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 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

ANOVA modeling Ref, ANOVA modeling.doc

Assumptions of ANOVA modeling Normality Independence Equality of variance

Process of one-way ANOVA hypothesis testing

Process of two-way ANOVA hypothesis testing

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

One-way ANOVA table (for total testing)

Two-way ANOVA table (for total testing)

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)

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