an individual $i$ through the log link

$$\ln\lambda_{ij}=\mathbf{z}_{i}^{\prime}\mathbf{\boldsymbol{\beta}}_{j},\label{eq:lambda}$$

which allows the flexibility to capture different shapes of the hazard function across time. Here, $\mathbf{z}_{i}=(1,\mathbf{x}_{i}^{\prime})$ is the original covariate vector of length $P$ augmented with a column of $1$, $\mathbf{\boldsymbol{\beta}}_{j}=(\mathbf{\beta}_{0j},\mathbf{\beta}_{1j},\cdots,\mathbf{\beta}_{Pj})$ represents the vector of regression coefficients, where the intercept $\mathbf{\beta}_{0j}=\ln(\lambda_{0j})$ is the log of the baseline hazard.

Partitioning time into discrete intervals generates interval-based data. The survival time $t_{i}$ breaks into several exposure times $t_{ij}=\max(0,\min(t_{i}-\tau_{j-1},\tau_{j}-\tau_{j-1}))$ which define the amount of time an individual $i$ is exposed to the occurence of the event during the interval $I_{j}$. The exposure time is equal to the length of $I_{j}$ (if individual $i$ survived through this interval), or it is equal to $t_{i}-\tau_{j-1}$ (if individual $i$ experienced the event or is censored in the interval $I_{j}$), otherwise it is equal to zero. Furthermore, the event indicator expands into a vector of binary variables $d_{ij}=1$ if the event occurs in interval $I_{j}$, and $d_{ij}=0$ if the individual is censored in or survives through the interval $I_{j}$.

With the assumption that covariates enter per (\[eq:lambda\]), the survival function for individual $i$ becomes

$$S\left(t_{i}|\boldsymbol{x}_{i}\right)=\exp\left(-\left[\sum_{j=1}^{h-1}\lambda_{ij}\left(\tau_{j}-\tau_{j-1}\right)\right]-\lambda_{ih}\left(t_{i}-\tau_{h-1}\right)\right),\,\textrm{if }\tau_{h-1}\leq t_{i}<\tau_{h},\,h\leq J,\label{eq:survival function}$$

and the likelihood function (\[eq:likelihood\]) can be factorized across the 