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ANOVA model

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

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ANOVA model

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  1. ANOVA model Comparison between groups

  2. Basic model • One-way ANOVA • Yin=μj+ein=μ+αj+ein, set μj=μ+αj • μ is the total mean, αj is the grouping effect, 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 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 • H0: μ1=μ2… =μj, for all j, (H0: α1=α2… =αj=0, for all j) • H1: at least one μunequal 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

  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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