two-way anova - up · experimental design two-way anova nested (hierarchical) anova observation:...

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Experimental Design Two-way ANOVA

Two-way ANOVA

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Two-way ANOVAOrthogonal (Factorial) ANOVA

Nested (Hierarchical) ANOVA

All possible combinations between factor levels are represented in the analysis

Levels of a hierarchically inferior factor are exclusive of the hierarchically superior factor

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Orthogonal Analysis of VarianceRecall the “Lakes” experiment

Suppose that we want to search for differences among Lakes, but we also want to test if male and female fish differ in average length

Two orthogonal factors

● Lakes (Fixed, with a=3 levels)● Sex (Fixed, with b=2 levels)

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Orthogonal Analysis of Variance

● L = Lakes (Fixed, with a=3 levels)● S = Sex (Fixed, with b=2 levels)● n = 5

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Orthogonal Analysis of Variance● L = Lakes (Fixed, with a=3 levels)● S = Sex (Fixed, with b=2 levels)● n = 5

Source of Variation

SS DF MS F P After

L

S

Error

Total

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Source of Variation

SS DF MS F P After

L

S

Error

Total

Degrees of Freedom

Lake → a-1 = 2Sex → b-1 = 1Error → ab(n-1) = 24Total → abn-1 = 29

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Source of Variation

SS DF MS F P After

L 2

S 1

Error 24

Total 29

Degrees of Freedom

Lake → a-1 = 2Sex → b-1 = 1Error → ab(n-1) = 24Total → abn-1 = 29

Something is missing!

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Interaction

Every orthogonal factor creates an interaction with any other orthogonal factor

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Source of Variation

SS DF MS F P After

L 2

S 1

L*S

Error 24

Total 29

Degrees of Freedom

L → a-1 = 2S → b-1 = 1L*S → (a-1)(b-1) = 2Error → ab(n-1) = 24Total → abn-1 = 29

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Source of Variation

SS DF MS F P After

L 2

S 1

L*S 2

Error 24

Total 29

Degrees of Freedom

L → a-1 = 2S → b-1 = 1L*S → (a-1)(b-1) = 2Error → ab(n-1) = 24Total → abn-1 = 29

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Orthogonal Analysis of VarianceInteraction

Measures the degree of dependency of one factor (X) on the other (Y) and vice-versa

If it is non-significantDifferences among levels of X do not depend on the levels of Y and vice-versa

If it is significantDifferences among levels of X depend on the levels of Y and vice-versa

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Orthogonal Analysis of VarianceNon-significant Interaction (α=0.05)

Source of Variation

SS DF MS F P After

L 1.650 2 0.825 13.333 0.000 Error

S 0.041 1 0.041 0.657 0.426 Error

L*S 0.090 2 0.045 0.731 0.492 Error

Error 1.485 24 0.062

Total 3.266 29

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Orthogonal Analysis of VarianceSignificant Interaction (α=0.05)

Source of Variation

SS DF MS F P After

L 1.650 2 0.825 13.333 0.000 Error

S 0.041 1 0.041 0.657 0.426 Error

L*S 0.090 2 0.045 0.731 0.012 Error

Error 1.485 24 0.062

Total 3.266 29

You cannot use these!

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Orthogonal Analysis of VarianceNo interaction Significant interaction

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Orthogonal Analysis of VarianceModel I (Factor A Fixed, Factor B Fixed)

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Orthogonal Analysis of VarianceModel II (Factor A Random, Factor B Random)

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Orthogonal Analysis of VarianceModel III (Factor A Fixed, Factor B Random)

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Nested (Hierarchical) ANOVAObservation: Eucalyptus forests strive in drier soils and apparently their pedofauna (soil fauna) is less diversified than that of Pine or Oak forests. Oak forests are likely to have the most diversified soil fauna.

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Xijk=μ+Ai+B(A)j(i)+εijk

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Nested ANOVA after orthogonalization

two-way anova - up · experimental design two-way anova nested (hierarchical) anova observation:...

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