Mixed Designs:Between and WithinPsy 420AinsworthMixed Between and Within Designs• Conceptualizing the Design▫ Types of Mixed Designs • Assumptions• Analysis▫ Deviation▫ Computation• Higher order mixed designs• Breaking down significant effectsConceptualizing the Design• This is a very popular design because you are combining the benefits of each design• Requires that you have one between groups IV and one within subjects IV• Often called “Split-plot” designs, which comes from agriculture• In the simplest 2 x 2 design you would haveConceptualizing the Design• In the simplest 2 x 2 design you would have subjects randomly assigned to one of two groups, but each group would experience 2 conditions (measurements)GRE - before GRE - afterS1S1S2S2S3S3S4S4S5S5S6S6S7S7S8S8S9S9S10S10KaplanPrincetonConceptualizing the Design• Advantages▫ First, it allows generalization of the repeated measures over the randomized groups levels▫ Second, reduced error (although not as reduced as purely WS) due to the use of repeated measures• Disadvantages▫ The addition of each of their respective complexitiesConceptualizing the Design Pretest Posttest S1 S1 S2 S2 S3 S3 S4 S4 Treatment Group S5 Treatment S5 S6 S6 S7 S7 S8 S8 S9 S9 Control Group S10 No Treatment S10 • Types of Mixed Designs▫ Other than the mixture of any number of BG IVs and any number of WS IVs…▫ Pretest Posttest Mixed Design to control for testing effectsAssumptions• Normality of Sampling Distribution of the BG IVs▫ Applies to the case averages (averaged over the WS levels)• Homogeneity of Variance▫ Applies to every level or combination of levels of the BG IV(s)Assumptions• Independence, Additivity, Sphericity▫ Independence applies to the BG error term ▫ But each WS error term confounds random variability with the subjects by effects interaction▫ So we need to test for sphericity instead; the test is on the average variance/covariance matrix (over the levels of the BG IVs)Assumptions• Outliers▫ Look for them in each cell of the design• Missing data▫ Causes the same problems that they did in the BG and WS designs separately Data points missing across the WS part can be estimated as discussed previously Missing data in the randomized groups part causes non-orthogonalityAnalysisb1b2b3S1S1S1S2S2S2S3S3S3S4S4S4S5S5S5S6S6S6S7S7S7S8S8S8S9S9S9S10S10S10S11S11S11S12S12S12S13S13S13S14S14S14S15S15S15Within Groupsa1a2a3Randomized GroupsSources of Variance• SST=SSBG+SSWS• What are the sources of variance?▫ A▫ S/A▫ B▫ AB▫ BxS/A▫ T• Degrees of freedom?Example – Books by Month• Example:▫ Imagine if we designed the previous research study concerning reading different novels over time▫ But instead of having everyone read all of the books for three months we randomly assign subjects to three different books and have them read for three monthsb1: Month 1b2: Month 2b3: Month 3S11 3 6 S1 = 3.333S21 4 8 S2 = 4.333S33 3 6 S3 = 4S45 5 7 S4 = 5.667S52 4 5 S5 = 3.667a1b1 = 2.4 a1b2 = 3.8 a1b3 = 6.4 a1 = 4.2S63 1 0 S6 = 1.333S74 4 2 S7 = 3.333S85 3 2 S8 = 3.333S94 2 0 S9 = 2S104 5 3 S10 = 4a2a2b1 = 4 a2b2 = 3 a2b3 = 1.4 a2 = 2.8S114 2 0 S11 = 2S122 6 1 S12 = 3S133 3 3 S13 = 3S146 2 1 S14 = 3S153 3 2 S15 = 2.667a3b1 = 3.6 a3b2 = 3.2 a3b3 = 1.4 a3 = 2.733b1 = 3.333 b2 = 3.333 b3 = 3.067 GM = 3.244B: MonthA: Type of NovelCase Meansa1: Science Fictiona2: Mysterya3: RomanceSums of Squares - Deviation/ * /222... . . ... .. ...222. ... . . ... .. ...222.. . . . .. . .Total A B AB S A B S Aijk j j k kjk jk j j k ki j ijk jk i kSS SS SS SS SS SSY Y n Y Y n Y Yn Y Y n Y Y n Y Yj Y Y Y Y j Y Y The total variability can be partitioned into A, B, AB, S/A, and B*S/A22 2 2. . ...22 2 2.. ...222. ... . . ... .. ...2. ...15*[ 4.2 3.244 2.8 3.244 2.733 3.244 ] 20.58315*[ 3.333 3.244 3.333 3.244 3.067 3.244 ] .7085A j jB k kAB jk jk j j k kjk jkSS n Y YSS n Y YSS n Y Y n Y Y n Y Yn Y Y2 2 22 2 22 2 2*[ 2.4 3.244 3.8 3.244 6.4 3.2444 3.244 3 3.244 1.4 3.2443.6 3.244 3.2 3.244 1.4 3.244 ] 92.71192.711 20.583 .708 71.420ABSS22 2 2/ .. . .2 2 2 2 22 2 2 2 2223*[ 3.333 4.2 4.333 4.2 4 4.25.667 4.2 3.667 4.2 3.333 2.8 1.333 2.8 3.333 2.83.333 2.8 2 2.8 2 2.733 3 2.733 3 2.7333 2.733 2.667 2.733 ] 26.400S A i jSS k Y Y22* / . .. . .22 2 2 2 2.2 2 2 2 22 2 2 2 22 2 2 2 22 2 21 2.4 1 2.4 3 2.4 5 2.4 2 2.43 3.8 4 3.8 3 3.8 5 3.8 4 3.86 6.4 8 6.4 6 6.4 7 6.4 5 6.43 4 4 4 5 4 4 4 4 41 3 4 3 3 3B S A ijk jk i jijk jkSS Y Y k Y YYY222 2 2 2 22 2 2 2 22 2 2 2 22 2 2 2 2*/2 3 5 30 1.4 2 1.4 2 1.4 0 1.4 3 1.44 3.6 2 3.6 3 3.6 6 3.6 3 3.62 3.2 6 3.2 3 3.2 2 3.2 3 3.20 1.4 1 1.4 3 1.4 1 1.4 2 1.4 63.663.6 26.4 37.2B S ASS2...2 2 2 2 22 2 2 2 22 2 2 2 22 2 21 3.244 1 3.244 3 3.244 5 3.244 2 3.2443 3.244 4 3.244 3 3.244 5 3.244 4 3.2446 3.244 8 3.244 6 3.244 7 3.244 5 3.2443 3.244 4 3.244 5 3.244 4 3.244Total ijkTotalSS Y YSS222 2 2 2 22 2 2 2 22 2 2 2 22 2 2 2 24 3.2441 3.244 4 3.244 3 3.244 2 3.244 5 3.2440 3.244 2 3.244 2 3.244 0 3.244 3 3.2444 3.244 2 3.244 3 3.244 6 3.244 3 3.2442 3.244 6 3.244 3 3.244 2 3.244 3 3.2440 3.2 2 2 2 2244 1 3.244 3 3.244 1 3.244 2 3.244 156.311b1: Month 1b2: Month 2b3: Month 3S11 3 6 S1 = 10S21 4 8 S2 = 13S33 3 6 S3 = 12S45 5 7 S4 = 17S52 4 5 S5 = 11a1b1 = 12 a1b2 = 19 a1b3 = 32 a1 = 63S63 1 0 S6 = 4S74 4 2 S7 = 10S85 3 2 S8 = 10S94 2 0 S9 = 6S104 5 3 S10 = 12a2b1 = 20 a2b2 = 15 a2b3 = 7 a2 = 42S114 2 0 S11 = 6S122 6 1 S12 = 9S133 3 3 S13 = 9S146 2 1 S14 = 9S153 3 2 S15 = 8a3b1 = 18 a3b2 = 16 a3b3 = 7 a3 = 41b1 = 50 b2 = 50 b3 = 46 Total = 146B: MonthCase Total A: Type of Novela1: Science Fictiona2: Mysterya3: RomanceSums of Squares - Computational What are the degrees of freedom? And convert them into the formulas▫ A = a - 1▫ S/A = a(s – 1) = as - a▫ B = b - 1▫ AB = (a – 1)(b – 1)▫ BxS/A = a(b – 1)(s – 1) ▫ …