7.2 The partial likelihood
Suppose there are no ties, and order the event times \(t_{(1)} < \cdots < t_{(r)}\), with \(i_j\) the subject who has the event at \(t_{(j)}\) and \(R_j\) the risk set just before it. Given that exactly one of the subjects in \(R_j\) has an event at \(t_{(j)}\), the probability that it is subject \(i_j\) is \[ \frac {h(t_{(j)} \mid x_{i_j})}{\sum _{l \in R_j}h(t_{(j)} \mid x_l)} = \frac {h_0(t_{(j)})e^{\beta ^{\mathsf T}x_{i_j}}}{\sum _{l \in R_j}h_0(t_{(j)})e^{\beta ^{\mathsf T}x_l}} = \frac {e^{\beta ^{\mathsf T}x_{i_j}}}{\sum _{l \in R_j}e^{\beta ^{\mathsf T}x_l}} , \] and the unknown baseline hazard cancels. Multiplying over the event times gives Cox’s partial likelihood \[ L_P(\beta ) = \prod _{j = 1}^r\frac {e^{\beta ^{\mathsf T}x_{i_j}}}{\sum _{l \in R_j}e^{\beta ^{\mathsf T}x_l}}, \qquad \ell _P(\beta ) = \sum _{j = 1}^r\Bigl (\beta ^{\mathsf T}x_{i_j} - \log \sum _{l \in R_j}e^{\beta ^{\mathsf T}x_l}\Bigr ) . \] It is not a full likelihood, since it ignores the information in the exact times between events, but Cox showed, and later work with counting processes proved, that maximising it gives consistent, asymptotically normal estimates, with standard errors from the inverse of the information \(-\partial ^2\ell _P/\partial \beta ^2\), as for an ordinary likelihood. Censored subjects enter only through the risk sets, as they should. The maximisation uses Newton–Raphson; \(\ell _P\) is concave, so the maximum is unique when it exists.
Example 7.2.1. Eight subjects have times \(2, 3, 5^+, 6, 8, 9^+, 11, 14\) and a binary covariate \(x = 1, 0, 1, 1, 0, 1, 0, 0\). At the first event (time \(2\), \(x = 1\)) all eight are at risk, four with \(x = 1\), so the factor is \(e^\beta /(4e^\beta + 4)\). At time \(3\) (\(x = 0\)) seven are at risk, three with \(x = 1\): \(1/(3e^\beta + 4)\). Continuing over the six events and maximising gives \(\hat \beta = 0.444\), \(\widehat {\text {HR}} = 1.56\) (Software 7.5.1).
Theorem 7.2.2. For a single binary covariate and no ties, the score test of \(\beta = 0\) in the Cox model is the log-rank test.
Proof. At \(\beta = 0\), \(\partial \ell _P/\partial \beta = \sum _j\bigl (x_{i_j} - \sum _{l \in R_j}x_l/|R_j|\bigr ) = \sum _j(d_{1j} - n_{1j}/n_j)\), which is \(O_1 - E_1\). The information at \(\beta = 0\) is \(\sum _j(n_{1j}/n_j)(1 - n_{1j}/n_j)\), the log-rank variance with \(d_j = 1\). The score statistic is \((O_1 - E_1)^2/V\). □
With ties the two differ slightly, according to how the ties are handled (the subject of the next section): for sex in lung, the score test gives \(10.325\) and the log-rank test \(10.327\).
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