I’m reading about Cox’s proportional hazards approach to (continuous) survival analysis and I’m finding it difficult to understand his argument for the derivation of the partial likelihood in his 1972 paper.

He states on page 6 of the pdf in the link that the probability to observe a failure at time $t_{(i)}$ on the individual $i$ (given that there is *exactly* one failure at $t_{(i)}$) is equal to $$frac{exp(z_{(i)}beta)}{sum_{lin R(t_{(i)})}exp(z_{(i)}beta)}.$$

Now I understand that the time-dependent part in the denominator is supposed to cancel against the time-dependent part in the numerator. However, the denominator itself surprises me, because it seems that this denominator is associated to the expected number of failures, rather than the probability to observe exactly one failure (which is what I would expect from the conditional probability that this partial likelihood is supposed to describe).

I’m wondering if, instead, I should view this partial likelihood as some approximation to the exact problem (and if so, why this approximation is justified).

Also, I’m curious if it’s still necessary with modern computers to actually split the full likelihood into partial likelihoods or if this is something that was mostly useful in the 70s.