Factorial ANOVA Notes

Factorial ANOVA

Factorial Designs

• Psychologists often investigate the effects of multiple independent variables (IVs) on a dependent variable (DV).
• Factorial Experimental Design: Research designs involving more than one IV.
  • Main Effect: The impact of one IV on the DV.
  • Interaction: The effect of one IV on the DV depends on the level of another IV.

Factorial Design Example

• Eskine, Kacinik, & Prinz, 2010:
  • Investigated whether moral judgments vary with the experience of gustatory disgust.
  • Examined if liberals and conservatives differ in making extreme moral judgments based on disgust.
• Method:
  • Participants identified as liberal or conservative.
  • Participants consumed a bitter beverage, a sweet beverage, or water.
  • They read scenarios depicting moral transgressions.
  • Participants rated the moral wrongness of each transgression.

2 x 3 Design

• Illustrates a design with two levels of one factor (e.g., Liberal/Conservative) and three levels of another (e.g., Bitter, Neutral/Water, Sweet).

Possible Outcomes

• Potential study outcomes:
  • Liberals and conservatives differ in moral judgments.
  • Extremity of moral judgments differs across taste conditions.
  • The impact of taste on moral judgments varies based on political ideology.
• The study could reveal one or more of these possibilities.

Main Effects and Interactions

• Factorial ANOVA is similar to One-way ANOVA but evaluates multiple effects.
• With two IVs:
  • Two main effects.
  • One interaction effect.
• Requires multiple Sum of Squares (SS) estimates and F-ratios.

Sum of Squares Total – SST

• Aims to explain the total variability in the outcome variable:
  • $SST = \sum(X - GMX)^2$

Sum of Squares Within (Error) – SSR

• Error term reflects the pooled variability within each group:
  • $SSR = \sum(Xi k - X k)^2$

Sum of Squares for the Model – SSM

• Systematic variation explained by the model represents the improvement over simply guessing the grand mean:
  • $SSM = \sum nk (X k - X GM)^2$

Sum of Squares for Main Effects – SSA and SSB

• The sum of squares for the model can be divided into the sum of squares for the main effects:
  • $SSA = \sum nk (X k - X GM)^2$
  • $SSB = \sum nk (X k - X GM)^2$

Sum of Squares for the Interaction – SSA x B

• The sum of squares for the model includes a component for the interaction:
  • $SSAxB = SSM - SSA - SSB$
  • Degrees of freedom for the interaction term: $df AxB = df A \times dfB$

Mean Squares and F-Ratios

• Formulas for Mean Squares (MS) and F-ratios:
  • $MSAxB = \frac{SSAxB}{df AxB}$
  • $MSA = \frac{SSA}{df A}$
  • $MSB = \frac{SSB}{df B}$
  • $MSR = \frac{SSR}{df R}$
  • $F = \frac{MSA}{MSR}$
  • $F = \frac{MSB}{MSR}$
  • $F = \frac{MSAxB}{MSR}$

Example Results: Descriptive Statistics

• Presents descriptive statistics for the extremity of moral judgment based on political orientation (Conservative, Liberal) and taste condition (Bitter, Water, Sweet).
• Includes Mean, Standard Deviation, and N for each group.

Example Results: Tests of Between-Subjects Effects

• Shows ANOVA results with:
  • Source, Type III Sum of Squares, df, Mean Square, F, Sig., Partial Eta Squared for Corrected Model, Intercept, ConLib (Political Orientation), Condition (Taste), ConLib*Condition (Interaction), and Error.
• Indicates the significance of main effects and interaction.

Example Results: Marginal Means

• Reports Estimated Marginal Means for:
  • Political Orientation (Conservative, Liberal).
  • Taste Condition (Bitter, Water, Sweet).
• Includes Mean, Std. Error, and 95% Confidence Intervals.

Factorial ANOVA – Interpreting and Evaluating Effects

• F ratios are omnibus tests.
• Significant Fs require follow-up with planned comparisons or post hoc tests.

Factorial ANOVA – Planned Comparisons for Main Effects

• Follow-up comparisons for main effects are similar to one-way ANOVA.
  • Logic and rules are identical.
  • Formulas are slightly different.
• If an IV has only two levels, planned comparisons are unnecessary; means can be directly compared.

Main Effect of A - Pairwise Main Comparisons

• $\psi A = X 1 - X 2$
  • $\psi$ is the same as before.
  • SSAcomp is the Sum of Squares A comparison.
  • n is the number of people in each CELL of the design (assumes equal n).
  • b is the number of levels of variable B.
  • The b would change to a and equal the number of levels of variable A if we were evaluating the main effect of B.
• SSAcomp = \frac{2(bn)(\psi A)^2}

Pairwise Main Comparisons (Continued)

• $MSAcomp = SScomp$
  • Note that, here, there is always 1 degree of freedom, so MScomp will always equal SScomp.
• $FAcomp = \frac{MSAcomp}{MSWithin}$
  • Note that we use the error term from our omnibus test here.

Main Effect of A – Complex Main Comparisons

• With coefficients, formulas change slightly.
• Incorporate the number of levels of B:
  • $\psi A = (c1)(X1)+ (c2)(X2 )+ (c3)(X3 )+…$
  • $SSAcomp = \frac{2(\sum c)^2}{(bn)}$

Interaction Effects

• Analyzing interaction effects is more complex.
• Requires a plan of attack.
• Goal: Fully explain the interaction in an interpretable way consistent with theory.
• Types of comparisons:
  • Simple Effects
  • Pairwise simple comparisons
  • Complex simple comparisons

Simple Effects

• The effect of one variable at a specific level of another.
  • E.g., the effectiveness of the taste manipulation for conservatives.
• Compute the sum of squares for the simple comparison:
  • $SS A \text{ comp at } bj = \sum nk (X k - X GM)^2$

Simple Effects (Continued)

• $MS A \text{ comp. at bj } = \frac{SS}{df}$
  • Degrees of freedom equal the number of groups minus 1 (i.e., k-1)
• F-ratio:
  • $F A \text{ comp. at bj } = \frac{MS}{MSWithin}$
  • Use the error term from the omnibus test.

Pairwise Simple Comparisons

• $\psi A = X 1 - X 2$
  • $SSAcomp = \frac{(\psi A)^2}{2n}$
  • Where n is the number of people in each cell you are comparing (assumes equal n).
• $FAcomp = \frac{MS}{MSWithin}$
  • We once again use the error term from our omnibus test.

Complex Simple Comparisons

• Need psi again:
  • $\psi = (c1)(X1)+ (c2)(X2 )+ (c3)(X3 )+…$
• F formula: $F = \frac{MS}{MSWithin}$, same omnibus error term.

Computation of Effect Sizes

• Report a measure of effect size for any significant effect.
• The most straightforward measure is eta-squared:
  • $\eta^2 = \frac{SSM}{SST}$