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Taking logarithms of the Cox partial likelihood givesDifferentiate each term, using the chain rule for the logarithm:This score function is the sum over failures of the observed covariate minus its current risk-weighted mean. Define that mean as ${\overline{z}}_{\beta }(x_i)$; observations with $v_i=0$ have no direct event term, although they occur in the relevant risk sets.

Let $R_i=\{j:x_j\ge x_i\}$ be the risk set at an observed event time $x_i$. In the Cox proportional-hazards model, conditional on one member of this set failing then and on the preceding history, the probability that it is individual $i$ isThe baseline hazard cancels. Multiplying these conditional contributions gives the Cox partial likelihoodThe no-ties assumption avoids the need for a tied-event approximation. This construction assumes independent censoring conditional on the modelled covariates. The printed phrase calling $z$ a scalar parameter is read as a scalar observed covariate $z_i$; $\beta$ is the unknown regression parameter.

In a proportional hazards model, individual $i$ hasThe exponential term is the hazard multiplier relative to the baseline hazard $h_0$. With $h_0$ unspecified, the Cox partial likelihood multiplies, over event times, the failing subject's multiplier divided by the sum of multipliers in the current risk set. After estimating $\beta$, the Breslow estimator is

A Stratified Cox model uses a separate baseline hazard $h_{0s}$ for each stratum but a common $\beta$. Its partial likelihood is the product of within-stratum partial likelihoods, and a separate integrated baseline hazard is estimated in each stratum.

In a matched pair, the only informative comparison occurs while both members remain at risk. If the exposed member fails first, the pair contributes $e^{\beta }/(1+e^{\beta })$; if the unexposed member fails first, it contributes $1/(1+e^{\beta })$. A censoring as the first recorded time gives no informative failure comparison, and any later one-person risk set contributes one. With only two observations per stratum, each stratum supplies almost no information about its arbitrary baseline hazard, so a useful common integrated baseline-hazard estimate is generally unavailable.

A proportional hazards model has $h(t\mid g)=h_0(t)r(g)$, with a time-independent hazard ratio. In the Cox proportional-hazards model, $r(g)=e^{\beta g}$ and the baseline hazard $h_0$ is unspecified. At each untied event time, conditioning on the identity of the person who fails given the current risk set produces the factorMultiplying these factors gives the Cox partial likelihood. It is called partial because it uses the conditional event ordering information that identifies $\beta$ while discarding the part of the full likelihood involving the unspecified baseline hazard.

Letand let $C$ be the common partial likelihood contribution from the first $n-3$ observations. Since $t_{n-2}>x_{n-2}$, the three possible complete-data tail orderings and their partial likelihoods areTheir sum isWhen individual $n-2$ is right-censored at $x_{n-2}$, that individual leaves the risk set before the event at $x_{n-1}$, so the directly calculated Cox partial likelihood is also $C,a/(a+b)$. Summing over the unobserved compatible event orderings therefore reproduces the censored-data partial likelihood.

For each observed event $i$ with $v_i=1$, let $R_i=\{j:x_j\ge x_i\}$ be its risk set. The Cox partial likelihood isMaximize it to obtain $\hat{\beta }$, estimate its variance from the observed partial information, and test $H_0:\beta =0$ using a partial likelihood-ratio test, Wald test, or score test. A positive fitted coefficient means the group with $z=1$ has the larger hazard.

A semiparametric proportional hazards model specifieswith finite-dimensional parameter $\beta$ but an unspecified baseline hazard $h_0$. A partial likelihood uses a component of the data likelihood that depends on $\beta$ while eliminating the nuisance function. In the Cox proportional-hazards model, conditioning on which member of each risk set experiences the event produces the Cox partial likelihood.

The Cox proportional-hazards model iswhere $h_0$ is an unspecified baseline hazard and $z_i$ indicates treatment. With no tied events, the Cox partial likelihood isEquality of the two hazard functions is $H_0:\beta =0$, testable with a likelihood-ratio, score, or Wald test from this likelihood.

At a distinct event time, the Schoenfeld function is the event subject's covariate minus the hazard-weighted covariate mean over the risk set, as a function of a trial coefficient $\beta$. Its evaluation at the fitted coefficient is a Schoenfeld residual. The sum of these functions is the Cox partial likelihood score function, and each is conditionally centered at zero at the true proportional-hazards coefficient.

At an untied event time, the Cox partial likelihood weights the current risk set by $w_{jk}=e^{{\beta }^T{z}_k}/{\sum }_{\ell \in R_j}{e}^{{\beta }^T{z}_{\ell }}$. Differentiation gives the displayed score function and the negative Hessian as a sum of weighted covariances. The score is the sum of Schoenfeld functions, so Schoenfeld residuals sum to zero at a finite interior unpenalized fit. At arbitrary trial coefficients or boundary fits, a zero sum is not a general identity.

In an untied Cox proportional-hazards model with a single subject remaining after all other observed exit times, that subject's event contributes the constant factor one to Cox partial likelihood. Replacing its terminal censoring time by a later event therefore preserves the fitted coefficients and every earlier Breslow estimator increment. At its new event, the subject's fitted cumulative hazard increases by one, cancelling the increase of one in its event indicator. Every martingale residual is unchanged. This property relies on the subject being alone in the terminal risk set and the same covariate history being retained.