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Tukey's test of additivity

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

${\displaystyle {\widehat {\mu }}={\bar {Y}}_{\cdot \cdot }}$

${\displaystyle {\widehat {\alpha }}_{i}={\bar {Y}}_{i\cdot }-{\bar {Y}}_{\cdot \cdot }}$

${\displaystyle {\widehat {\beta }}_{j}={\bar {Y}}_{\cdot j}-{\bar {Y}}_{\cdot \cdot }}$

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,

${\displaystyle {\widehat {\gamma }}_{ij}=Y_{ij}-({\widehat {\mu }}+{\widehat {\alpha }}_{i}+{\widehat {\beta }}_{j}),}$

the fitted values

${\displaystyle {\widehat {Y}}_{ij}={\widehat {\mu }}+{\widehat {\alpha }}_{i}+{\widehat {\beta }}_{j}+{\widehat {\gamma }}_{ij}\equiv Y_{ij}}$

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

Tukey therefore proposed a more constrained interaction model of the form

${\displaystyle \operatorname {E} Y_{ij}=\mu +\alpha _{i}+\beta _{j}+\lambda \alpha _{i}\beta _{j}}$

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[edit]

To carry out Tukey's test, set

${\displaystyle SS_{A}\equiv n\sum _{i}({\bar {Y}}_{i\cdot }-{\bar {Y}}_{\cdot \cdot })^{2}}$

${\displaystyle SS_{B}\equiv m\sum _{j}({\bar {Y}}_{\cdot j}-{\bar {Y}}_{\cdot \cdot })^{2}}$

${\displaystyle SS_{AB}\equiv {\frac {(\sum _{ij}Y_{ij}({\bar {Y}}_{i\cdot }-{\bar {Y}}_{\cdot \cdot })({\bar {Y}}_{\cdot j}-{\bar {Y}}_{\cdot \cdot }))^{2}}{\sum _{i}({\bar {Y}}_{i\cdot }-{\bar {Y}}_{\cdot \cdot })^{2}\sum _{j}({\bar {Y}}_{\cdot j}-{\bar {Y}}_{\cdot \cdot })^{2}}}}$

${\displaystyle SS_{T}\equiv \sum _{ij}(Y_{ij}-{\bar {Y}}_{\cdot \cdot })^{2}}$

${\displaystyle SS_{E}\equiv SS_{T}-SS_{A}-SS_{B}-SS_{AB}}$

Then use the following test statistic ^[2]

${\displaystyle {\frac {SS_{AB}/1}{MS_{E}}}.}$

Under the null hypothesis, the test statistic has an F distribution with 1, q degrees of freedom, where q = mn − (m + n) is the degrees of freedom for estimating the error variance.

See also[edit]

• Tukey's range test for multiple comparisons

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