Group Sequential Methods

When to Use This Skill

• Designing group sequential trials with interim analyses
• Implementing alpha spending functions
• Setting futility stopping rules
• Calculating information fractions
• Using sim_gs_n() for GS simulations
• Integrating with gsDesign2 package

Fundamental Concepts

Group Sequential Design

A group sequential design allows for:

• Early stopping for efficacy: If treatment effect is larger than expected
• Early stopping for futility: If treatment effect is unlikely to reach significance
• Reduced expected sample size: When treatment effect is present

Information Fraction

Information fraction at analysis k:

I_k / I_K = (events at analysis k) / (total planned events)

For time-to-event trials, information ≈ number of events.

Type I Error Spending

The key constraint is that the design controls the overall Type I error at the planned alpha level. Spending functions define cumulative alpha spending over information time, and boundaries are derived using the joint distribution of sequential test statistics. They are not obtained by simply assigning independent nominal alpha levels to each look.

Alpha Spending Functions

O'Brien-Fleming (OBF)

Properties:

• Conservative at early analyses
• Nearly full alpha at final analysis
• Difficult to stop early
• Maintains nominal Type I error

Formula:

α*(t) = 2 - 2Φ(z_{α/2} / √t)

When to Use:

• Want maximum power at final analysis
• Early efficacy stopping unlikely
• Regulatory preference for conservative early bounds

Pocock

Properties:

• Equal spending at each analysis
• Easier to stop early
• Inflated final alpha
• Lower power at final analysis

Formula:

α*(t) = α × log(1 + (e-1)t)

When to Use:

• Early stopping is a priority
• Treatment effect expected to be large
• Willing to sacrifice final analysis power

Hwang-Shih-DeCani (HSD)

Properties:

• Flexible family indexed by γ
• γ = -4: Similar to OBF
• γ = 1: Similar to Pocock
• γ = 0: Linear (Pocock-like)

Formula:

α*(t) = α × (1 - e^{-γt}) / (1 - e^{-γ})

When to Use:

• Want flexibility between OBF and Pocock
• Customized spending pattern needed

Spending Function Comparison

Futility Boundaries

Binding Futility

• If futility boundary crossed, trial MUST stop
• Affects Type I error calculation
• More powerful than non-binding

Non-Binding Futility

• Crossing futility boundary is advisory
• Trial can continue at investigator discretion
• Conservative: assumes no early stopping for futility in Type I error

Beta-Spending for Futility

Similar to alpha-spending, but for Type II error:

β*(t) = spending function × β

simtrial GS Implementation

create_cut() - Define Analysis Timing

# Interim Analysis 1
ia1_cut <- create_cut(
  planned_calendar_time = 20,        # Minimum 20 months
  target_event_overall = 100,        # Target 100 events
  max_extension_for_target_event = 24,  # Wait up to 24 months for events
  min_n_overall = 200,               # At least 200 enrolled
  min_followup = 12                  # 12 months minimum follow-up
)

# Interim Analysis 2
ia2_cut <- create_cut(
  planned_calendar_time = 32,
  target_event_overall = 200,
  max_extension_for_target_event = 34,
  min_time_after_previous_analysis = 10  # At least 10 months after IA1
)

# Final Analysis
fa_cut <- create_cut(
  planned_calendar_time = 45,
  target_event_overall = 350
)

sim_gs_n() - Run GS Simulations

library(simtrial)
library(gsDesign2)

# Define enrollment
enroll_rate <- define_enroll_rate(
  duration = c(4, 12),
  rate = c(10, 30)
)

# Define failure rates
fail_rate <- define_fail_rate(
  duration = c(3, 100),
  fail_rate = log(2)/9,
  hr = c(1, 0.6),
  dropout_rate = 0.001
)

# Run simulation
results <- sim_gs_n(
  n_sim = 1000,
  sample_size = 400,
  enroll_rate = enroll_rate,
  fail_rate = fail_rate,
  test = wlr,
  cut = list(ia1 = ia1_cut, ia2 = ia2_cut, fa = fa_cut),
  weight = fh(rho = 0, gamma = 0)
)

Integration with gsDesign2

library(gsDesign2)

# Design with gsDesign2
design <- gs_design_ahr(
  enroll_rate = define_enroll_rate(duration = c(4, 12), rate = c(10, 30)),
  fail_rate = define_fail_rate(
    duration = c(3, 100),
    fail_rate = log(2)/9,
    hr = c(1, 0.6),
    dropout_rate = 0.001
  ),
  alpha = 0.025,
  beta = 0.1,
  analysis_time = c(24, 36, 48),
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfHSD, param = -4, total_spend = 0.1)
) |> to_integer()

# Simulate with design object
sim_results <- sim_gs_n(
  n_sim = 1000,
  sample_size = max(design$analysis$n),
  enroll_rate = design$enroll_rate,
  fail_rate = design$fail_rate,
  test = wlr,
  cut = NULL,  # Auto-generated from design
  original_design = design,
  weight = fh(rho = 0, gamma = 0)
)

Bound Updates with sim_gs_n()

When using original_design, sim_gs_n() can compute updated bounds:

# Results include planned and updated bounds
results <- sim_gs_n(
  # ... parameters ...
  original_design = design,
  ia_alpha_spending = "min_planned_actual",  # Conservative
  fa_alpha_spending = "full_alpha"           # Spend full alpha at FA
)

# Output includes:
# - planned_upper_bound, planned_lower_bound
# - updated_upper_bound, updated_lower_bound

Alpha Spending Options:

Different Tests Across Analyses

# Different tests at each analysis
ia1_test <- create_test(wlr, weight = fh(rho = 0, gamma = 0))
ia2_test <- create_test(wlr, weight = fh(rho = 0, gamma = 0.5))
fa_test <- create_test(wlr, weight = mb(delay = 6, w_max = Inf))

results <- sim_gs_n(
  n_sim = 1000,
  sample_size = 400,
  enroll_rate = enroll_rate,
  fail_rate = fail_rate,
  test = list(ia1 = ia1_test, ia2 = ia2_test, fa = fa_test),
  cut = list(ia1 = ia1_cut, ia2 = ia2_cut, fa = fa_cut)
)

Common GS Patterns

Two-Look Design (1 IA + FA)

# IA at 50% information, FA at 100%
ia_cut <- create_cut(target_event_overall = 150)  # 50%
fa_cut <- create_cut(target_event_overall = 300)  # 100%

sim_gs_n(
  n_sim = 1000,
  sample_size = 400,
  test = wlr,
  cut = list(ia = ia_cut, fa = fa_cut),
  weight = fh(0, 0)
)

Three-Look Design (2 IA + FA)

# Standard 33%, 67%, 100% information
ia1_cut <- create_cut(target_event_overall = 100)
ia2_cut <- create_cut(target_event_overall = 200)
fa_cut <- create_cut(target_event_overall = 300)

Event-Driven with Calendar Constraints

# Events-based but with minimum calendar time
ia_cut <- create_cut(
  target_event_overall = 150,
  planned_calendar_time = 18,  # At least 18 months
  max_extension_for_target_event = 24  # Max 24 months
)

Operating Characteristics

Key Metrics to Evaluate

1. Power: P(reject H0 | H1 true)
2. Type I Error: P(reject H0 | H0 true)
3. Expected Sample Size: E[N] under H0 and H1
4. Expected Events: E[events] at each analysis
5. Stopping Probabilities: P(stop at analysis k)

Simulation Summary

# Summarize simulation results
results_summary <- results |>
  group_by(analysis) |>
  summarise(
    mean_events = mean(event),
    mean_z = mean(z),
    power = mean(z < qnorm(0.025)),  # One-sided
    .groups = "drop"
  )

Best Practices

1. Information Fraction: Target evenly spaced (e.g., 50%, 100% or 33%, 67%, 100%)
2. Alpha Spending: OBF is default for most regulatory submissions
3. Futility: Use non-binding to preserve flexibility
4. Validation: Compare simulated power to gsDesign analytical results
5. Documentation: Record all boundary calculations for regulatory submission
6. Parallelization: Use plan("multisession") for large simulations

Regulatory Considerations

• Pre-specify number and timing of interim analyses
• Pre-specify spending function and parameters
• Document stopping rules clearly in protocol
• Consider DSMB recommendations for unblinded reviews
• Maintain blinding for operational team