Why is it called a one-degree-of-freedom test?

It isolates one targeted component from the additive residual variation.

Does it compare every pair of means?

No; that is a different Tukey procedure.

Can it miss interactions?

Yes; its targeted form is limited.

AI Glossary / Statistics and probability / Intermediate

Tukey's test of additivity

Definition

Tukey's test of additivity is a one-degree-of-freedom test for a particular form of nonadditivity in a two-way layout. It examines whether the response contains a multiplicative component related to the row and column effects, beyond their additive combination. It is different from Tukey's HSD procedure for pairwise mean comparisons.

What it means

An additive model assumes that the effect of one factor is the same across levels of the other factor. For example, the difference between two settings would remain constant across batches, apart from random error.
With only one observation per row-column combination, a fully unrestricted interaction cannot be separated from error without additional assumptions. Tukey's test targets a specific interaction pattern using one degree of freedom rather than attempting to estimate every possible interaction.

How it works

Arrange observations in a complete r-by-c table. Fit the additive model Yij = mu + ai + bj + error, with row and column effects centered so each set sums to zero.
Estimate the grand mean, row effects, and column effects from the table means. Subtract the additive fitted values to obtain residuals eij. These residuals contain departures from additivity as well as random error.
Construct zij as the product of the estimated row and column effects. This is the direction of the multiplicative pattern being examined. The augmented form includes a term proportional to ai times bj.
Project the additive residuals onto that product pattern. In the usual balanced calculation, the nonadditivity sum of squares is [sum(eij zij)] squared divided by sum(zij squared), provided the denominator is nonzero.
Subtract that component from the additive residual sum of squares. The targeted component uses one degree of freedom, while the remaining component has (r-1)(c-1)-1 degrees of freedom. A positive remaining degree count and usable residual variance are necessary for the usual F calculation.
Form the ratio of the nonadditivity mean square to the remaining residual mean square. Under the additive normal independent equal-variance model, the classical reference uses an F distribution with one numerator degree of freedom and the stated denominator degrees of freedom.
Recognize the narrow alternative. A different interaction pattern can escape this test, particularly if it is poorly aligned with the product of the main effects. If either estimated main-effect vector is zero, the simple product-based calculation becomes degenerate.
Interpret evidence with residual plots and the study design. Rejection suggests the additive model misses the targeted pattern, but does not identify a causal mechanism. Nonrejection does not prove that every interaction is absent. Transformations can sometimes reduce nonadditivity, but must be scientifically meaningful and evaluated rather than chosen mechanically.

Three practical examples

A production analyst examines whether setting effects vary with batch effects in a complete unreplicated two-way table.
A reviewer interprets the additivity option in a reliability-analysis report without confusing it with an item-mean comparison.
An experimenter uses a targeted diagnostic to decide whether a future study needs replication to investigate interactions more fully.

Think of it this way

Think of adding two volume controls. An additive system gives the same increase from one control regardless of the other setting. A multiplicative departure means the effect of one control becomes stronger or weaker in proportion to the other. The test looks for that particular pattern, not every possible malfunction.

Worked example

Consider an original three-by-three table with rows [8.2, 8.6, 10.2], [7.6, 10.8, 11.6], and [8.2, 10.6, 14.2]. The grand mean is 10; row means are 9, 10, 11; column means are 8, 10, 12.
The estimated row effects are -1, 0, 1, and column effects are -2, 0, 2. Their products zij have squared sum 16. After removing the additive fitted values, the residual squared sum is 5.44, and sum(eij zij) = 8.
The nonadditivity sum of squares is 8 squared/16 = 4. The remaining sum of squares is 5.44 - 4 = 1.44. The additive residual has (3-1)(3-1) = 4 degrees of freedom; removing one leaves 3.
The F statistic is 4/(1.44/3) = 8.3333. This demonstrates the calculation and its one-versus-three degrees of freedom. A significance decision requires the appropriate reference distribution and chosen level. The constructed table does not itself establish independent normal errors or validate the model assumptions.

Illustrative case study

Fictional case: Brookhaven Manufacturing records one output measurement for each combination of three settings and three batches. Engineers notice that settings seem to separate more strongly in high-output batches.
The analyst fits the additive model and examines the targeted multiplicative departure. Regardless of that diagnostic's result, the team plans replicated measurements for the next study so that broader interaction patterns and measurement variability can be investigated with better information.

Common misconceptions and mistakes

Do not confuse additivity with equality of row means or column means. Large main effects can coexist with perfect additivity. This is not a comprehensive test for every interaction, nor is it a reliability coefficient. One observation per cell does not suddenly provide unrestricted interaction estimates. A nonsignificant result should not be described as proof that an additive model is true.

Business use

The test helps interpret legacy two-way analyses and identify a specific weakness in an additive model. It can inform model revision or experimental redesign, especially when replication is limited. Reports should state the targeted interaction form and the lack of a general interaction assessment rather than presenting a single diagnostic as complete model validation.

Check your understanding

In the example, how much residual variation is assigned to the targeted component?

Answer: Four squared units.

How many denominator degrees of freedom remain?

Answer: Three.

Would nonrejection demonstrate no interaction of any kind?

Answer: No.

Related terms

Additive model
interaction
two-way ANOVA
residual sum of squares
multiplicative effect
unreplicated design
Tukey's HSD

Learn next

Fit a two-way additive table and inspect its residual matrix. Compare a product-of-effects interaction with a different interaction pattern, then consider how replication changes what can be estimated.

Sources

IBM SPSS Statistics, Reliability Analysis statistics, Tukey additivity option: https://www.ibm.com/docs/en/spss-statistics/32.0.0?topic=analysis-reliability-statistics R additivityTests package, Tukey Additivity Test: https://search.r-project.org/CRAN/refmans/additivityTests/html/tukey.test.html NIST research paper, Partitioning of interaction in analysis of variance: https://nvlpubs.nist.gov/nistpubs/jres/73b/jresv73bn4p309_a1b.pdf Peter McCullagh, University of Chicago course notes, sections 13.5.1-13.5.2, pp. 187-188: https://www.stat.uchicago.edu/~pmcc/courses/stat349/Projects.pdf

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