ANOVA model
Comparison between groups
Basic model
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One-way ANOVA
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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
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Yijn=μ+αi+βj+(αβ)ij+eijn
βj is the second grouping effect, (αβ)ij is
the interaction between the first and
second factor
ANOVA modeling
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Ref, ANOVA modeling.doc
Assumptions of ANOVA modeling
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Normality
Independence
Equality of variance
Process of one-way ANOVA
hypothesis testing
Process of two-way ANOVA
hypothesis testing
Types of comparison
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Validity testing of total model
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The pair-wise comparison
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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)
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
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The testing for some particular comparisons
One-way ANOVA table (for
total testing)
Two-way ANOVA table (for
total testing)
Degree of freedom
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DFM=j-1 (j=the number of groups; the types
of experiments, etc.)
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DFE=(n-1)-(j-1)=n-j
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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)
Two-way DFE=(n-1)-(ab-1)=n-ab
DFT=n-1 (n=the total sample size)
Interaction between groups
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Plot the cell mean value along the two
dimensions and watch out for the
intersection