Mathematical Equivalence

The hazard rate for individual \(i\) in time interval \(j\) is:

\[\lambda_{ij} = \lambda_j \exp(X_i \beta)\]

where \(\lambda_j\) is the baseline hazard for interval \(j\). In a survival model, the log-likelihood contribution of an observation is given by:

\[\log L_{ij} = d_{ij} \log(\lambda_{ij}) - \int_{t \in I_j} \lambda_{ij} dt\]

Under the assumption that \(\lambda_j\) is constant over the interval duration \(\Delta t_{ij}\), the integral simplifies to:

\[\log L_{ij} = d_{ij} (\log(\Delta t_{ij}) + \log(\lambda_j) + X_i \beta) - (\Delta t_{ij} \lambda_j e^{X_i \beta})\]

This is identical (up to a constant \(\log(\Delta t_{ij})\)) to the log-likelihood of a Poisson distribution \(\text{Poisson}(\mu_{ij})\)