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09_2way_1 - Two-way ANOVA(CRF-pq design(example(pp 367-369...

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1 (example) (pp. 367-369 in Kirk) Dependent variable: Attitude toward minority group following the course. Factor A: Type of beat ( p = 3) 1 = upper class; 2 = middle class; 3 = inner city Factor B: Length of the course ( q = 3) 1 = 5 hours; 2 = 10 hours; 3 = 15 hours Data Summary: 1 2 3 mean 1 33.00 35.00 38.00 35.33 2 30.00 31.00 36.00 32.33 3 20.00 40.00 52.00 37.33 mean 27.67 35.33 42.00 35.00 Factor B Factor A Two-way ANOVA (CRF- pq design) (model) , ) ( ) ( jk i jk k j ijk Y ε + αβ + β + α + µ = where Y ijk is the score for the i th experimental unit in the jk th treatment combination, µ is the grand mean of the scores, α j is the treatment effect of the j th level of the first factor, β k is the treatment effect of the k th level of the second factor, ( αβ ) jk is the interaction effect of α j and β k ., and ε i ( jk ) is the error effect for Y ijk . ε i ( jk ) ~ N (0, σ 2 ε ). i = (1, …. , n ); j = (1, … , p ); k = (1, …. , q )
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2 Usually, both α j and β k are fixed effects. Accordingly, ( αβ ) jk is a fixed effect as well. • RB- p design is a special case of CRF- pq design when the second factor is a nuisance factor and a random effect . (estimation of parameters) ( ) ˆ ˆ ˆ ˆ ˆ ˆ ( ) ( ) ˆ ˆ ˆ ˆ [ ( ) ] j j k k jk jk j k jk j k i jk ijk j k jk ijk jk Y Y Y Y Y Y Y Y Y Y Y Y Y ••• • • ••• •• ••• • • •• ••• µ = α = β = αβ = − µ + α +β = + ε = − µ + α +β + αβ = (Partition of sum of squares?)
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