---
slug: type_iii_error
name: Type III Error
one_liner: "A framing-level inferential error: giving a correct, low-error answer to the WRONG question — the machinery is sound but the target question is mis-specified, so precision is spent on a mis-framed or irrelevant problem."
type: prime
schema_version: 2
status: density_v2_draft
categories: [structural, statistical, decision_theoretic]
origin_domain: statistics
alternate_origin_domains: [decision_analysis, policy_analysis, research_design, systems_engineering]
aliases: [wrong_problem_error, problem_misspecification, right_answer_wrong_question, solving_the_wrong_problem, error_of_the_third_kind]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-14 lattice/symmetry completion) — the FRAMING-LEVEL error obtained by extending the hypothesis-testing error family {Type I false positive, Type II false negative} one level UP: an error located in PROBLEM FORMULATION rather than in the inference given the formulation. Where Types I/II presuppose a correctly-specified question and err WITHIN the test, a Type III error executes the test flawlessly but answers a mis-specified, mis-framed, or irrelevant question — 'the right answer to the wrong problem'. Discovered via op-14 on 2026-07-04. MODERATE hold (verified absent from corpus; cited construct — Kimball 1957; Mitroff & Featheringham 'solving the wrong problem'). STATUS: HOLD pending owner keep/cut."
- "NAME-CONTESTED: 'Type III error' has multiple non-standard definitions in the literature — some authors mean 'correct rejection of H0 for the wrong reason', others 'wrong-sign / wrong-direction error', and some treat the label as a semi-joke. The STRUCTURE (framing-level error) is real and well-cited; the NAME is not stable. Strongly consider renaming to `wrong_problem_error` or `problem_misspecification` at incorporation. BE CANDID about this in owner review."
- "Created 2026-07-04; STUB ONLY (frontmatter + Core Idea + Notes). SI/SF/similarity are PROPOSER ESTIMATES pending triangulation. Body sections (Structural Signature onward) NOT written."
- "Phase-C action: re-similarity-check vs `type_i_type_ii_errors` (the sibling dyad this completes), `Goodharts_law` / optimizing-the-wrong-metric (adjacent), problem-framing / framing primes, and `mcnamara_fallacy` if present; confirm it is the genuine framing-level complement (error of the question itself, not error within the test) and not a restatement of framing or Goodhart; draw a lattice/sibling edge to `type_i_type_ii_errors` + composition/adjacency edges to framing and `Goodharts_law`; write a v1 draft and sync the one_liner across v1/v2 per CLAUDE.md before merge."
substrate_independence:
  composite_substrate_independence: 3
  domain_breadth: 4
  structural_abstraction: 3
  transfer_evidence: 3
  reasoning: 'The bare structure — a sound inference procedure with low within-test error applied to a mis-specified or mis-framed target question, so that a correct answer is produced to the wrong problem — is substrate-neutral wherever a formulation step precedes an inference step. It recurs in statistics/experimental design (a well-powered test of the wrong hypothesis), decision and policy analysis (a rigorously-answered but wrongly-posed policy question), systems engineering (optimizing the wrong objective), and research design (methodologically clean studies of the wrong construct). domain_breadth scores 4: these are structurally-unlike siblings sharing the two-level formulation-then-inference shape. structural_abstraction is 3 (not higher) because the concept is defined by CONTRAST with the Type I/II dyad and leans on the presupposition of an inference/testing frame; the "wrong question" residue carries a mild evaluative/pragmatic cast (what counts as the "right" question is context-relative). transfer_evidence is 3: named and cited across statistics, decision analysis, and policy, but under an unstable label with competing definitions, weakening the clean-transfer evidence.'
similarity_to_nearest_existing_prime:
  slug: type_i_type_ii_errors
  similarity: 0.62
structural_framed:
  label: mixed
  aggregate: 0.47
  criteria: {vocab_travels: 0.55, evaluative_weight: 0.5, institutional_origin: 0.35, human_practice_bound: 0.5, import_vs_recognize: 0.45}
  rationale: 'Reads MIXED. The core shape — a sound inference machine aimed at a mis-specified question, so a correct answer lands on the wrong problem — is recognized rather than imported and travels into decision analysis, policy, and engineering. But framing residue is heavier than for a pure structural prime: (1) the concept is parasitic on the hypothesis-testing frame it extends (it is defined as the level-2 complement of Type I/II), so vocab_travels and import_vs_recognize sit mid-range; (2) what counts as the "wrong" question is evaluative and practice-relative — the error only exists against a normative standard for the right question — keeping evaluative_weight and human_practice_bound near 0.5. The underlying two-level (formulation vs inference) structure is substrate-neutral, but the label and its diagnosis are bound to human problem-framing practice, so it does not read as cleanly structural as its parent dyad.'
phase_b_passes_completed: []
---
# Type III Error

## Core Idea

A Type III error is giving a *correct, low-error answer to the wrong question*: the inference machinery is sound, but the target question is mis-specified, mis-framed, or irrelevant, so precision is spent in the wrong place. Where a **Type I error** (false positive) and a **Type II error** (false negative) are errors that occur *within* a correctly-specified test — the question is fixed, and the error is a wrong verdict on it — a Type III error sits one level *up*: the test is executed flawlessly (low Type I and Type II rates), but it answers a problem that was never the right problem to solve. The classic gloss is Mitroff & Featheringham's: "solving the wrong problem precisely." The load-bearing distinction is between the *inference layer* (getting the right answer given the formulation) and the *formulation layer* (having the right formulation in the first place); the Type I/II dyad lives entirely in the inference layer and presupposes the formulation is correct, so it structurally *cannot represent* an error in the formulation itself. Type III names exactly that gap.

The distinctive anatomy has three parts that must co-occur: (1) the inference machinery is *sound* — were the question the right one, the answer would be trustworthy (low within-test error); (2) the *target question is mis-specified* — the hypothesis tested, objective optimized, or construct measured is not the one that matters; and (3) consequently *precision is wasted on the wrong target* — effort and rigor amplify rather than correct the framing mistake. This is why a Type III error is insidious in a way Types I/II are not: all the internal quality signals (power, significance, calibration) look good, because the failure is invisible to any check that presupposes the question. The error is located in the *choice of question*, one level above the space where the standard error dyad is defined.

Read structurally, the same framing-level error appears wherever a formulation step precedes an inference step, and the siblings are genuinely unlike. In *statistics and experimental design*, it is a well-powered, clean test of the wrong hypothesis. In *decision and policy analysis*, it is a rigorously-answered but wrongly-posed question — the right answer to the wrong policy question, delivered with confidence. In *systems engineering and optimization*, it is optimizing the wrong objective — where the machinery converges perfectly onto a target that does not capture what was wanted (kin to Goodhart's law and metric-substitution failures). In *research design*, it is a methodologically impeccable study of the wrong construct. Across these, the shared abstract shape is a two-level structure — formulation then inference — where the error lives at the formulation level while the inference level performs correctly; the statistical "Type III" is one species of this genus, and the structurally-unlike siblings show the genus is the bare framing-level error.

## Notes

**Provenance.** Produced by operator 14 (lattice / symmetry completion), extending the hypothesis-testing error family {Type I, Type II} with a *framing-level* member. The Type I/II dyad partitions errors *within* a fixed, correctly-specified test (false positive / false negative); op-14 asks what error lives one level up, at the specification of the question itself — yielding the "right answer to the wrong problem" structure. Surfaced 2026-07-04. Cited construct: A.W. Kimball 1957 ("A bias in certain classes of statistical estimation") is usually credited with the term "error of the third kind"; Mitroff & Featheringham developed the influential "solving the wrong problem" reading. Claimed siblings: mis-specified hypotheses in experimental design, wrongly-posed but rigorously-answered policy questions in decision analysis, optimizing-the-wrong-objective in engineering (adjacent to Goodhart), and clean studies of the wrong construct in research design. Verified absent from the corpus. Manifest: MODERATE hold.

**For owner evaluation (keep/cut).** BE CANDID — this is a moderate, not strong, hold, and the case cuts both ways. The case FOR: the *structure* is genuine and cited (Mitroff's "solving the wrong problem"), it completes the error family with the *framing layer that the Type I/II dyad structurally cannot express* (the dyad presupposes a fixed question; Type III is the error of the question itself), and it is high-leverage — framing/specification errors dominate real-world failure and no current prime names "correct answer to the wrong problem." The case AGAINST: (1) **the NAME is contested** — "Type III error" has inconsistent definitions in the literature (some authors mean "correct rejection of H0 for the wrong reason", others "wrong-sign / wrong-direction error"), and some treat the label as a semi-joke; the concept is real but its name is not stable, so a clearer name like `wrong_problem_error` or `problem_misspecification` may be preferable. (2) **Overlap with framing and Goodhart** — it is adjacent to problem-framing primes (the error is a framing failure) and to `Goodharts_law` / optimizing-the-wrong-objective (a special case where the mis-specified target is a proxy metric); the discriminating claim is that Type III is the *general* framing-level error (any wrong question), of which metric-substitution is one instance, and framing is the broader faculty of which this names a specific failure mode. Sharp keep/cut question: *keep the STRUCTURE (a framing-level inferential error, distinct from within-test error) under a clear name, or fold it into framing?* — the proposer recommends KEEPING the structure (it is the lattice complement the Type I/II dyad demands and framing does not supply), most likely under a renamed slug (`wrong_problem_error`), rather than folding into framing.

**At incorporation.** Re-run similarity against `type_i_type_ii_errors` (the sibling dyad this completes — screen that Type III is genuinely the framing-level complement and not a third within-test error), `Goodharts_law` and any optimizing-the-wrong-metric prime (adjacent special case), the problem-framing / framing primes (broader faculty), and `mcnamara_fallacy` if present (quantifying-the-wrong-thing sibling). If kept: draw a lattice/sibling edge to `type_i_type_ii_errors` (the {Type I, Type II, Type III} family), plus adjacency/specialization edges to `Goodharts_law` (metric-substitution as a special case) and framing (Type III as a named failure mode of framing). Resolve the NAME question before merge — strongly consider renaming to `wrong_problem_error` or `problem_misspecification` given the literature's contested use of "Type III." Add a missing-edge note flagging the Goodhart and framing relationships explicitly. Write a v1 draft and sync the one_liner across v1 and v2 per CLAUDE.md before merge.
