Mathematical Equivalence
========================


The hazard rate for individual :math:`i` in time interval :math:`j` is:

.. math::
    \lambda_{ij} = \lambda_j \exp(X_i \beta)

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

.. math::
    \log L_{ij} = d_{ij} \log(\lambda_{ij}) - \int_{t \in I_j} \lambda_{ij} dt

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

.. math::
    \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 :math:`\log(\Delta t_{ij})`) to the
log-likelihood of a Poisson distribution :math:`\text{Poisson}(\mu_{ij})`