ANOVA Overview of Major Designs

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

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

Randomized Block ANOVA SourceSSdfMSFeta p Between subjects Within subjects Sessions <.001 Error term Total

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

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

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.

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.

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

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

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

Completely Randomized Analysis of Variance SourceSSdfMSFetap Between conditions Within conditions Total

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

Randomized Block Analysis of Variance SourceSSdfMSFeta p Between subjects Within subjects A <.001 Error term Total

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

Two-Way Analysis of Variance SourceSSdfMSFetap Between conditions Treatment A Treatment B Interaction AB Within conditions Total

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

Randomized Block Factorial Analysis of Variance SourceSSdfMSFeta p Between subjects Within subjects A <.001 B <.001 AB <.001 Error term Total

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).

Split Plot Factorials - Mixed Designs SourceSSdfMSFeta p Between subjects 729 A Error Within subjects B AB Error C AC Error BC ABC Error Total 26039