Factorial Within SubjectsPsy 420 AinsworthFactorial WS Designs Analysis Factorial – deviation and computationalPower, relative efficiency and sample sizePower, relative efficiency and sample size Effect size Missing data Specific ComparisonsExample for Deviation Approach b1: Science Fiction b2: Mystery Case Means a1: Month 1 a2: Month 2 a3: Month 3 a1: Month 1 a2: Month 2 a3: Month 3 S1 1 3 6 5 4 1 S1 = 3.333 S1 1 3 6 5 4 1 S1 = 3.333 S2 1 4 8 8 8 4 S2 = 5.500 S3 3 3 6 4 5 3 S3 = 4.000 S4 5 5 7 3 2 0 S4 = 3.667 S5 2 4 5 5 6 3 S5 = 4.167 Treatment Means a1b1 = 2.4 a2b1 = 3.8 a3b1 = 6.4 a1b2 = 5 a2b2 = 5 a3b2 = 2.2 GM = 4.133Analysis – Deviation What effects do we have? A B AB S AS BS ABS TAnalysis – Deviation DFs DFA= a – 1 DFB= b – 1B DFAB= (a – 1)(b – 1) DFS= (s – 1) DFAS= (a – 1)(s – 1) DFBS= (b – 1)(s – 1) DFABS= (a – 1)(b – 1)(s – 1) DFT= abs - 1Sums of Squares - Deviation()()()2 22... . . ... .. ...Total A B AB S AS BS ABSijk j j k kSS SS SS SS SS SS SS SSY Y n Y Y n Y Y= + + + + + +− = − + − +∑ ∑ ∑ The total variability can be partitioned into A, B, AB, Subjects and a Separate Error Variability for Each Effect()()()( ) ( )( )( )( )( )( )... . . ... .. ...2 22. ... . . ... .. ...22 2.. ... . . . .. ...2. . . .. ..ijk j j k kjk jk j j k ki ij j ii k j iY Y n Y Y n Y Yn Y Y n Y Y n Y Yjk Y Y k Y Y jk Y Yj Y Y jk Y Y− = − + − + + − − − − −   + − + − − − +  + − − −∑ ∑ ∑∑ ∑ ∑∑ ∑ ∑( )( )( )( ) ()2.2 2 22. .. ... . . . . . .ijk jk i ij j i k jY Y jk Y Y k Y Y j Y Y +   + − − − − − − −  ∑ ∑∑ ∑ ∑ ∑Analysis – Deviation Before we can calculate the sums of squares we need to rearrange the data When analyzing the effects of A (Month) you need to AVERAGE over the effects of B (Novel) and vice versa The data in its original form is only useful for calculating the AB and ABS interactionsFor the A and AS effects Remake data, averaging over B a1: Month 1 a2: Month 2 a3: Month 3 Case Means S1 3 3.5 3.5 S1 = 3.333 S2 4.5 6 6 S2 = 5.500 S3 3.5 4 4.5 S3 = 4.000 S4 4 3.5 3.5 S4 = 3.667 S5 3.5 5 4 S5 = 4.167 Treatment Means a1 = 3.7 a2 = 4.4 a3 = 4.3 GM = 4.133()( ) ( ) ( )( )( ) ( ) ( )( ) ( )()()22 2 2. . ...22 2 2.. ...2 22210*[ 3.7 4.133 4.4 4.133 4.3 4.133 ] 2.867(3*2)*[ 3.333 4.133 5.5 4.133 4 4.1333.667 4.133 4.167 4.133 16.467A j jS iSS n Y YSS jk Y Y= − = − + − + − == − = − + − + − ++ − + − =∑∑∑ ∑()()( )22. . . .. ...2. . .2*[AS ij j iij jSS k Y Y jk Y Yk Y Y= − − − =− =∑ ∑∑( ) ( ) ( ) ( ) ()( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 2 2 2 22 2 2 2 22 2 2 2 23 3.7 4.5 3.7 3.5 3.7 4 3.7 3.5 3.73.5 4.4 6 4.4 4 4.4 3.5 4.4 5 4.43.5 4.3 6 4.3 4.5 4.3 3.5 4.3 4 4.3 ] 20.620.6 16.467 4.133ASSS− + − + − + − + − ++ − + − + − + − + − ++ − + − + − + − + − == − =For B and BS effects Remake data, averaging over A b1: Science Fiction b2: Mystery Case Means S1 3.333 3.333 S1 = 3.333 S1 3.333 3.333 S1 = 3.333 S2 4.333 6.667 S2 = 5.500 S3 4.000 4.000 S3 = 4.000 S4 5.667 1.667 S4 = 3.667 S5 3.667 4.667 S5 = 4.167 Treatment Means b1 = 4.200 b2 = 4.067 GM = 4.133()( ) ( )( ) ( )( )( ) ( ) ( )()()()()22 2.. ...2 2. .. .. ...22 2 2. ..2 2 2 215*[ 4.2 4.133 4.067 4.133 ] .1333*[ 3.333 4.2 4.333 4.2 4 4.25.667 4.2 3.667 4.2 3.333 4.067 6.667 4.067B k kBS i k k ii k kSS n Y YSS j Y Y jk Y Yj Y Y= − = − + − == − − − =− = − + − + − ++ − + − + − + − +∑∑ ∑∑()()()()( ) ( )2 25.667 4.2 3.667 4.2 3.333 4.067 6.667 4.0674 4.067 1.667 4.067+ − + − + − + − ++ − + − +( )( )22.. ...4.667 4.067 ] 5016.46750 16.467 33.533S iBSSS jk Y YSS− == − == − =∑For AB and ABS effects Use the data in its original form b1: Science Fiction b2: Mystery Case Means a1: Month 1 a2: Month 2 a3: Month 3 a1: Month 1 a2: Month 2 a3: Month 3 S 1 3 6 5 4 1 S= 3.333 S1 1 3 6 5 4 1 S1 = 3.333 S2 1 4 8 8 8 4 S2 = 5.500 S3 3 3 6 4 5 3 S3 = 4.000 S4 5 5 7 3 2 0 S4 = 3.667 S5 2 4 5 5 6 3 S5 = 4.167 Treatment Means a1b1 = 2.4 a2b1 = 3.8 a3b1 = 6.4 a1b2 = 5 a2b2 = 5 a3b2 = 2.2 GM = 4.133()()()( )( ) ( ) ( )( ) ( ) ( )2 22. ... . . ... .. ...22 2 2. ...2 2 25*[ 2.4 4.133 3.8 4.133 6.4 4.1335 4.133 5 4.133 2.2 4.133 ] 67.467AB jk jk j j k kjk jkSS n Y Y n Y Y n Y Yn Y Y= − − − − − =− = − + − + −+ − + − + − =∑ ∑ ∑∑( )( )2. . ...2.. ...2.867.13367.467 2.867 .1j j Ak k BABn Y Y SSn Y Y SSSS− = =− = == − −∑∑33 64.467=()( )()()( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )()()()()2 2 22. .. ... . . . . . .22 2 2.2 2 2 22 2 2 22 2 2 21 2.4 1 2.4 3 2.45 2.4 2 2.4 3 3.8 …