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.
Core Idea¶
Tukey’s trend test assesses an ordered-factor trend using a family of regressions with alternative dose scores.[^ref-f0d692866baa] The basic arithmetic, ordinal and arithmetic–logarithmic scores yield correlated slope statistics against a basic null of all three slopes being zero.[^ref-f0d692866baa] The modern multiple marginal models adjustment estimates their dependency and provides joint inference, replacing an earlier heuristic correction.[^ref-f0d692866baa] This avoids relying on one score or selecting the strongest unadjusted result, without making highest-dose elimination or a safe-dose finding a necessary output.
Scope of Application¶
Use the instrument where meaningful dose scores, outcome models, aligned experimental units and justified joint calibration are available:
- Single continuous endpoint: the paper reconstructs individual canine albumin observations from earlier summary statistics.[^ref-f0d692866baa]
- Joint continuous/count endpoints: the rat litter-weight and litter-size analysis uses shared dam-level records without establishing a dose-response relation.[^ref-f0d692866baa]
- Covariate-adjusted families: fitted models retain endpoint-specific explanatory variables in the joint assessment.[^ref-f0d692866baa]
- Laboratory variation: mixed-model liver-weight analysis accounts for laboratory structure.[^ref-f0d692866baa]
Clarity¶
Separate uncertain response shape from multiple opportunities to obtain favorable evidence. Alternative scores offer different modeled relations, but the strongest score does not identify the true biological curve. Keep the 1985 approach distinct from the source-backed modern empirical-correlation adjustment.[^ref-f0d692866baa] A nonsignificant adjusted result is still a test application and does not prove no effect, safety or a dose threshold.
Manages Complexity¶
Track the prespecified score/model family, target slopes, uncertainty and dependency together. This condenses multiple endpoint and shape questions without pretending that all outcomes have identical scales or nuisance parameters. An estimated correlation matrix cannot validate an unsuitable outcome model or repair experimental units that were incorrectly aligned. Nor can adjustment for a conveniently selected subset account for omitted alternatives that helped generate the favorable result, so successful computation by itself does not confirm a valid testing context.
Abstract Reasoning¶
Compare coding-sensitive slope evidence, then interpret the family-adjusted result under declared sidedness; the worked applications use two-sided inference.[^ref-f0d692866baa] Equivalent scores can produce perfectly correlated statistics that need not add multiplicity cost.[^ref-f0d692866baa] Check zero-dose handling rather than taking log zero: the arithmetic–logarithmic control convention uses the first distinct positive doses.[^ref-f0d692866baa] Finally scrutinize the reference-law assumptions, because the proposed finite-sample multivariate-t treatment, including averaged degrees of freedom where models differ, is approximate rather than a universal exact error guarantee.[^ref-f0d692866baa]
Knowledge Transfer¶
The instrument transfers literally across suitable outcomes when ordered scores, compatible models, aligned units and joint calibration remain meaningful, including applications with qualified negative inferential results.[^ref-f0d692866baa] Statistical Test supplies its procedure genus: null model, discriminating statistic, reference comparison and declared interpretation. Tukey’s construction adds the alternative dose-score family and modern dependency adjustment; general multiplicity concerns do not identify that specific procedure. An informal narrative that something is trending has no corresponding sampling/reference structure and supports only an analogy, even if its vocabulary sounds similar.
Relationships to Other Abstractions¶
Current abstraction Tukey’s trend test Domain-specific
Parents (1) — more general patterns this builds on
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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 (97th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2721 abstractions)
Nearest neighbors
- Latent growth modeling — 0.77
- Structural Break — 0.77
- Statistical Conclusion Validity — 0.76
- Multi-vari chart — 0.76
- Anderson–Darling test — 0.76
Computed from structural-signature embeddings · 2026-10-17
References¶
[^ref-f0d692866baa]: 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.