``ties``
--------

- Available in: CoxPH
- Hyperparameter: no

Description
~~~~~~~~~~~

This option configures approximation method for handling ties in the partial likelihood. This can be either **efron** (default) or **breslow**).

Of the two approximations, Efron's produces results closer to the exact combinatoric solution than Breslow's. Under this approximation, the partial likelihood and log partial likelihood are defined as:

 :math:`PL(\beta) = \prod_{m=1}^M \frac{\exp(\sum_{j \in D_m} w_j\mathbf{x}_j^T\beta)}{\big[\prod_{k=1}^{d_m}(\sum_{j \in R_m} w_j \exp(\mathbf{x}_j^T\beta) - \frac{k-1}{d_m} \sum_{j \in D_m} w_j \exp(\mathbf{x}_j^T\beta))\big]^{(\sum_{j \in D_m} w_j)/d_m}}`

 :math:`pl(\beta) = \sum_{m=1}^M \big[\sum_{j \in D_m} w_j\mathbf{x}_j^T\beta - \frac{\sum_{j \in D_m} w_j}{d_m} \sum_{k=1}^{d_m} \log(\sum_{j \in R_m} w_j \exp(\mathbf{x}_j^T\beta) - \frac{k-1}{d_m} \sum_{j \in D_m} w_j \exp(\mathbf{x}_j^T\beta))\big]`

Under Breslow's approximation, the partial likelihood and log partial likelihood are defined as:

 :math:`PL(\beta) = \prod_{m=1}^M \frac{\exp(\sum_{j \in D_m} w_j\mathbf{x}_j^T\beta)}{(\sum_{j \in R_m} w_j \exp(\mathbf{x}_j^T\beta))^{\sum_{j \in D_m} w_j}}`

 :math:`pl(\beta) = \sum_{m=1}^M \big[\sum_{j \in D_m} w_j\mathbf{x}_j^T\beta - (\sum_{j \in D_m} w_j)\log(\sum_{j \in R_m} w_j \exp(\mathbf{x}_j^T\beta))\big]`


Related Parameters
~~~~~~~~~~~~~~~~~~

none

Example
~~~~~~~

.. example-code::
   .. code-block:: r

    library(h2o)
    h2o.init()
    # import the heart dataset
    heart <- h2o.importFile("http://s3.amazonaws.com/h2o-public-test-data/smalldata/coxph_test/heart.csv")

    # set the predictor name and response column
    x <- "age"
    y <- "event" 

    # set the start and stop columns
    start <- "start"
    stop <- "stop"

    # train your model
    coxph.h2o <- h2o.coxph(x=x, event_column=y, 
                           start_column=start, stop_column=stop, 
                           ties="breslow", training_frame=heart.hex)

    # view the model details
    coxph.h2o
    Model Details:
    ==============

    H2OCoxPHModel: coxph
    Model ID:  CoxPH_model_R_1527700369755_2 
    Call:
    "Surv(start, stop, event) ~ age"

          coef exp(coef) se(coef)    z     p
    age 0.0307    1.0312   0.0143 2.15 0.031

    Likelihood ratio test=5.17  on 1 df, p=0.023
    n= 172, number of events= 75