Survival Analysis
Two-group survival sample size calculation using the log-rank test under exponential survival and uniform patient accrual. Enter the group-specific hazard rates, significance level, power, accrual period, follow-up period, and allocation ratio to obtain the required number of events and total enrollment.
This calculator uses the standard Schoenfeld event-count approach for a two-group log-rank test under proportional hazards, followed by the exponential-survival/uniform-accrual conversion from the required number of events to the required number of enrolled subjects. This method uses this approach for survival designs with exponential data and uniform accrual, including accrual and follow-up as design parameters.
Let p1 and p2 be the allocation proportions, with p1 + p2 = 1. For a target hazard ratio HR = λ1/λ2, the required total number of events is
This is the Schoenfeld (1981) large-sample event requirement for the log-rank test. With equal allocation, p1p2 = 0.25. The required event count is therefore driven primarily by the magnitude of the hazard-ratio effect, alpha, power, and allocation ratio.
Subjects are assumed to enter uniformly during an accrual period Ta, followed by an additional follow-up period Tf. Thus the total study duration is Ta + Tf. Under an exponential survival distribution with constant hazard λi, the average probability that a subject in group i experiences the event before the study ends is
This is the accrual-averaged event probability: patients enrolled early have longer potential observation than patients enrolled near the end of accrual.
The overall expected event probability is the allocation-weighted average of the two group-specific probabilities:
The continuous enrollment estimate is rounded upward to the next whole participant. The resulting total is then divided according to the specified allocation ratio.
The calculation assumes two independent groups, proportional hazards, constant exponential hazard rates within each group, uniform accrual, independent administrative censoring at the end of the planned study, and no additional loss to follow-up. It is intended for planning a conventional fixed-sample two-group log-rank design, not for group sequential, non-inferiority, equivalence, competing-risk, or non-proportional-hazards designs.
The validation target below reproduces the numerical example reported in the relevant methodological literature for the Schoenfeld and Wu proportional-hazards log-rank procedure. this method specifies two-sided α = 0.05, 90% power, uniform accrual of 1 time unit, 2 additional time units of follow-up, hazard rates 0.17834 and 0.34657, and equal allocation. This yields 37 events in Group 1, 59 events in Group 2, 96 total events, and 102 subjects per group (204 total).
Because the event-count calculation is a continuous large-sample approximation, the unrounded event requirement is 95.215 in the independent calculation; rounding the resulting enrollment gives the same 204-subject target reported by this method. The this method example's displayed integer event counts are 37 + 59 = 96.
Schoenfeld, D.A. (1981). “The asymptotic properties of nonparametric tests for comparing survival distributions.” Biometrika, 68(1), 316–319.
Schoenfeld, D.A. (1983). “Sample-size formula for the proportional-hazards regression model.” Biometrics, 39(2), 499–503.
this method / the software. Logrank Tests with Proportional Hazards (Schoenfeld and Wu). this method Sample Size Software. The documented validation example specifies uniform accrual, exponential survival, 90% power, α = 0.05, hazard rates 0.17834 and 0.34657, accrual time 1, follow-up time 2, and reports 96 required events and 204 total subjects.