Bivariate Data Analysis: Two-Way ANOVA

Factorial Design Fundamentals

• Factor: Categorical independent variable (IV) used for grouping participants.

• Level: Specific categories contained within a factor.

• Design Types:

  • 1-Way: $1$ IV.

  • 2-Way: $2$ IVs (e.g., $2 \times 2$ or $2 \times 3$).

  • Factorial: Multiple categorical IVs (e.g., $2 \times 3 \times 2$).

• Comparisons:

  • Within-Group (Paired/Repeated): Multiple data points per person in $1$ group.

  • Between-Group (Independent): $1$ data point per person across $2+$ distinct groups.

ANOVA Taxonomy & Components

• ANOVA Variants:

  • 1-Way Independent: $1$ between-groups IV.

  • 2-Way Independent: $2$ between-groups IVs.

  • 2-Way Repeated: $2$ within-groups IVs.

  • 2-Way Mixed: $1$ between-groups IV and $1$ within-groups IV.

• Effects:

  • Main Effect: Direct effect of $1$ IV on DV (evaluated for $\text{IV}_1$ and $\text{IV}_2$).

  • Interaction Effect ($\text{IV}_1 \times \text{IV}_2$): Effect of $1$ IV changes across levels of another IV.

  • Post-Hoc Tests: Follow-up tests to locate specific group mean differences.

• Interpretation Rule: If interaction is significant, interpret main effects cautiously.

Mathematical Framework for Two-Way ANOVA

• Sum of Squares ($SS$) Formulas:

  • Total: $SS_T = SS_M + SS_R$

  • Model: $SS_M = SS_A + SS_B + SS_{A \times B}$

  • Residual: $SS_R = \sum_{k=1}^{k} \sum_{i=1}^{n_k} (x_{ik} - \bar{x}_k)^2$

• Degrees of Freedom ($df$) Calculations:

  • $df_A = \text{Levels of IV}_1 - 1$

  • $df_B = \text{Levels of IV}_2 - 1$

  • $df_{A \times B} = df_A \times df_B$

  • $df_R = \text{Total Groups} \times (\text{Persons per Cell} - 1)$

• Calculated Example Values:

  • $SS$ Values: $SS_M = 44.1252$, $SS_R = 2.58$, $SS_A = 22.445$, $SS_B = 20.44$, $SS_{A \times B} = 1.24$

  • $df$ Values: $df_A = 1$, $df_B = 2$, $df_{A \times B} = 2$, $df_R = 12$

  • Mean Squares ($MS = \frac{SS}{df}$): $MS_A = 22.445$, $MS_B = 10.22$, $MS_{AB} = 0.62$, $MS_R = 0.215$

  • F-Ratios ($F = \frac{MS_{\text{Effect}}}{MS_R}$):

  • Factor A: $F_A = \frac{22.445}{0.215} = 104.4$ $\rightarrow$ $F(1, 12) = 104.4$

  • Factor B: $F_B = \frac{10.22}{0.215} = 47.53$ $\rightarrow$ $F(2, 12) = 47.53$

  • Interaction: $F_{AB} = \frac{0.62}{0.215} = 2.88$ $\rightarrow$ $F(2, 12) = 2.88$

JASP Empirical Analysis and Reporting

• Scenario (PSYC311_Data_4):

  • DV: Satisfaction

  • $\text{IV}_1$: Tipping Method (Cash vs. Credit)

  • $\text{IV}_2$: Time of Day (Morning vs. Evening)

• Main Effects:

  • Tipping Method: Significant, $F(1, 357) = 88.553$, $p < .001$

  • Time of Day: Not significant, $F(1, 357) = 0.556$, $p = .457$

• Interaction Effect:

  • Significant, $F(1, 357) = 710.817$, $p < .001$

• Post-Hoc Summary (Tukey HSD):

  • Morning: Credit > Cash ($p < .001$)

  • Evening: Cash > Credit ($p < .001$)

  • Cash: Evening > Morning ($p < .001$)

  • Credit: Morning > Evening ($p < .001$)

  • Cross-Group: Evening Cash > Morning Credit ($p < .001$); Morning Cash > Evening Credit ($p < .001$)