It can be applied when there are no replicated values in the data set, a situation in which it is impossible to directly estimate a fully general non-additive regression structure and still have information left to estimate the error variance.
The most common setting for Tukey's test of additivity is a two-way factorial analysis of variance (ANOVA) with one observation per cell.
The response variable Yij is observed in a table of cells with the rows indexed by i = 1,..., m and the columns indexed by j = 1,..., n. The rows and columns typically correspond to various types and levels of treatment that are applied in combination.
The additive model can be generalized to allow for arbitrary interaction effects by setting EYij = μ + αi + βj + γij.
Thus there are no remaining degrees of freedom to estimate the variance σ2, and no hypothesis tests about the γij can performed.