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ANOVA model Comparison between groups. 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.

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Presentation on theme: "ANOVA model Comparison between groups. 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."— Presentation transcript:

1 ANOVA model Comparison between groups

2 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

3 ANOVA modeling Ref, ANOVA modeling.doc

4 Assumptions of ANOVA modeling Normality Independence Equality of variance

5 Process of one-way ANOVA hypothesis testing

6 Process of two-way ANOVA hypothesis testing

7 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

8 One-way ANOVA table (for total testing)

9 Two-way ANOVA table (for total testing)

10 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)

11 Interaction between groups Plot the cell mean value along the two dimensions and watch out for the intersection


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