4.15 Three Factor Factorial Designs • The complete interaction model for a three-factor completely randomized design is: yijkl = (35) { µ is the baseline mean, { τi, βj, and γk are the main factor effects for A, B, and C, respectively. { (τβ)ij,(τγ)ik and (βγ)jk are the two-factor interaction effects for interactions AB, AC, and BC, respectively. { (τβγ)ijk are the three-factor interaction effects for the ABC interaction. th th { eijkl is the random error of the k observation from the (i; j; k) treatment. 2 We assume eijkl ∼ IIDN(0; σ ). For now, we will also assume all effects are fixed. 4.15.1 Partitioning the total sum of squares Pa Pb Pc Pn 2 • SST = i=1 j=1 k=1 l=1(yijkl − y····) • Note that we can rewrite yijkl as yijkl = µb + τbi + βbj + γbk + (τβc)ij + (τγc)ik + (βγc)jk + (τβγd )ijk + eijkl where µb = y···· τbi = yi··· − y···· βbj = y·j·· − y···· γbk = y··k· − y···· (τβc)ij = yij··−yi···−y·j··+y···· (τγc)ik = yi·k·−yi···−y··k·+y···· (βγc)jk = y·jk·−y·j··−y··k·+y···· (τβγd )ijk = yijk· − yij·· − yi·k· − y·jk· + yi··· + y·j·· + y··k· − y···· eijkl = yijkl − yijk· • Substitution of these estimates into SST yields a b c n X X X X 2 SST = (yijkl − y····) i=1 j=1 k=1 l=1 a b c n X X X X 2 = τbi + βbj + γbk + (τβc)ij + (τγc)ik + (βγc)jk + (τβγd )ijk + eijkl i=1 j=1 k=1 l=1 a b c n a b c n a b c n X X X X 2 X X X X 2 X X X X = τbi + βbj + γbk i=1 j=1 k=1 l=1 i=1 j=1 k=1 l=1 i=1 j=1 k=1 l=1 a b c n a b c n a b c n a b c n X X X X 2 X X X X 2 X X X X 2 X X X X 2 + (τβc)ij + (τγc)ik + (βγc)jk + (τβγd )ijk i=1 j=1 k=1 l=1 i=1 j=1 k=1 l=1 i=1 j=1 k=1 l=1 i=1 j=1 k=1 l=1 a b c n a b c n X X X X 2 X X X X + eijkl + (all cross-products of parameter estimates) i=1 j=1 k=1 l=1 i=1 j=1 k=1 l=1 Through tedious algebra, it can be shown PPPP cross-products of parameter estimates = 0. Simplification of the multiple summations produces: a b c a b a c b c X 2 X 2 X X X 2 X X 2 X X 2 SST = bcn τbi + acn βbj + abn γbk + cn (τβc)ij + bn (τγc)ik + an (βγc)jk i=1 j=1 k=1 i=1 j=1 i=1 k=1 j=1 k=1 a b c a b c n X X X 2 X X X X 2 +n (τβγd )ijk + eijkl i=1 j=1 k=1 i=1 j=1 k=1 l=1 = SSA + SSB + SSC + SSAB + SSAC + SSBC + SSABC + SSE 161 Ii = τbi Jj = βbj Kk = γbk (IJ)ij = τβcij (IK)ik = τγcik (JK)jk = βγcjk (IJK)ijk = τβγd ijk Eijkl = eijkl 174 162 Three-Factor Factorial Designs: Fixed Factors A, B, C Three Factor Factorial Example • In a paper production process, the effects of percentage of hardwood concentration in raw wood pulp, the vat pressure, and the cooking time on the paper strength were studied. 175 • There were a = 3 levels of hardwood concentration (CONC = 2%, 4%, 8%). There were b = 3 levels of vat pressure (PRESS = 400, 500, 650). There were c = 2 levels of cooking time (TIME = 3 hours, 4 hours). 163 • A three factor factorial experiment with n = 2 replicates was run. The order of data collection was completely randomized. • We assume all three factors are fixed. • The experimental data are in the table below. Cooking time 3.0 hours Cooking time 4.0 hours Hardwood Pressure Pressure Concentration 400 500 650 400 500 650 2 196.6 197.7 199.8 198.4 199.6 200.6 196.0 196.0 199.4 198.6 200.4 200.9 4 198.5 196.0 198.4 197.5 198.7 199.6 197.2 196.9 197.6 198.1 198.0 199.0 6 197.5 195.6 197.4 197.6 197.0 198.5 196.6 196.2 198.1 198.4 197.8 199.8 • From the SAS output, the p-value = .2903 is not significant for the test of the equality of the three factor interaction effects: H0 : τβγijk = 0 for all i; j; k vs H1 : τβγijk 6= 0 for some i; j; k • The p-values of .0843, .0146, and .0750 for the CONC*TIME, CONC*PRESS, and TIME*PRESS two factor interactions are all significant at the α = :10 level indicating the interpretation of the significant main effects for CONC (p-value=.0009), TIME (p-value< :0001), and PRESS (p-value< :0001) may be masked. To understand the effects in the model, we need to examine the interaction plots. THREE FACTOR ANALYSIS OF VARIANCE The GLM Procedure Dependent Variable: strength Sum of Source DF Squares Mean Square F Value Pr > F Model 17 59.72888889 3.51346405 9.61 <.0001 Error 18 6.58000000 0.36555556 Corrected Total 35 66.30888889 R-Square Coeff Var Root MSE strength Mean 0.900767 0.305274 0.604612 198.0556 Source DF Type III SS Mean Square F Value Pr > F conc 2 7.76388889 3.88194444 10.62 0.0009 time 1 20.25000000 20.25000000 55.40 <.0001 conc*time 2 2.08166667 1.04083333 2.85 0.0843 press 2 19.37388889 9.68694444 26.50 <.0001 conc*press 4 6.09111111 1.52277778 4.17 0.0146 time*press 2 2.19500000 1.09750000 3.00 0.0750 conc*time*press 4 1.97333333 0.49333333 1.35 0.2903 164 THREE FACTOR ANALYSIS OF VARIANCETHREE FACTOR ANALYSIS OF VARIANCE The GLM Procedure The GLM Procedure Distribution of strength Distribution of strength 201 201 200 200 199 199 THREE FACTOR ANALYSIS OF VARIANCE THREE FACTOR ANALYSIS OF VARIANCE h h t t g The GLM Procedureg The GLM Procedure n n e e r r t Distribution of t strength Distribution of strength s 198 s 198 201 201 200 200 197 197 199 199 h h t t g g n n e e r r t t s 198 s 198 THREE FACTOR ANALYSIS OF VARIANCE 196 197 196 197 196 The GLM Procedure 196 400 400 500 500 650 2 650 2 4 8 8 press conc Distributiopnr eosfs strength conc strength strength 201 Level of Level of press N Mean Std Dev conc N Mean Std Dev 400 12 197.583333 0.85687947strength 2 12 198.666667strength1.75620113 500 12 197.491667 1.50903843 4 12 197.958333 0.98669175 Level of650 12 199.091667 1.12043687 Level of 8 12 197.541667 1.12448641 200 press N Mean Std Dev conc N Mean Std Dev THREE FACTOR ANALYSIS OF VARIANCE 400 12 197.583333 0.85687947 2 12 198.666667 1.75620113 500 12 197.491667 1.50903843 4 12The197.958333 GLM Procedure0.98669175 199 650 12 199.091667 1.12043687 8 12 197.541667 1.12448641 THREE FACTOR ANALYSIS OF VARIANCE THREE FACTOR ANALYSIS OF VARIANCE h t g Distribution of strength The GLM Procedure The GLM Procedure n e r t Distribution of strength 201 Distribution of strength s 198 201 201 200 200 200 197 199 199 h h t t g g n n e e r r t t s 198 s 198 199 196 197 197 h t 196 g 196 n e r t 2 400 2 500 2 400 2 26 5050 0 2 650 44 440000 4 500 4 6 550 00s 8 4001988 450 06508 650 8 400 8 500 8 63 50 4 conc*press time conc*press strength strength Level of Level of Level of conc press N Mean Std Dev time N Mean Std Dev 2 400 4 197.400000 1.29614814strength197 3 18 197.305556 1.20219027 2 500 4 198.425000 1.97378655 4 18 198.805556 1.12431533 Level of 2 Level650 of 4 200.175000 0.69462220 conc 4 press400 4 N197.825000 0.58523500Mean Std Dev 4 500 4 197.400000 1.19163753 2 4 400 650 4 198.6500004 197.4000000.85440037 1961.29614814 8 400 4 197.525000 0.73654599 2 8 500 500 4 196.6500004 198.4250000.95742711 1.97378655 8 650 4 198.450000 1.00829890 2 650 4 200.175000 0.69462220 3 4 4 400 4 197.825000 0.58523500 time 4 500 4 197.400000 1.19163753 strength 4 650 4 198.650000 0.85440037 Level of 8 400 4 197.525000 0.73654599 time N Mean Std Dev 8 500 4 196.650000 0.95742711 3 18 197.305556 1.20219027 8 650 4 198.450000 1.00829890 4 18 198.805556 1.12431533 165 THREE FACTOR ANALYSIS OF VARIANCE THREE FACTOR ANALYSIS OF VARIANCE The GLM Procedure The GLM Procedure Distribution of strength Distribution of strength 201 201 200 200 199 199 THREE FACTOR ANALYSIS OF VARIANCE THREE FACTOR ANALYSIS OF VARIANCE h h t t g The GLM Procedure g The GLM Procedure n n e e r r t Distribution of strength t Distribution of strength s 198 s 198 201 201 200 200 197 197 199 199 h h t t g g n n e e r r t t s 198 s 198 196 197 196 197 196 196 3 400 3 500 3 400 3 5003 6503 650 4 400 4 4004 500 4 650 42 5 300 42 2 6 3450 2 4 4 3 3 4 4 8 34 4 8 4 8 3 8 4 time*press conc*time time*press conc*time strength strength Level of Level of Level of Level of time press N Mean Std Dev conc time N Mean Std Dev 3 400 6 197.066667 strength0.87559504 2 3 6 197.583333 1.68572437strength 3 500 6 196.400000 0.76681158 2 4 6 199.750000 1.06160256 Level of Level3 of 650 6 198.450000 0.96695398 Level of Level4 3 of 6 197.433333 0.94798031 time press4 400N 6 198.100000Mean0.45607017 Std Dev conc time4 4 N6 198.483333 0.76267075Mean Std Dev 4 500 6 198.583333 1.24966662 8 3 6 196.900000 0.92951600 3 400 4 6506 197.0666676 199.733333 0.915787460.87559504 2 38 4 66198.183333197.5833330.96419224 1.68572437 3 500 6 196.400000 0.76681158 2 4 6 199.750000 1.06160256 3 650 6 198.450000 0.96695398 4 3 6 197.433333 0.94798031 4 400 6 198.100000 0.45607017 4 4 6 198.483333 0.76267075 4 500 6 THREE198.583333 FACTOR1.24966662 ANALYSIS8 OF 3VARIANCE6 196.900000 0.92951600 4 650 6 199.733333 0.91578746 8 4 6 198.183333 0.96419224 The GLM Procedure Distribution of strength 201 200 199 h t g n e r t s 198 197 196 2 2 2 2 2 2 4 4 4 4 4 4 8 8 8 8 8 8 3 3 3 4 4 4 3 3 3 4 4 4 3 3 3 4 4 4 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 00 00 50 00 00 50 00 00 50 00 00 50 00 00 50 00 00 50 conc*time*press strength Level of Level of Level of conc time press N Mean Std Dev 166 2 3 400 2 196.300000 0.42426407 2 3 500 2 196.850000 1.20208153 2 3 650 2 199.600000 0.28284271 2 4 400 2 198.500000 0.14142136 2 4 500 2 200.000000 0.56568542 2 4 650 2 200.750000 0.21213203 4 3 400 2 197.850000 0.91923882 4 3 500 2 196.450000 0.63639610 4 3 650 2 198.000000 0.56568542 4 4 400 2 197.800000 0.42426407 4 4 500 2 198.350000 0.49497475 4 4 650 2 199.300000 0.42426407 8 3 400 2 197.050000 0.63639610 8 3 500 2 195.900000 0.42426407 Check of Normality Assumption The UNIVARIATE Procedure Variable: resid Moments N 36 Sum Weights 36 Mean 0 Sum Observations 0 Std Deviation 0.43358967 Variance 0.188 Skewness 0 Kurtosis -1.1102611 Uncorrected SS 6.58 Corrected SS 6.58 Coeff Variation .