h_{i} (t) = h_{0} (t) exp (β^{T} z_{i}),

(1)

where

h_{0} (t)

is an unspecified common baseline hazard and

z_{i}

is the individual's covariate vector. The hazard ratio for two fixed covariate vectors is

exp [β^{T} (z_{i} - z_{k})]

, constant in time. Assume independent subjects and independent censoring conditional on their covariates, and use the question's absence of tied events.

Let

t_{j}

be an event time,

i_{j}

its failing subject, and

R_{j}

the risk set immediately before that time. In a short interval of length

d t

, the conditional probability that subject

i

fails, given one failure among those at risk, is

\frac{h _{i} ( t _{j} ) d t + o ( d t )}{\sum _{k \in R_{j}} h _{k} ( t _{j} ) d t + o ( d t )} ⟶ \frac{e ^{β^{T} z_{i}}}{\sum _{k \in R_{j}} e ^{β^{T} z_{k}}} .

(2)

The baseline hazard cancels. Multiplying the successive conditional event-identity factors gives the Cox partial likelihood for

β

L_{p} (β) = j \prod \frac{e ^{β^{T} z_{i_{j}}}}{\sum _{k \in R_{j}} e ^{β^{T} z_{k}}}, ℓ_{p} (β) = j \sum β^{T} z_{i_{j}} - lo g k \in R_{j} \sum e^{β^{T} z_{k}} .

(3)

This is a partial likelihood, not the full survival likelihood for the unspecified

h_{0}

. Risk sets evolve with the observed history; the derivation uses their successive conditional hazards, rather than assuming the risk sets themselves are independent.

Define