8.2 Tied event times

The partial likelihood assumed that no two events occur at the same time. Recorded times are rounded, to days or months, so ties are common: the veteran data have \(31\) death times shared with another death. Three methods handle them.

• Breslow: use the full risk set in the denominator for each of the \(d_j\) tied events. Simple, but biased towards zero when ties are many.
• Efron: as the tied subjects’ events are thought of as happening one after another in unknown order, remove on average a fraction of them from the risk set for each successive event. Much more accurate, and R’s default.
• Exact: average over all the possible orderings of the tied events. Exact for the continuous-time model but expensive with many ties.

In the veteran model the treatment coefficient is \(0.257\) by Breslow, \(0.262\) by Efron and \(0.263\) by the exact method (Software 8.6.2): with moderate ties the methods hardly differ, and Breslow’s is the one furthest from the exact answer. Software packages differ in their defaults (SAS and Stata use Breslow), which explains small discrepancies between them.

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