Partial likelihood function in PH models

Based on equation (3.18), one can derive the probability that all spells completed at \(t_j\) ends in the \(j^{\text{th}}\) failure time, such that:

\[\begin{equation} \begin{aligned} \mathcal{L}_{p,\ t_j} & = \mathbb{P}\big[T_1 = t_j, \dots, T_{d_j} = t_j \ | \ R(t_j)\big] \\\\ & = \Pi_{m \in D(t_j)} \ \mathbb{P}\big[T_m = t_j | R(t_j) \big] \\\\ & = \Pi_{m \in D(t_j)} \ \frac{\phi(\mathrm{x_m}, \beta)}{\sum_{l \in R(t_j)} \phi(\mathrm{x_l}, \beta)} \\\\ & = \Pi_{m \in D(t_j)} \ \phi(\mathrm{x_m}, \beta) \times \Pi_{m \in D(t_j)} \ \frac{1}{\sum_{l \in R(t_j)} \phi(\mathrm{x_l}, \beta)} \\\\ \mathcal{L}_{p,\ t_j} & = \frac{\Pi_{m \in D(t_j)} \ \phi(\mathrm{x_m}, \beta)}{\Big[\sum_{l \in R(t_j)} \phi(\mathrm{x_l}, \beta)\Big]^{d_j}} \end{aligned} \tag{6.6} \end{equation}\]

The joint probability over the \(k\) ordered discrete failure times then becomes:

\[\begin{equation} \begin{aligned} \mathcal{L}_p & = \Pi_{j=1}^{k} \ \mathcal{L}_{p,\ t_j} \\\\ \mathcal{L}_p & = \Pi_{j=1}^{k} \ \frac{\Pi_{m \in D(t_j)} \ \phi(\mathrm{x_m}, \beta)}{\Big[\sum_{l \in R(t_j)} \phi(\mathrm{x_l}, \beta)\Big]^{d_j}} \end{aligned} \tag{6.7} \end{equation}\]