Three-Way ANOVA

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Three-Way ANOVA

Definition

Three-Way ANOVA (Analysis of Variance) examines the influence of three independent variables on a continuous dependent variable. It tests main effects and interactions between factors. It tests:

Main effects of each factor
Interaction effects between factors
The three-way interaction between all three factors

This test is an extension of the two-way ANOVA, adding a third factor to the model.

Model

Y_{ijkl} = \mu + A_i + B_j + C_k + (AB)_{ij} + (AC)_{ik} + (BC)_{jk} + (ABC)_{ijk} + \epsilon_{ijkl}

Where:

$Y_{ijkl}$ = dependent variable value for the $l$ th observation in the group $i,j,k$
$\mu$ = overall mean
$A_i, B_j, C_k$ = main effects of factors $A, B, C$
$(AB)_{ij}, (AC)_{ik}, (BC)_{jk}$ = two-way interaction effects
$(ABC)_{ijk}$ = three-way interaction
$\epsilon_{ijkl}$ = error (residual) term

Test statistic for each factor:

F = \frac{MS_{Factor}}{MS_{Error}}

Key Assumptions

Independence: Observations are independent

Normality: Residuals are normally distributed

Homogeneity: Equal variances across groups

Practical Example

Step 1: State the Data

Investigating the effects of teaching method, gender, and grade level on test scores:

Dependent Variable: Test Score
Factor A: Teaching Method (Lecture, GroupWork, Mixed)
Factor B: Gender (Male, Female)
Factor C: Grade Level (9th, 10th)
Sample Size: 60 students

Teaching Method	Gender	Grade	Test Scores
Lecture	Male	9th	98
Lecture	Male	9th	84
Lecture	Male	10th	98
GroupWork	Male	9th	72
GroupWork	Male	10th	72
...

For the complete dataset, please refer to the code examples below.

Step 2: State Hypotheses

Main Effects:

$H_0^A$ : No effect of teaching method
$H_0^B$ : No effect of gender
$H_0^C$ : No effect of grade level

Interactions:

$H_0^{AB}$ : No method × gender interaction
$H_0^{AC}$ : No method × grade interaction
$H_0^{BC}$ : No gender × grade interaction
$H_0^{ABC}$ : No three-way interaction

Step 3: ANOVA Results

Source	SS	df	MS	F	p-value
Method	68	2	34.22	0.403	0.671
Gender	180	1	180.27	2.123	0.152
Grade	38	1	38.40	0.452	0.504
Method:Gender	66	2	32.92	0.388	0.681
Method:Grade	218	2	108.95	1.283	0.286
Gender:Grade	4	1	4.27	0.050	0.824
Method:Gender:Grade	137	2	68.52	0.807	0.452
Residuals	4076	48	84.91

Step 4: Conclusions

No significant main effect of Method (F(2,48) = 1.283, p = .286)
No significant main effect of Gender (F(1,48) = 0.452, p = .504)
No significant main effect of Grade (F(1,48) = 0.388, p = .681)
No significant two-way interactions: Method:Gender (F(2,48) = 0.388, p = .681), Method:Grade (F(2,48) = 1.283, p = .286), Gender:Grade (F(1,48) = 0.050, p = .824)
No significant three-way interaction between Method:Gender:Grade (F(2,48) = 0.807, p = .452)

Code Examples

1library(tidyverse)
2
3set.seed(42)
4
5# Data preparation
6data <- tibble(
7  Method = rep(c("Lecture", "GroupWork", "Mixed"), each = 20),
8  Gender = rep(rep(c("Male", "Female"), each = 10), 3),
9  Grade = rep(c("9th", "10th"), each = 5, times = 6),
10  Score = round(runif(60, min = 70, max = 100))
11)
12
13# Run 3-way ANOVA
14model <- aov(Score ~ Method * Gender * Grade, data = data)
15summary(model)

Python

1import pandas as pd
2import numpy as np
3import statsmodels.api as sm
4from statsmodels.stats.anova import anova_lm
5
6np.random.seed(42)
7
8# Create example data
9data = pd.DataFrame({
10    'Method': np.repeat(['Lecture', 'GroupWork', 'Mixed'], 20),
11    'Gender': np.tile(np.repeat(['Male', 'Female'], 10), 3),
12    'Grade': np.tile(np.repeat(['9th', '10th'], 5), 6),
13    'Score': np.random.randint(70, 100, 60)
14})
15
16# Fit the model
17model = sm.OLS.from_formula('Score ~ Method * Gender * Grade', data=data).fit()
18
19# Get ANOVA table
20anova_table = anova_lm(model, typ=3)
21print(anova_table)

Post-Hoc Analysis

For significant effects:

Tukey HSD: Compare all pairs of levels
Simple Effects Analysis: Examine one factor at levels of others

Related Calculators

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Three-Way ANOVA

Calculator

1. Load Your Data

2. Select Columns & Options

Learn More

Three-Way ANOVA

Definition

Model

Key Assumptions

Practical Example

Step 1: State the Data

Step 2: State Hypotheses

Step 3: ANOVA Results

Step 4: Conclusions

Code Examples

Post-Hoc Analysis

Related Calculators

One-Way ANOVA Calculator

Two-Way ANOVA Calculator

Independent T-Test Calculator

Repeated Measures ANOVA Calculator

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