42
that this subject i is dependently censored through time c. For c > xi, we have
= ffaffa,xt)dx
⅛i∖c) r∞ f.( ∖ j
Jxi ffa,zfadx
= Pr(C,i ≥ c∖Ci > xi,Ti = xi)
_ PfaCi ≥ c,Ti = xi)
Pr(C,i ≥ xi,Ti = xi)
= ɪ - Hu{Fifai),Gi(c)-a}
1 - Hu{Fifai),Gifai)-a}' '
where Hu(a,b∙,a) = дн^’В?).1 .
u ∖ ’ ’ / OU / ʌ , ,4
I (u,υ)=(α,o)
Let Ei fa fa represent the piece of mass that a failed subject i losses at dependent
censoring time Xj. Again, for Xj > xi, we define that
EfaXj) Qifaj-1) Qifaj')∙
(3.8)
Note that there are a few restrictions of above notations listed in Section 3.3.3.
3.3.3 Partial likelihood functions
In order to consider the event and dependent censoring simultaneously, we introduce
copula-based indicator functions Di fa fa and Ei fa fa, which can take any value be-
tween 0 and 1; while traditional indicators can only take two values, either 0 or 1.
For a subject with dependent censoring, we assume that its contribution to the likeli-
hood function is decreasing gradually at each observed event time point, represented
by Pi (i) and Di fa fa. Intuitively, this subject is going to fail gradually rather than
immediately, as the traditional survival analysis assumes.
We define an extended Cox partial likelihood function for an event as follows:
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