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CPS2055 Asanao S. et al.
If = = 1, then

0, >

= { 1, < .

If = 0, = 1, then

0, >

̂
̂
= Pr( < | > ) = { 1 − ( ) , < .

̂
( )

If = 1, = 0, then

̂
( )

̂
̂
= Pr( > | > ) = { ( ) , > .

1, <

If = = 0, then

̂
1 ( )
̂ , >

̂
= Pr( < | > > ) = 2 ( )

,

̂
( )

1 − , <

̂
{ 2 ( )

For in the case of = = 0, it is assumed that if the subject with the

shorter censored value of lives as long as the time in paired subject, the
remaining conditional probability of concordance is 1/2.
Korn and Simon (1990) proposed the measure based on the rank
correlation between observed and predicted survival times:
2
̂
= ∑ ( > ) ,

( − 1)
,
where
∗
−
∗−
∗−
−
= Pr ̂ ( < ) = ∑[1 − ̂ ( )][ ̂ ( ) − ̂ ( )] + [1 − ̂ ( )] ̂ ( ),

∗
≤

∗
, ,⋯ are the ascending-ordered event times, and represents just before
−
∗
1
2
.

2.2 Splitting criteria based on measures for concordance probability
We define a tree-structured model as . The tree-structured model is
constructed by the splitting rules of the covariate space and the nodes that
are subsets of the resulting spaces. Let be a node in tree . If the node does
not exist in the bottom layer of the tree, we call it an internal node. Each
internal node has a splitting rule to separate that node. Although there are
several splitting methods for obtaining the splitting rules, the most popular
one is the dichotomize method. The splitting rule of can be induced by any
question of the form `` ∈ ?'', where is called the child node of . The

counterpart of that is obtained by division of is also called as the child

node of . That is, the splitting rule divides the internal node into two child
nodes, and , and is called the parent node of and . The most widely

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