Partial likelihood isolates the relative-risk parameter by conditioning on which member of a risk set fails. The baseline hazard cancels because every individual in that instantaneous comparison shares the same unknown multiplier.
Risk-set conditioning
Theorem (Cox partial likelihood). Suppose event times are continuous, so ties occur with probability zero. Conditional on one event at time \(t\) and on the risk set \(\mathcal R(t)=\{j:Y_j(t)=1\}\), the probability that subject \(i\in\mathcal R(t)\) fails is \[ P\{i\text{ fails at }t\mid\mathcal R(t),\text{ one failure}\} =\frac{e^{\beta^\top Z_i(t)}} {\sum_{j\in\mathcal R(t)}e^{\beta^\top Z_j(t)}}. \] Hence, if \(i_k\) fails at \(t_k\), the partial likelihood is \[ L_p(\beta) =\prod_{k=1}^{D} \frac{e^{\beta^\top Z_{i_k}(t_k)}} {\sum_jY_j(t_k)e^{\beta^\top Z_j(t_k)}}. \]
Proof. Over \([t,t+\mathrm{d}t)\), subject \(i\)'s event probability is \(Y_i(t)\lambda_0(t)e^{\beta^\top Z_i(t)}\mathrm{d}t+o(\mathrm{d}t)\). Conditional on exactly one event among those at risk, divide this probability by the sum over the risk set. The common factors \(\lambda_0(t)\mathrm{d}t\) cancel. Multiplying the successive conditional probabilities yields \(L_p\).
Conditioning statement. This argument conditions locally on the event occurrence and the predictable risk set. It does not condition globally on all event times in arbitrary censoring models. The rigorous justification is either a profile-likelihood calculation for baseline-hazard jumps or the multiplicative-intensity likelihood.
Profile likelihood derivation
Proposition. Treat \(\Lambda_0\) as a step function with jump \(\alpha_k\) at each distinct event time \(t_k\). If there is one failure at each \(t_k\), maximizing the full likelihood over \(\alpha_k\) gives \[ \widehat\alpha_k(\beta) =\frac{1}{S_n^{(0)}(\beta,t_k)},\qquad S_n^{(0)}(\beta,t)=\sum_{j=1}^nY_j(t)e^{\beta^\top Z_j(t)}. \] Substitution into the full likelihood yields \(L_p(\beta)\) up to a parameter-free factor.
Proof. The part involving \(\alpha_k\) is \[ \log\alpha_k -\alpha_kS_n^{(0)}(\beta,t_k). \] Its strictly concave maximizer is \(1/S_n^{(0)}(\beta,t_k)\). The remaining \(\beta\)-dependent factor at \(t_k\) is \[ \frac{e^{\beta^\top Z_{i_k}(t_k)}}{S_n^{(0)}(\beta,t_k)}. \] Multiplication over \(k\) gives partial likelihood.
Risk-set moments, score, and information
Definition. For \(k=0,1,2\), let \[ S_n^{(k)}(\beta,t) =\sum_{i=1}^nY_i(t)e^{\beta^\top Z_i(t)} Z_i(t)^{\otimes k}, \] where \(z^{\otimes0}=1\), \(z^{\otimes1}=z\), and \(z^{\otimes2}=zz^\top\). Define \[ \bar Z_n(\beta,t)=\frac{S_n^{(1)}(\beta,t)}{S_n^{(0)}(\beta,t)} \] and \[ V_n(\beta,t) =\frac{S_n^{(2)}(\beta,t)}{S_n^{(0)}(\beta,t)} -\bar Z_n(\beta,t)\bar Z_n(\beta,t)^\top. \] These are the exponentially weighted mean and covariance of the current risk set.
Theorem. The partial log-likelihood, score, and observed information are \[ \ell_p(\beta) =\sum_i\int_0^\tau \left[\beta^\top Z_i(t)-\log S_n^{(0)}(\beta,t)\right] \mathrm{d}N_i(t), \] \[ U_n(\beta) =\frac{\partial\ell_p(\beta)}{\partial\beta} =\sum_i\int_0^\tau \{Z_i(t)-\bar Z_n(\beta,t)\}\,\mathrm{d}N_i(t), \] \[ I_n(\beta) =-\frac{\partial^2\ell_p(\beta)} {\partial\beta\partial\beta^\top} =\int_0^\tau V_n(\beta,t)\,\mathrm{d}N(t). \] Thus \(\ell_p\) is concave.
Proof. Differentiate \(\log S_n^{(0)}\): \[ \nabla\log S_n^{(0)}=\frac{S_n^{(1)}}{S_n^{(0)}}=\bar Z_n, \qquad \nabla^2\log S_n^{(0)}=V_n. \] The matrix \(V_n\) is a weighted covariance and is positive semidefinite. Differentiating under the finite sums and event integrals gives the formulas and makes the Hessian \(-I_n\) negative semidefinite.
Corollary. At the true parameter \(\beta_0\), \[ U_n(\beta_0) =\sum_i\int_0^\tau \{Z_i(t)-\bar Z_n(\beta_0,t)\}\,\mathrm{d}M_i(t), \] where \[ M_i(t)=N_i(t)-\int_0^t Y_i(u)e^{\beta_0^\top Z_i(u)}\,\mathrm{d}\Lambda_0(u). \] Hence the Cox score is a martingale integral with conditional mean zero.
Proof. Replace \(\mathrm{d}N_i\) by \(\mathrm{d}M_i+Y_i e^{\beta_0^\top Z_i}\mathrm{d}\Lambda_0\). The compensator term is \[ \left[ S_n^{(1)}(\beta_0,t) -\bar Z_n(\beta_0,t)S_n^{(0)}(\beta_0,t) \right]\mathrm{d}\Lambda_0(t)=0. \]
Concavity, uniqueness, and separation
Proposition. If, for every nonzero \(a\in\mathbb R^p\), there is an event time at which the risk-set values \(a^\top Z_j(t)\) are not all equal, then \(I_n(\beta)\) is positive definite and \(\ell_p\) is strictly concave. Any finite solution of \(U_n(\beta)=0\) is therefore the unique maximizer.
Proof. For nonzero \(a\), \[ a^\top I_n(\beta)a =\int\operatorname{Var}_{w_{\beta,t}}\{a^\top Z(t)\}\,\mathrm{d}N(t), \] where \(w_{\beta,t}\) are positive risk-set weights. The assumed risk-set variation makes at least one integrand strictly positive. Hence the quadratic form is positive, so the Hessian is negative definite.
A finite maximizer can still fail to exist. If a direction \(a\) ranks every observed failure above every comparable nonfailure in its risk set, then \(\ell_p(\beta+ca)\) increases toward a finite supremum as \(c\to\infty\). This is the survival analogue of complete separation in logistic regression. Penalization or a carefully specified prior regularizes the problem but changes the estimand.
A two-risk-set calculation
Example. Three subjects have scalar covariates \(Z=(0,1,2)\). Subject \(3\) fails first; subject \(1\) fails second after subject \(3\) has left. Then \[ L_p(\beta) =\frac{e^{2\beta}}{1+e^\beta+e^{2\beta}} \frac{1}{1+e^\beta}. \] The score is \[ U(\beta) =2-\frac{e^\beta+2e^{2\beta}}{1+e^\beta+e^{2\beta}} -\frac{e^\beta}{1+e^\beta}. \] Each event compares its failing subject only with those still at risk. A subject's contribution can therefore change across event times even when its covariate is fixed.
Tied event times
If \(d\) subjects fail at the same recorded time \(t\), the continuous-time no-ties likelihood no longer specifies their latent ordering. Three common treatments are:
Definition. Let \(D(t)\) be the tied failure set and \(R(t)\) the risk set.
- The exact discrete partial likelihood uses \[ \frac{\exp\{\beta^\top\sum_{i\in D(t)}Z_i\}} {\sum_{A\subset R(t),\,|A|=d} \exp\{\beta^\top\sum_{j\in A}Z_j\}}. \]
- The Breslow approximation uses \[ \frac{\exp\{\beta^\top\sum_{i\in D(t)}Z_i\}} {\{S_n^{(0)}(\beta,t)\}^{d}}. \]
- The Efron approximation successively removes fractions of the tied failure weight: \[ \frac{\exp\{\beta^\top\sum_{i\in D(t)}Z_i\}} {\prod_{\ell=0}^{d-1} \left[S_n^{(0)}(\beta,t) -\frac{\ell}{d}\sum_{i\in D(t)}e^{\beta^\top Z_i}\right]}. \]
Breslow is adequate when ties are sparse; Efron better approximates unknown orderings when ties are moderate; exact discrete likelihood is principled for genuinely discrete event times but can be computationally expensive. The choice is part of the statistical model and should be reported.
Score, Wald, and likelihood-ratio inference
For testing \(H_0:R\beta=r\), the three regular statistics are \[ \begin{aligned} W&=(R\widehat\beta-r)^\top \{R I_n(\widehat\beta)^{-1}R^\top\}^{-1} (R\widehat\beta-r),\\ S&=U_R(\widetilde\beta)^\top I_R(\widetilde\beta)^{-1}U_R(\widetilde\beta),\\ LR&=2\{\ell_p(\widehat\beta)-\ell_p(\widetilde\beta)\}, \end{aligned} \] where \(\widetilde\beta\) maximizes under \(H_0\). Under regularity and a fixed-dimensional null of codimension \(q\), all converge to \(\chi_q^2\). Their equivalence follows from the same local quadratic expansion used on the next page.