Survival Analysis
Sample size planning for a two-arm survival study under a common-shape Weibull model. Specify the expected median survival times, Weibull shape, accrual and follow-up periods, significance level, power, and allocation ratio. The calculation uses the parametric Schoenfeld formulation described by Wu for randomized survival trials under the Weibull model.
This calculator uses a two-arm parametric survival sample-size formulation for a common-shape Weibull model. The Weibull survival function in group j is parameterized by its median survival time \(m_j\) and common shape parameter \(\kappa\):
The corresponding proportional-hazards ratio implied by the two Weibull medians is:
For uniform entry over an accrual period \(t_a\), followed by an additional fixed follow-up period \(t_f\), the probability that a subject in group \(j\) experiences the event before the administrative end of the study is obtained by integrating the Weibull survival function over the possible entry times:
The integral is evaluated numerically in the calculator. If there is no accrual period, the event probability reduces to \(1-S_j(t_f)\).
Let \(\pi\) denote the treatment-to-control allocation ratio, so that \(n_T=\pi n_C\). With a two-sided significance level \(\alpha\), target power \(1-\beta\), and the Weibull hazard ratio \(\Delta\), the required continuous control-group sample size under the Schoenfeld parametric formulation is:
The corresponding treatment-group size is \(n_T=\pi n_C\). The calculator rounds the required sample sizes upward so that the final design does not fall below the requested asymptotic power.
Interpretation of the Weibull shape: κ = 1 corresponds to the exponential model and a constant hazard. κ > 1 represents an increasing hazard over time, while κ < 1 represents a decreasing hazard. Because the shape parameter enters both the Weibull event probabilities and the implied hazard ratio, it can materially affect the required sample size.
The implementation was checked against the numerical example reported by Wu for a randomized two-arm survival trial. The published sample-size table uses a common Weibull shape of \(\kappa=1\), a median-survival ratio \(R=m_T/m_C=1.5\), 5% two-sided significance level, 90% power, uniform accrual over 5 time units, and 2 additional time units of follow-up. With equal allocation, the Schoenfeld parametric calculation gives a continuous requirement of approximately 144.33 patients per group, which rounds up to 145 patients per group, or 290 patients total.
This validation target is important because it checks the full implementation rather than only the Weibull distribution function: the event-probability integration, allocation adjustment, normal quantiles, Weibull treatment effect, and final rounding must all be consistent with the published formulation.
This calculator assumes two independent treatment groups, a common known or externally estimated Weibull shape parameter, uniform patient entry during the accrual period, no loss to follow-up, and administrative censoring at the end of the planned study. The parametric test is based on the large-sample normal approximation described by Wu. It is not a simulation-based power calculation.
The treatment effect is represented through the ratio of the two Weibull median survival times. Under the common-shape Weibull proportional-hazards formulation, this corresponds to a constant hazard ratio. If the two groups have different Weibull shape parameters, or if survival curves are expected to cross, this calculation should not be interpreted as a proportional-hazards design.
Wu, J. (2013/2015). Power and Sample Size for Randomized Phase III Survival Trials under the Weibull Model. Journal of Biopharmaceutical Statistics, 25, 16–28. The paper derives parametric Weibull sample-size formulas for randomized two-arm survival trials, including the Schoenfeld formulation, uniform entry, and the calculation of event probabilities under Weibull survival.
Heo, M., Faith, M. S., & Allison, D. B. (1998). Power and sample size for survival analysis under the Weibull distribution when the whole lifespan is of interest. Mechanisms of Ageing and Development, 102(1), 45–53. doi:10.1016/S0047-6374(98)00010-4. This paper developed an earlier proportional-Weibull extension of survival power and sample-size calculations and established the importance of the Weibull shape parameter.
NCSS/PASS documentation. PASS provides extensive survival-analysis sample-size procedures and Weibull-based survival procedures. The PASS documentation describes the survival procedures, their technical formulas, examples, and validation material. See the official PASS Sample Size Software Documentation for the current procedure documentation.