Planning enrollment and follow-up for survival designs
In gsDesign: Group Sequential Design

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Overview

Survival trial planning connects three time quantities:

enrollment duration;
minimum follow-up after the last participant enrolls; and
total study duration.

The identity is

total study duration = enrollment duration + minimum follow-up.

gsSurv() can solve one component of this plan while deriving a powered group sequential design. gsSurvCalendar() uses the same enrollment model but fixes analysis times on the calendar. gsSurvPower() answers the reverse question: given a fixed operational plan, what power will it achieve? Finally, toInteger() converts continuous expected events and enrollment to an integer-compatible plan.

Choose the workflow by the question

The appropriate function depends first on whether the objective is to derive a powered design, evaluate a fixed plan, or quantify variation during trial execution.

| Question | Workflow | What is fixed or solved | | --- | --- | --- | | What event-driven design achieves target power? | gsSurv() | Solves the required sample size, enrollment, or follow-up component | | What design achieves target power at specified calendar looks? | gsSurvCalendar() | Fixes calendar analysis times and solves the powered design | | What power does a specified plan achieve under a scenario? | gsSurvPower() | Keeps supplied enrollment, failure, treatment-effect, and timing assumptions fixed | | How variable are analysis dates and operating characteristics in execution? | Simulation, such as simtrial | Generates trial realizations under stochastic enrollment, failure, dropout, and timing |

gsSurvPower() is the bridge between initial design and simulation. It is well-suited to rapid deterministic scenario grids: vary enrollment rates, failure rates, dropout, hazard ratios, or operational timing rules and compare the resulting expected analysis times, events, and power. Its calculations use expected enrollment and event accumulation, however, so event- or enrollment-triggered analysis times are expected times. Use simulation when the distribution of those times—or the chance that competing operational rules determine an analysis—is important.

For detailed combinations of calendar floors, event targets, minimum spacing, enrollment plus follow-up, and extension deadlines, see vignette("gsSurvPower").

library(gsDesign)

lambdaC <- log(2) / 12
hr <- 0.7

# Four enrollment periods with equal relative rate increments.
gamma_ramp <- 1:4
R_ramp <- rep(1, 4)

Every nSurv object contains identical scalar values n and N for total expected enrollment. Every gsSurv object contains N as a vector of cumulative total expected enrollment at each analysis. It is the row total of the control and experimental enrollment components:

x$N == rowSums(x$eNC) + rowSums(x$eNE)

Pattern 1: fix study duration and minimum follow-up

This is the most common planning pattern. Specifying T and minfup fixes the enrollment duration at T - minfup. The values in gamma describe the relative ramp-up shape; gsSurv() scales all rates proportionally to power the trial. When the supplied R periods do not fill the enrollment duration, the last period is extended.

fixed_duration <- gsSurv(
  k = 3,
  lambdaC = lambdaC,
  hr = hr,
  T = 26,
  minfup = 12,
  gamma = gamma_ramp,
  R = R_ramp
)

data.frame(
  period = seq_along(fixed_duration$R),
  duration = fixed_duration$R,
  rate = as.vector(fixed_duration$gamma)
)

fixed_duration$N

Here enrollment lasts 14 months. The first three ramp-up periods last one month each and the fourth rate continues through the remaining 11 months.

Pattern 2: fix rates and minimum follow-up

Set T = NULL to keep gamma fixed and solve how long enrollment must remain open. The final R period is extended to obtain the required sample size; the earlier ramp-up periods are unchanged.

fixed_rates <- gsSurv(
  k = 3,
  lambdaC = lambdaC,
  hr = hr,
  T = NULL,
  minfup = 12,
  gamma = gamma_ramp,
  R = R_ramp
)

data.frame(
  period = seq_along(fixed_rates$R),
  duration = fixed_rates$R,
  rate = as.vector(fixed_rates$gamma)
)

c(
  enrollment_duration = sum(fixed_rates$R),
  minimum_follow_up = fixed_rates$minfup,
  total_duration = max(fixed_rates$T)
)

Absolute rates of 1, 2, 3, and 4 participants per month are deliberately low, so this example produces a long enrollment duration. In practice, multiply the ramp by realistic site-level or program-level rates.

Pattern 3: fix enrollment and solve follow-up

With both T = NULL and minfup = NULL, enrollment rates and their durations are fixed. gsSurv() solves the follow-up duration needed to power the trial. This option can fail when the fixed enrollment plan is over-powered even with almost no follow-up, or under-powered regardless of follow-up.

fixed_enrollment <- gsSurv(
  k = 3,
  lambdaC = lambdaC,
  hr = hr,
  T = NULL,
  minfup = NULL,
  gamma = 50 * gamma_ramp,
  R = R_ramp
)

c(
  enrollment_duration = sum(fixed_enrollment$R),
  minimum_follow_up = fixed_enrollment$minfup,
  total_duration = max(fixed_enrollment$T)
)

When this solve is infeasible, revise the fixed enrollment plan, target power, or event assumptions rather than interpreting the error as a numerical failure.

Calendar-time analyses

Use gsSurvCalendar() when interim analyses are specified as months from the start of enrollment. The final calendar time and minfup imply the enrollment duration, while the four-period ramp-up is scaled to power the trial.

calendar_design <- gsSurvCalendar(
  calendarTime = c(12, 18, 26),
  lambdaC = lambdaC,
  hr = hr,
  minfup = 12,
  gamma = gamma_ramp,
  R = R_ramp
)

data.frame(
  analysis_month = calendar_design$T,
  expected_events = calendar_design$n.I,
  expected_enrollment = calendar_design$N
)

Use gsSurv() instead when analyses are defined by event or information fractions rather than calendar dates.

Power for a fixed operational plan

gsSurvPower() does not resize enrollment to hit target power. It evaluates power for the supplied rates, durations, treatment effect, and analysis timing. For example, the following sensitivity analysis evaluates 80% of the planned enrollment rates at the original calendar analysis times.

slower_enrollment <- gsSurvPower(
  x = fixed_duration,
  gamma = 0.8 * fixed_duration$gamma,
  plannedCalendarTime = fixed_duration$T
)

c(
  planned_power = 1 - fixed_duration$beta,
  slower_enrollment_power = slower_enrollment$power
)

Use targetEvents = fixed_duration$n.I instead of plannedCalendarTime when event counts, rather than dates, remain fixed and the analysis dates may move.

Integer event and enrollment plans

Design calculations use expected counts and can therefore be non-integer. Apply toInteger() after deriving the design to obtain integer event targets and a final total enrollment compatible with the randomization allocation.

integer_design <- toInteger(fixed_duration)

data.frame(
  analysis = seq_len(integer_design$k),
  events = integer_design$n.I,
  enrollment = integer_design$N
)

The input ratio is experimental-to-control randomization. For example, ratio = 1 produces allocation-compatible even totals; ratio = 2 produces totals compatible with 2:1 randomization.

Stratified enrollment

For a stratified design, matrix columns identify strata. Align the columns of the control hazards, dropout rates, and enrollment rates. The example below uses two strata with different control medians and enrollment contributions.

lambda_strata <- matrix(log(2) / c(10, 16), nrow = 1)
gamma_strata <- cbind(
  0.6 * gamma_ramp,
  0.4 * gamma_ramp
)

stratified_design <- gsSurv(
  k = 3,
  lambdaC = lambda_strata,
  hr = hr,
  eta = matrix(c(0.001, 0.001), nrow = 1),
  T = 26,
  minfup = 12,
  gamma = gamma_strata,
  R = R_ramp
)

data.frame(
  analysis = seq_len(stratified_design$k),
  control = rowSums(stratified_design$eNC),
  experimental = rowSums(stratified_design$eNE),
  total = stratified_design$N
)

The same matrix conventions apply to gsSurvCalendar() and gsSurvPower(). For final operational planning, inspect both N and the stratum-specific eNC and eNE matrices before applying toInteger().