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ANOVA model Comparison between groups
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Basic model One-way ANOVA Y in =μ j+ e in =μ+α j +e in, set μ j =μ+α j μ is the total mean, α j is the grouping effect, e in is the residuals of model Two-way ANOVA Y ijn =μ+α i +β j +(αβ) ij +e ijn β j is the second grouping effect, (αβ) ij is the interaction between the first and second factor
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ANOVA modeling Ref, ANOVA modeling.doc
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Assumptions of ANOVA modeling Normality Independence Equality of variance
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Process of one-way ANOVA hypothesis testing
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Process of two-way ANOVA hypothesis testing
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Types of comparison Validity testing of total model H 0 : μ 1 =μ 2 … =μ j, for all j, (H 0 : α 1 =α 2 … =α j =0, for all j) H 1 : at least one μ unequal to others ( H 1 : at least one α≠ 0) The pair-wise comparison H 0 : μ i =μ i ’, for any group i and i≠i ’ The sequential cell mean comparison (for two- or more factor-way ANOVA) H 0 : μ ij =μ i ’ j ’, for any cell ij and (i≠i ’ or j≠j ’ ) The contrast comparison The testing for some particular comparisons
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One-way ANOVA table (for total testing)
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Two-way ANOVA table (for total testing)
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Degree of freedom DFM=j-1 (j=the number of groups; the types of experiments, etc.) Two-way DFM= ab-1 DFA=a-1 (a=the number of A type groups) DFB=b-1 (b=the number of B type groups) DFAB=(a-1)(b-1) DFE=(n-1)-(j-1)=n-j Two-way DFE=(n-1)-(ab-1)=n-ab DFT=n-1 (n=the total sample size)
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Interaction between groups Plot the cell mean value along the two dimensions and watch out for the intersection
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