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.