# 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"

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