ANOVA model

ANOVA model

Comparison between groups

Basic model One-way ANOVA

Yin=μj+ein=μ+αj+ein, set μj=μ+αj μ is the total mean, αj is the grouping effe

ct, ein is the residuals of model Two-way ANOVA

Yijn=μ+αi+βj+(αβ)ij+eijn βj is the second grouping effect, (αβ)ij is t

he 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

H0: μ1=μ2… =μj, for all j, (H0: α1=α2… =αj=0, for all j) H1: at least one μ unequal to others (H1: at least one α

≠0) The pair-wise comparison

H0: μi=μi’, for any group i and i≠i’ The sequential cell mean comparison (for two- or more

factor-way ANOVA) H0: μij=μi’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