Page 171 - Contributed Paper Session (CPS) - Volume 7

CPS2055 Asanao S. et al.
Let = ( ) be the estimated risk score for the subject with , and we

assume the high value of represents the high risk of the event. There is a

lot of measures to quantify how well the score predicts the distribution of

the failure time, like explained variation-based measures and discrimination-
based measures. The C-index is one of the representative measure among
them. Let (u) be the conditional survival probability for :

() = Pr( > | = ).

For two independent subjects ( ,H ) and ( ,H ), if the following inequalities

is satisfied, the pair is said to be concordant at :
̂
̂
< and () < ()

or
̂
̂
< and () < () .

In a same meaning, the pair is said to be concordant if the following is satisfied:
< and >

or
> and < .

Based on the concordant pairs, the concordance probability is defined as
follows:
= Pr (H > H | < , < ),

where is the max time point to evaluate . When the event time may be
censored, the estimation of is not simple. Harrell et al. (1996) proposed the
measure that is derived as Kendall’s tau for censored survival data. The
measure uses only usable pairs in the observed sample:
∑ ( < , ≤ )( > )

,

̂
= ,

∑ ( < , ≤ )

,
where ∑ = ∑ ∑ ≠ . This measure becomes the consistent estimator if there
,

are no censoring. However, it may depend on the censoring distribution.
Therefore, it is no clear what the effect of censoring.
As the modification of the censoring bias of Harrell’s C-index, Uno et al.
(2011) proposed a new measure:
−2
̂
∑ {( )} ( < , ≤ )( > )

,

̂
= −2 ,

̂
∑ {( )} ( < , ≤ )
,

̂
where (. ) is the Kaplan-Meier estimator for the censoring distribution
() = Pr( > ).
As the other modification of the bias of Harrell’s C-index, Begg et al. (2000)
proposed the measure that uses the all pairs including the unusable pairs. In
their approach, the concordance values for unusable pairs are replaced by
estimates of conditional expectation of it:
2
̂
= ( − 1) ∑ ( > ) ,

,
where is defined as follows:

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