43
= Λ f ⅛)exp(⅜ )θ'faj
j W ⅛l∑Lι^(¾)expW)/
and,
L∞(∕5) = ∏<⅝
J=I
ππ( ⅜-)exp(⅜ }d^
⅛MlΣLι^(¾)θχp(^)∫
where ʧɔ(/?) is the likelihood function for the time point xi.
To make the above equation well-defined, we need to make several adjustments.
First, we set P⅛(¾∙) = 1 for xi ≤ k. Because by definition, P⅛(¾) represents the
probability of the subject к survived at time Xj for Xj > k, given that it survives until
time k. That is for Xj ≤ k, the subject k survives, i.e. Pk(xj) = 1 ■
Second, please note that observations have been sorted in ascending order already.
That is, for i = 1,..., n, the subject i failed at time xi. Therefore, for a failed subject
j, we set Pi(xj) = 0 for j > i.
Third, regarding Di(xj) for a failed subject i, instead of using Equation 3.6, we
set Di(xi) = 1 and Di(xj) = 0 for j > i. That is, the subject i losses all of its unit
mass when it fails at time xi. As a result, a failed subject contributes only one time
in the extended Cox partial likelihood function.
The counterpart of the extended Cox partial likelihood function is shown below,