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	~~My name~~ is ~~Dee Holifield~~. ~~I life~~ in ~~Werkendam~~ (~~Netherlands~~).<br><br>~~My web page~~: ~~[https~~://~~prsewing~~.~~com~~/~~Explaining~~-~~Straightforward~~-~~Secrets~~-Of-~~Essay~~-~~Writing~~.~~htm buy essays cheap~~]		In [[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=232–242 \| jstor=3001938}}</ref> named for [[John Tukey]], is an approach used in two-way ANOVA ([[regression analysis]] involving two qualitative factors) to assess whether the factor variables are additively related to the [[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]] 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]] (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'' = 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''<sub>''ij''</sub> = ''μ'' + ''α''<sub>''i''</sub> + ''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>
			\hat{\mu} = \bar{Y}_{\cdot\cdot}
			</math>

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

			:<math>
			\hat{\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> = ''μ'' + ''α''<sub>''i''</sub> + ''β''<sub>''j''</sub> + ''γ''<sub>''ij''</sub>. However after fitting the natural estimator of γ''<sub>''ij''</sub>,

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

			the fitted values

			:<math>
			\hat{Y}_{ij} = \hat{\mu} + \hat{\alpha}_i + \hat{\beta}_j + \hat{\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>
			EY_{ij} = \mu + \alpha_i + \beta_j + \lambda\alpha_i\beta_j
			</math>

			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==

			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})}{\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>

			<br>
			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''', 63–85. </ref>

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

			<br>
			Under the null hypothesis, the test statistic has an [[F-distribution\|''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==
			:*[[Tukey's range test]] for multiple comparisons

			{{More footnotes\|date=February 2010}}

			==References==
			<references/>

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

en>Xnn: /* A simple example for an evasive boolean function */ generarl -> general - don't skip preview!

2012-01-19T16:57:03Z

A simple example for an evasive boolean function: generarl -> general - don't skip preview!

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My name is Dee Holifield. I life in Werkendam (Netherlands).<br><br>My web page: [https://prsewing.com/Explaining-Straightforward-Secrets-Of-Essay-Writing.htm buy essays cheap]

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