The Cox model is semiparametric: regression effects are finite-dimensional, while the baseline hazard remains an unrestricted increasing-function parameter. Its usefulness comes from a cancellation inside risk sets, not from pretending the baseline hazard is absent.
Conditional proportional hazards
Definition. For covariate vector \(Z\in\mathbb R^p\), the Cox proportional hazards model is \[ \lambda(t\mid Z)=\lambda_0(t)e^{\beta^\top Z}, \qquad \beta\in\mathbb R^p, \] where \(\lambda_0\) is an unknown nonnegative baseline hazard. More generally, for a predictable process \(Z(t)\), \[ \lambda_i(t\mid\mathcal F_{t-}) =Y_i(t)\lambda_0(t)e^{\beta^\top Z_i(t)}. \] The cumulative baseline hazard is \(\Lambda_0(t)=\int_0^t\lambda_0(u)\,\mathrm{d}u\).
Proposition. With fixed \(Z=z\), \[ \Lambda(t\mid z)=e^{\beta^\top z}\Lambda_0(t), \qquad S(t\mid z)=S_0(t)^{e^{\beta^\top z}}, \quad S_0(t)=e^{-\Lambda_0(t)}. \] For two covariate values \(z_1,z_0\), \[ \frac{\lambda(t\mid z_1)}{\lambda(t\mid z_0)} =e^{\beta^\top(z_1-z_0)}, \] which is constant in time.
Proof. Integrate the conditional hazard: \[ \Lambda(t\mid z)=\int_0^t\lambda_0(u)e^{\beta^\top z}\,\mathrm{d}u =e^{\beta^\top z}\Lambda_0(t). \] Exponentiating its negative gives the survival relation. Cancellation of \(\lambda_0(t)\) gives the hazard ratio.
Interpretation. A coefficient \(\beta_j\) is a conditional log hazard ratio holding the other modeled covariates fixed. It is not a ratio of event probabilities, a ratio of mean survival times, or generally a marginal causal effect. Hazard ratios are noncollapsible: integrating over a prognostic covariate can change the hazard ratio even without confounding.
Example (binary exposure). If \(Z\in\{0,1\}\), then \[ \lambda(t\mid Z=1)=e^\beta\lambda(t\mid Z=0),\qquad S(t\mid Z=1)=S(t\mid Z=0)^{e^\beta}. \] When \(\beta=\log2\), the instantaneous hazard doubles at every time, but the difference \(S(t\mid1)-S(t\mid0)=S_0(t)^2-S_0(t)\) varies with \(t\). A constant hazard ratio therefore does not imply a constant survival-probability difference.
Full likelihood and the nuisance baseline
Proposition. Under independent right censoring, the event-law likelihood for time-independent covariates is \[ L(\beta,\Lambda_0) =\prod_{i=1}^n \left\{e^{\beta^\top Z_i}\,\mathrm{d}\Lambda_0(X_i)\right\}^{\Delta_i} \exp\{-e^{\beta^\top Z_i}\Lambda_0(X_i)\}. \] In counting-process notation, \[ \ell(\beta,\Lambda_0) =\sum_{i=1}^n\int_0^\tau \{\log\mathrm{d}\Lambda_0(t)+\beta^\top Z_i(t)\}\,\mathrm{d}N_i(t) -\sum_{i=1}^n\int_0^\tau Y_i(t)e^{\beta^\top Z_i(t)}\,\mathrm{d}\Lambda_0(t). \]
Proof. Substitute \(\lambda(t\mid Z)=\lambda_0(t)e^{\beta^\top Z}\) and \(\Lambda(t\mid Z)=e^{\beta^\top Z}\Lambda_0(t)\) into the censored-data likelihood from page 1. The stochastic-integral expression records a log-hazard contribution at each observed event and subtracts integrated exposure.
The baseline hazard is not estimable at ordinary pointwise density rates without smoothness assumptions. The cumulative baseline hazard, however, is estimable at root-\(n\) rate on intervals with positive risk; this is why the model is naturally parameterized by \((\beta,\Lambda_0)\).
Identifiability and coding
Proposition. An intercept in \(\beta^\top Z\) is not identifiable separately from \(\lambda_0\). If \(Z=(1,W^\top)^\top\), then for every \(c\), \[ \lambda_0(t)e^{\beta_0+\beta_W^\top W} =\{\lambda_0(t)e^c\} e^{(\beta_0-c)+\beta_W^\top W}. \]
Proof. The displayed reparameterization leaves every conditional hazard unchanged. Hence the observation law cannot distinguish the two parameter values.
Software therefore omits an intercept from the Cox linear predictor. Identifiability of the remaining coefficients requires risk-set variation: no nonzero vector \(a\) may make \(a^\top Z_i(t)\) identical for all subjects at risk throughout all informative event times.
Time scale and baseline meaning
The baseline hazard absorbs arbitrary evolution in the chosen analysis time. Age, time since treatment, and calendar time define different risk sets and different conditional questions. Changing time scale is not a harmless relabeling when entry and covariates depend on that scale.
Proposition (monotone time transformation). Let \(u=g(t)\) be strictly increasing and continuously differentiable. If \(T^\star=g(T)\), then \[ \lambda^\star(u\mid Z) =\lambda(g^{-1}(u)\mid Z)\frac{\mathrm{d}g^{-1}(u)}{\mathrm{d}u}. \] Thus a Cox model remains proportional after a common monotone time transformation, with the same \(\beta\) and a transformed baseline hazard.
Proof. The transformed survival function is \(S^\star(u\mid Z)=S(g^{-1}(u)\mid Z)\). Differentiating \(-\log S^\star\) gives the formula; the Jacobian factor is common to all covariate values and enters only the baseline.
Stratification and time-varying effects
Definition (stratified Cox model). For stratum \(K=k\), \[ \lambda(t\mid Z,K=k)=\lambda_{0k}(t)e^{\beta^\top Z}. \] Each stratum has its own baseline hazard, while \(\beta\) is shared. Risk-set comparisons occur only within strata.
Definition (nonproportional effect). A time-varying coefficient model has \[ \lambda(t\mid Z)=\lambda_0(t)\exp\{\beta(t)^\top Z\}. \] A finite-dimensional version uses \(\beta(t)=\sum_{k=1}^K\theta_kb_k(t)\). This is distinct from a time-dependent covariate \(Z(t)\): the former changes the effect of a covariate, while the latter changes the covariate history itself.
Model boundary. Stratification controls a categorical variable without estimating its hazard ratio. Interaction with time estimates nonproportional effects. Neither device repairs omitted confounding or informative censoring. Those are assumptions about the observation and causal structures, not about baseline flexibility.
Connection with piecewise exponential models
Proposition. Partition time into intervals \(I_k\) and assume \(\lambda_0(t)=\alpha_k\) on \(I_k\). Conditional on total exposure \[ E_{ik}=\int_{I_k}Y_i(t)\,\mathrm{d}t, \] the event count \(D_{ik}=\int_{I_k}\mathrm{d}N_i(t)\) has log-likelihood \[ \sum_{i,k}\left[ D_{ik}\{\log\alpha_k+\beta^\top Z_i\} -\alpha_kE_{ik}e^{\beta^\top Z_i} \right], \] the kernel of a Poisson log-linear model with offset \(\log E_{ik}\).
Proof. Insert the piecewise constant intensity \(Y_i(t)\alpha_ke^{\beta^\top Z_i}\) into the counting-process likelihood. The event integral counts \(D_{ik}\), and the compensator integral equals \(\alpha_kE_{ik}e^{\beta^\top Z_i}\).
As the partition is refined to the distinct event times and the \(\alpha_k\)'s are profiled out, the resulting likelihood for \(\beta\) approaches Cox partial likelihood. This supplies one concrete route from an ordinary full likelihood to the semiparametric construction.