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 <div class="title">Cox-Proportional Hazards Regression</div>  </div>
 <div class="ingroups"><a class="el" href="group__grp__suplearn.html">Supervised Learning</a></div></div>
 <div class="contents">
 <div id="dynsection-0" onclick="return toggleVisibility(this)" class="dynheader closed" style="cursor:pointer;">
   <img id="dynsection-0-trigger" src="closed.png" alt="+"/> Collaboration diagram for Cox-Proportional Hazards Regression:</div>
 <div id="dynsection-0-summary" class="dynsummary" style="display:block;">
 </div>
 <div id="dynsection-0-content" class="dyncontent" style="display:none;">
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 <dl class="user"><dt><b>About:</b></dt><dd>Proportional-Hazard models enable the comparison of various survival models. These survival models are functions describing the probability of an 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>Generally, applications 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>
 <dl class="user"><dt><b>Input:</b></dt><dd></dd></dl>
 <p>The training data is expected to be of the following form:<br/>
  </p>
 <pre>{TABLE|VIEW} <em>sourceName</em> (
     ...
     <em>dependentVariable</em> FLOAT8,
     <em>independentVariables</em> FLOAT8[],
     ...
 )</pre><p> Note: Dependent Variables refer to the time of death. There is no need to pre-sort the data. Additionally, all the data is assumed</p>
 <dl class="user"><dt><b>Usage:</b></dt><dd><ul>
 <li>Get vector of coefficients \( \boldsymbol \beta \) and all diagnostic statistics:<br/>
  <pre>SELECT * FROM <a class="el" href="cox__prop__hazards_8sql__in.html#a9c04e1fd1353bb3cfb942b6251851179">cox_prop_hazards</a>(
     '<em>sourceName</em>', '<em>dependentVariable</em>', '<em>independentVariables</em>'
     [, <em>numberOfIterations</em> [, '<em>optimizer</em>' [, <em>precision</em> ] ] ]
 );</pre> Output: Output: <pre>coef | log_likelihood | std_err | z_stats | p_values  | condition_no | num_iterations
                                                ...
 </pre></li>
 <li>Get vector of coefficients \( \boldsymbol \beta \):<br/>
  <pre>SELECT (<a class="el" href="cox__prop__hazards_8sql__in.html#a9c04e1fd1353bb3cfb942b6251851179">cox_prop_hazards</a>('<em>sourceName</em>', '<em>dependentVariable</em>', '<em>independentVariables</em>')).coef;</pre></li>
 <li>Get a subset of the output columns, e.g., only the array of coefficients \( \boldsymbol \beta \), the log-likelihood of determination: <pre>SELECT coef, log_likelihood
 FROM <a class="el" href="cox__prop__hazards_8sql__in.html#a9c04e1fd1353bb3cfb942b6251851179">cox_prop_hazards</a>('<em>sourceName</em>', '<em>dependentVariable</em>', '<em>independentVariables</em>');</pre></li>
 </ul>
 </dd></dl>
 <dl class="user"><dt><b>Examples:</b></dt><dd></dd></dl>
 <ol type="1">
 <li>Create the sample data set: <div class="fragment"><pre class="fragment">
 sql&gt; SELECT * FROM data;
       val   | time
 ------------|--------------
  {0,1.95}   | 35
  {0,2.20}   | 28
  {1,1.45}   | 32
  {1,5.25}   | 31
  {1,0.38}   | 21
 ...
 </pre></div></li>
 <li>Run the cox regression function: <div class="fragment"><pre class="fragment">
 sql&gt; SELECT * FROM cox_prop_hazards('data', 'val', 'time', 100, 'newto', 0.001);
 ---------------|--------------------------------------------------------------
 coef           | {0.881089349817059,-0.0756817768938055}
 log_likelihood | -4.46535157957394
 std_err        | {1.16954914708414,0.338426252282655}
 z_stats        | {0.753356711368689,-0.223628410729811}
 p_values       | {0.451235588326831,0.823046454908087}
 condition_no   | 12.1135391339082
 num_iterations | 4

 </pre></div></li>
 </ol>
 <dl class="user"><dt><b>Literature:</b></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>
 <dl class="note"><dt><b>Note:</b></dt><dd>Source and column names have to be passed as strings (due to limitations of the SQL syntax).</dd></dl>
 <dl class="see"><dt><b>See also:</b></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 SQL functions) </dd></dl>
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	<dl class="user"><dt><b>About:</b></dt><dd>Proportional-Hazard models enable the comparison of various survival models. These survival models are functions describing the probability of an 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>Generally, applications 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>
	<dl class="user"><dt><b>Input:</b></dt><dd></dd></dl>
	<p>The training data is expected to be of the following form:<br/>
	</p>
	<pre>{TABLE\|VIEW} <em>sourceName</em> (
	...
	<em>dependentVariable</em> FLOAT8,
	<em>independentVariables</em> FLOAT8[],
	...
	)</pre><p> Note: Dependent Variables refer to the time of death. There is no need to pre-sort the data. Additionally, all the data is assumed</p>
	<dl class="user"><dt><b>Usage:</b></dt><dd><ul>
	<li>Get vector of coefficients \( \boldsymbol \beta \) and all diagnostic statistics:<br/>
	<pre>SELECT * FROM <a class="el" href="cox__prop__hazards_8sql__in.html#a9c04e1fd1353bb3cfb942b6251851179">cox_prop_hazards</a>(
	'<em>sourceName</em>', '<em>dependentVariable</em>', '<em>independentVariables</em>'
	[, <em>numberOfIterations</em> [, '<em>optimizer</em>' [, <em>precision</em> ] ] ]
	);</pre> Output: Output: <pre>coef \| log_likelihood \| std_err \| z_stats \| p_values \| condition_no \| num_iterations
	...
	</pre></li>
	<li>Get vector of coefficients \( \boldsymbol \beta \):<br/>
	<pre>SELECT (<a class="el" href="cox__prop__hazards_8sql__in.html#a9c04e1fd1353bb3cfb942b6251851179">cox_prop_hazards</a>('<em>sourceName</em>', '<em>dependentVariable</em>', '<em>independentVariables</em>')).coef;</pre></li>
	<li>Get a subset of the output columns, e.g., only the array of coefficients \( \boldsymbol \beta \), the log-likelihood of determination: <pre>SELECT coef, log_likelihood
	FROM <a class="el" href="cox__prop__hazards_8sql__in.html#a9c04e1fd1353bb3cfb942b6251851179">cox_prop_hazards</a>('<em>sourceName</em>', '<em>dependentVariable</em>', '<em>independentVariables</em>');</pre></li>
	</ul>
	</dd></dl>
	<dl class="user"><dt><b>Examples:</b></dt><dd></dd></dl>
	<ol type="1">
	<li>Create the sample data set: <div class="fragment"><pre class="fragment">
	sql> SELECT * FROM data;
	val \| time
	------------\|--------------
	{0,1.95} \| 35
	{0,2.20} \| 28
	{1,1.45} \| 32
	{1,5.25} \| 31
	{1,0.38} \| 21
	...
	</pre></div></li>
	<li>Run the cox regression function: <div class="fragment"><pre class="fragment">
	sql> SELECT * FROM cox_prop_hazards('data', 'val', 'time', 100, 'newto', 0.001);
	---------------\|--------------------------------------------------------------
	coef \| {0.881089349817059,-0.0756817768938055}
	log_likelihood \| -4.46535157957394
	std_err \| {1.16954914708414,0.338426252282655}
	z_stats \| {0.753356711368689,-0.223628410729811}
	p_values \| {0.451235588326831,0.823046454908087}
	condition_no \| 12.1135391339082
	num_iterations \| 4

	</pre></div></li>
	</ol>
	<dl class="user"><dt><b>Literature:</b></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>
	<dl class="note"><dt><b>Note:</b></dt><dd>Source and column names have to be passed as strings (due to limitations of the SQL syntax).</dd></dl>
	<dl class="see"><dt><b>See also:</b></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 SQL functions) </dd></dl>
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