ANOVA Demystified: Part 2 – Understanding Two-Way ANOVA

In our previous post, we explored One-Way ANOVA, which allows us to compare the means of multiple groups based on one independent variable (factor). But what if we want to analyze the impact of two factors simultaneously?

This is where Two-Way ANOVA comes into play! It helps us determine:
If each factor has a significant effect on the dependent variable.
If there’s an interaction effect between the two factors.

Let’s break it down in an easy-to-understand way!

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When to Use Two-Way ANOVA?

Use Two-Way ANOVA when:
You have two categorical independent variables (factors).
Your dependent variable is numerical (continuous).
You want to test if each factor affects the outcome individually and whether their combination creates an interaction effect.

Example:
Imagine you’re a data scientist analyzing exam scores based on:

• Study Method: Online vs. In-Person
• Practice Hours: Low vs. High

A Two-Way ANOVA helps answer:
Does study method impact scores?
Does practice time impact scores?
Do study method & practice time interact? (e.g., does online learning work best only with high practice hours?)

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Understanding Factors and Levels

In Two-Way ANOVA, each factor has multiple levels:

Factor	Levels
Study Method	Online, In-Person
Practice Hours	Low, High

This creates four groups:
Online & Low Practice
Online & High Practice
In-Person & Low Practice
In-Person & High Practice

Two-Way ANOVA helps us see how each factor alone and together impacts exam scores.

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Main Effects vs. Interaction Effects

Two-Way ANOVA tests for:

Main Effect of Factor A: Does Study Method affect exam scores?
Main Effect of Factor B: Does Practice Hours affect exam scores?
Interaction Effect (A × B): Does the combination of Study Method & Practice Hours create a unique effect?

Example of Interaction:

• If both study methods perform the same regardless of practice time: No interaction.
• If online learning only works well when students practice a lot: There’s an interaction!

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The Two-Way ANOVA Model & Hypotheses

Two-Way ANOVA follows this model:

$$Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} + \epsilon_{ijk}$$

Where:

• $Y_{ijk}$ = observed value for subject $k$ in group $i, j$
• $\mu$ = overall mean
• $\alpha_i$ = effect of Factor A (Study Method)
• $\beta_j$ = effect of Factor B (Practice Hours)
• $(\alpha\beta)_{ij}$ = interaction effect
• $\epsilon_{ijk}$ = error term

Hypotheses for Two-Way ANOVA

• H₀ (Factor A): No difference between study methods.
• H₀ (Factor B): No difference between practice time levels.
• H₀ (Interaction A × B): No interaction between study method & practice time.

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The ANOVA Table Explained

Source of Variation	Sum of Squares (SS)	Degrees of Freedom (df)	Mean Square (MS)	F-Statistic	p-value
Factor A (Study Method)	$SS_A$	$a-1$	$MS_A =\\ \dfrac{SS_A}{(a-1)}$	$F_A =\\ \dfrac{MS_A}{MS_{res}}$	p-value
Factor B (Practice Time)	$SS_B$	$b-1$	$MS_B =\\ \dfrac{SS_B}{(b-1)}$	$F_B =\\ \dfrac{MS_B}{MS_{res}}$	p-value
Interaction (A × B)	$SS_{AB}$	$(a-1)(b-1)$	$MS_{AB} =\\ \dfrac{SS_{AB}}{((a-1)(b-1))}$	$F_{AB} =\\ \dfrac{MS_{AB}}{MS_{res}}$	p-value
Error (Within Groups)	$SS_{res}$	$N-ab$	$MS_{res} =\\ \dfrac{SS_{res}}{(N-ab)}$	–	–
Total	$SS_{tot}$	$N-1$	–	–	–

How to Interpret the Table?

If p < 0.05 for Factor A: Study method affects exam scores.
If p < 0.05 for Factor B: Practice time affects exam scores.
If p < 0.05 for Interaction: The effect of study method depends on practice time.

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What Graph to Use?

Interaction Plot – Shows how factors combine to impact the dependent variable.
Boxplots – Compare groups visually.
Mean Plot with Error Bars – Displays group means with confidence intervals.

Example:

If lines in an interaction plot cross, there’s an interaction, one factor’s effect depends on the other!

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How to Interpret & Report Two-Way ANOVA Results?

State the Hypothesis:
“We analyzed the effects of study method and practice hours on exam scores.”

Report F-Statistics & p-values:
“Two-Way ANOVA found a significant main effect of study method, F(1, 36) = 8.42, p = 0.006, and practice hours, F(1, 36) = 12.91, p = 0.002. However, the interaction effect was non-significant, F(1, 36) = 1.89, p = 0.18.”

What Does This Mean?
“Study method and practice hours independently affect scores, but their combination does not create an additional effect.”

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Assumptions of Two-Way ANOVA

Before running Two-Way ANOVA, check these assumptions:

Normality – The dependent variable should be normally distributed within each group. (Shapiro-Wilk test)
Homogeneity of Variance – Groups should have similar variances. (Levene’s test)
Independence – Observations must be independent (Random sampling ensures this)

What if assumptions are violated?

• Use non-parametric alternatives like the Friedman Test.

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Post-Hoc Comparisons: Tukey’s HSD

If ANOVA finds significant effects, Tukey’s test helps identify which specific groups differ.

Example Interpretation:
“Post-hoc Tukey’s test showed that high practice time led to significantly higher scores (p = 0.01), but study method differences were only significant for low practice hours (p = 0.04).”

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Final Thoughts

Two-Way ANOVA helps uncover how multiple factors interact to influence an outcome. Always check for interaction effects – sometimes, the real story lies in how variables combine, not just their individual effects!