Fine & Gray Model

T: failure times. C: censoring times. Observe $X=min(T,C), \Delta=I(T\le C)$ and Z.

Subdistribution hazard for $T*=I(\epsilon=1)T+(1-I(\epsilon=1)\infty$:

\[\begin{aligned} \lambda_1(t;Z) = lim_{\Delta t\to 0}\frac{1}{\Delta t} Pr(t\le T\le t+\Delta t , \epsilon=1 | T\ge t\cup (T\le t)\cap \epsilon\ne1, Z) =(dF_1(t;Z)/dt) / (1-F_1(t;Z))=-dlog(1-F_1(t;Z))/dt \end{aligned}\]

Risk set associated with $\mathrm{\lambda}_1$ is unnatural: those who failed from causes other than 1 prior to time t are not at risk at t.

Complete data (no censoring)

Risk set at the time of failure for j-th individual: $R_i={j: (T_j\ge T_i)\cup (T_j\le T_i \cap \epsilon_j \ne 1)}$. Those who has not failed from cause of interest by time t.

\[N_i(t)=I(T_i\le t, \epsilon_i=1), Y_i(t)=1-N_i(t-)\]

• Log-partial likelihood (same as Cox model):

\[log(L(\beta))=\sum_{i=1}^n I(\epsilon_i=1) (Z_i^T(T_i)\beta - log(\sum_{j\in R_i} exp(Z_j^T(T_i)\beta)))\]

• Score function: \(U_1(\beta) = \sum_{i=1}^n \int_0^\infty \left(Z_i(s) - \frac{\sum_j Y_j(s)Z_j(s)exp(Z_j^T(s)\beta)}{\sum_j Y_j(s)exp(Z_j^T(s)\beta)}\right)dN_i(s)\)

\[=\sum_{i=1}^n \int_0^\infty \left[Z_i(s) - \frac{\sum_j Y_j(s)Z_j(s)exp(Z_j^T(s)\beta)}{\sum_j Y_j(s)exp(Z_j^T(s)\beta)}\right]dM^1_i(s,\beta)\]

where $M_i^{1}(t,\beta) = N_i(t) - \int^t_0 Y_i(u)\lambda_{10}(u)exp(Z_i^T(u)\beta)$.

$M_i^1(t,\beta_0)$ is martingale under $\mathcal{F}^1(t) = \sigma{ N_i(u), Y_i(u)Z_i(u), u\le t, i=1,…,n }$.