I am doing some self study on Cox regression, and am trying to figure out how we can derive the partial likelihood for the Cox model from the full likelihood. Generally, I know that to get a partial likelihood, we can just use proportionality, but here, this doesn't seem as intuitive.
Consider the Cox full log likelihood: $$l(\theta) = \sum_{i=1}^n \delta_i log h_i(T_i;\theta) - \int_0^{T_{i}}h_i(s;\theta)ds$$
where $h_i(T_i;\theta)$ is the hazard: $h_i(T_i;\theta) = h_0(t)exp(\gamma^Tw_i)$.
The corresponding partial log likelihood would be:
$$ pl(\gamma) = \sum_{i=1}^n \delta_i \bigg[\gamma^Tw_i - log \bigg\{\sum_{T_j\geq T_i}exp(\gamma^tw_j) \bigg\} \bigg] $$
Of course, the first apparent step I see is to substitute in the hazard function into the full likelihood:
$$l(\theta) = \sum_{i=1}^n \delta_i log (h_0(t)exp(\gamma^Tw_i)) - \int_0^{T_{i}}h_0(t)exp(\gamma^Tw_i)ds$$
$$ = \sum_{i=1}^n \delta_i \bigg[\gamma^Tw_i+ log (h_0(t))\bigg] - \int_0^{T_{i}}h_0(t)exp(\gamma^Tw_i)ds$$
From here, any advice in the derivation or intuition would be much appreciated!
