Page 175 - Contributed Paper Session (CPS) - Volume 7

CPS2055 Asanao S. et al.
̂
̂
the sub-tree . For example, if we use the for splitting step, then ( ) is

ℎ

given by
∑ () ( < )( < ) + 0.5 ∑ () ( < )( = )

̂
= ,∈ℒ ,∈ℒ ,

∑ ,∈ℒ () ( < )

̂
∗
∗
∗
where = ∑ {1 − ( )}, , … . , are unique event times in ℒ − ℒ () ,

1

=1

̂
̃

′
and (. ) is the KaplanMeier estimate in ∈ ( ) , which ∈ ℒ () is

included. Finally, we select the subtree that satisfy
1
̂
′
arg max { ∑ ( ( ))}.

3. Result
We present simple simulation studies to examine the properties of the
proposed approach in several situations. As the first simulation, we used the
following model to generate the data to compare the performance in terms
of splitting criterion. As the quantitative covariate, = (100 × )/

300 for = 1, ⋯ ,300. The true splitting rule is given by “ ≤ 50?”. Then, in

each child node, the exponential models with parameter = 1.0 and =

0.5 are specified as the distribution of failure times, respectively. The censoring
times are generated from uniform distribution as the censoring rate become
about 50%. Simulations are repeated 1,000 times. The results of the simulation
are shown in Table 1. Table 1 lists the average values and standard deviations
of the selected thresholds as the splitting rules. Table 1 lists the average values
and standard deviations of the selected thresholds as the splitting rules.
Table 1: The average and standard deviation of the split points.
Criterion Ave. selected threshold (std.)
̂ 49.6 (6.1)

̂ 50.6 (5.8)

̂ 50.3 (5.4)

̂ 48.8 (7.1)

Log  rank test 47.0 (11.9)

To compare the performance of a tree obtained by using each criterion,
we used the following model to generate data. The four categorical
covariates , ⋯ , are generated from a discrete uniform distribution with
4
1
{1,2,3,4}. The failure and censoring times are generated from exponential
distributions with parameter and 0.37, respectively. The model of is given

by
0.4, ∈ {1,2} ∩ ∈ {1,2}
1 2
0.7, ∈ {1,2} ∩ ∈ {3,4} ∩ ∈ {1,2}
3
1
2
= 1.0, ∈ {1,2} ∩ ∈ {3,4} ∩ ∈ {3,4}.

2
1
1.3, ∈ {3,4} ∩ ∈ {1,2} 3
1 3
{1.6, ∈ {3,4} ∩ ∈ {3,4}
3
1
162 | I S I W S C 2 0 1 9