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Single-step parametric maxT inference

Algorithms AIMech:SingleStepParametricMaxTInference DRAFT

A simultaneous-inference procedure for a declared family of linear functions of model parameters that standardizes joint estimates and uses the maximum absolute statistic under a centered multivariate normal or t reference to compute multiplicity-adjusted tests and a common critical value for simultaneous intervals. Exact or asymptotic validity depends on the joint sampling law and covariance estimation.

Identity Notes & Concept Scope No normalized identity match among all 442 pinned AIO and 562 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches include ignored and hidden files in the recovery, older discovery and original Documents checkouts, returning 218, 0 and 0 lines. Candidate and diagnostic hits were inspected through canonical records and pending source notes; long report/search-index lines and generated mirrors were classified but not all reread in full. The semantic assessment covers 67 AIO rows and 233 native names/definitions: 61 and 220 prior rows were verified unchanged, with six new AIO rows and thirteen new native definitions read. Fifteen complete AIO nodes, incident edges and generated records and ten native context records were read or verified against prior readings. This general fixed-family normal/t maxT procedure is distinct from the existing Bayesian separator-conflict specialization, INLA computation, random latent score checks, quadratic Wald tests, resampling/permutation maxT and closed-testing stepwise procedures. The local label names the procedure, not the multcomp package or every multiple-testing algorithm.

Claims & Grounded Evidence

7 Claims

Scope #fixed-linear-family

Choose a finite, fixed family of estimable linear functions v=K*theta and null values m, with K a fixed k-by-p matrix. Use a parameter estimator theta_hat and its joint sampling covariance estimate S_hat; the comparison estimates and covariance are v_hat=K*theta_hat and V_hat=K*S_hat*K^T. K need not be square or invertible, and rows need not be zero-sum contrasts. This procedure consumes joint estimates and covariance; it does not itself define the model-fitting algorithm.

supports Source: Simultaneous Inference in General Parametric Models · Locator: Section 2, Equations 1–3, PDF pp. 4–5 / printed pp. 2–3; Section 4, PDF pp. 8–10 / printed pp. 6–8

Mechanism #standardization-and-redundancy

For positive finite diagonal comparison variances, define D=diag(V_hat), t=D^(-1/2)*(v_hat-m) and R_hat=D^(-1/2)*V_hat*D^(-1/2). Redundant comparisons can make R_hat singular without preventing a max-absolute statistic. Interpret its reference as a valid joint Gaussian or t probability law, including singular laws, rather than assuming an ordinary full-dimensional density exists. Zero-variance comparisons cannot be standardized and require separate deterministic/support handling.

supports Source: Simultaneous Inference in General Parametric Models · Locator: Section 2, Equation 4, PDF pp. 4–5 / printed pp. 2–3; Section 3, Equation 5, PDF p. 6 / printed p. 4; redundant Tukey matrix, PDF p. 9 / printed p. 7

Mechanism #exact-and-asymptotic-references

The general reference is a centered multivariate normal law with the consistently estimated correlation. Its asymptotic justification requires a multivariate sampling central limit theorem and consistent covariance estimation for the fixed comparison family. An exact normal reference is available for exact normal estimates with known covariance. An exact multivariate t reference additionally requires the appropriate common independent chi-square variance scaling, as in a correctly specified homoscedastic Gaussian linear model. Arbitrary estimated standard errors or a chosen degrees of freedom do not create an exact t law.

supports Source: Simultaneous Inference in General Parametric Models · Locator: Section 2, Equations 1–4, PDF pp. 4–5 / printed pp. 2–3; exact cases in Section 3, PDF p. 6 / printed p. 4; linear-model example, PDF pp. 8–9 / printed pp. 6–7

Mechanism #joint-rectangle-adjustment

For the chosen centered joint reference Z, define G(c)=P(max_j|Z_j|<=c). The two-sided global maxT tail is 1-G(max_j|t_j|), and the single-step adjusted tail for comparison j is 1-G(|t_j|), using the same full declared family for every component. Under that same continuous reference each adjusted value is at least the corresponding marginal two-sided tail. The joint rectangle probability retains dependence; treating correlated comparisons as independent changes the procedure. A quadratic Wald statistic uses a different rejection region and is not interchangeable with the absolute maximum.

supports Source: Simultaneous Inference in General Parametric Models · Locator: Sections 3.1–3.2, Equation 5 and adjusted p-values, PDF pp. 6–7 / printed pp. 4–5

Mechanism #simultaneous-intervals-and-error-control

Choose c_alpha so G(c_alpha)=1-alpha. The simultaneous two-sided intervals are v_hat_j plus or minus c_alpha*sqrt(V_hat_jj); reject an individual equality when its null value falls outside its interval. A valid joint law of the centered estimation errors gives simultaneous coverage and bounds the probability of any true-null rejection by alpha for the declared family. The normal plug-in result is asymptotic under the stated conditions, not automatic finite-sample or uniform-in-parameter control. Pointwise normality of separate estimates without their valid joint covariance is insufficient.

supports Source: Simultaneous Inference in General Parametric Models · Locator: Section 2 sampling limit, PDF pp. 4–5 / printed pp. 2–3; Section 3.2, PDF p. 7 / printed p. 5

Limitation #procedure-boundaries

Single-step means a common joint critical value is used without successively removing rejected hypotheses. Closed-testing or stepwise improvements and resampling/permutation maxT need separate constructions and assumptions. One-sided tests require the corresponding one-sided joint region and quantile, rather than applying the two-sided absolute maximum unchanged. Data-dependent family selection, very large or growing families, weak identification and invalid covariance estimates require additional justification; this fixed-family procedure is not a general repair for adaptive data dredging.

supports Source: Simultaneous Inference in General Parametric Models · Locator: Section 3.2 and stepwise discussion, PDF pp. 7–8 / printed pp. 5–6; fixed K and sampling conditions, Section 2, PDF pp. 4–5 / printed pp. 2–3

Limitation #implementation-and-interpretation

The framework applies to model classes whose estimators meet its joint-law conditions, including suitable regression, generalized linear, mixed and survival models. A software interface accepting coefficient and covariance outputs does not validate those conditions. Numerical normal/t rectangle integration introduces computational approximation. Adjusted tails concern testing errors, not posterior probabilities that hypotheses are true, effect sizes, causality or universal model quality. The source examples and package interface were read without executing multcomp or reproducing the applications.

supports Source: Simultaneous Inference in General Parametric Models · Locator: Sections 4–6, PDF pp. 8–22 / printed pp. 6–20; numerical integration discussion, Section 3, PDF pp. 6–7 / printed pp. 4–5

⚠️ Curated Limitations & Negative Results

This is the general parametric single-step maxT procedure, not the narrower Bayesian separator-conflict construction, every maxT resampling method, an omnibus Wald test, step-down testing or a software-package instance.
The asymptotic result uses fixed family dimension and a fixed contrast matrix with estimable positive-variance comparisons and a consistent joint covariance. It does not establish high-dimensional, adaptive-selection or uniformly valid inference without further theory.
Multivariate t exactness needs the model-specific common variance-scaling law and correct residual degrees of freedom. The report has ambiguous n-p-1 notation alongside a p counting all coefficients; use the actual model rank rather than copying that notation.
The archived report labels a displayed interval as 1-2alpha while specifying the joint critical probability as 1-alpha. This record uses the coherent latter convention: G(c_alpha)=1-alpha gives joint coverage 1-alpha under the valid reference.
The evidence is the complete 25-page February 15, 2008 LMU Technical Report 019, with PDF page n+2 matching printed page n. Journal metadata is identity context; this PDF is not assumed byte-identical to the journal article or a later package vignette.
No multcomp installation or implementation-source audit, fitted model, application outcome, benchmark or empirical calibration study was performed. Constructed known-covariance Gaussian examples illustrate the procedure and dependence boundaries only.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:DataDredgingBias` Data Dredging Bias	related_match	The pinned AIO bias node names apparent significance from testing many hypotheses. This procedure addresses that multiplicity risk for a declared comparison family under a valid joint reference. The algorithm is related to the risk it addresses, not a subtype or equivalent of the bias, and it does not automatically correct unaccounted adaptive selection or every form of data dredging.

Evidence Sources & Primary Literature

1 Source

Source ID	Publication / Document	Type / Access	Reference & Checksum
`hothorn2008`	Simultaneous Inference in General Parametric Models Torsten Hothorn, Frank Bretz, Peter Westfall	primary paper full_text	ID: https://doi.org/10.5282/ubm/epub.2120 sha256:e5a1a1e1790b…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-13T02:13:16Z

Updated: 2026-10-13T02:13:16Z

Source File: data/curated/SingleStepParametricMaxTInference.yaml