FACTORIAL DESIGNS Two Factor Factorial Designs

1 4 FACTORIAL Factor FACTORIAL DESIGNS Atwo- Factor factorialdesign is an experimental design in which data is collected for all possiblecombinations of the levels of the two factors of interest. If equal sample sizes are taken for each of the possible Factor combinations then the design is abalanced two- Factor factorialdesign. A balanceda bfactorial design is a FACTORIAL design for which there arealevels of factorA,blevelsof factorB, andnindependent replications taken at each of thea btreatment combinations. Thedesign size isN=abn. The effect of a Factor is defined to be the average change in the response associated with a change inthe level of the Factor . This is usually called amain effect. If the average change in response across the levels of one Factor are not the same at all levels of theother Factor , then we say there is aninteractionbetween the (ifnij=n)Cell(i,j)yij = nijk=1yijkyij =yij /nij=yij /nithlevel ofAyi = bj=1 nijk=1yijkyi =yi / bj=1nij=yi /bnjthlevel ofBy j = ai=1 nijk=1yijky j =y j / ai=1nij=y j /anOverally = ai=1 bj=1 nijk=1yijky =y / ai=1 bj=1nij=y /abnwherenijis the number of observations in cell (i,j).

2 EXAMPLE(A 2 2 balanced design ): A virologist is interested in studying the effects ofa= 2 differentculture media (M) andb= 2 different times (T) on the growth of a particular virus. She performs abalanced design withn= 6 replicates for each of the 4M Ttreatment combinations. TheN= 24measurements were taken in a completely randomized order. The results:THE DATAMM edium 1 Medium 21221 23 2025 24 29 Thours22 28 2626 25 271837 38 3531 29 30hours39 38 3634 33 35 TOTALST= 1T= 2T= 12y11 = 140y12 = 156y1 = 296T= 18y21 = 223y22 = 192y2 = 415y 1 = 363y 2 = 348y = 711i= Level ofT j= Level ofMk= Observation numberyijk=kthobservation from theithlevel ofTandjthlevel ofMMEANSM= 1M= 2T= 12y11 = = 26y1 = 18y21 = = 32y2 = 1 = 2 = = The effect of changingTfrom 12 to 18 hours on the response depends on the level ofM.

3 For medium 1, theTeffect = = For medium 2, theTeffect =32 26 = The effect on the response of changingMfrom medium 1 to 2 depends on the level ofT. ForT= 12 hours, theMeffect =26 ForT= 18 hours, theMeffect =32 =125 If either of these pairs of estimated effects are significantly different then we say there exists asignificant interactionbetween factorsMandT. For the 2 2 design example: If is significantly different than 6 for theMeffects, then we have a significantM , If is significantly different than for theTeffects, then we have a significantM Tinteraction. There are two ways of defining an interaction between two factorsAandB: If the average change in response between the levels of factorAis not the same at all levels offactorB, then aninteractionexists between factorsAandB. The lack of additivity of factorsAandB, or the nonparallelism of the mean profiles ofAandB, is called theinteractionofAandB.

4 When we assume there is no interaction betweenAandB, we say the effects areadditive. Aninteraction plotortreatment means plotis a graphical tool for checking for potentialinteractions between two factors . To make an interaction plot,1. Calculate the cell means for alla bcombinations of the levels Plot the cell means against the levels of Connect and label means the same levels of factorB. The roles ofAandBcan be reversed to make a second interaction plot. Interpretation of the interaction plot: Parallel lines usually indicate no significant interaction. Severe lack of parallelism usually indicates a significant interaction. Moderate lack of parallelism suggests a possible significant interaction may exist. Statistical significance of an interaction effect depends on the magnitude of theMSE:For smal values of theMSE, even small interaction effects (less nonparallelism) may be significant.

5 When anA Binteraction is large, the corresponding main effectsAandBmay have little practicalmeaning. Knowledge of theA Binteraction is often more useful than knowledge of the main effect. We usually say that a significant interaction can maskthe interpretation of significant main is, the experimenter must examine the levels of one Factor , sayA, at fixed levels of the otherfactor to draw conclusions about the main effect ofA. It is possible to have a significant interaction between two factors , while the main effects for bothfactors are not significant. This would happen when the interaction plot shows interactions in differentdirections that balance out over one or both factors (such as an X pattern). This type of interaction,however, is Interaction Model Theinteraction modelfor a two- Factor completely randomized design is:yijk=(22)where is the baseline mean, iis theithfactorAeffect, jis thejthfactorBeffect,( )ijis the (i,j)thA Binteraction effect, ijkis the random error of thekthobservation from the (i,j) assume ijk IID N(0, 2).

6 For now, we will also assume all effects are fixed. If ( )ijis removed from (22), we would have theadditive model:yijk= + i+ j+ ijk(23) If we impose the constraintsa i=1 i=b j=1 j= 0a i=1( )ij= 0 for alljandb j=1( )ij= 0 for alli,(24)then the least squares estimates of the model parameters are = i= j= ij= If we substitute these estimates into (22) we getyijk= + i+ j+ ij+eijk=y + (yi y ) + (y j y ) + (yij yi y j +y ) +eijkwhereeijkis thekthresidual from the treatment (i,j)thcell, andeijk= For the 2 2 design ,y = = = 1= 2= Assuming the constraints in (24), 1= = 2= = 1= = 2= = 11= + = 12= 26 + = 21= + = 22= 32 + = Forms for the Twoway ANOVAE xample: Consider a completely randomized 2 3 FACTORIAL design withn= 2 replications for each of thesix combinations of the two factors (AandB).

7 The following table summarizes the results:FactorAFactorBLevelsLevels12311 , 24 , 65 , 623 , 55 , 74 , 6 Model:yijk= + i+ j+ ( )ij+ ijkfori= 1,2j= 1,2,3k= 1,2 and ijk N(0, 2) Assume (i) 2i=1 i= 0(ii) 3j=1 j= 0(iii) 3j=1( )ij= 0 fori= 1,2(iv) 2i=1( )ij= 0 forj= 1,2,3 Thus, for the main effect constraints, we have 2= 1and 3= 1 2. The interaction effect constraints can be written in terms of just 11and 12: 12= 22= 13= 23= Thus, the reduced form of model matrixXrequires only 6 columns: , 1, 1, 2, 11and 12. 1 1 2 11 12X= 11101011101011010111010111 1 1 1 111 1 1 1 11 110 101 110 101 1010 11 1010 11 1 1 1111 1 1 111 y= 124656355746 X X= 120 0 0 0 00 12 0 0 0 000 8 4 0 000 4 8 0 000 0 0 8 400 0 0 4 8 X y= 54-6-101-6-3 (X X) 1=112 1 000000 100000 02 1000 0 12000 0002 10 000 12 (X X) 1X y= 1 1 2 11 12 Thus, 2= 1= 3= 1 2= 21= 11= 22= 12= 0 13= 11 12= 23= 11+ 12= Approach.

8 Keeping1 +a+b+ (a b)Columns 1 2 1 2 3 11 12 13 21 22 23X= 1101001000001101001000001100100100001100 1001000011000100100011000100100010110000 0100101100000100101010000010101010000010 1010010000011010010000010110000000000001 1100000000000011100000000000011100000010 0100000000010010000000001001 y= 1246563557460000000 X X= 126 64 4 42 2 2 2 2 267 12 2 22 2 2 0 0 061 72 2 20 0 0 2 2 242 25 1 12 0 0 2 0 042 21 5 10 2 0 0 2 042 21 1 50 0 2 0 0 222 02 0 04 1 1 1 0 022 00 2 01 4 1 0 1 022 00 0 21 1 4 0 0 120 22 0 01 0 0 4 1 120 20 2 00 1 0 1 4 120 20 0 20 0 1 1 1 4 X y= 5424301122213101181210 (X X) 1=1180 86-45 -45-20 -20 -20-6 -6 -6 -6 -6 -6-4570 2000 -0-10 -10 -10 10 10 10-4520 70-00010 10 10 -10 -10 -10-200 -080 -10 -10-30 15 15 -30 15 15-2000-10 80 -1015 -30 15 15 -30 15-20-00-10 -10 8015 15 -30 15 15 -30-6-10 10-30 15 1576 -14 -14 -4 -4 -4-6-10 1015 -30 15-14 76 -14 -4 -4 -4-6-10 1015 15 -30-14 -14 76 -4 -4 -4-610 -10-30 15 15-4 -4 -4 76 -14 -14-610 -1015 -30 15-4 -4 -4 -14 76 -14-610 -1015 15 -30-4 -4 -4 -14 -14 76 (X X) 1X y= = 1 2 1 2 3 11 12 13 21 22 23 for an ANOVA SSA=nba i=1(yi y )2= the sum of squares for factorA(df=a 1)MSA=SSA/(a 1) = the mean square for factorA SSB=nab j=1(y j y )2= the sum of squares for factorB(df=b 1)MSB=SSB/(b 1)

9 = the mean square for factorB SSAB=na i=1b j=1[(yij y ) (yi y ) (y j y )]2=na i=1b j=1(yij yi y j +y )2= theA Binteraction sum of squares (df= (a 1)(b 1))MSAB=SSAB/(a 1)(b 1)= the mean square for theA Binteraction SSE= ai=1 bj=1 nk=1(yijk yij )2= the error sum of squares (df=ab(n 1))MSE=SSE/ab(n 1)= the mean square error SST=a i=1b j=1n k=1(yijk y )2= the total sum of squares (df=abn 1) the total sum of squares is partitioned into components corresponding to the terms in the model:a i=1b j=1n k=1(yijk y )2=nba i=1(yi y )2+nab j=1(y j y )2+na i=1b j=1(yij yi y j +y )2+r i=1ni j=1(yij yi )2OR The alternateSSformulas for the balancedtwo FACTORIAL design are:SST=a i=1b j=1n k=1y2ijk y2 abnSSA=a i=1y2i bn y2 abnSSB=b j=1y2 j an y2 abnSSAB=a i=1b j=1y2ij n SSA SSB y2 abnSSE=SST SSA SSB SSAB The alternateSSformulas for the unbalancedtwo FACTORIAL design are:SST=a i=1b j=1nij k=1y2ijk y2 NSSA=a i=1y2i ni y2 NSSB=b j=1y2 j n j y2 NSSAB=a i=1b j=1y2ij nij SSA SSB y2 NSSE=SST SSA SSB SSAB whereN= ai=1 bj=1nij, ni = bj=1nij, n j= ai= Two- Factor FACTORIAL ANOVA TableSource ofSum 1 MSA=SSA/(a 1)FA=MSA/MSEBSSBb 1 MSB=SSB/(b 1)FB=MSB/MSEA BSSAB(a 1)(b 1)MSAB=SSAB/(a 1)(b 1)FA B=MSAB/MSEE rrorSSEab(n 1)MSE=SSE/(ab(n 1)) TotalSStotalabn 1 For the unbalanced case, replaceab(n 1) withN abfor the forSSEand replaceabn 1 withN 1for the forSStotalwhereN= ai=1 bj= on Interpreting the ANOVA TestH0: ( )11= ( )12= = ( )abvs.

10 H1: at least one ( )ij6= ( )i j first. If this test indicates that there is not a significant interaction, then continue testing the hy-potheses for the two main effects:H0: 1= 2= = : at least one i6= i H0: 1= 2= = : at least one j6= j If this test indicates that there is a significant interaction, then the interpretation of significantmain effects hypotheses can be masked. To draw conclusions about a main effect, we will fixthe levels of one Factor and vary the levels of the other. Using this approach (combined withinteraction plots) we may be able to provide an interpretation of main effects. If we assume the constraints in (24), then the hypotheses can be rewritten as:H0: ( )11= ( )12= = ( )ab= 0 : at least one ( )ij6= 0H0: 1= 2= = a= 0 : at least one i6= 0H0: 1= 2= = b= 0 : at least one j6= for a2 2 FACTORIAL design Example We will now use SAS to analyze the 2 2 FACTORIAL design data discussed 1 Medium 21221 23 2025 24 29 Thours22 28 2626 25 271837 38 3531 29 30hours39 38 3634 33 35132 ANOVA and Estimation of Effects for a 2x2 DesignThe GLM ProcedureDependent Variable: growthANOVA and Estimation of Effects for a 2x2 DesignThe GLM ProcedureDependent Variable: growthSourceDFSum ofSquaresMean SquareF ValuePr> <.