**Factorial Analysis of Variance** 46-511 Between Groups Fixed Effects Designs

**Two-Way ANOVA Example:(Yerkes – Dodson Law)** Factor B: Arousal Factor A: Task Difficulty

**Partitioning Variance** Factor B: Arousal Variation among means on A represent effect of A Factor A: Task Difficulty Variation among people treated the same = error Variation among means on B represents effect of B Leftover variation = interaction

**Partitioning Variance: Interaction** Factor B: Arousal Factor A: Task Difficulty Dependence of means on levels of both A & B represents the effect of an interaction.

**Or Graphically…**

**In words** • Types of Effects vs. 1-way • Main Effect for A • Main Effect for B • Interaction (A x B) • Structural Model: XIJK = μ++++IJK • Partitioning Variance/Sums of Squares • First, total variance: • Between Groups: • Thus Total is:

**Sums of Squares Between** Definitional Formula Variation of cell means around grand mean, weighted by n. Computational Formula • Computational formulae: • More accurate for hand calculation • Easier to work • Less intuitive

**Sums of Squares A** Definitional Formula Variation of row means around grand mean, weighted by n times the number of levels of B, or q. Computational Formula

**Sums of Squares B** Definitional Formula Variation of column means around grand mean, weighted by n times the number of levels of A, or p. Computational Formula

**Sums of Squares AxB** Definitional Formula Computational Formula SSAxB = Variation of cell means around grand mean, that cannot be accounted for by effects of A or B alone.

**Sums of Squares Within (Error)** Definitional Formula Computational Formula SSW = Variation of individual scores around cell mean.

**Numerical Example**

**Degrees of Freedom** • df between = k – 1; or, (kA x kB – 1) • df A = kA – 1 • df B = kB – 1 • df A x B = dfbetween – dfA – dfB • dfW = k(n-1)

**Source Table**

**More Digression on Interactions** • Ways to talk about interactions • Scores on the DV depend upon levels of both A and B • The effect of A is moderated by B • The effect of B is moderated by A • There is a multiplicative effect for A and B

**More Digresions (cont’d)No effect whatsoever…**

**Main effects for A and B…**

**Graphically…**

**Interaction significant also…**

**Graphically…**

**Further Analyses on Main Effects** • Contrasts • Planned Comparisons • Post-Hoc Methods • In the presence of a significant interaction

**Further Analyses on Interaction** • What it means • Simple (Main) Effects • Contrasts • Partial Interactions • Contrasts • Simple Comparisons / Post-Hoc Methods • How to get q

**Simple Main Effects Analysis**

**Simple Main Effects** Sum of Squares Formula: F Ratio: df = dfj,dfw:

**Partial Interaction Analysis**

**In Class Exercise**

**Based on two pieces of information** 1)

**Compute simple main effects** 2)

**Effects** A B C A x B A x C B x C A x B x C A Vague Example DV = Treatment Outcome Factor A: Gender Factor B: Age (14 or 17) Factor C: Treatment 3-Way ANOVA

**Results**

**Significant Two-Way Interaction**

**Significant Three-Way Interaction**

**Other Stuff** • Higher order models (4-way, 5-way, etc.) • Unequal Cell Sizes and SS Type • Use of contrast coefficients • Short-Cuts using SPSS • Custom Models in SPSS • Observed Power