Partial likelihood estimates relative risk without specifying the baseline hazard. Absolute survival prediction requires putting that nuisance parameter back. The natural target is the cumulative baseline hazard, whose estimator is a weighted Nelson-Aalen process evaluated at the fitted regression coefficient.
Breslow estimator
Definition. For \[ S_n^{(0)}(\beta,t)=\sum_{i=1}^nY_i(t)e^{\beta^\top Z_i(t)}, \] the Breslow estimator of the cumulative baseline hazard is \[ \widehat\Lambda_0(t;\beta) =\int_0^t\frac{\mathrm{d}N(u)}{S_n^{(0)}(\beta,u)} =\sum_{t_j\le t}\frac{d_j}{S_n^{(0)}(\beta,t_j)}. \] The fitted estimator is \(\widehat\Lambda_0(t)=\widehat\Lambda_0(t;\widehat\beta)\).
Theorem (profile NPMLE). If the baseline cumulative hazard is allowed to jump only at observed event times and ties are handled by the Breslow convention, then for fixed \(\beta\) its maximizing jump at \(t_j\) is \[ \Delta\widehat\Lambda_0(t_j;\beta) =\frac{d_j}{S_n^{(0)}(\beta,t_j)}. \]
Proof. The full log-likelihood terms involving the jump \(\alpha_j=\Delta\Lambda_0(t_j)\) are \[ d_j\log\alpha_j-\alpha_jS_n^{(0)}(\beta,t_j). \] This strictly concave function has derivative \(d_j/\alpha_j-S_n^{(0)}(\beta,t_j)\), whose zero gives the stated jump. Summing the jumps gives the estimator.
Example. With a binary covariate and fitted coefficient \(\widehat\beta=\log2\), suppose an event time has three unexposed and two exposed subjects at risk, with one observed failure. The weighted risk-set size is \[ 3e^0+2e^{\log2}=7, \] so the baseline cumulative hazard jumps by \(1/7\). An unexposed subject's fitted cumulative hazard also jumps by \(1/7\), while an exposed subject's jumps by \(2/7\).
Consistency and first-order expansion
Theorem. Under the regularity conditions of the preceding page, \[ \sup_{t\le\tau} |\widehat\Lambda_0(t)-\Lambda_0(t)| \xrightarrow{P}0. \] Moreover, \[ \sqrt n\{\widehat\Lambda_0(t)-\Lambda_0(t)\} =\mathbb W_n(t)-B(t)^\top \sqrt n(\widehat\beta-\beta_0)+o_P(1) \] uniformly in \(t\in[0,\tau]\), where \[ \mathbb W_n(t) =\frac1{\sqrt n}\sum_{i=1}^n \int_0^t\frac{\mathrm{d}M_i(u)} {s^{(0)}(\beta_0,u)} \] and \[ B(t)=\int_0^t \frac{s^{(1)}(\beta_0,u)} {s^{(0)}(\beta_0,u)} \,\mathrm{d}\Lambda_0(u) =\int_0^t e(\beta_0,u)\,\mathrm{d}\Lambda_0(u). \]
Proof. At \(\beta_0\), \[ \widehat\Lambda_0(t;\beta_0)-\Lambda_0(t) =\int_0^t \frac{\sum_i\mathrm{d}M_i(u)} {S_n^{(0)}(\beta_0,u)}. \] Uniform positive risk and the martingale law of large numbers give consistency. Taylor-expand \(\widehat\Lambda_0(t;\beta)\) in \(\beta\): \[ \frac{\partial}{\partial\beta} \widehat\Lambda_0(t;\beta) =-\int_0^t \frac{S_n^{(1)}(\beta,u)} {\{S_n^{(0)}(\beta,u)\}^2}\,\mathrm{d}N(u). \] At \(\beta_0\), this derivative converges uniformly to \(-B(t)\). Replacing normalized empirical risk-set sums by \(s^{(k)}\) in the martingale term gives \(\mathbb W_n(t)\), and the remainder is \(o_P(1)\).
Joint Gaussian limit
Theorem. The pair \[ \left( \sqrt n(\widehat\beta-\beta_0), \sqrt n(\widehat\Lambda_0-\Lambda_0) \right) \] converges jointly to a centered Gaussian element in \(\mathbb R^p\times D[0,\tau]\). An influence function for \(\Lambda_0(t)\) is \[ \phi_{\Lambda,t}(O_i) =\int_0^t\frac{\mathrm{d}M_i(u)} {s^{(0)}(\beta_0,u)} -B(t)^\top\phi_\beta(O_i). \]
Proof. Stack the martingale-integral representations of \(\widehat\beta\) and \(\widehat\Lambda_0\). Their predictable covariance functions converge uniformly, and their jumps vanish. The multivariate martingale FCLT gives joint Gaussian convergence. Substitution of the influence representation for \(\widehat\beta\) yields the displayed \(\phi_{\Lambda,t}\).
Proposition (plug-in covariance). A consistent covariance estimator is obtained from empirical influence values \[ \widehat\phi_{\Lambda,t,i} =\int_0^t \frac{\mathrm{d}\widehat M_i(u)} {\mathbb S_n^{(0)}(\widehat\beta,u)} -\widehat B(t)^\top\widehat\phi_{\beta,i}, \] using \[ \widehat{\operatorname{Cov}} \{\widehat\Lambda_0(s),\widehat\Lambda_0(t)\} =\frac1{n^2}\sum_{i=1}^n \widehat\phi_{\Lambda,s,i} \widehat\phi_{\Lambda,t,i}. \]
Conditional survival prediction
Definition. For a fixed covariate vector \(z\), \[ \widehat S(t\mid z) =\exp\{-\widehat\Lambda_0(t)e^{\widehat\beta^\top z}\}. \] For a predictable future path \(z(u)\), \[ \widehat S(t\mid z(\cdot)) =\exp\left\{ -\int_0^t e^{\widehat\beta^\top z(u)} \,\mathrm{d}\widehat\Lambda_0(u) \right\}, \] provided the full path is meaningful and available for prediction.
Theorem (delta-method influence function). Let \[ H_z(t)=e^{\beta_0^\top z}\Lambda_0(t), \qquad S_z(t)=e^{-H_z(t)}. \] Then \[ \phi_{H_z,t} =e^{\beta_0^\top z} \{\phi_{\Lambda,t}+\Lambda_0(t)z^\top\phi_\beta\}, \] and \[ \phi_{S_z,t}=-S_z(t)\phi_{H_z,t}. \]
Proof. The derivative of \((\beta,\Lambda)\mapsto e^{\beta^\top z}\Lambda(t)\) in direction \((h,g)\) is \[ e^{\beta^\top z}\{g(t)+\Lambda(t)z^\top h\}. \] The derivative of \(h\mapsto e^{-h}\) is \(-e^{-h}\). Apply the functional delta method to the joint asymptotic expansion.
Prediction targets and calibration
Three quantities must not be conflated:
- the linear predictor \(\eta=\beta^\top z\), which ranks relative hazard;
- the absolute risk \(1-S(t\mid z)\), which depends on \(\Lambda_0\) and a time horizon;
- the restricted mean survival time \[ \mu_z(\tau)=\int_0^\tau S(t\mid z)\,\mathrm{d}t, \] which summarizes survival on the time scale.
Proposition. If \(\widehat S(\cdot\mid z)\) converges uniformly and is bounded by one, then \[ \widehat\mu_z(\tau) =\int_0^\tau\widehat S(t\mid z)\,\mathrm{d}t \xrightarrow{P}\mu_z(\tau). \] Its influence function is \[ \phi_{\mu_z,\tau} =\int_0^\tau\phi_{S_z,t}\,\mathrm{d}t. \]
Proof. Uniform convergence directly bounds the integral error by \(\tau\|\widehat S-S\|_\infty\). The integration map is continuous and linear on \(\ell^\infty[0,\tau]\), so applying it to the process influence expansion gives the second statement.
Discrimination assesses ranking, whereas calibration compares predicted and observed risks. A high concordance statistic does not guarantee calibrated absolute risk. External prediction also requires transportability of both relative effects and baseline incidence; re-estimating only \(\Lambda_0\) can recalibrate the overall event rate when relative effects remain stable.
Pointwise and simultaneous uncertainty
Pointwise intervals follow from the influence variance at a fixed \(t\), preferably on a monotone transform such as \(\log\{-\log S(t\mid z)\}\). A simultaneous band on \([t_1,t_2]\) requires a critical value for \[ \sup_{t\in[t_1,t_2]} \left| \frac{\sqrt n\{\widehat S(t\mid z)-S(t\mid z)\}} {\widehat\sigma_z(t)} \right|. \] It can be approximated by multiplier draws \[ \frac1{\sqrt n}\sum_{i=1}^n \xi_i\widehat\phi_{S_z,t,i}, \qquad E\xi_i=0,\quad E\xi_i^2=1, \] conditional on the data. This preserves the covariance across time and is therefore preferable to a collection of unrelated pointwise intervals.