Cox proportional-hazards model

The Cox model writes $h(t\mid z)=h_0(t)e^{{\beta }^{T}z}$ with an unspecified baseline hazard.

A likelihood-based analogue of the coefficient of determination compares the fitted and intercept-only log-likelihoods ${\ell }_1$ and ${\ell }_0$:For a Cox proportional-hazards model, these are maximized partial likelihood logarithms with and without the covariates. The expression measures improvement in fit, rather than the fraction of variation in event times explained. Its attainable maximum can be below one. It does not test the proportional hazards assumption or establish predictive calibration.

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.

If the risk set has two subjects with covariate zero and one with covariate one, its hazard-weighted mean is $e^{\beta }/(2+e^{\beta })$. A zero-covariate event has Schoenfeld function $-e^{\beta }/(2+e^{\beta })$, while a one-covariate event has function $2/(2+e^{\beta })$. At the true coefficient their probabilities are $2/(2+e^{\beta })$ and $e^{\beta }/(2+e^{\beta })$, so the conditional expected function is zero.

A proportional hazards assumption test assesses whether fitted covariate coefficients remain constant over time. A common implementation uses trends in scaled Schoenfeld residuals, with separate covariate tests and a joint global test. Failure to reject is compatible with proportional hazards, but does not establish the correct covariate functional form or independent censoring.

A time interaction extends a constant-coefficient Cox proportional-hazards model by a covariate term $z_{k}g(t)$ and tests whether its coefficient is zero. The time function is evaluated at the current event/risk-set time, not each person's eventual outcome time. Scaled Schoenfeld residuals provide related graphical and score-test diagnostics for coefficient trends.

At an untied event time, a Schoenfeld residual is the event subject's covariate vector minus its fitted risk-weighted average over the risk set. Under a correctly specified Cox proportional-hazards model, these residuals have no systematic mean trend with event time. They are defined at event times, unlike Cox–Snell residuals which can be evaluated at both events and censoring times.

A scaled Schoenfeld residual rescales a Schoenfeld residual using fitted information so that its trend can diagnose departures from a constant coefficient in a Cox proportional-hazards model. A smooth trend against transformed event time estimates a time-dependent coefficient pattern; a flat trend supports the working proportional hazards assumption test.

The Cox partial likelihood conditions each event on its risk set, eliminating the baseline hazard.

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 a discrete-time model with independent event indicators whose odds are $a{u}_i$, condition on exactly $d$ events in the risk set $R$. The common factor $a^d$ cancels and the conditional probability of the event set $D$ is the displayed ratio. The denominator is an elementary symmetric polynomial and can be evaluated by recursion rather than enumerating every subset. This conditional model is distinct from summing a continuous-time rank likelihood over all unobserved within-tie orders.

For $d$ events tied within a risk set, put $u_i=\exp ({\beta }^T{z}_i)$, $S={\sum }_{i\in R}{u}_i$ and $U_D={\sum }_{i\in D}{u}_i$. The Efron approximation to Cox partial likelihood removes the average tied-event hazard multiplier from the denominator at each successive event. Unlike the Breslow approximation for tied event times, it accounts approximately for depletion of the risk set within the group. Factors independent of $\beta$ can be dropped for estimation. This is an approximation for coarsened continuous event times, not a full interval-observation likelihood.

When $d_j$ events occur at the same recorded time, the Breslow approximation to Cox partial likelihood uses the factor $\exp ({\sum }_{i\in D_j}{x}_i^T\beta )/({\sum }_{i\in R_j}{e}^{x_i^T\beta }{)}^{d_j}$, where $R_j$ is the risk set and $D_j$ the event set. It approximates the ordering of tied events rather than treating their within-tie order as known.

Assume independent individual event times with positive time-constant hazard multipliers and a common baseline hazard whose cumulative hazard tends to infinity. Transforming each event time by that cumulative hazard then gives independent exponential event times with corresponding rates, so all event orders exist. Their complete-order probabilities select each next label proportionally to its remaining multiplier. Marginalizing over the position of a deleted label leaves the order law of the other exponential times, hence the same sequential formula without that label. This proves the deletion identity used when unobserved ranks are summed out. It does not supply a likelihood function for censoring times; ignoring censoring in observed survival analysis still needs independent censoring.

The Breslow estimator of the baseline cumulative hazard in a Cox proportional-hazards model adds the observed number of events at each event time divided by the sum of fitted relative risks in its risk set.

A stratified Cox model gives each stratum its own baseline hazard while sharing regression coefficients.

Separate first-event episodes from all later episodes with a stratum indicator. A Stratified Cox model permits one baseline hazard for first events and another for recurrent events while retaining a common treatment coefficient. Later episodes can share their baseline; stratifying each recurrence number separately is a stronger model than the first-versus-later distinction alone.

A stratum is a subgroup assigned its own nuisance baseline.