.OnSelectItem(4)"><span class="SelectionMark"> </span>Groups</a></div>

	<!-- iframe showing the search results (closed by default) -->
	<div id="MSearchResultsWindow">
	<iframe src="javascript:void(0)" frameborder="0"
	name="MSearchResults" id="MSearchResults">
	</iframe>
	</div>

	<div class="header">
	<div class="headertitle">
	<div class="title">Cox-Proportional Hazards 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"> <ul>
	<li class="level1">
	<a href="#about">About</a> </li>
	<li class="level1">
	<a href="#help">Online Help</a> </li>
	<li class="level1">
	<a href="#training">Training Function</a> </li>
	<li class="level1">
	<a href="#output">Output Table</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>
	</ul>
	</div><p><a class="anchor" id="about"></a></p>
	<dl class="section user"><dt>About</dt><dd>Proportional-Hazard models enable the comparison of various survival models. These survival models are functions describing the probability of a one-item event (prototypically, this event is death) with respect to time. The interval of time before death occurs is the survival time. Let T be a random variable representing the survival time, with a cumulative probability function P(t). Informally, P(t) is the probability that death has happened before time t.</dd></dl>
	<p><a class="anchor" id="help"></a></p>
	<dl class="section user"><dt>Online Help</dt><dd></dd></dl>
	<p>To display a brief summary of the Cox Proportional Hazards module, call the cox_prop_hazards() function with no argument: </p>
	<pre class="fragment">SELECT madlib.cox_prop_hazards();
	</pre><p><a class="anchor" id="train"></a></p>
	<dl class="section user"><dt>Training Function</dt><dd></dd></dl>
	<p>Following is the syntax for the cox_prop_hazards() training function: </p>
	<pre class="fragment">cox_prop_hazards(
	source_table,
	out_table,
	dependent_variable,
	independent_variable,
	right_censoring_status)
	)
	</pre><p> <b>Arguments</b> </p>
	<dl class="arglist">
	<dt>source_table </dt>
	<dd>TEXT. The name of the table containing input data. </dd>
	<dt>out_table </dt>
	<dd>TEXT. The name of the table where the output model is saved. </dd>
	<dt>dependent_variable </dt>
	<dd>TEXT. A string containing the name of a column that contains an array of numeric values, or a string expression in the format 'array[1, x1, x2, x3]', where <em>x1</em>, <em>x2</em> and <em>x3</em> are column names. Dependent variables refer to the time of death. There is no need to pre-sort the data. </dd>
	<dt>independent_variable </dt>
	<dd>TEXT. The name of the independent variable. </dd>
	<dt>right_censoring_status (optional) </dt>
	<dd>TEXT, default: TRUE for all observations. A string containing an expression that evaluates to the right-censoring status for the observation—TRUE if the observation is not censored and FALSE if the observation is censored. The string could contain the name of the column containing the right-censoring status, a fixed Boolean expression (i.e., 'true', 'false', '0', '1') that applies to all observations, or a Boolean expression such as 'column_name < 10' that can be evaluated for each observation. </dd>
	</dl>
	<p><a class="anchor" id="notes"></a></p>
	<dl class="section user"><dt>Notes</dt><dd><ul>
	<li>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"').</li>
	<li>The <a class="el" href="cox__prop__hazards_8sql__in.html#a3310cf98478b7c1e400e8fb1b3965d30" title="Compute cox-regression coefficients and diagnostic statistics. ">cox_prop_hazards_regr()</a> function has been deprecated, and cox_prop_hazards() should be used instead.</li>
	</ul>
	</dd></dl>
	<p><a class="anchor" id="output"></a></p>
	<dl class="section user"><dt>Output Table</dt><dd></dd></dl>
	<p>The output is saved in the table named by the <em>out_table</em> argument. It has the following columns: </p>
	<dl class="arglist">
	<dt>coef </dt>
	<dd>FLOAT8[]. Vector of the coefficients. </dd>
	<dt>std_err </dt>
	<dd>FLOAT8[]. Vector of the standard error of the coefficients. </dd>
	<dt>stats </dt>
	<dd>FLOAT8[]. Vector of the statistics of the coefficients. </dd>
	<dt>p_values </dt>
	<dd>FLOAT8[]. Vector of the p-values of the coefficients. </dd>
	</dl>
	<p><a class="anchor" id="examples"></a></p>
	<dl class="section user"><dt>Examples</dt><dd><ol type="1">
	<li>Create an input data set: <pre class="fragment">DROP TABLE IF EXISTS sample_data;
	CREATE TABLE sample_data (
	id INTEGER NOT NULL,
	grp DOUBLE PRECISION,
	wbc DOUBLE PRECISION,
	timedeath INTEGER,
	status BOOLEAN
	);
	COPY sample_data FROM STDIN DELIMITED BY '\|';
	0 \| 0 \| 1.45 \| 35 \| t
	1 \| 0 \| 1.47 \| 34 \| t
	3 \| 0 \| 2.2 \| 32 \| t
	4 \| 0 \| 1.78 \| 25 \| t
	5 \| 0 \| 2.57 \| 23 \| t
	6 \| 0 \| 2.32 \| 22 \| t
	7 \| 0 \| 2.01 \| 20 \| t
	8 \| 0 \| 2.05 \| 19 \| t
	9 \| 0 \| 2.16 \| 17 \| t
	10 \| 0 \| 3.6 \| 16 \| t
	11 \| 1 \| 2.3 \| 15 \| t
	12 \| 0 \| 2.88 \| 13 \| t
	13 \| 1 \| 1.5 \| 12 \| t
	14 \| 0 \| 2.6 \| 11 \| t
	15 \| 0 \| 2.7 \| 10 \| t
	16 \| 0 \| 2.8 \| 9 \| t
	17 \| 1 \| 2.32 \| 8 \| t
	18 \| 0 \| 4.43 \| 7 \| t
	19 \| 0 \| 2.31 \| 6 \| t
	20 \| 1 \| 3.49 \| 5 \| t
	21 \| 1 \| 2.42 \| 4 \| t
	22 \| 1 \| 4.01 \| 3 \| t
	23 \| 1 \| 4.91 \| 2 \| t
	24 \| 1 \| 5 \| 1 \| t
	\.
	</pre></li>
	<li>Run the cox regression function: <pre class="fragment">SELECT * FROM madlib.cox_prop_hazards(
	'sample_data',
	'sample_cox',
	'timedeath',
	'ARRAY[grp,wbc]',
	'status');
	</pre></li>
	<li>View the results of the regression: <pre class="fragment">\x on
	SELECT * FROM sample_cox;
	</pre> Results: <pre class="fragment">-[ RECORD 1 ]-----------------------------------------
	coef \| {2.54449137803027,1.67183255057665}
	std_err \| {0.677308807341768,0.387308633304678}
	z_stats \| {3.75676700265663,4.31653830257251}
	p_values \| {0.000172122613528057,1.58495189046891e-05}
	</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="cox__prop__hazards_8sql__in.html" title="SQL functions for cox proportional hazards. ">cox_prop_hazards.sql_in</a> documenting the functions</dd></dl>
	<p><a class="anchor" id="Literature"></a></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] John Fox: Cox Proportional-Hazards Regression for Survival Data, Appendix to An R and S-PLUS companion to Applied Regression Feb 2012, <a href="http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-cox-regression.pdf">http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-cox-regression.pdf</a></p>
	<p>[2] Stephen J Walters: What is a Cox model? <a href="http://www.medicine.ox.ac.uk/bandolier/painres/download/whatis/cox_model.pdf">http://www.medicine.ox.ac.uk/bandolier/painres/download/whatis/cox_model.pdf</a></p>
	<p><a class="anchor" id="background"></a></p>
	<dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
	<p>Generally, proportional-hazard models start with a list of \( \boldsymbol n \) observations, each with \( \boldsymbol m \) covariates and a time of death. From this \( \boldsymbol n \times m \) matrix, we would like to derive the correlation between the covariates and the hazard function. This amounts to finding the parameters \( \boldsymbol \beta \) that best fit the model described below.</p>
	<p>Let us define:</p>
	<ul>
	<li>\( \boldsymbol t \in \mathbf R^{m} \) denote the vector of observed dependent variables, with \( n \) rows.</li>
	<li>\( X \in \mathbf R^{m} \) denote the design matrix with \( m \) columns and \( n \) rows, containing all observed vectors of independent variables \( \boldsymbol x_i \) as rows.</li>
	<li>\( R(t_i) \) denote the set of observations still alive at time \( t_i \)</li>
	</ul>
	<p>Note that this model <b>does not</b> include a <b>constant term</b>, and the data cannot contain a column of 1s.</p>
	<p>By definition, </p>
	<p class="formulaDsp">
	\[ P[T_k = t_i \| \boldsymbol R(t_i)] = \frac{e^{\beta^T x_k} }{ \sum_{j \in R(t_i)} e^{\beta^T x_j}}. \,. \]
	</p>
	<p>The <b>partial likelihood </b>function can now be generated as the product of conditional probabilities: </p>
	<p class="formulaDsp">
	\[ \mathcal L = \prod_{i = 1}^n \left( \frac{e^{\beta^T x_i}}{ \sum_{j \in R(t_i)} e^{\beta^T x_j}} \right). \]
	</p>
	<p>The log-likelihood form of this equation is </p>
	<p class="formulaDsp">
	\[ L = \sum_{i = 1}^n \left[ \beta^T x_i - \log\left(\sum_{j \in R(t_i)} e^{\beta^T x_j }\right) \right]. \]
	</p>
	<p>Using this score function and Hessian matrix, the partial likelihood can be maximized using the <b> Newton-Raphson algorithm</b>. <b>Breslow's method</b> is used to resolved tied times of deaths. The time of death for two records are considered "equal" if they differ by less than 1.0e-6</p>
	<p>The inverse of the Hessian matrix, evaluated at the estimate of \( \boldsymbol \beta \), can be used as an <b>approximate variance-covariance matrix </b> for the estimate, and used to produce approximate <b>standard errors</b> for the regression coefficients.</p>
	<p class="formulaDsp">
	\[ \mathit{se}(c_i) = \left( (H)^{-1} \right)_{ii} \,. \]
	</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 condition number is computed as \( \kappa(H) \) 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>
	</div><!-- contents -->
	</div><!-- doc-content -->
	<!-- start footer part -->
	<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
	<ul>
	<li class="footer">Generated on Thu Jan 9 2014 20:35:40 for MADlib by
	<a href="http://www.doxygen.org/index.html">
	<img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.4 </li>
	</ul>
	</div>
	</body>
	</html>