1. ANOVA Overview of Major Designs

2. Between or Within Subjects Between-subjects (completely randomized) designs –Subjects are nested within treatment conditions –By nested we mean that subjects are observed under only a single condition of the study Within-subjects (randomized block) designs –Subjects are crossed by treatment conditions –By crossed we mean that subjects are observed under two or more conditions of the study

3. When to Use Repeated Measures Within-subjects designs are an advantage when 1.Scores under one condition are correlated with scores under another condition 2.When examining the effects of practice on performance of a learning task, or the effects of age in a longitudinal study of development 3.That in which a series of tests or subtests is to be administered to a group of subjects

4. Randomized Block ANOVA SourceSSdfMSFeta p Between subjects 13.35 3 Within subjects Sessions 60.67 230.334.00.98<.001 Error term 2.65 6.44 Total 76.67 11

5. Fixed Effects Fixed factors are those in which we have selected particular levels of the factor in question not by random sampling but on the basis of our interest in those particular effects. –Cannot view these levels as representative –Cannot generalize to other levels –Examples: most manipulated variables, organismic variables, time, sessions, subtests

6. Random Effects Random factors are those in which we view the levels of the factor as having been randomly sampled from a larger population of such levels. –The most common random factor is subjects. –If subjects are not randomized we cannot generalize to others

7. Error Terms in Four Designs The appropriate choice of an error term in a repeated measures design depends on the fixed and random effects of within sampling units and between sampling units. The effects (fixed or random) we want to test are properly tested by dividing the MS for that effect by the MS for a random source of variation.

8. Aggregating Error Terms When the number of df per error term is small, insignificant interactions can be aggregated with the error term to produce a pooled error term with more df. Once we compute an aggregated (pooled) error term, it replaces all the individual error terms that contributed to its computation.

9. Assumptions 1.Independence of errors 2.Normality 3.Homogeneity of variance including the sphericity assumption (homogeneity-of- variance-of-differences)

10. What’s in a Name? Choosing the appropriate statistic or design involves an understanding of –The number of independent variables and levels –The nature of assignment of subjects to treatment levels –The number of dependent variables The source table for an analysis of variance describes the partition of the total sum of squares

11. Between Subjects Completely Randomized ANOVA One independent variable with two or more levels Subjects completely randomly assigned to treatment levels Also called One-Way ANOVA

12. Completely Randomized Analysis of Variance SourceSSdfMSFetap Between conditions57319.07.60.86.01 Within conditions2082.5 Total77117.0

13. Within Subjects Randomized Block ANOVA One independent variable with two or more levels Uses repeated measures of matching Also called –One-Way with Repeated Measures ANOVA

14. Randomized Block Analysis of Variance SourceSSdfMSFeta p Between subjects 13.35 3 Within subjects A 60.67 230.334.00.98<.001 Error term 2.65 6.44 Total 76.67 11

15. Between Subjects Completely Randomized Factorial ANOVA Two or more independent variables each with two or more levels Subjects are completely randomly assigned to all treatment combinations Also called –Two-Way or Higher Order Analysis of Variance

16. Two-Way Analysis of Variance SourceSSdfMSFetap Between conditions57319.0 7.60.86.01 Treatment A27127.010.80.76.01 Treatment B27127.010.80.76.01 Interaction AB 31 3.0 1.20.36.30 Within conditions208 2.5 Total7711 7.0

17. Within Subjects Randomized Block Factorial Analysis of Variance Two or more independent variables each with two or more levels All treatment combinations use repeated measures or matching. Also called –Two-Way or Higher Order Repeated Measures Analysis of Variance

18. Randomized Block Factorial Analysis of Variance SourceSSdfMSFeta p Between subjects4.93 1 Within subjects A.523 1 15.8.27<.001 B1.174 2.58717.8.39<.001 AB.193 2.009622.92.09<.001 Error term1.78 54.00329 Total8.60 60

19. Mixed Analysis of Variance Split-Plot Factorial Two or more independent variables each with two or more levels At least one variable is completely randomized (between subjects) At least one variable is randomized block (within subjects).

20. Split Plot Factorials - Mixed Designs SourceSSdfMSFeta p Between subjects 729 A 40140.0010.00.75.013 Error 3284.00 Within subjects 18830 B 90190.040.00.91.0002 AB 010.00.001.00 Error 1882.25 C 40140.0020.00.85.002 AC 010.00.001.00 Error 1682.00 BC 10110.005.71.65.043 ABC 010.00 1.00 Error 1481.75 Total 26039