Cox-Proportional Hazards Regression

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 the occurrence of death can be called 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.

- Training Function

Following is the syntax for the coxph_train() training function:

coxph_train( source_table, output_table, dependent_variable, independent_variable, right_censoring_status, strata, optimizer_params )

**Arguments**

- source_table
- TEXT. The name of the table containing input data.
- output_table
TEXT. The name of the table where the output model is saved. The output is saved in the table named by the

*output_table*argument. It has the following columns:coef FLOAT8[]. Vector of the coefficients. loglikelihood FLOAT8. Log-likelihood value of the MLE estimate. std_err FLOAT8[]. Vector of the standard error of the coefficients. stats FLOAT8[]. Vector of the statistics of the coefficients. p_values FLOAT8[]. Vector of the p-values of the coefficients. hessian FLOAT8[]. The Hessian matrix computed using the final solution. num_iterations INTEGER. The number of iterations performed by the optimizer. Additionally, a summary output table is generated that contains a summary of the parameters used for building the Cox model. It is stored in a table named <output_table>_summary. It has the following columns:

source_table The source table name. dependent_variable The dependent variable name. independent_variable The independent variable name. right_censoring_status The right censoring status strata The stratification columns num_processed The number of rows that were actually used in the computation. num_missing_rows_skipped The number of rows that were skipped in the computation due to NULL values in them. - dependent_variable
- 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
*x1*,*x2*and*x3*are column names. Dependent variables refer to the time of death. There is no need to pre-sort the data. - independent_variable
- TEXT. The name of the independent variable.
- right_censoring_status (optional)
- 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.
- strata (optional)
- VARCHAR, default: NULL, which does not do any stratifications. A string of comma-separated column names that are the strata ID variables used to do stratification.
- optimizer_params (optional)
VARCHAR, default: NULL, which uses the default values of optimizer parameters: max_iter=100, optimizer=newton, tolerance=1e-8, array_agg_size=10000000, sample_size=1000000. It should be a string that contains 'key=value' pairs separated by commas. The meanings of these parameters are:

- max_iter — The maximum number of iterations. The computation stops if the number of iterations exceeds this, which usually means that there is no convergence.
- optimizer — The optimization method. Right now, "newton" is the only one supported.
- tolerance — The stopping criteria. When the difference between the log-likelihoods of two consecutive iterations is smaller than this number, the computation has already converged and stops.
- array_agg_size — To speed up the computation, the original data table is cut into multiple pieces, and each pieces of the data is aggregated into one big row. In the process of computation, the whole big row is loaded into memory and thus speed up the computation. This parameter controls approximately how many numbers we want to put into one big row. Larger value of array_agg_size may speed up more, but the size of the big row cannot exceed 1GB due to the restriction of PostgreSQL databases.
- sample_size — To cut the data into approximate equal pieces, we first sample the data, and then find out the break points using this sampled data. A larger sample_size produces more accurate break points.

- Proportional Hazards Assumption Test Function

The cox_zph() function tests the proportional hazards assumption (PHA) of a Cox regression.

Proportional-hazard models enable the comparison of various survival models. These PH models, however, assume that the hazard for a given individual is a fixed proportion of the hazard for any other individual, and the ratio of the hazards is constant across time. MADlib does not currently have support for performing any transformation of the time to compute the correlation.

The cox_zph() function is used to test this assumption by computing the correlation of the residual of the coxph_train() model with time.

Following is the syntax for the cox_zph() function:

cox_zph(cox_model_table, output_table)

**Arguments**

- cox_model_table
TEXT. The name of the table containing the Cox Proportional-Hazards model.

- output_table
- TEXT. The name of the table where the test statistics are saved. The output table is named by the
*output_table*argument and has the following columns:covariate TEXT. The independent variables. rho FLOAT8[]. Vector of the correlation coefficients between survival time and the scaled Schoenfeld residuals. chi_square FLOAT8[]. Chi-square test statistic for the correlation analysis. p_value FLOAT8[]. Two-side p-value for the chi-square statistic.

Additionally, the residual values are outputted to the table named *output_table*_residual. The table contains the following columns:

<dep_column_name> | FLOAT8. Time values (dependent variable) present in the original source table. |
---|---|

residual | FLOAT8[]. Difference between the original covariate values and the expectation of the covariates obtained from the coxph_train model. |

scaled_residual | Residual values scaled by the variance of the coefficients. |

- Notes

- Table names can be optionally schema qualified (current_schemas() is used if a schema name is not provided) and 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—'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"'.
- The cox_prop_hazards_regr() and cox_prop_hazards() functions have been deprecated; coxph_train() should be used instead.

- Prediction Function
- The prediction function is provided to calculate the linear predictionors, risk or the linear terms for the given prediction data. It has the following syntax:
coxph_predict(model_table, source_table, id_col_name, output_table, pred_type, reference)

**Arguments**

- model_table
TEXT. Name of the table containing the cox model.

- source_table
TEXT. Name of the table containing the prediction data.

- id_col_name
TEXT. Name of the id column in the source table.

- output_table
TEXT. Name of the table to store the prediction results in. The output table is named by the

*output_table*argument and has the following columns:id TEXT. The id column name from the source table. predicted_result DOUBLE PRECISION. Result of prediction based of the value of the prediction type parameter. - pred_type
TEXT, OPTIONAL. Type of prediction. This can be one of 'linear_predictors', 'risk', or 'terms'. DEFAULT='linear_predictors'.

- 'linear_predictors' calculates the dot product of the independent variables and the coefficients.
- 'risk' is the exponentiated value of the linear prediction.
- 'terms' correspond to the linear terms obtained by multiplying the independent variables with their corresponding coefficients values (without further calculating the sum of these terms)

- reference
- TEXT, OPTIONAL. Reference level to use for centering predictions. Can be one of 'strata', 'overall'. DEFAULT='strata'. Note that R uses 'sample' instead of 'overall' when referring to the overall mean value of the covariates as being the reference level.

- Examples

- Create an input data set.
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 WITH DELIMITER '|'; 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 \.

- Run the Cox regression function.
DROP TABLE IF EXISTS sample_cox, sample_cox_summary; SELECT madlib.coxph_train( 'sample_data', 'sample_cox', 'timedeath', 'ARRAY[grp,wbc]', 'status' );

- View the results of the regression.
\x on SELECT * FROM sample_cox;

Results:-[ RECORD 1 ]--+---------------------------------------------------------------------------- coef | {2.54407073265254,1.67172094779487} loglikelihood | -37.8532498733 std_err | {0.677180599294897,0.387195514577534} z_stats | {3.7568570855419,4.31751114064138} p_values | {0.000172060691513886,1.5779844638453e-05} hessian | {{2.78043065745617,-2.25848560642414},{-2.25848560642414,8.50472838284472}} num_iterations | 5

- Computing predictions using cox model. (This example uses the original data table to perform the prediction. Typically a different test dataset with the same features as the original training dataset would be used.)
\x off -- Display the linear predictors for the original dataset DROP TABLE IF EXISTS sample_pred; SELECT madlib.coxph_predict('sample_cox', 'sample_data', 'id', 'sample_pred'); SELECT * FROM sample_pred;

id | predicted_value ----+-------------------- 0 | -2.97110918125034 4 | -2.41944126847803 6 | -1.5167119566688 8 | -1.96807661257341 10 | 0.623090856508638 12 | -0.58054822590367 14 | -1.04863009128623 16 | -0.714285901727259 18 | 2.01061924317838 20 | 2.98327228490375 22 | 3.85256717775708 24 | 5.507570916074 1 | -2.93767476229444 3 | -1.71731847040418 5 | -1.09878171972008 7 | -2.03494545048521 9 | -1.78418730831598 15 | -0.881457996506747 19 | -1.53342916614675 11 | 0.993924357027849 13 | -0.343452401208048 17 | 1.02735877598375 21 | 1.19453087076323 23 | 5.35711603077246 (24 rows)

-- Display the relative risk for the original dataset DROP TABLE IF EXISTS sample_pred; SELECT madlib.coxph_predict('sample_cox', 'sample_data', 'id', 'sample_pred', 'risk'); SELECT * FROM sample_pred;

id | predicted_value ----+-------------------- 1 | 0.0529887971503509 3 | 0.179546963459175 5 | 0.33327686110022 7 | 0.130687611255372 9 | 0.167933483703554 15 | 0.414178600294289 19 | 0.215794402223054 11 | 2.70181658768287 13 | 0.709317242984782 17 | 2.79367735395696 21 | 3.30200833843654 23 | 212.112338046551 0 | 0.0512464372091503 4 | 0.0889713146524469 6 | 0.219432204682557 8 | 0.139725343898993 10 | 1.86468261037506 12 | 0.559591499901241 14 | 0.350417460388247 16 | 0.489541567796517 18 | 7.46794038691975 20 | 19.7523463121038 22 | 47.1138577624204 24 | 246.551504798816 (24 rows)

- Run the test for Proportional Hazards assumption to obtain correlation between residuals and time.
SELECT madlib.cox_zph( 'sample_cox', 'sample_zph_output' );

- View results of the PHA test.
\x on SELECT * FROM sample_zph_output;

Results:-[ RECORD 1 ]----------------------------------------- covariate | ARRAY[grp,wbc] rho | {0.00237308357328641,0.0375600568840431} chi_square | {0.000100675718191977,0.0232317400546175} p_value | {0.991994376850758,0.878855984657948}

- Technical Background

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.

Let us define:

- \( \boldsymbol t \in \mathbf R^{m} \) denote the vector of observed dependent variables, with \( n \) rows.
- \( X \in \mathbf R^{m} \) denote the design matrix with \( m \) columns and \( n \) rows, containing all observed vectors of independent variables \( \boldsymbol x_i \) as rows.
- \( R(t_i) \) denote the set of observations still alive at time \( t_i \)

Note that this model **does not** include a **constant term**, and the data cannot contain a column of 1s.

By definition,

\[ P[T_k = t_i | \boldsymbol R(t_i)] = \frac{e^{\beta^T x_k} }{ \sum_{j \in R(t_i)} e^{\beta^T x_j}}. \,. \]

The **partial likelihood **function can now be generated as the product of conditional probabilities:

\[ \mathcal L = \prod_{i = 1}^n \left( \frac{e^{\beta^T x_i}}{ \sum_{j \in R(t_i)} e^{\beta^T x_j}} \right). \]

The log-likelihood form of this equation is

\[ L = \sum_{i = 1}^n \left[ \beta^T x_i - \log\left(\sum_{j \in R(t_i)} e^{\beta^T x_j }\right) \right]. \]

Using this score function and Hessian matrix, the partial likelihood can be maximized using the ** Newton-Raphson algorithm**. **Breslow's method** 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

The inverse of the Hessian matrix, evaluated at the estimate of \( \boldsymbol \beta \), can be used as an **approximate variance-covariance matrix ** for the estimate, and used to produce approximate **standard errors** for the regression coefficients.

\[ \mathit{se}(c_i) = \left( (H)^{-1} \right)_{ii} \,. \]

The Wald z-statistic is

\[ z_i = \frac{c_i}{\mathit{se}(c_i)} \,. \]

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

\[ p_i = \Pr(|Z| \geq |z_i|) = 2 \cdot (1 - F( |z_i| )) \]

where \( Z \) is a standard normally distributed random variable.

The condition number is computed as \( \kappa(H) \) during the iteration immediately *preceding* 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.

- Literature

A somewhat random selection of nice write-ups, with valuable pointers into further literature:

[1] John Fox: Cox Proportional-Hazards Regression for Survival Data, Appendix to An R and S-PLUS companion to Applied Regression Feb 2012, http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-cox-regression.pdf

[2] Stephen J Walters: What is a Cox model? http://www.medicine.ox.ac.uk/bandolier/painres/download/whatis/cox_model.pdf

- Notes

If number of ties in the source table is very large, a memory allocation error may be raised. The limitation is about \((10^8 / m)\), where \(m\) is number of featrues. For instance, if there are 100 featrues, the number of ties should be fewer than 1 million.

- Related Topics

File cox_prop_hazards.sql_in documenting the functions