Tukey’s trend test¶

Tests an ordered-factor trend across alternative dose codings, with joint inference that accounts for the correlated family of slope statistics.

Version: v1 · 2026-10-17 · History
Domain-specific #: 14343
Domain group: Formal Sciences
Origin domain: Experimental Design & Statistics
Subdomain: Dose Response Inference → Experimental Design & Statistics
Aliases: Tukey trend test

Core Idea¶

Tukey’s trend test assesses an ordered-factor trend through a family of marginal regression models using alternative dose scores. Its basic construction uses actual dose, dose-level order and an arithmetic–logarithmic score. Each produces a slope statistic; inference must account for the fact that these statistics come from the same observations. The detailed realization here is the modern multiple marginal models (MMM) adjustment described by Schaarschmidt, Ritz and Hothorn.^[1]

The approach avoids committing the entire trend question to one dose-response shape. It also avoids declaring a trend from whichever unadjusted model happens to look strongest. The 2022 method estimates the joint correlation from marginal-model score contributions and replaces earlier heuristic adjustment with an asymptotic correction. This modern machinery should not be attributed without qualification to the original 1985 proposal.^[1]

Sequentially deleting the highest dose and seeking a no-significant-trend dose is not a necessary step or output of this retained joint test. A statistical trend result is also not a measurement of a safe dose, proof of a biological mechanism, or identification of the true response curve.

Structural Signature¶

Ordered-factor response observations — Supply known dose levels and response observations on identified experimental units. Outcome families and covariates must match the modeled question; differently scaled endpoints remain possible.^[1]
Alternative dose codings — Define the prespecified trend family. The basic arithmetic, ordinal and arithmetic–logarithmic scores encode different candidate relations without choosing the strongest one after inspection.^[1]
Marginal trend fits — Produce the dose-related slope estimates and their uncertainty. In the basic linear realization, each statistic is estimated slope divided by its standard error.^[1]
Joint dependence and calibration — Align experimental units across model score contributions, estimate the family’s correlation and choose the justified joint reference law. Treating different units as matching rows changes the covariance problem.^[1]
Family-adjusted trend inference — Interpret the statistics within the declared testing family, including sidedness and the model assumptions. Adjustment is constitutive here, rather than a cosmetic addition to a selected favorable raw result.^[1]

For positive doses the logarithmic score is defined directly. With a zero control, the source extrapolates its score from the first positive doses:

[ x_{\mathrm{arilog},1}=\log d_2-\frac{d_2-d_1}{d_3-d_2}(\log d_3-\log d_2),\qquad d_1=0. ] ^[1]

The expression requires defined, distinct positive \(d_2,d_3\); it does not take \(\log 0\). In the basic family the global null sets all three slopes to zero. The alternative permits at least one nonzero slope, and the worked applications use two-sided inference. A two-sided joint rule must preserve both directions, rather than silently selecting the largest signed statistic.^[1]

What It Is Not¶

One favorable unadjusted regression. Model choice from the same observations belongs to the testing context.
An intrinsically nonparametric procedure. Linear, generalized and mixed model realizations have declared statistical assumptions.
Tukey’s pairwise range or additivity test. The target here is a dose-score slope family, not pairwise group differences or a product-shaped interaction.
Siegel–Tukey dispersion testing. Alternate-extreme ranks in two samples perform a different operation.
A universal finite-sample error guarantee. The modern joint law is asymptotic; its proposed multivariate-t handling, including averaged degrees of freedom where models differ, is an approximation.^[1]
A safe-dose locator by definition. Nonrejection does not establish no effect or safety.
A method that identifies the true curve from its winning score. Evidence for a coded slope is conditional on that modeled relation.

Scope of Application¶

The instrument travels where an ordered factor, meaningful alternative score models, aligned units and an appropriate joint calibration are available:

Single continuous response analysis, as in the paper’s reconstructed canine serum-albumin application.^[1]
Differently scaled joint endpoints, as in the rat litter-weight and litter-size application.^[1]
Covariate-adjusted trend families, when the fitted models and their shared-unit score contributions support the desired adjustment.^[1]
Multi-laboratory mixed-model trends, as in the paper’s liver-weight application with laboratory variation.^[1]

These are positive modeled uses, rather than generic labels for fields that might benefit. Correlation must be estimated for the actual family and experimental-unit structure. A nonlinear, dependent or sparse dataset does not become adequately modeled merely because a program can return adjusted p-values.

Clarity¶

Separate uncertainty about shape from uncertainty about evidence. Several score models may be plausible before analysis; the testing family accounts for the opportunity to obtain a favorable result among them. It does not determine which curve is substantively true.

Also separate the named historical approach from this modern adjustment. The underlying multi-score idea predates the empirical MMM correlation calculation described in the retained source.^[1] Calling every version correlation-adjusted in the same way would hide a material methodological change.

Finally, keep a failed rejection distinct from a failed application. The rat analysis is a real use of the instrument even though its joint results did not establish the dose-response relation.^[1] That conclusion remains different from proving an absence of effects.

Manages Complexity¶

The method compresses a collection of shape and endpoint questions into a declared family of models, target parameters, standard errors and a joint dependency structure. An analyst can inspect which choices enlarge the family and which simply reproduce existing statistics. It supports comparison without pretending that every marginal model has the same outcome scale or identical nuisance parameters.

The reduction has a boundary. An estimated correlation matrix does not validate the outcome models or rescue misaligned experimental units. Nor does a family-adjusted p-value repair a post hoc family definition that omitted unfavorable alternatives. Those are questions about the construction of the test, not merely its numerical output.

Abstract Reasoning¶

First compare what changes under the alternative codings. A dose score can change the evidence for a linear slope without proving that the underlying response is linear on that scale. Then inspect the adjusted family result rather than promoting the smallest raw p-value to the overall answer.

Redundancy supplies another useful test. Scores equivalent up to a rescaling can generate perfectly correlated slope statistics; the paper explains why such identical models need not add multiplicity cost.^[1] The number of listed fits alone therefore does not determine the penalty.

Finally check calibration before interpreting extremeness. The finite-sample multivariate-t proposal is not an exact guarantee for every generalized or mixed-model combination.^[1] A surprising result should trigger questions about sample regime, model assumptions, unit alignment and sidedness before claims about a new biological threshold.

Knowledge Transfer¶

The instrument is literally reusable across suitable response substrates. Replacing albumin with a count endpoint changes the outcome model while preserving ordered scores, marginal trend fits and joint-family inference. The paper’s endpoint and laboratory examples demonstrate these substitutions under their declared statistical models.^[1]

The transfer condition is statistical structure, rather than membership in toxicology. Another ordered exposure can support the method when its scores, experimental units and outcome models are meaningful. A chronological narrative with no sampling model or an informal claim that something is trending supplies an analogy at most.

The wider obligation to account for a testing family belongs to the general multiplicity problem. Tukey’s specific dose-score construction and the modern MMM reference calculation remain additional commitments; they do not become universal properties of every multiple comparison procedure.

Examples¶

Canonical¶

The published canine albumin example analyzes simulated individual observations designed to reproduce earlier summary statistics. It fits the three dose-score models to one continuous endpoint and compares their correlated slope evidence with adjusted results.^[1] This is a worked reconstruction, not access to the original raw dog observations.

Mapped back: The reconstructed dose/albumin observations fill ordered-factor response observations; arithmetic, ordinal and arithmetic–logarithmic scores fill alternative dose codings; their regression slopes and standard errors fill marginal trend fits; common observation rows and the estimated correlation fill joint dependence and calibration; adjusted slope evidence fills family-adjusted trend inference. The strongest score does not identify the true biological curve.

Applied / In Practice¶

The paper’s rat-dam application combines continuous litter weight, weight models with covariates and an overdispersed count model for litter size, each under the alternative dose codings. Nine marginal models enter the joint assessment; the reported results do not establish a dose-response relation for those endpoints.^[1]

Mapped back: Shared dam-level dose and outcome records fill ordered-factor response observations; the score family fills alternative dose codings; continuous and count fits fill marginal trend fits; aligned units and within/across-model dependency fill joint dependence and calibration; the qualified nonsignificant assessment fills family-adjusted trend inference. A negative inferential result still instantiates the instrument and does not prove safety.

Structural Tensions¶

Shape coverage versus family penalty. Prespecifying more nonidentical score alternatives can make the analysis responsive to more relations, but those opportunities belong to the joint testing family. A smaller family can require less adjustment while missing a poorly represented shape. The cost depends on dependence: perfectly redundant statistics need not add a penalty.^[1] Diagnostic: Which scores are justified before inspection, and which supply distinct evidence rather than duplicates?

Outcome flexibility versus calibration stability. Suitable count, covariate and mixed models can preserve important features of different endpoints. Forcing one simple model can misrepresent them, yet a diverse family may have different degrees of freedom and less dependable finite-sample approximation. The proposed reference law therefore needs scrutiny in the actual sample regime.^[1] Diagnostic: Are the model and joint-law assumptions supported for these observations, rather than inferred from successful software execution?

Structural–Framed Character¶

This is a structural statistical instrument with a methodological frame. Evaluative weight: statistical evidence does not itself endorse the outcome, identify a safe dose or establish practical importance. Human practice: the analyst defines scores, family, sidedness and model assumptions; automation can carry out the same formal operations. Institutional origin: the identity is a named statistical construction, rather than a legally created validity condition. Vocabulary travel: ordered exposure and modeled response can replace toxicological doses and endpoints when the statistical prerequisites survive. Import versus recognition: explicit models and a test family are imposed for inference, while dependency in observations constrains their calibration.

Its character: a reusable formal instrument whose inference travels under declared statistical prerequisites, retaining its dose-score and reference-law choices.

Structural Core vs. Domain Accent¶

Skeletal relation. Several correlated evidence statistics must be interpreted within the joint testing context that generated them.

Domain-bound mechanism. The named method constructs alternative dose-score regressions, targets their slopes and, in the retained modern realization, calibrates their dependency through MMM score contributions. These are load-bearing methodological choices.

Future-Prime question. A generic correlated-evidence/family relation would need to survive removing statistical score, model and reference conventions; informal collections of several observations do not suffice. Multiple Comparisons Correction already addresses the statistical family context, so wider reach cannot be established merely by repeating that problem in another applied science.

Why the named entry does not clear the Prime bar. Literal cross-outcome instrument use retains score construction, fitted-model assumptions and a joint reference convention. It does not establish a new general pattern outside statistical inference.

This entry is a kind of Statistical Test.

Statistical Test is the procedure genus: a null model, discriminating statistic, justified reference comparison and declared evidence interpretation remain present. Tukey’s construction adds an ordered factor, alternative dose-score regressions and joint slope-family calibration; a statistical test can use another statistic without those additions.

Hypothesis Testing (Null vs. Alternative) concerns calibrated evidence against a declared null. Multiple Comparisons Correction concerns interpretation of a whole test family rather than one selected p-value. Both describe wider statistical commitments; neither identifies the specific dose-score construction by itself. The outcome model, coding family and joint calibration distinguish this named instrument within that setting.

Relationships to Other Abstractions¶

Current abstraction Tukey’s trend test Domain-specific

Parents (1) — more general patterns this builds on

Tukey’s trend test is a kind of Statistical Test Domain-specific

Tukey’s trend test is a statistical test specialized by alternative ordered-dose score regressions and joint slope-family inference.

Hierarchy path (1) — routes to 1 parentless root

Tukey’s trend test → Statistical Test

Neighborhood in Abstraction Space¶

Tukey’s trend test sits in a sparse region of the domain-specific corpus (97^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2721 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-17

Not to Be Confused With¶

One prespecified linear trend test: one score and one slope do not constitute the basic multi-score family.
Tukey HSD: pairwise mean comparisons use a different target and reference construction.
Tukey’s additivity test: a restricted product interaction in a two-factor table is not this trend family.
Siegel–Tukey test: pooled alternate-extreme dispersion ranks differ from dose-score slope fits.
MCPMod: another approach to dose-response model uncertainty; the retained source discusses contrasts representing candidate nonlinear models, not a universal generalization of every Tukey implementation.^[1]
Sequential dose elimination: an optional historical procedure does not define every modern joint trend assessment.

References¶

[1] Frank Schaarschmidt, Christian Ritz and Ludwig A. Hothorn. (2022). The Tukey trend test: Multiplicity adjustment using multiple marginal models. Biometrics 78(2), 789–797; first published online 9 February 2021. Full original methods article consulted, including §§2.1–2.5, Equation 1, §§3.1–3.3 and Tables 1–2. The original 1985 implementation is described through this paper; its full text was not independently inspected. The modern adjustment is asymptotic, the finite-sample multivariate-t treatment is approximate, and the canine individual observations are reconstructed from earlier summary statistics. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w ↩x