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anova3_2page - Factorial Experiments and Quality...

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Factorial Experiments and Quality Improvement III March 13, 2007 2 3 experiments: introduction 2 3 experiments have three factors, each at two levels. Y ijk = μ ij + ijk = μ + α i + β j + γ k + ( αβ ) ij + ( αγ ) ij + ( βγ ) jk + ( αβγ ) ijk + ijk Main effects and interactions sum to 0 in any subscript , with other subscripts held fixed.
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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
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Three way interaction ( αβγ ) 111 = ( μ 111 - μ 121 - μ 121 + μ 221 ) - ( μ 112 - μ 122 + μ 122 - 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. Visualizing the effects (Empty slide for drawing)
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Visualizing the effects (Empty slide for drawing) Representation of effects run A B C AB AC BC ABC 1 + + + + + + + 2 - + + - - + - 3 + - + - + - - 4 - - + + - - + 5 + + - + - - - 6 - + - - + - + 7 + - - - - + + 8 - - - + + + - Runs would be put in a random order
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2 3 experiments: example 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.)
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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) ANOVA table anova(lmfit) Analysis of Variance Table Response: performance Df Sum Sq Mean Sq F value Pr(>F) temp 1 0.03062 0.03062 0.8183 0.392103 wt 1 0.00562 0.00562 0.1503 0.708359 delay 1 0.88360 0.88360 23.6099 0.001257 ** temp:wt 1 0.10562 0.10562 2.8223 0.131474 temp:delay 1 0.65610 0.65610 17.5311 0.003050 **
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