Tukey's test of additivity

Introduction

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 Y_ij 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 EY_ij = μ + α_i + β_j, where the α_i and β_j are unknown constant values. The unknown model parameters are usually estimated as

${\displaystyle \hat{\mu }={\overline{Y}}_{\cdot \cdot }}$

${\displaystyle {\hat{\alpha }}_i={\overline{Y}}_{i\cdot }-{\overline{Y}}_{\cdot \cdot }}$

${\displaystyle {\hat{\beta }}_j={\overline{Y}}_{\cdot j}-{\overline{Y}}_{\cdot \cdot }}$

where Y_i• is the mean of the i^th row of the data table, Y_•j is the mean of the j^th 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 EY_ij = μ + α_i + β_j + γ_ij. However, after fitting the natural estimator of γ_ij,

${\displaystyle {\hat{\gamma }}_{ij}=Y_{ij}-(\hat{\mu }+{\hat{\alpha }}_i+{\hat{\beta }}_j),}$

the fitted values

${\displaystyle {\hat{Y}}_{ij}=\hat{\mu }+{\hat{\alpha }}_i+{\hat{\beta }}_j+{\hat{\gamma }}_{ij}\equiv Y_{ij}}$

fit the data exactly. Thus there are no remaining degrees of freedom to estimate the variance σ², and no hypothesis tests about the γ_ij can performed.

Tukey therefore proposed a more constrained interaction model of the form

${\displaystyle \mathrm{E}{Y}_{ij}=\mu +{\alpha }_i+{\beta }_j+\lambda {\alpha }_i{\beta }_j}$

By testing the null hypothesis that λ = 0, we are able to detect some departures from additivity based only on the single parameter λ.