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Repeated Measures ANOVA

Repeated Measures ANOVA. Between Subjects Variance Is Partitioned From the Error Variance Conceptual Approach Using The Source Table. Total Variability. Effect Variability (MS Between). Error Variability (MS Within). The Source Table. Keeps track of all data in complex ANOVA designs

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Repeated Measures ANOVA

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  1. Repeated Measures ANOVA • Between Subjects Variance Is Partitioned From the Error Variance • Conceptual Approach • Using The Source Table Anthony Greene

  2. Total Variability Effect Variability(MS Between) ErrorVariability(MS Within) The Source Table • Keeps track of all data in complex ANOVA designs • Source of SS, df, and Variance (MS) • Partitioning the SS, df and MS • All variability is attributable toeffect differences or error (all unexplained differences) Anthony Greene

  3. Source Table for 1-Way ANOVA Anthony Greene

  4. Source Table for 1-Way ANOVA Effect Variability Error Variability

  5. Repeated Measures ANOVA Treatment 1 Treatment 2 Treatment 3 Measurement 1 Measurement 2 Measurement 3 M1 M3 M2 Anthony Greene

  6. An Example of a Repeated Measures Design

  7. Repeated Measures ANOVA • Recall the differences between a t-test a tD-test: • The same group is used in the experimental and control group via the before-after design • Thereby individual differences are removed from the error term • Identical concepts to repeated measures ANOVA • Individual Variation is computed and subtracted from the error term Anthony Greene

  8. Total Variability { Stage 1 Effect Variability(MS Between) MS Within { Individual Variability ErrorVariability Stage 2 Partitioning Variability For Repeated Measures ANOVA Anthony Greene

  9. Total Variability { Stage 1 Effect Variability(MS Between) MS Within Numerator of F-ratio { Individual Variability ErrorVariability Stage 2 Denominator of F-ratio Partitioning Variability For Repeated Measures ANOVA

  10. Source Table For Repeated Measures ANOVA Tests of Within-Subjects Effects Source SS df MS F p FACTOR1 30.0 3 10.0 6.3 .005 Error 23.5 15 1.5 Subj. 72.5 5 Within 96.0 20 Total 126.0 23 Source SS df MS F p FACTOR1 30.0 3 10.0 2.1 n.s. Error 23.5 15 Subj. 72.5 5 Within 96.0 20 4.8 Total 126.0 23 Anthony Greene

  11. Source Table For Repeated Measures ANOVA Tests of Within-Subjects Effects Source SS df MS F Sig. FACTOR1 30.0 3 10.0 6.3 .005 Error 23.5 15 1.5 Subj. 72.5 5 Within 96.0 20 Total 126.0 23 Anthony Greene

  12. Source Table For Repeated Measures ANOVA Tests of Within-Subjects Effects Source SS df MS F Sig. FACTOR1 30.0 3 10.0 6.3 .005 Error 23.5 15 1.5 Subj. 72.5 5 Within 96.0 20 Total 126.0 23 Anthony Greene

  13. Source Table For Repeated Measures ANOVA Tests of Within-Subjects Effects Source SS df MS F Sig. FACTOR1 30.0 3 10.0 6.3 .005 Error 23.5 15 1.5 Subj. 72.5 5 Within 96.0 20 Total 126.0 23 Anthony Greene

  14. Source Table For Repeated Measures ANOVA Tests of Within-Subjects Effects Source SS df MS F Sig. FACTOR1 30.0 3 10.0 6.3 .005 Error 23.5 15 1.5 Subj 72.5 5 Within 96.0 20 Total 126.0 23 Tests of Within-Subjects Contrasts Source SS df MS F Sig. Linear 4.8 1 4.8 1.5 .263 Quadratic 24.0 1 24.0 18.4 .008 Cubic 1.2 1 1.2 3.1 .136 Anthony Greene

  15. Doing the Computations

  16. Total Variability { Stage 1 Effect Variability(MS Between) MS Within { Individual Variability ErrorVariability Stage 2 • New computation: • Compute SS for P and deg freedom • Subtract those from SS w/in and d.f. w/in • This leaves SS error and d.f. error Anthony Greene

  17. Tests of Within-Subjects Effects Source SS df MS F Sig. FACTOR1 30.0 3 Error Subj. STAGE 1 Within 96.0 20 Total 126.0 23 Source SS df MS F Sig. FACTOR1 30.0 3 Error Subj. 72.5 5 STAGE 2 Within 96.0 20 Total 126.0 23 Source SS df MS F Sig. FACTOR1 30.0 3 10.0 6.3 .005 Error 23.5 15 1.5 Subj. 72.5 5 Within 96.0 20 Total 126.0 23 Anthony Greene

  18. Repeated Measures ANOVA Source d.f. SS MS F Between 11.67 Within 12 Subject 3 3 Residual Total 15 51 Anthony Greene

  19. Repeated Measures ANOVA Source d.f. SS MS F Between 3 11.67 Within 12 Subject 3 3 Residual Total 15 51 Anthony Greene

  20. Repeated Measures ANOVA Source d.f. SS MS F Between 3 35 11.67 Within 12 Subject 3 3 Residual Total 15 51 Anthony Greene

  21. Repeated Measures ANOVA Source d.f. SS MS F Between 3 35 11.67 Within 12 16 Subject 3 3 Residual Total 15 51 Anthony Greene

  22. Repeated Measures ANOVA Source d.f. SS MS F Between 3 35 11.67 Within 12 16 Subject 3 3 Residual 9 Total 15 51 Anthony Greene

  23. Repeated Measures ANOVA Source d.f. SS MS F Between 3 35 11.67 Within 12 16 Subject 3 3 Residual 9 13 Total 15 51 Anthony Greene

  24. Repeated Measures ANOVA Source d.f. SS MS F Between 3 35 11.67 Within 12 16 Subject 3 3 Residual 9 13 1.44 Total 15 51 Anthony Greene

  25. Repeated Measures ANOVA Source d.f. SS MS F Between 3 35 11.67 9.0 Within 12 16 Subject 3 3 Residual 9 13 1.44 Total 15 51 Anthony Greene

  26. Repeated-Measures ANOVA An experimenter wishes to examine whether a certain therapy for Attention Deficit Disorder is effective. Four ADD children, 12 years old, are selected. The dependent measure is a standard test of attention wherein the child hears a narrative and is then asked questions about its content Anthony Greene

  27. Repeated-Measures ANOVA Compute the treatment totals, T, and the grand total, G Anthony Greene

  28. Repeated-Measures ANOVA Count n for each treatment, the total N, and k Anthony Greene

  29. Repeated-Measures ANOVA Compute the treatment means Anthony Greene

  30. Repeated-Measures ANOVA (5-3)2=4 (2-3)2=1 (4-3)2=1 (1-3)2=4 sum Compute the treatment SSs Anthony Greene

  31. Repeated-Measures ANOVA Compute all X2s and sum them Anthony Greene

  32. Repeated-Measures ANOVA Add in Person totals column Anthony Greene

  33. Repeated-Measures ANOVA Compute SSTotal SSTotal= X2 – G2/N SSTotal=62 Anthony Greene

  34. Repeated-Measures ANOVA Compute SSbetween SSbetween= T2/n – G2/N SSbetween=8 Anthony Greene

  35. Repeated-Measures ANOVA Compute SSWithin SSWithin= SSi SSWithin = 54 Anthony Greene

  36. Repeated-Measures ANOVA Compute SSBtw sub SSBtw Sub= P2/k – G2/N SSBtw Sub=22.66 SSerror = SSwithin-SSbtw sub = 54-22.66 = 31.34 Anthony Greene

  37. Repeated-Measures ANOVA Determine d.f.s d.f. Within=N-k d.f. Between=k-1 d.f. Total=N-1 d.f. error = (N-k) – (n-1) Anthony Greene

  38. Repeated-Measures ANOVA Ready to move it to a source table Anthony Greene

  39. Repeated Measures ANOVA Source SS d.f. MS F Between 8 2 4 0.766 Within 54 9 Subject 22.66 3 Error 31.34 6 5.22 Total 62 11 Compare to critical value of F(2,6) = 5.14 Fail to reject (no significant difference) Anthony Greene

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