Page 173 - Contributed Paper Session (CPS) - Volume 7

CPS2055 Asanao S. et al.
used splitting rule consists of a single covariate ( = 1, ⋯ , ). If is a

quantitative variable, then the rule becomes `` ≤ ?'', where is a threshold.

If is a categorical variable with the set of possible values ℱ , then the rule

becomes `` ∈ ℱ ?'', where ⊂ ℱ .

The CART algorithm for constructing the tree-structured model comprises
splitting, pruning, and selection. In the splitting step, covariates space are
recursively divided based on the optimal splitting rules and the maximum-size
tree is constructed. To determine the optimal splitting rule of a node into
0
and , we evaluate all the possible splitting rules for . In order to build the

model with measures for concordance probability, we assume the following
to dichotomize the node : has higher risk than , we evaluate the

concordance probabilities from = 0 to = max{ ; = 1, ∈ }, and

the contribution of the pair (,), where = to the estimate of is 0.5.

Under these assumptions, the splitting criterions based on the measures
̂ ̂ ̂ ̂
, , and are given by as follows:

(i) The criterion based on Harrell’s C
∑ ∑ ( < ) + 0.5{∑ ( < ) + ∑ ( < )}

̂
= ∈ ∈ ∑ ,∈ ,∈

( < )

,∈
(ii) The criterion based on Uno’s approach
−2 −2 −2

∑ ∈ ∑ ∈ { ̂ ( )} ( < ) + 0.5{∑ ,∈ { ̂ ( )} ( < ) + ∑ ,∈ { ̂ ( )} ( < )}
̂ = −2

∑ ,∈ { ̂ ( )} ( < )

̂
where (. )is the Kaplan-Meier estimator for the censoring distribution based

on the samples included in .
(iii) The criterion based on Begg’s approach
2 ( − 1) + ( − 1)
̂
= ( − 1) {∑ ∑ + 4 },

∈ ∈
where is the number of samples included in the node . is defined as

follows: If = = 1, then

0, >

= { 1, < .

If = 0, = 1, then

If = 1, = 0, then

If = = 0, then

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