Mixed Designs: Between And Within - California State University, Northridge
Mixed Designs: Between and Within Psy 420 Ainsworth
Mixed Between and Within Designs Conceptualizing the Design Types of Mixed Designs Assumptions Analysis Deviation Computation Higher order mixed designs Breaking down significant effects
Conceptualizing 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 have
Conceptualizing 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 - after S1 S1 Kaplan Princeton S2 S2 S3 S3 S4 S4 S5 S5 S6 S6 S7 S7 S8 S8 S9 S9 S10 S10
Conceptualizing 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 complexities
Conceptualizing the Design Treatment Group Control Group Pretest Posttest S1 S1 S2 S2 S3 Treatment S3 S4 S4 S5 S5 S6 S6 S7 S7 S8 No Treatment S8 S9 S9 S10 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 effects
Assumptions 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-orthogonality
Analysis Within Groups b1 b2 b3 Randomized Groups a1 a2 a3 S1 S1 S1 S2 S2 S2 S3 S3 S3 S4 S4 S4 S5 S5 S5 S6 S6 S6 S7 S7 S7 S8 S8 S8 S9 S9 S9 S10 S10 S10 S11 S11 S11 S12 S12 S12 S13 S13 S13 S14 S14 S14 S15 S15 S15
Sources 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 months
S1 B: Month b1: b2: b3: Month 1 Month 2 Month 3 Case Means 1 3 6 S1 3.333 S2 1 4 8 S2 4.333 a1: Science Fiction S3 3 3 6 S3 4 S4 5 5 7 S4 5.667 S5 2 4 5 S5 3.667 A: Type of Novel a1b1 2.4 a1b2 3.8 a1b3 6.4 a2: Mystery a3: Romance a1 4.2 S6 3 1 0 S6 1.333 S7 4 4 2 S7 3.333 S8 5 3 2 S8 3.333 S9 4 2 0 S9 2 S10 4 5 3 S10 4 a2 a2b1 4 S11 4 2 0 S11 2 S12 2 6 1 S12 3 S13 3 3 3 S13 3 S14 6 2 1 S14 3 S15 3 3 2 S15 2.667 a2b2 3 a2b3 1.4 a3b1 3.6 a3b2 3.2 a3b3 1.4 a2 2.8 a3 2.733 b1 3.333 b2 3.333 b3 3.067 GM 3.244
Sums of Squares - Deviation The total variability can be partitioned into A, B, AB, S/A, and B*S/A SSTotal SS A SS B SS AB SS S / A SS B*S / A Yijk Y. 2 n j Y. j. Y. n jk Y. jk Y. j Yi. Y. j. 2 2 2 nk Y.k Y. n j Y. j. Y. Yijk Y. jk 2 2 j 2 nk Y.k Y. Yi. Y.k . 2 2
SS A SS B SS AB n j Y. j . Y. nk Y.k Y. 2 2 n jk Y. jk Y. n jk Y. jk Y. 4 3.244 2 3.6 3.244 2 15*[ 4.2 3.244 15*[ 3.333 3.244 2 n j Y. j . Y. 5 *[ 2.4 3.244 3 3.244 2 2 2 3.2 3.244 2 2 2 nk Y.k Y. 2 2 2.733 3.244 ] 20.583 2 2 2 2 3.067 3.244 ] .708 2 6.4 3.244 1.4 3.244 ] 92.711 SS AB 92.711 20.583 .708 71.420 2 3.333 3.244 3.8 3.244 1.4 3.244 2 2.8 3.244 2
SS S / A k Yi. Y. j . 5.667 4.2 3.333 2.8 3 2.733 2 2 2 2 3*[ 3.333 4.2 3.667 4.2 2 2.8 2 2 4.333 4.2 3.333 2.8 2 2.733 2 2 2 2.667 2.733 ] 26.400 2 2 1.333 2.8 3 2.733 2 2 4 4.2 2 3.333 2.8 3 2.733 2 2
SS B*S / A Yijk Y. jk Yijk Y. jk 3 3.8 6 6.4 3 4 1 3 2 2 4 3.6 2 3.2 0 1.4 1 2.4 4 3.8 4 3 2 2 2 2 2 2 8 6.4 4 4 2 0 1.4 SS B*S / A 2 2 2 2 2 1.4 k 2 2 2 6 3.2 2 2 2 2 1.4 3 3.6 3 3.2 3 1.4 63.6 26.4 37.2 5 3.8 2 2 2 2 2 2 2 0 1.4 6 3.6 2 3.2 1 1.4 2 2 2 2 2 2 5 6.4 4 4 5 3 5 2.4 4 3.8 2 7 6.4 4 4 2 3 2 3 2.4 2 6 6.4 2 2 2 3 3.8 3 3 2 Yi. Y. j . 1 2.4 5 4 2 3.6 1 1.4 2 2 2 2 2 3 1.4 3 3.6 3 3.2 2 1.4 2 2 2 63.6 2 2.4 2
SSTotal SSTotal Yijk Y. 1 3.244 3 3.244 6 3.244 3 3.244 1 3.244 0 3.244 4 3.244 2 3.244 0 3.244 2 2 2 2 2 2 2 2 2 2 1 3.244 4 3.244 8 3.244 4 3.244 4 3.244 2 3.244 2 3.244 6 3.244 1 3.244 2 2 2 2 2 2 2 2 2 3 3.244 3 3.244 6 3.244 5 3.244 3 3.244 2 3.244 3 3.244 3 3.244 3 3.244 2 2 2 2 2 2 2 2 2 5 3.244 2 5 3.244 2 7 3.244 2 4 3.244 2 2 3.244 0 3.244 2 6 3.244 2 2 3.244 1 3.244 2 2 2 2 3.244 4 3.244 2 2 5 3.244 2 4 3.244 5 3.244 2 2 3 3.244 2 3 3.244 2 3 3.244 2 3.244 2 2 156.311
S1 B: Month b1: b2: b3: Month 1 Month 2 Month 3 Case Total 1 3 6 S1 10 S2 1 4 8 S2 13 a1: Science Fiction S3 3 3 6 S3 12 S4 5 5 7 S4 17 S5 2 4 5 S5 11 A: Type of Novel a1b1 12 a1b2 19 a1b3 32 a2: Mystery S6 3 1 0 S6 4 S7 4 4 2 S7 10 S8 5 3 2 S8 10 S9 4 2 0 S9 6 S10 4 5 3 S10 12 a2b1 20 a2b2 15 a3: Romance a1 63 a2b3 7 a2 42 S11 4 2 0 S11 6 S12 2 6 1 S12 9 S13 3 3 3 S13 9 S14 6 2 1 S14 9 S15 3 3 2 S15 8 a3b1 18 a3b2 16 b1 50 b2 50 a3b3 7 a3 41 b3 46 Total 146
Sums 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) T abs – 1 or N - 1
df A a 1 3 1 2 df S / A df B a ( s 1) 3(5 1) 12 b 1 3 1 2 df AB (a 1)(b 1) (3 1)(3 1) df BxS / A dfT 4 a (b 1)( s 1) 3(3 1)(5 1) abs 1 N 1 3(3)(5) 1 44 24
Results – ANOVA summary table
Higher order mixed designs
Breaking down significant effects Between Groups IV(s) If you have a significant BG main effect(s) they need to be broken down to find which levels are different The comparisons are done the same way as completely BG comparisons The BG comparison error term is the same for all BG comparisons F nY ( 2 w jY j ) / MS S / AB w 2 j SS( reg . X j ) MS S / AB
Breaking down significant effects Within Groups Variables If a WG main effect is significant it also needs to be followed by comparisons WG comparisons differ from BG variables in that a separate error term needs to be generated for each comparison Instead of the Fcomp formula you would actually rearrange the data into a new data set
Breaking down significant effects Example
Breaking down significant effects B b1 b3 S1 1 6 S1 7 S2 1 8 S2 9 a1 S3 3 6 S3 9 S4 5 7 S4 12 S5 S6 2 5 S5 7 a1b1 12 a1b3 32 a1 44 3 0 S6 3 S7 4 2 S7 6 a2 S8 5 2 S8 7 S9 4 0 S9 4 S10 S11 4 3 S10 7 a2b1 20 a2b3 7 a2 27 4 0 S11 4 S12 2 1 S12 3 a3 S13 3 3 S13 6 S14 6 1 S14 7 S15 3 2 S15 5 a3b1 18 a3b3 7 a3 25 T 96
Breaking down significant effects Interactions Purely BG interactions can be treated with simple effects, simple contrasts and interaction contrasts using the Fcomp formula, the same error term each time Purely WG and mixed BG/WG interactions require a new error term for each simple effect, simple contrast and interaction contrast (leave it to SPSS)
Mixed Between and Within Designs Conceptualizing the Design Types of Mixed Designs Assumptions Analysis Deviation Computation Higher order mixed designs Breaking down significant effects. Conceptualizing the Design This is a very popular design because you are
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1. Introduction The lme4 package (Bates, Maechler, Bolker, and Walker 2014a) for R (R Core Team 2015) provides functions to fit and analyze linear mixed models, generalized linear mixed models and nonlinear mixed models. In each of these names, the term “mixed” or, more fully, “mixed
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