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1406:{\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}}}}
40:
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
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By testing the null hypothesis that Ī» = 0, we are able to detect some departures from additivity based only on the single parameter Ī».
733:{\displaystyle {\widehat {Y}}_{ij}={\widehat {\mu }}+{\widehat {\alpha }}_{i}+{\widehat {\beta }}_{j}+{\widehat {\gamma }}_{ij}\equiv Y_{ij}}
45:
proposed by Tukey has one degree of freedom under the null hypothesis, hence this is often called "Tukey's one-degree-of-freedom test."
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fit the data exactly. Thus there are no remaining degrees of freedom to estimate the variance Ļ, and no hypothesis tests about the
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82:. The rows and columns typically correspond to various types and levels of treatment that are applied in combination.
588:{\displaystyle {\widehat {\gamma }}_{ij}=Y_{ij}-({\widehat {\mu }}+{\widehat {\alpha }}_{i}+{\widehat {\beta }}_{j}),}
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Alin, A. and Kurt, S. (2006). āTesting non-additivity (interaction) in two-way ANOVA tables with no replicationā.
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845:{\displaystyle \operatorname {E} Y_{ij}=\mu +\alpha _{i}+\beta _{j}+\lambda \alpha _{i}\beta _{j}}
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The additive model can be generalized to allow for arbitrary interaction effects by setting
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1074:{\displaystyle SS_{B}\equiv m\sum _{j}({\bar {Y}}_{\cdot j}-{\bar {Y}}_{\cdot \cdot })^{2}}
965:{\displaystyle SS_{A}\equiv n\sum _{i}({\bar {Y}}_{i\cdot }-{\bar {Y}}_{\cdot \cdot })^{2}}
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277:{\displaystyle {\widehat {\alpha }}_{i}={\bar {Y}}_{i\cdot }-{\bar {Y}}_{\cdot \cdot }}
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366:{\displaystyle {\widehat {\beta }}_{j}={\bar {Y}}_{\cdot j}-{\bar {Y}}_{\cdot \cdot }}
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are unknown constant values. The unknown model parameters are usually estimated as
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The most common setting for Tukey's test of additivity is a two-way factorial
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17:
1506:{\displaystyle SS_{T}\equiv \sum _{ij}(Y_{ij}-{\bar {Y}}_{\cdot \cdot })^{2}}
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involving two qualitative factors) to assess whether the factor variables (
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Tukey therefore proposed a more constrained interaction model of the form
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Tukey, John (1949). "One degree of freedom for non-additivity".
1714:) is the degrees of freedom for estimating the error variance.
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57:(ANOVA) with one observation per cell. The response variable
188:{\displaystyle {\widehat {\mu }}={\bar {Y}}_{\cdot \cdot }}
1606:{\displaystyle SS_{E}\equiv SS_{T}-SS_{A}-SS_{B}-SS_{AB}}
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is observed in a table of cells with the rows indexed by
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Under the null hypothesis, the test statistic has an
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1677:{\displaystyle {\frac {SS_{AB}/1}{MS_{E}}}.}
1784:Learn how and when to remove this message
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1845:Statistical Methods in Medical Research
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1616:Then use the following test statistic
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1728:for multiple comparisons
1768:more precise citations.
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1866:Analysis of variance
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49:Introduction
24:, named for
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15:
1766:introducing
1860:Categories
1811:Biometrics
1797:References
1749:references
26:John Tukey
18:statistics
1585:−
1569:−
1553:−
1537:≡
1489:⋅
1486:⋅
1479:¯
1469:−
1441:∑
1437:≡
1386:⋅
1383:⋅
1376:¯
1366:−
1358:⋅
1351:¯
1332:∑
1316:⋅
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