No; it evaluates trends through dose scores.

Does using ranks make the whole procedure nonparametric?

No; the fitted model and inference still have assumptions.

Can two scales duplicate each other?

Yes, as ordinal and logarithmic scores do in the example.

AI Glossary / Statistics and probability / Intermediate

Tukey's trend test

Definition

Tukey's trend test is an approach to detecting a dose-response trend by examining regression slopes under several prespecified dose scales, commonly arithmetic, ordinal, and logarithmic scales. A multiplicity-adjusted version evaluates the correlated evidence jointly. It helps when the response shape is uncertain, without treating the most favorable of several analyses as though it were the only test performed.

What it means

A response may change approximately linearly with the actual dose, with the order of dose groups, or with the logarithm of dose. Committing to one scale can miss a pattern that another plausible scale describes better.
The Tukey-Ciminera-Heyse approach considers several such possibilities. The term refers to a trend procedure, not Tukey's HSD pairwise comparison test. The regression-based implementation is not automatically distribution-free merely because one of its predictors uses ordered scores.

How it works

Define the ordered exposure levels and the outcome. Dose can represent a controlled input such as concentration, processing intensity, or another quantitative setting. Specify whether the scientific question concerns an increasing trend, a decreasing trend, or either direction.
Create the dose scores before examining which gives the smallest p-value. Arithmetic scoring uses the actual numerical doses, preserving their unequal spacing. Ordinal scoring assigns equally spaced scores to ordered dose levels, regardless of physical spacing.
Logarithmic scoring uses log dose for positive levels. A zero-dose control requires a documented special convention because log zero is undefined. Adding an arbitrary constant changes the model; implementations may use a specific arithmetic-logarithmic interpolation instead.
Fit the appropriate regression model for each scoring choice. In a basic continuous-outcome analysis, each fit yields a slope estimate and its standard error. Generalized or mixed-model extensions need assumptions and covariance methods suitable for their outcomes and dependence structure.
Standardize each slope by its estimated standard error. For an increasing-trend alternative, a maximum over the directional statistics summarizes the strongest evidence among the candidate scales. A two-sided version requires corresponding two-sided joint calibration.
Account for correlations between the statistics, because the same units and outcomes appear in every fit. Multiple-marginal-model methods estimate their joint covariance and obtain adjusted p-values or simultaneous intervals. The largest ordinary t statistic cannot simply be read against a single unadjusted t reference.
Recognize redundant scales. If doses are equally spaced after taking logarithms, the ordinal and logarithmic predictors differ only by an affine transformation. Their slope tests can then be identical, so treating them as independent tests misrepresents the evidence.
Examine the observed response pattern and model fit alongside the trend result. A significant slope on one scale does not prove every adjacent group increases, determine the correct biological or operational mechanism, or identify an optimal setting. Multiple endpoints and additional searched models expand the family requiring consideration.

Three practical examples

A manufacturing experiment examines output across increasing input concentrations while allowing several plausible trend shapes.
A model-performance study compares response across controlled resource levels and documents which score scales were considered.
A reviewer checks whether a report adjusted for trying arithmetic, ordinal, and logarithmic dose models.

Think of it this way

Think of viewing the same rising landscape with three horizontal rulers: one marks actual distance, one marks equally spaced checkpoints, and one marks multiplicative distance. Each can make the slope look different, but they are three views of the same observations, not three independent experiments.

Worked example

Consider an original demonstration with positive input levels 1, 2, 4, and 8, and two outcomes at each level: [1,3], [3,5], [5,7], and [7,9]. Use ordinal scores 0, 1, 2, 3; their shift from scores 1 through 4 does not change a slope test with an intercept.
On the ordinal scale, the fitted slope is 2, the residual sum of squares is 8, and the centered predictor sum of squares is 10. With eight observations and two fitted coefficients, the residual mean square is 8/6. The slope standard error is sqrt((8/6)/10), approximately 0.36515, giving t approximately 5.47723.
Using actual input levels gives slope 0.8, residual sum of squares 11.2, and t approximately 4.44008. The base-two logarithms of 1,2,4,8 are exactly the ordinal scores, so their slope test is identical. Natural logarithms only rescale the slope and its standard error together.
The maximum directional statistic is therefore 5.47723. This is not an adjusted p-value; completing the joint trend test requires the dependence-aware reference calculation and valid model assumptions.

Illustrative case study

Fictional case: Bracken Materials studies four increasing processing intensities. Analysts notice that the log-scale fit looks more convincing than the arithmetic fit. Instead of reporting only its unadjusted result, they document the prespecified candidate scales and perform joint inference.
They also report the response plot and practical differences between settings. A detectable trend does not establish which setting is best once operating costs and product defects are considered.

Common misconceptions and mistakes

Arithmetic scoring does not replace unequal actual doses with equally spaced numbers; ordinal scoring does that. Logarithmic scoring cannot directly include zero. Testing three scales and reporting only the smallest unadjusted p-value ignores multiplicity. The statistics are correlated, not independent replications. A positive trend statistic is not a proof of strict monotonicity at every observed or unobserved dose.

Business use

The method supports controlled experiments where several simple dose-response shapes are plausible. Define the candidate transformations, direction, endpoints, and covariance treatment before selecting results. Report adjusted inference and effect interpretation together, and avoid translating a statistical trend directly into a recommendation about an optimal operating level.

Check your understanding

Which two score systems give identical tests in the example?

Answer: Ordinal and logarithmic.

Why is the maximum t statistic not a final p-value?

Answer: Its reference distribution must account for selecting among correlated tests.

Related terms

Logarithmic transformation
Dose-response relationship
multiple marginal models
regression slope
multiplicity adjustment
ordinal score
Williams trend test

Learn next

Compare dose-score transformations on a plotted experiment. Then study how joint covariance affects adjusted inference when candidate models use the same outcomes.

Sources

Schaarschmidt and colleagues, The Tukey trend test: Multiplicity adjustment using multiple marginal models: https://onlinelibrary.wiley.com/doi/10.1111/biom.13442 Hothorn, Ritz and Schaarschmidt, package-author examples and methodology: https://stat.ethz.ch/CRAN/web/packages/tukeytrend/vignettes/ExamplesTukeytrendtest.pdf

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