Preface

We consider a 2-arm group sequential trial with an experimental and control group and a time-to-event endpoint. Such trials are common, for example, in cardiovascular disease and oncology. For instance, a drug for improving lipids could be compared to a standard drug to see if cardiovascular endpoints such as heart attacks or death could be prevented. In oncology trials, a new treatment might be compared to a control to see if survival or time-to-progression might be extended. Such trials can be large and complex. This technical report documents software in the gsDesign R package for designing such trials. In fact, the report is intended as a tool to thoroughly document the sample size functionality in the gsDesign R package that is supported by the nSurv() and gsSurv() functions. Some common issues that arise that are dealt with by this software are:

• All designs assume a proportional hazards model.
• Piecewise constant enrollment rates are allowed given that enrollment may be slow initially during study ramp-up.
• Piecewise exponential failure rates are allowed to allow changing outcome incidence over time.
• Piecewise exponential censoring rates allow changing dropout rates by treatment group and over time.
• A patient population may be stratified to allow subgroups with different underlying enrollment rates as well as differing incidence rates for endpoints and censoring.
• The software computes expected study durations and timing of interim analyses as well as the expected enrollment and number of events at interim analyses.
• Statistical bounds are given in terms of Z-value cutoffs for test statistics as well as approximate hazard ratios required to cross such bounds; additional computations available for more general group sequential designs are also available from other gsDesign package routines.

In the following, we assume a given Type I error, timing of interim analyses, endpoint event rates and censoring rates. There are two general approaches to carrying out calculations.

1. Assuming fixed accrual duration and minimum follow-up duration determine the accrual rate and sample size required to provide the desired power.
2. Assume given accrual rates over time and vary accrual or follow-up duration to achieve a desired power.

A group sequential design is generally defined by Type I and Type II error, the number and timing of interim analyses and the boundaries at interim analyses. For many endpoint types, timing is linear in the number of evaluable observations. For time-to-event endpoints, timing is related to the number of endpoint events available for analysis, which is generally a non-linear function of accrual rates and duration, endpoint event rates, censoring rates and duration of follow-up after accrual termination. Lachin and Foulkes (1986) have provided a general formulation for calculating the expected number of events over time in the scenarios described above. They also derive the sample size method used here for the case where enrollment and study durations are fixed, which requires deriving enrollment rates required to achieve the desired Type I error and power. Their methods for calculating power are extended to compute power for similarly complex designs, but where enrollment or follow-up duration are varied to achieve power given fixed enrollment rates; such methods are analogous to those of Kim and Tsiatis (1990).

This document is organized based on supporting the following calculations:

• Computing the probability that any individual entering the study will have an endpoint observed as a function of random enrollment, event, and censoring times. This calculation, presented in Section 2, is key to all other quantities considered. The routine provided in the gsDesign package documented in Section 2 is eEvents().
• Designing a fixed (single analysis) trial with a time-to-event endpoint is presented in Section 3 where the nSurv() routine in the gsDesign package is defined.
• Designing a group sequential trial with a time-to-event endpoint is presented in Section 4, where the gsDesign function gsSurv() is defined. Supporting routines in gsDesign to compute the expected number of events at a given point in time (nEventsIA()) or the time required for the expected number of events to reach a given proportion of the final planned number (tEventsIA()).

Each section presents extensive examples to show how to use the key routines.

Summary of updates

• August, 2015. Updated nEventsIA() to work correctly for stratified populations when the simple = TRUE option is chosen. This also impacted tEventsIA() for stratified populations.
• August, 2023 (ONGOING)
  • Updated gsSurv() to enable power computation
  • Created gsSurvCalendar() to enable calendar-based interim analysis planning
  • Incorporated revisions of code that have been put into gsDesign package
  • Fixing stratified sample size computation
  • Added EDC0, EDC1 to output of gsSurv() to match help file output specification.
• April, 2025. Migrate from Sweave document to Quarto book.

Kim, Kyungmann, and Anastasios A. Tsiatis. 1990. “Study Duration for Clinical Trials with Survival Response and Early Stopping Rule.” Biometrics 46: 81–92.

Lachin, John M., and Mary A. Foulkes. 1986. “Evaluation of Sample Size and Power for Analyses of Survival with Allowance for Nonuniform Patient Entry, Losses to Follow-up, Noncompliance, and Stratification.” Biometrics 42: 507–19.