multiplicity-methodslisted

# 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 ```r 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 ```r 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.