anova3_article

anova3_article - 1 Three-factor experiments 2 3 experiments...

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Unformatted text preview: 1 Three-factor experiments 2 3 experiments: introduction 2 3 experiments have three factors, each at two levels. Y ijk` = μ ijk + ijk` = μ + α i + β j + γ k + ( αβ ) ij + ( αγ ) ik + ( βγ ) jk + ( αβγ ) ijk + ijk` Main effects and interactions sum to 0 in any subscript , with other subscripts held fixed. Overall mean and main effects μ = μ 111 + μ 112 + μ 121 + μ 122 + μ 211 + μ 212 + μ 221 + μ 222 8 α 1 = ( μ 111 + μ 112 + μ 121 + μ 122 )- ( μ 211 + μ 212 + μ 221 + μ 222 ) 8 β 1 = ( μ 111 + μ 112 + μ 211 + μ 212 )- ( μ 121 + μ 122 + μ 221 + μ 222 ) 8 and similarly for γ 1 Two-way interactions ( αβ ) 11 = ( μ 111 + μ 112 )- ( μ 121 + μ 122 )- ( μ 211 + μ 212 ) + ( μ 221 + μ 222 ) 8 and similarly for ( αγ ) 11 and ( βγ ) 11 Three way interaction ( αβγ ) 111 = ( μ 111- μ 121- μ 211 + μ 221 )- ( μ 112- μ 122- μ 212- μ 222 ) 8 = { ( αβ ) 11 when C=1 } - { ( αβ ) 11 when C=2 } 8 The three-way interaction is • the change in the (AB) interaction as C changes from 1 to 2. • the change in the (AC) interaction as B changes from 1 to 2. • the change in the (BC) interaction as A changes from 1 to 2. All three changes are equal! Visualizing the effects (Empty slide for drawing) Visualizing the effects (Empty slide for drawing) Representation of effects run A B C AB AC BC ABC mean 1 + + + + + + + μ 222 2- + +-- +- μ 122 3 +- +- +-- μ 212 4- - + +-- + μ 112 5 + +- +--- μ 221 6- +-- +- + μ 121 7 +- --- + + μ 211 8- - - + + +- μ 111 For the experiment: runs would be put in a random order 1.1 Lipton example 2 3 experiments: example 2 • Lipton carried out an experiment with 5 factors, each at two levels. – there were 16 “runs” • As we will see later, only three factors had effects • Ignore the factors without effects – ⇒ 2 3 experiment with two replicates Lipton 2 3 experiment: factors Factors in this analysis: • Temperature (0=water cooled, 1=ambient temperature) • batch weight (1500 lb, 2000 lb) • delay (1 day, 7 day) Response = performance (std. dev.) R code lipton.data = read.table(’lipton.txt’,header=TRUE) attach(lipton.data) lmfit = lm(performance ~ temp*wt*delay) pdf(’lipton_boxplot_temp_delay.pdf’) boxplot(performance~temp*delay,xlab=’temp*delay’, ylab=’performance’) pdf(’lipton_boxplot_wt_delay.pdf’) boxplot(performance~wt*delay,xlab=’wt*delay’, ylab=’performance’) graphics.off() detach(lipton.data) 3 ANOVA table...
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anova3_article - 1 Three-factor experiments 2 3 experiments...

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