# 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:

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

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

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

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

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

## Example¶

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
heart_coxph <- h2o.coxph(x = x, event_column = y,
start_column = start, stop_column = stop,
ties = "breslow", training_frame = heart)

# view the model details
heart_coxph
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

import h2o
from h2o.estimators.coxph import H2OCoxProportionalHazardsEstimator
h2o.init()

# import the heart dataset
heart = h2o.import_file("http://s3.amazonaws.com/h2o-public-test-data/smalldata/coxph_test/heart.csv")

# set the parameters
heart_coxph = H2OCoxProportionalHazardsEstimator(start_column="start",
stop_column="stop",
ties="breslow")

# train your model
heart_coxph.train(x="age", y="event", training_frame=heart)

# view the model details
heart_coxph