= Solution
Because $t^*>t_b>t_a$, the three possible complete event orders are $t_a<t^*<t_c<t_d$, $t_a<t_c<t^*<t_d$, and $t_a<t_c<t_d<t^*$. Write $K=\phi_a/(\phi_a+\phi_b+\phi_c+\phi_d)$ and $V=\phi_b+\phi_c+\phi_d$. Their <partial likelihoods>, with the final singleton factor omitted, are respectively
$$
L_1=K\frac{\phi_b}{V}\frac{\phi_c}{\phi_c+\phi_d},\qquad L_2=K\frac{\phi_c}{V}\frac{\phi_b}{\phi_b+\phi_d},\qquad L_3=K\frac{\phi_c}{V}\frac{\phi_d}{\phi_b+\phi_d}.
$$
Adding first $L_2$ and $L_3$ gives $K\phi_c/V$, and therefore
$$
L_1+L_2+L_3=K\frac{\phi_c}{V}\left(\frac{\phi_b}{\phi_c+\phi_d}+1\right)=K\frac{\phi_c}{\phi_c+\phi_d}.
$$
Thus
$$
\boxed{L_1+L_2+L_3=L_{a,b,c,d}.}
$$
This is <Cox rank-likelihood deletion consistency>: summing out the unobserved position of $b$ leaves the relative order information in the observed events. It is an algebraic marginalization of complete-order <probabilities> under time-constant <hazard multipliers>. It does not include the <probability density function> of the actual censoring time, establish the distribution of $t^*$ given all observed times, or justify informative censoring. Ignoring the censoring mechanism in the survival analysis still requires <independent censoring> given the modeled <covariates>.
Back to article page
