bio-clinical-biostatistics-multiplicity-graphical

What this skill does

## Version Compatibility

Reference examples tested with: R `gMCP` 0.8.16+, `graphicalMCP` 0.2+, `gatekeeping`, `multcomp`, `multxpert`; Python `statsmodels` 0.14+ for basic FDR/FWER methods.

Before using code patterns, verify installed versions match. If versions differ:
- R: `packageVersion('<pkg>')` then `?function_name`
- Python: `pip show <package>` then `help(module.function)`

If code throws ImportError, AttributeError, or TypeError, introspect the installed package and adapt the example to match the actual API rather than retrying.

# Multiplicity Control for Confirmatory Trials

**"Design the multiplicity strategy for my trial"** -> Specify a closed-testing procedure (graphical, gatekeeping, hierarchical, or step-down Bonferroni-Holm) that controls family-wise error rate at the trial-wide level across primary endpoints, key secondary endpoints, and subgroup analyses, with provable strong FWER control.

## The Foundational Theorem -- Closed Testing Is Necessary

**Marcus, Peritz & Gabriel 1976 *Biometrika* 63:655:** a hypothesis H_I (I ⊆ {1,...,m}) is rejected iff every intersection hypothesis ∩_{J⊇I} H_J is rejected by a valid α-level local test. Strong FWER control holds for ANY choice of local tests.

**Goeman, Hemerik & Solari 2021 *Ann Stat* 49:1218** tightens this: closed testing is not merely sufficient — it is *necessary* for admissibility under FDP/FWER/k-FWER. **Every admissible multiplicity procedure is equivalent to some closed test.** Graphical procedures, gatekeepers, Hommel, fixed-sequence, fallback — all are closed tests in disguise.

**FWER vs FDR philosophical divide:**

- **FWER:** P(any false positive among m tests) — regulatory standard for confirmatory inference (agency wants to bound per-trial false-positive rate)
- **FDR:** Expected proportion of false discoveries among rejections — exploratory standard (genomics, fMRI, biomarker screens) where many true positives expected

Confirmatory clinical trials use FWER essentially universally.

## Algorithmic Taxonomy

| Procedure | Type | FWER control | Power profile | Use case |
|-----------|------|--------------|---------------|----------|
| Bonferroni | Single-step | Yes, any dependence | Conservative; loses 30-50% power vs Hommel under positive dependence | Very small m; worst-case dependence |
| Holm 1979 | Step-down | Yes, any dependence | Better than Bonferroni; uniformly dominates | Default for any dependence pattern |
| Hochberg 1988 | Step-up | Yes under PRDS (Sarkar 1998) | Better than Holm under PRDS | Positive correlation; verify PRDS |
| Hommel 1988 | Step-up via closed tests | Yes under PRDS | Uniformly dominates Hochberg by 1-3% | Whenever Hochberg is valid |
| Fixed-sequence (hierarchical) | Sequential | Yes, any dependence | Full alpha for first; subsequent zero if any fail | When clear priority ordering; "key secondary" labelling |
| Parallel gatekeeping (Dmitrienko 2003) | Multi-family | Yes | Family-by-family; secondary tested if any primary rejects | Primary family + secondary family |
| Serial gatekeeping | Sequential families | Yes | Strict: family k tested only if ALL of family k-1 reject | Co-primary + secondary tiers |
| Mixed gatekeeping (Dmitrienko-Tamhane 2008) | Combination | Yes | Combines closed-testing local procedures across families | Complex hierarchies |
| Graphical procedures (Bretz-Maurer 2009) | Closed-test as directed graph | Yes by construction | Flexible; allocate alpha to hypotheses via graph weights | Modern standard for confirmatory SAPs |
| Graphical + Simes/parametric (Bretz et al 2011) | Closed-test with non-Bonferroni local tests | Yes when Simes valid | Gains power under correlation | Complex co-primary + key secondary + subgroup hierarchies |
| Maurer-Bretz 2013 entangled graphs | Memory-augmented graphs | Yes by construction | Alpha propagation depends on origin | Parent-descendant constraints |
| Benjamini-Hochberg 1995 | FDR | FDR controlled at level q | Higher power than FWER | Exploratory only; NOT for confirmatory regulatory |

**Postdoc reading list:**

- Marcus R, Peritz E, Gabriel KR 1976 *Biometrika* 63:655 (closed testing — the foundation)
- Goeman JJ, Hemerik J, Solari A 2021 *Ann Stat* 49:1218 (closed testing necessary for admissibility)
- Holm S 1979 *Scand J Stat* 6:65 (step-down Bonferroni)
- Hochberg Y 1988 *Biometrika* 75:800 (step-up Simes)
- Hommel G 1988 *Biometrika* 75:383 (closed Simes; dominates Hochberg)
- Sarkar SK 1998/2008 *Ann Stat* (PRDS for Hochberg validity)
- Bretz F, Maurer W, Brannath W, Posch M 2009 *Stat Med* 28:586 (graphical procedures — foundational paper)
- Bretz F, Posch K, Glimm E, Klinglmueller F, Maurer W, Rohmeyer K 2011 *Biom J* 53:894 (Simes/parametric extensions)
- Maurer W, Bretz F 2013 *Stat Med* 32:1739 (entangled graphs / memory)
- Dmitrienko A, Offen WW, Westfall PH 2003 *Stat Med* 22:2387 (parallel gatekeeping)
- Dmitrienko A, Tamhane AC, Wiens B 2008 *Biom J* (mixed/multistage gatekeeping)
- FDA 2022 *Multiple Endpoints in Clinical Trials* Final Guidance (October 2022)
- Pocock SJ, Ariti CA, Collier TJ, Wang D 2012 *Eur Heart J* (win-ratio)

## Decision Tree by Scenario

| Scenario | Recommended procedure | Why |
|----------|----------------------|-----|
| 2 co-primary endpoints (both must succeed) | No alpha split needed; per-endpoint alpha-level test; cite FDA 2022 | Co-primary doesn't split alpha; inflates n via joint power |
| 2 multiple primary endpoints (any-wins) | Graphical procedure or Holm with weights | Alpha must be allocated; graphical is flexible |
| 1 primary + 2 key secondary endpoints | Hierarchical (serial gatekeeping) OR graphical with alpha propagation | Modern SAPs favour graphical |
| 1 primary + 3 secondary + 4 subgroup analyses | Graphical procedure via gMCP with pre-specified weights | Complex hierarchies benefit from graph visualisation |
| Primary endpoint + tipping-point sensitivity | No multiplicity adjustment needed for sensitivity | Sensitivity is "what if" not "another claim" |
| Many exploratory biomarker subgroups | Benjamini-Hochberg FDR | Exploratory; not for label claims |
| Win-ratio composite (cardiology) | Single test; no multiplicity | Composite captures multiple events in single hierarchy |
| Subgroup analysis (pre-specified) | Graphical alpha allocation; small budget (≤20%) per Dane 2019 | Confirmatory subgroup discovery requires explicit allocation |
| Adaptive trial with treatment arm dropping | Combination tests (Bauer-Köhne 1994) + closed testing | See clinical-biostatistics/adaptive-designs |
| Group-sequential with multiple endpoints | gsDesign or rpact with multivariate alpha spending | Hierarchical alpha across both time and endpoints |

## Bretz-Maurer Graphical Procedures -- The Modern Standard

**The Bretz-Maurer-Brannath-Posch 2009 *Stat Med* 28:586 framework recast weighted Bonferroni-Holm closed tests as directed weighted graphs:**

- Vertices = elementary null hypotheses with local weights summing to 1
- Directed edges = alpha-propagation rule (when a hypothesis is rejected, its weight redistributes to descendants per edge weights)
- The graph IS the procedure: a single visual fully specifies a closed-test procedure across primary, key secondary, and subgroup hierarchies

### gMCP R package

```r
library(gMCP)

# Construct a graph for primary + 2 key secondary endpoints
# Primary endpoint at full alpha; if rejected, alpha propagates equally to secondaries
hypotheses <- c('Primary', 'Sec1', 'Sec2')
weights <- c(1, 0, 0)  # initial alpha all on primary
# Transition matrix: rows = source, columns = target
# When Primary rejects, weight 0.5 goes to each secondary; when Sec1/Sec2 rejects, alpha returns
transitions <- matrix(c(
    0,    0.5,  0.5,
    0,    0,    1,
    0,    1,    0
), nrow = 3, byrow = TRUE, dimnames = list(hypotheses, hypotheses))

graph <- graphMCP(m = transitions, weights = weights, hnames = hypotheses)
# Note: in current gMCP, the graph constructor is `graphMCP(m=, weights=, hnames=)`;
# `m