Tukey's Test of Additivity¶
Test a one-degree product-of-main-effects departure from additivity in a two-way response table, especially when each cell has one observation.
Core Idea¶
Tukey's test of additivity assesses whether a two-way response table departs from a row-plus-column additive model in a specific, product-shaped direction. With one observation per cell, an unrestricted interaction uses all the variation needed to estimate error. Tukey's one-degree contrast instead tests whether an interaction proportional to the row effect times the column effect improves the fit, comparing that component with the residual variation left over.[ref-5c648d8d475d][ref-1fafab51a31a]
Scope of Application¶
The classic case is an unreplicated, complete two-factor table: varieties by blocks for yield, or temperature by pressure for impurity. Montana State's \(3\times5\) impurity example reports one nonadditivity degree of freedom, seven residual degrees and \(p=0.5660\), a nonrejection of this product-shaped alternative rather than proof of all-form additivity. The error comparison requires \(ab-a-b>0\) for \(a\) row and \(b\) column levels; a \(2\times2\) unreplicated table has no such denominator.[^ref-1fafab51a31a]
Clarity¶
This is not Tukey HSD for comparing group means or the Siegel–Tukey dispersion test. A rejection supports a particular product-shaped nonadditivity; a nonrejection does not prove all possible interactions absent. The test does not prescribe a response transformation, though one may be investigated later.[^ref-6e807cb3621c]
Manages Complexity¶
The procedure compresses a broad interaction space to one interpretable direction and reserves the rest of the additive-model residual variation for an error comparison. That makes a restricted question testable without pretending an unreplicated table can identify arbitrary interaction and error separately.[^ref-5c648d8d475d]
Abstract Reasoning¶
Identify the two factors, response, replication and completeness. Fit the additive model, obtain row and column effect estimates, form their cellwise products, and assess the one-degree residual component along that direction against the remaining mean square. Interpret the F result only for this declared alternative and check whether the residual assumptions and degrees of freedom are defensible.[ref-5c648d8d475d][ref-1fafab51a31a]
Knowledge Transfer¶
Agricultural and industrial tables share the same roles—two crossed factors, a response, additive baseline, product contrast and residual comparison—without sharing a substantive mechanism. Statistical Test is the accepted strict genus; Two-Way Analysis of Variance supplies model context, not an additional accepted parent.
[^ref-5c648d8d475d]: Purdue University statistics lecture, Tukey's Test for Additivity. [^ref-1fafab51a31a]: Montana State University, “Tukey's Test for Nonadditivity,” §4.12. [^ref-6e807cb3621c]: The R Journal, original research article on nonadditivity tests in unreplicated two-way studies, §6.1.
Relationships to Other Abstractions¶
Current abstraction Tukey's Test of Additivity Domain-specific
Parents (1) — more general patterns this builds on
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Tukey's Test of Additivity is a kind of Statistical Test Domain-specific
Tukey's one-degree additivity procedure is a statistical test.
Hierarchy path (1) — routes to 1 parentless root
- Tukey's Test of Additivity → Statistical Test
Neighborhood in Abstraction Space¶
Tukey's Test of Additivity sits in a sparse region of the domain-specific corpus (99th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Lack-of-Fit Sum of Squares — 0.75
- Variance function — 0.75
- Residual Sum of Squares — 0.74
- Truncated Regression Model — 0.74
- Factor Regression Model — 0.74
Computed from structural-signature embeddings · 2026-10-08