Two-Way ANOVA

Analyze effects of two independent variables and their interaction on a dependent variable.

Two-Way ANOVA Overview & Factorial Design

What it tests:

Examines the effects of two independent categorical variables (factors) on a continuous dependent variable, including their interaction effects and main effects simultaneously.

Three Hypotheses Tested:

• • Main Effect A: Factor A group means are equal across all levels
• • Main Effect B: Factor B group means are equal across all levels
• • Interaction A×B: No interaction between the two factors

When to Use:

• • Two categorical independent variables
• • Continuous dependent variable
• • Multiple subjects per factor combination
• • Need to test interaction effects
• • Factorial experimental design

Key Assumptions:

• • Normal distribution within groups
• • Equal variances across all groups
• • Independence of observations
• • Random sampling from populations
• • Adequate sample sizes per cell

F-Statistic & Sum of Squares Decomposition

Total sum of squares decomposition

SSA

= sum of squares for Factor A

SSB

= sum of squares for Factor B

SSAB

= sum of squares for the A×B interaction

SSE

= sum of squares error (within cells)

F-statistic for a main effect or interaction

MSeffect

= mean square for the factor/interaction under test

MSE

= mean square error

Effect size (eta squared)

SSeffect

= sum of squares for the effect of interest

SST

= total sum of squares

Reading the Two-Way ANOVA Interaction Plot

Interpretation Guidelines

Interaction Effects Priority:

• • Significant Interaction: Main effects interpretation depends on interaction pattern - examine simple effects
• • No Interaction: Main effects can be interpreted independently
• • Always examine interaction plots for practical significance, even if not statistically significant

Effect Size Interpretation (η²):

Small Effect

η² ≈ 0.01

1% of variance explained

Medium Effect

η² ≈ 0.06

6% of variance explained

Large Effect

η² ≈ 0.14

14% of variance explained

sixsigmacenter.com Implementation Guide

The Design of Experiments module at sixsigmacenter.com automates two-way ANOVA computations, including sum of squares decomposition, F-tests, and interaction plots.

Automated Features:

• • Automatic sum of squares and F-statistic computation
• • Interaction plot generation
• • Post-hoc comparisons (Tukey HSD) when effects are significant
• • Effect size (η²) reporting for all effects

Platform Advantages:

• • Eliminates manual sum of squares calculation errors
• • Clear visual identification of interaction effects
• • Assumption checking (normality, equal variances)
• • Export-ready ANOVA tables

Example Data Layout for Two-Way ANOVA

Factorial design with response variable and two categorical factors (A and B). Sample Size Recommendation: Minimum 5-10 observations per cell (factor combination) for reliable two-way ANOVA results and interaction detection.

This is how your spreadsheet or CSV file should be structured before you upload it. Match the column names, keep one row per observation, and leave the analysis to the platform.