Multiplicity Methods

When to Use This Skill

• Selecting appropriate multiplicity adjustment procedures
• Implementing gatekeeping for primary/secondary endpoints
• Designing graphical testing procedures
• Optimizing truncation parameters (gamma)
• Ensuring FWER control in multi-arm/multi-endpoint trials

Fundamental Concepts

Family-Wise Error Rate (FWER)

FWER = P(reject at least one true null hypothesis)

Multiplicity adjustments control FWER at level α (typically 0.025 one-sided or 0.05 two-sided).

Closed Testing Principle

A hypothesis H_i can be rejected at level α if and only if all intersection hypotheses containing H_i are rejected at level α.

This principle underlies most powerful multiplicity procedures.

Single-Step Procedures

Bonferroni

Method: Reject H_i if p_i ≤ α × w_i (where Σw_i = 1)

Properties:

• Most conservative
• Valid under any dependence
• Simple implementation

MultAdjProc(proc = "BonferroniAdj",
            par = parameters(weight = c(0.5, 0.5)))

Step-Down Procedures

Holm Procedure

Method:

1. Order p-values: p_(1) ≤ p_(2) ≤ ... ≤ p_(m)
2. Reject H_(j) if p_(j) ≤ α/(m - j + 1) for all j ≤ i

Properties:

• More powerful than Bonferroni
• Valid under any dependence
• Consonant and coherent

MultAdjProc(proc = "HolmAdj",
            par = parameters(weight = c(0.6, 0.4)))

Fixed-Sequence Procedure

Method: Test hypotheses in predetermined order; stop at first non-rejection.

Properties:

• Maximum power for first hypothesis
• Zero power for later hypotheses if early ones fail
• Useful for clear hierarchy

MultAdjProc(proc = "FixedSeqAdj")
# Tests in order defined in AnalysisModel

Step-Up Procedures

Hochberg Procedure

Method:

1. Order p-values: p_(1) ≤ p_(2) ≤ ... ≤ p_(m)
2. Find largest j where p_(j) ≤ α × j/m
3. Reject all H_(i) with p_(i) ≤ α × j/m

Properties:

• More powerful than Holm
• Requires positive dependence (PRDS) or independence
• Step-up → starts from largest p-value

MultAdjProc(proc = "HochbergAdj",
            par = parameters(weight = c(0.5, 0.5)))

Hommel Procedure

Method: More complex step-up based on Simes' inequality

Properties:

• Most powerful step-up procedure
• Requires PRDS or independence
• Computationally more intensive

MultAdjProc(proc = "HommelAdj")

Graphical Procedures

Chain Procedure

Generalizes fixed-sequence with flexible weight transfer.

Components:

• Initial weights: w = (w_1, ..., w_m), Σw_i = 1
• Transition matrix: G where G_ij = weight transferred from H_i to H_j upon rejection

Algorithm:

1. Test each H_i at level α × w_i
2. Upon rejecting H_j, update: w_i ← w_i + w_j × G_ji, w_j ← 0

# Equal split with full transfer
MultAdjProc(
  proc = "ChainAdj",
  par = parameters(
    weight = c(0.5, 0.5),
    transition = matrix(c(0, 1,
                          1, 0), 2, 2, byrow = TRUE)
  )
)

Fallback Procedure

Special case of chain where rejected hypothesis passes weight to next in sequence.

MultAdjProc(
  proc = "FallbackAdj",
  par = parameters(weight = c(0.5, 0.3, 0.2))
)

Gatekeeping Procedures

Parallel Gatekeeping

For trials with primary and secondary endpoint families where secondary can only be tested if at least one primary is rejected.

Structure:

• Family F_1 (primary): Must reject at least one to "open the gate"
• Family F_2 (secondary): Tested only after gate opens

Components:

• family: List of hypothesis indices per family
• proc: Procedure for each family
• gamma: Truncation parameter (0 = Bonferroni, 1 = Holm within family)

MultAdjProc(
  proc = "ParallelGatekeepingAdj",
  par = parameters(
    family = families(
      family1 = c(1, 2),     # Primary (H1, H2)
      family2 = c(3, 4)      # Secondary (H3, H4)
    ),
    proc = families(
      family1 = "HolmAdj",
      family2 = "HolmAdj"
    ),
    gamma = families(
      family1 = 0.8,         # Truncation for primary
      family2 = 1            # Full Holm for secondary
    )
  ),
  tests = tests("Primary1", "Primary2", "Secondary1", "Secondary2")
)

Multiple-Sequence Gatekeeping

For complex hierarchies with multiple sequences of hypotheses.

Example: Two doses (High, Low) each with primary and secondary endpoints.

MultAdjProc(
  proc = "MultipleSequenceGatekeepingAdj",
  par = parameters(
    family = families(
      family1 = c(1, 2),     # Primary: DoseH, DoseL
      family2 = c(3, 4)      # Secondary: DoseH, DoseL
    ),
    proc = families(
      family1 = "HolmAdj",
      family2 = "HolmAdj"
    ),
    gamma = families(
      family1 = 0.8,
      family2 = 1
    )
  )
)

Mixture Gatekeeping

Combines serial and parallel gatekeeping components.

Components:

• serial: Matrix indicating serial relationships
• parallel: Matrix indicating parallel relationships

MultAdjProc(
  proc = "MixtureGatekeepingAdj",
  par = parameters(
    family = families(family1 = c(1), family2 = c(2, 3)),
    proc = families(family1 = "BonferroniAdj", family2 = "HolmAdj"),
    gamma = families(family1 = 1, family2 = 0.8),
    serial = matrix(c(0, 0, 0,
                      1, 0, 0,
                      1, 0, 0), 3, 3, byrow = TRUE),
    parallel = matrix(c(0, 0, 0,
                        0, 0, 0,
                        0, 1, 0), 3, 3, byrow = TRUE)
  )
)

Parametric Procedures

Normal Parametric

Uses correlation structure for more powerful testing when test statistics are multivariate normal.

# Correlation from study design
corr.matrix <- matrix(c(1.0, 0.5, 0.5, 1.0), 2, 2)

MultAdjProc(
  proc = "NormalParamAdj",
  par = parameters(
    corr = corr.matrix,
    weight = c(0.5, 0.5)
  )
)

Truncation Parameter (γ) Optimization

Role of γ

• γ = 0: Bonferroni within family (most conservative)
• γ = 1: Holm within family (most powerful)
• 0 < γ < 1: Trade-off between error spending and power

Optimization Strategy

1. Start with γ = 1 for all families
2. If simulated Type I error exceeds α, reduce γ for gatekeeper families
3. Binary search for optimal γ that maximizes power while controlling FWER

# Compare multiple gamma values
gamma.values <- c(0.5, 0.6, 0.7, 0.8, 0.9, 1.0)

for (g in gamma.values) {
  mult.adj <- MultAdjProc(
    proc = "ParallelGatekeepingAdj",
    par = parameters(
      family = families(family1 = c(1, 2), family2 = c(3, 4)),
      proc = families(family1 = "HolmAdj", family2 = "HolmAdj"),
      gamma = families(family1 = g, family2 = 1)
    )
  )
  # Run CSE and record power
}

Procedure Selection Guide

By Hypothesis Structure

Structure	Recommended Procedure
Independent hypotheses	Holm or Hochberg
Strict hierarchy	Fixed-Sequence
Primary/Secondary	Parallel Gatekeeping
Multiple doses × endpoints	Multiple-Sequence
Complex dependencies	Graphical (Chain)

By Dependence Structure

Dependence	Valid Procedures
Any	Bonferroni, Holm
PRDS/Independent	Hochberg, Hommel
Known correlation	NormalParamAdj

By Power Priority

Priority	Procedure
First hypothesis	Fixed-Sequence
Equal priority	Holm with equal weights
Weighted priority	Graphical with weights

Common Patterns

Two Primary + Two Secondary

# H1, H2 = primary; H3, H4 = secondary
MultAdjProc(
  proc = "ParallelGatekeepingAdj",
  par = parameters(
    family = families(family1 = c(1, 2), family2 = c(3, 4)),
    proc = families(family1 = "HolmAdj", family2 = "HolmAdj"),
    gamma = families(family1 = 0.8, family2 = 1)
  )
)

Three Doses vs Placebo

# All pairwise comparisons with equal weight
MultAdjProc(
  proc = "HolmAdj",
  par = parameters(weight = c(1/3, 1/3, 1/3))
)

Hierarchical Endpoints

# Primary → Key Secondary → Other Secondary
MultAdjProc(proc = "FixedSeqAdj")

Graphical with Recycling

# Two primary with full recycling
MultAdjProc(
  proc = "ChainAdj",
  par = parameters(
    weight = c(0.5, 0.5),
    transition = matrix(c(0, 1,
                          1, 0), 2, 2, byrow = TRUE)
  )
)

FWER Validation

Always validate FWER control under the global null:

# Set all treatment effects to null
null.data.model <- DataModel() +
  OutcomeDist(outcome.dist = "NormalDist") +
  SampleSize(100) +
  Sample(id = "Control", outcome.par = parameters(mean = 0, sd = 1)) +
  Sample(id = "Treatment", outcome.par = parameters(mean = 0, sd = 1))

# Check rejection rate ≤ alpha
null.results <- CSE(null.data.model, analysis.model, evaluation.model,
                    SimParameters(n.sims = 100000, proc.load = "full", seed = 123))

# DisjunctivePower under null = simulated FWER
# Should be ≤ 0.025 (one-sided)

Best Practices

1. Start Conservative: Begin with Holm/Bonferroni, add complexity as needed
2. Validate FWER: Always check Type I error under global null
3. Document Hierarchy: Clearly specify hypothesis ordering rationale
4. Optimize γ: Use simulation to find optimal truncation parameters
5. Consider Correlation: Use parametric methods when correlation is known
6. Plan Pre-Specification: Multiplicity strategy must be pre-specified in SAP