View source for Tukey's test of additivity

In [[Statistics|statistics]], '''Tukey's test of additivity''',<ref>{{cite journal | doi=10.2307/3001938 | last=Tukey | first=John | journal=Biometrics | title=One degree of freedom for non-additivity | volume=5 | issue=3 | year=1949 | pages=232&ndash;242 | jstor=3001938}}</ref> named for [[Biography:John Tukey|John Tukey]], is an approach used in two-way ANOVA ([[Regression analysis|regression analysis]] involving two qualitative factors) to assess whether the factor variables ([[Categorical variable|categorical variables]]) are additively related to the [[Expected value|expected value]] of the response variable.  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 [[Test statistic|test statistic]] proposed by Tukey has one degree of freedom under the null hypothesis, hence this is often called "Tukey's one-degree-of-freedom test."
==Introduction==

The most common setting for Tukey's test of additivity is a two-way factorial [[Analysis of variance|analysis of variance]] (ANOVA) with one observation per cell.  The response variable ''Y''<sub>''ij''</sub> is observed in a table of cells with the rows indexed by ''i''&nbsp;=&nbsp;1,...,&nbsp;''m'' and the columns indexed by ''j''&nbsp;=&nbsp;1,...,&nbsp;''n''.  The rows and columns typically correspond to various types and levels of treatment that are applied in combination.

The [[Additive model|additive model]] states that the expected response can be expressed ''EY''<sub>''ij''</sub>&nbsp;=&nbsp;''μ''&nbsp;+&nbsp;''α''<sub>''i''</sub>&nbsp;+&nbsp;''β''<sub>''j''</sub>, where the ''α''<sub>''i''</sub> and ''β''<sub>''j''</sub> are unknown constant values.  The unknown model parameters are usually estimated as

:<math>
\widehat{\mu} = \bar{Y}_{\cdot\cdot}
</math>

:<math>
\widehat{\alpha}_i = \bar{Y}_{i\cdot} - \bar{Y}_{\cdot\cdot}
</math>

:<math>
\widehat{\beta}_j = \bar{Y}_{\cdot j} - \bar{Y}_{\cdot\cdot}
</math>

where <span style="text-decoration: overline;">''Y''</span><sub>''i''•</sub> is the mean of the ''i''<sup>th</sup> row of the data table, <span style="text-decoration: overline;">''Y''</span><sub>•''j''</sub> is the mean of the ''j''<sup>th</sup> column of the data table, and <span style="text-decoration: overline;">''Y''</span><sub>••</sub> is the overall mean of the data table.

The additive model can be generalized to allow for arbitrary interaction effects by setting  ''EY''<sub>''ij''</sub>&nbsp;=&nbsp;''μ''&nbsp;+&nbsp;''α''<sub>''i''</sub>&nbsp;+&nbsp;''β''<sub>''j''</sub>&nbsp;+&nbsp;''γ''<sub>''ij''</sub>.  However, after fitting the natural estimator of ''γ''<sub>''ij''</sub>,

:<math>
\widehat{\gamma}_{ij} = Y_{ij} - (\widehat{\mu} + \widehat{\alpha}_i + \widehat{\beta}_j),
</math>

the fitted values

:<math>
\widehat{Y}_{ij} = \widehat{\mu} + \widehat{\alpha}_i + \widehat{\beta}_j + \widehat{\gamma}_{ij} \equiv Y_{ij}
</math>

fit the data exactly.  Thus there are no remaining degrees of freedom to estimate the variance σ<sup>2</sup>, and no hypothesis tests about the ''γ''<sub>''ij''</sub> can performed.

Tukey therefore proposed a more constrained interaction model of the form

:<math>
\operatorname{E} Y_{ij} = \mu + \alpha_i + \beta_j + \lambda\alpha_i\beta_j
</math>

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

==Method==

To carry out Tukey's test, set

:<math>
SS_A \equiv n \sum_{i} (\bar{Y}_{i \cdot}-\bar{Y}_{\cdot\cdot})^2 
</math>

:<math>
SS_B \equiv m \sum_{j} (\bar{Y}_{\cdot j} - \bar{Y}_{\cdot\cdot})^2
</math>

:<math>
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}
</math>

:<math>
SS_T \equiv \sum_{ij} (Y_{i j} - \bar{Y}_{\cdot\cdot})^2
</math>

:<math>
SS_E \equiv SS_T - SS_A - SS_B - SS_{AB}
</math>

Then use the following test statistic <ref>Alin, A. and Kurt, S. (2006). “Testing non-additivity (interaction) in two-way ANOVA tables with no replication”. ''Statistical Methods in Medical Research'' '''15''', 63&ndash;85.</ref>

:<math>
\frac{SS_{AB}/1}{MS_E}.
</math>

Under the null hypothesis, the test statistic has an [[F-distribution|''F'' distribution]] with 1,&nbsp;''q'' degrees of freedom, where ''q''&nbsp;=&nbsp;''mn''&nbsp;&minus;&nbsp;(''m''&nbsp;+&nbsp;''n'') is the degrees of freedom for estimating the error variance.

==See also==
:*[[Tukey's range test]] for multiple comparisons


==References==
<references/>

[[Category:Analysis of variance]]
[[Category:Statistical tests]]

{{Sourceattribution|Tukey's test of additivity}}