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1. Advanced Research Methods I11/17/08 ANALYSIS: Repeated Measure ANOVA

2. REVIEW Representation of Linear Effects in ANOVA • One-Factor ANOVA: Score of a subject i within group j: Xij = μ + βj + εij Recall: Partitioning of score variations Xij – M = (X – mj) + (mj – M) Within group Between Group  Xij = M + (mj – M) + (X – mj) Overall Mean Group (Treatment) Effect Error (due to factors not accounted by the model)

3. REVIEW Representation of Linear Effects in ANOVA • Factorial ANOVA (Two Factor): Score of a subject i within cell jk (belonging to row j and column k representing factors B and A, respectively) Xijk = μ + βj + αk + γjk + εijk Partitioning of Score Variation: Xijk = M + ( mj – M) + (mk – M) + (mjk – mj – mk + M) + (Xijk – mjk) Error (due to factors not included in the model) Effect due to Interaction between A and B Effect due to Factor B (Row) Effect due to Factor A (Column) Overall Mean

4. Review: Factorial ANOVAEstimates of Variance Components SS df MS Variances Estimated by the MS Row (B) R – 1 SS/df Column (A) C – 1 A x B (R – 1)(C – 1) Within nRC – RC Var(ε) Total nRC - 1

5. The Within-Group Error Variance Var(ε) • Represents Factors Unexplained by the Model (Individual differences on the Dependent Variables) • Determines Power of the F-test; so reducing Var(ε)  Improving Power • How?  By Including factors that are theoretically expected to “explain” (account for) the Var(ε): • ε = θ + ε’  Var(ε ) = Var(θ)* + Var(ε’ )  Var(ε’ ) < Var(ε) • One special way to address the issue: Including measures of the dependent variable ε = ρ + ε’  Var(ε ) = Var(ρ)* + Var(ε’ )  Subjects (people) as a factor in ANOVA *This variance component may also include variances due to interaction effects

6. Repeated Measure ANOVA As Two-Way Factorial ANOVA T1 T2 T1 T2 x11 x11 x12 x21 x21 x22 x31 x31 x32 x42 x52 x62 Repeated measure t-test or Two-factor ANOVA (Factor A includes 2 levels and Factor B, being subjects, include 3 levels) or Repeated measure ANOVA Independent Sample t-test or One-way ANOVA (Factor A includes 2 levels: T1, T2). n = 3; k = 2

7. Repeated Measure ANOVA As Two-Way Factorial ANOVA T1 T2 T3 T1 T2 T3 x11 x11 x12 x13 x21 x21 x22 x23 x31 x31 x32 x33 x42 x52 x62 x73 x83 x93 Two-Factor ANOVA with Subjects as the second factor or Repeated-Measure ANOVA with three levels of Factor A One-Factor ANOVA (Factor A includes 3 levels: T1, T2, T3). n = 3

8. Repeated Measures ANOVA • Designs that include subjects as a factor • Also called: Within-subject designs • Research Design (single factor repeated measures) X1 O1 X2 O2 X3 O3 ……. • If there are only two levels (X1 and X2 corresponding to O1 and O2)  T-test for repeated measures (matched-samples T-test) • Advantages: - Very powerful thanks to smaller within-group/cell error variance (Var(ε)) - Efficient use of sample  maximizing sample size. • Disadvantages: - Order Effect (Time as a confounding factor)  Solution randomizing subjects on time - Strong assumptions (sphericity – discussed later)

9. Repeated Measures ANOVA An Example Treatment & Time Confounded Randomizing Time-Treatment to eliminate the problem

10. Repeated Measures ANOVA as Two-Factor ANOVA Score of a subject i within level k of factor A (belonging to row i and column k representing the subject factor and factor A, respectively) Xik = μ + i+ αk + γik + εik Partitioning of Score Variation: Xik = M + ( mi – M) + (mk – M) + (mik – mi – mk + M) + (Xik – mik) Pre-existing individual differences on the DV Measure (Row) Effect due to Interaction between subjects and treatment Effect due to Factor A (Column) Random Error Overall Mean n=1  Xik = mik  these two sources of variation cannot be partitioned

11. Repeated Measures ANOVAEstimates of Variance Components SS df MS Variances Estimated by the MS Row (Subjects) n – 1 SS/df Column (A) C – 1 Subjects*A (n – 1)(C – 1) Within Confounded with Interaction (Subject*A) Total nC - 1 Note: Compared to two-factor ANOVA: i = j, n= 1, R = n here.

12. Repeated Measures ANOVAEstimates of Variance Components df Variances Estimated F by the MS Row (Subjects) n – 1 MSRow /MSSxA Column (A) C – 1 MSColumn /MSSxA Subjects*A (n – 1)(C – 1) Within Confounded with A x B 0 If there is no subject-treatment interaction (γik=0)  MSSxA estimates Var(ε)  Use it as Within-Cell error variance to compute the F-ratios

13. An Example Example 1: Page 1169 in Neter et al. (Additional Reading) – Open the Excel File attached

14. Assumptions • Normal distribution of scores under each treatment • Independence of scores within treatment • No subject by Treatment interaction • Sphericity

15. Understanding the Assumptions • Sphericity: Equal variance of difference scores between any two treatment conditions: For all k1 ,k2 ,k3 ,k4 (k1 ,k2 ,k3 ,k4 = 1, 2, 3, 4…..): Var(Xk1- Xk2) = Var(Xk1- Xk3) = Var(Xk1- Xk4) = Var(Xk2- Xk3)……

16. When the Sphericity Assumption Is Not Met • Adjusted degree of freedom  critical value (Huynh-Feldt or Greenhouse-Geisser) • Multivariate Approach Effect Size: Partial Eta Square

17. Two Within-Subject Factor ANOVA • Designs: e.g. Factors A (2 levels), B (3 levels) A1B1 A1B2 A1B3 A2B1 A2B2 A2B3 O1 x O2 x O3 x O4 x O5 x O6 In general: • Factor A has a levels; • Factor B has b levels; • Sample size = n; • Mean of subject i (across all levels of A and B): mi • Mean of level j of factor B (across all subjects and levels of A): mj • Mean of level k of factor A (across all subjects and levels of B): mk • Mean of cell jk (across all subjects in the cell): mjk

18. Two Within-Subject Factor ANOVA SS df MS Estimated Variances Subjects n – 1 SS/df Factor A a – 1 Factor B b - 1 A x B (a – 1)(b – 1) Within (n – 1)(ab – 1) (Error)

19. One Between-Subject Factor and One Within-Subject Factor ANOVA • A is the within-subject factor with a levels • B is the between-subject factor with b levels • There are n subjects in each level of Factor B (total number of subjects = n*b)

20. One Between-Subject and one Within-Subject Factor ANOVA SS df MS Estimated Variances Subjects b(n -1) SS/df Factor A a – 1 Factor B b - 1 A x B (a – 1)(b – 1) Within b(n-1)(a-1) (Error)

21. One Between-Subject and one Within-Subject Factor ANOVA F-ratio: • FSubjects = MSSubjects / MSWithin • FFactorA = MSA / MSWithin • FFactor B = MSB / MSSubjects • FAB = MSAB / MSWithin

22. One Between-Subject Factor and One Within-Subject Factor ANOVA E.g. Treatment – Control, Pre-Post Measure O1 X O2 O3 O4 Within-subject Factor: Pre-post test (Time) Between-subject Factor: Treatment Interaction Effect  Treatment effect.

23. One Between-Subject Factor and One Within-Subject Factor ANOVA Treatment – Control, Pre-Post Measure • Can be seen as including pretest measure (x) as a factor • Within-subject factor : Testing the difference (d) between pre and post tests • Interaction: Testing the difference of d across the Between-subject factor (Treatment and Between groups). x1 y1 d1 = y1 – x1 x2 y2 d2 = y2 – x2 . Treatment 1 md1, Var(ed) xn yn xn+1 yn+1 . . Treatment 2 md2, Var(ed) x2n y2n d2n = y2n – x2n