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 when1. Scores under one condition are correlated with

scores under another condition2. 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

Source SS df MS F eta p

Between subjects 13.35 3Within subjects Sessions 60.67 2 30.33 4.00 .98 <.001 Error term 2.65 6 .44Total 76.67 11

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 errors2. Normality3. 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

Source SS df MS F eta p

Between conditions 57 3 19.0 7.60 .86 .01

Within conditions 20 8 2.5

Total 77 11 7.0

Within SubjectsRandomized Block ANOVA

• One independent variable with two or more levels

• Uses repeated measures of matching• Also called

– One-Way with Repeated Measures ANOVA

Randomized BlockAnalysis of Variance

Source SS df MS F eta p

Between subjects 13.35 3Within subjects A 60.67 2 30.33 4.00 .98 <.001 Error term 2.65 6 .44Total 76.67 11

Between SubjectsCompletely 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

Source SS df MS F eta p

Between conditions 57 3 19.0 7.60 .86 .01 Treatment A 27 1 27.0 10.80 .76 .01 Treatment B 27 1 27.0 10.80 .76 .01 Interaction AB 3 1 3.0 1.20 .36 .30Within conditions 20 8 2.5Total 77 11 7.0

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

Source SS df MS F eta pBetween subjects 4.93 1Within subjects A .523 1 .523 15.8 .27 <.001 B 1.174 2 .587 17.8 .39 <.001

AB .193 2 .00962 2.92 .09 <.001

Error term 1.78 54 .00329

Total 8.60 60

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

Source SS df MS F eta pBetween subjects 72 9 A 40 1 40.00 10.00 .75 .013 Error 32 8 4.00Within subjects 188 30 B 90 1 90.0 40.00 .91 .0002 AB 0 1 0.00 0.00 .00 1.00 Error 18 8 2.25 C 40 1 40.00 20.00 .85 .002 AC 0 1 0.00 0.00 .00 1.00 Error 16 8 2.00 BC 10 1 10.00 5.71 .65 .043 ABC 0 1 0.00 0.00 0.00 1.00 Error 14 8 1.75Total 260 39