Family-Wise Error Rate

The probability that a declared family of simultaneous hypothesis tests contains at least one false rejection, with weak or strong control determined by which configurations of true nulls are covered.

Version

v3 · 2026-09-06 · History

Domain-specific #

1824

Origin domain

statistics

Subdomain

multiple hypothesis testing

Aliases

FWER, Familywise error rate, Family-wise Type I error rate

Core Idea

For a declared family of hypothesis tests, the family-wise error rate is the probability of making at least one Type I error: Π\((V\ge1)\), where (V) is the number of false rejections among true null hypotheses. A procedure controls FWER at level α when that probability does not exceed α under the specified guarantee.

Weak control applies only under the global null. Strong control applies under every configuration of true and false nulls. The family boundary is load-bearing because moving tests into or out of it changes the event whose probability is controlled.

Scope of Application

FWER is used when even one false positive in a coherent claim family is costly or simultaneous coverage is required. Holm's sequentially rejective procedure controls strong FWER without dependence restrictions and uniformly improves on single-step Bonferroni. Closed testing supplies a general framework for strong control through intersection hypotheses.

Clarity

Define the family before inspecting results; state confirmatory versus exploratory status, weak or strong control, dependence conditions, adjusted p-values or critical values, and whether selection occurred across analyses. Report the procedure separately from the criterion.

Manages Complexity

FWER turns many local false-positive risks into one global event: “any false claim in this family.” This supplies an interpretable simultaneous guarantee but spends error budget conservatively as families grow.

Abstract Reasoning

1. Define the inferential decision and smallest coherent selection family.
2. Enumerate its hypotheses and planned selection rule.
3. Choose the desired error criterion and level.
4. Select a procedure valid under the dependence structure.
5. Compute adjusted thresholds or p-values.
6. Reject according to the joint rule.
7. Interpret the guarantee over truth configurations.
8. Avoid silently redefining the family after seeing outcomes.

Knowledge Transfer

The portable pattern is treat any one failure across a declared decision family as the adverse event and bound its probability jointly. It transfers to simultaneous confidence systems. The proposed immediate parent is Risk.

Relationships to Other Abstractions

Current abstraction Family-Wise Error Rate Domain-specific

Parents (1) — more general patterns this builds on

• Family-Wise Error Rate is a kind of Risk Prime

  Risk is the proposed immediate parent.

Hierarchy paths (3) — routes to 3 parentless roots

• Family-Wise Error Rate → Risk → Uncertainty

Show alternative paths (2)

Neighborhood in Abstraction Space

Family-Wise Error Rate sits in a sparse region of the domain-specific corpus (92^nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

• Null distribution — 0.79
• Jeffreys-Lindley Paradox — 0.79
• False Discovery Rate — 0.78
• Benjamini–Hochberg Procedure — 0.78
• Exact test — 0.78

Computed from structural-signature embeddings · 2026-09-08