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 states that the expected response can be expressed
EYij =
μ +
αi +
βj, where the
αi and
βj are unknown constant values. The unknown model parameters are usually estimated as : \widehat{\mu} = \bar{Y}_{\cdot\cdot} : \widehat{\alpha}_i = \bar{Y}_{i\cdot} - \bar{Y}_{\cdot\cdot} : \widehat{\beta}_j = \bar{Y}_{\cdot j} - \bar{Y}_{\cdot\cdot} where
Yi• is the mean of the
ith row of the data table,
Y•
j is the mean of the
jth column of the data table, and
Y•• is the overall mean of the data table. The additive model can be generalized to allow for arbitrary interaction effects by setting
EYij =
μ +
αi +
βj +
γij. However, after fitting the natural estimator of
γij, : \widehat{\gamma}_{ij} = Y_{ij} - (\widehat{\mu} + \widehat{\alpha}_i + \widehat{\beta}_j), the fitted values : \widehat{Y}_{ij} = \widehat{\mu} + \widehat{\alpha}_i + \widehat{\beta}_j + \widehat{\gamma}_{ij} \equiv Y_{ij} fit the data exactly. Thus there are no remaining degrees of freedom to estimate the variance σ2, and no hypothesis tests about the
γij can performed. Tukey therefore proposed a more constrained interaction model of the form : \operatorname{E} Y_{ij} = \mu + \alpha_i + \beta_j + \lambda\alpha_i\beta_j By testing the null hypothesis that λ = 0, we are able to detect some departures from additivity based only on the single parameter λ. ==Method==