ABSTRACT
Institute of Medical Biometry and Medical Informatics, University Medical Center Freiburg
28.1 Clinical trials and time-to-event data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 28.1.1 Binomial sample size formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 28.1.2 Noncensored time-to-event endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 28.1.3 Exponential model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
28.2 Basic formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 28.2.1 Schoenfeld’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 28.2.2 Alternative formula by Freedman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576
28.3 Sample size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 28.3.1 Parametric estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 28.3.2 Nonparametric approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 28.3.3 Competing risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
28.4 Data example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 28.4.1 4D trial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
28.4.1.1 Two-state model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 28.4.1.2 Competing risks analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583
28.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 28.5.1 Multi-arm survival trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 28.5.2 Test for non-inferiority/superiority and equivalence . . . . . . . . . . . . . . . . . . 587 28.5.3 Prognostic factors and/or non-randomized comparisons . . . . . . . . . . . . . . 588 28.5.4 Left truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 28.5.5 Proportional subdistribution modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 28.5.6 Cluster-randomized trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 28.5.7 Cox regression with a time-varying covariate . . . . . . . . . . . . . . . . . . . . . . . . . 591
28.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592
In clinical research, investigators are often interested in the occurrence of certain events such as disease progression, relapse or death. One objective may be to evaluate the effect of a new treatment or a new drug on the prevention or reduction of such undesired events. The time to the occurrence of an event is referred to as “time-to-event.” When the event is death, the time-to-event is the patient’s survival time, hence analysis of time-to-event data is
analysis.” Statistical methods for analysis of time-to-event data are different from those used for continuous outcomes, e.g., comparing means or proportions, as time-to-event is usually subject to censoring. In the presence of censoring for some patients the exact event time is not known, but only that the event time is larger than an observed censoring time. Furthermore normality assumptions of standard statistical methods such as the t-test usually do not hold for time-to-event data.
