Three-way ANOVA

Created:January 6, 2025

Last Updated:April 2, 2025

This calculator helps you analyze the effects of three independent variables (factors) on a dependent variable. It calculates main effects and interaction effects between factors, helping you understand how multiple categorical variables influence your outcome measure. It checks assumptions, provides a detailed ANOVA table, and an APA-style report. for a quick example.

Important data requirements for three-way ANOVA:

Each factor (A, B, C) must have at least 2 levels (unique values).
Each cell (combination of factor levels) should have at least 3 observations for reliable interaction estimates and Shapiro-Wilk test.
The minimum data points needed is 2×2×2×3 = 24 (2 levels for each of the 3 factors, with 3 observations per cell).

Calculator

1. Load Your Data

2. Select Columns & Options

Select Factor A:

Select Factor B:

Select Factor C:

Select Dependent Variable:

Significance Level:

Related Calculators

One-Way ANOVA Calculator

Two-Way ANOVA Calculator

Independent T-Test Calculator

Repeated Measures ANOVA Calculator

Learn More

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}}

Dataset Requirements

Sample Size: Aim for 5-10 observations per cell (each unique factor combination). Larger samples (> 20 per cell) improve power to detect interactions.

Balanced Design: Equal or near-equal sample sizes across all factor combinations strengthen statistical power and robustness.

Data Structure: Complete factorial design with all factor combinations represented. Categorical independent variables (ideally 2-4 levels each) and a continuous dependent variable.

Independence: Observations should come from different subjects/units with random assignment to treatment combinations.

Effect Detection: Consider expected effect sizes when determining sample size. Interactions typically require larger samples to detect than main effects.

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

library(tidyverse)

set.seed(42)

# Data preparation
data <- tibble(
  Method = rep(c("Lecture", "GroupWork", "Mixed"), each = 20),
  Gender = rep(rep(c("Male", "Female"), each = 10), 3),
  Grade = rep(c("9th", "10th"), each = 5, times = 6),
  Score = round(runif(60, min = 70, max = 100))
)

# Run 3-way ANOVA
model <- aov(Score ~ Method * Gender * Grade, data = data)
summary(model)

Python

import pandas as pd
import numpy as np
import statsmodels.api as sm
from statsmodels.stats.anova import anova_lm

np.random.seed(42)

# Create example data
data = pd.DataFrame({
    'Method': np.repeat(['Lecture', 'GroupWork', 'Mixed'], 20),
    'Gender': np.tile(np.repeat(['Male', 'Female'], 10), 3),
    'Grade': np.tile(np.repeat(['9th', '10th'], 5), 6),
    'Score': np.random.randint(70, 100, 60)
})

# Fit the model
model = sm.OLS.from_formula('Score ~ Method * Gender * Grade', data=data).fit()

# Get ANOVA table
anova_table = anova_lm(model, typ=3)
print(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

Verification

Three-way ANOVA

Created:January 6, 2025

Last Updated:April 2, 2025

Important data requirements for three-way ANOVA:

Each factor (A, B, C) must have at least 2 levels (unique values).
Each cell (combination of factor levels) should have at least 3 observations for reliable interaction estimates and Shapiro-Wilk test.
The minimum data points needed is 2×2×2×3 = 24 (2 levels for each of the 3 factors, with 3 observations per cell).

Calculator

1. Load Your Data

2. Select Columns & Options

Select Factor A:

Select Factor B:

Select Factor C:

Select Dependent Variable:

Significance Level:

Related Calculators

One-Way ANOVA Calculator

Two-Way ANOVA Calculator

Independent T-Test Calculator

Repeated Measures ANOVA Calculator

Learn More

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}}

Dataset Requirements

Sample Size: Aim for 5-10 observations per cell (each unique factor combination). Larger samples (> 20 per cell) improve power to detect interactions.

Balanced Design: Equal or near-equal sample sizes across all factor combinations strengthen statistical power and robustness.

Data Structure: Complete factorial design with all factor combinations represented. Categorical independent variables (ideally 2-4 levels each) and a continuous dependent variable.

Independence: Observations should come from different subjects/units with random assignment to treatment combinations.

Effect Detection: Consider expected effect sizes when determining sample size. Interactions typically require larger samples to detect than main effects.

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

library(tidyverse)

set.seed(42)

# Data preparation
data <- tibble(
  Method = rep(c("Lecture", "GroupWork", "Mixed"), each = 20),
  Gender = rep(rep(c("Male", "Female"), each = 10), 3),
  Grade = rep(c("9th", "10th"), each = 5, times = 6),
  Score = round(runif(60, min = 70, max = 100))
)

# Run 3-way ANOVA
model <- aov(Score ~ Method * Gender * Grade, data = data)
summary(model)

Python

import pandas as pd
import numpy as np
import statsmodels.api as sm
from statsmodels.stats.anova import anova_lm

np.random.seed(42)

# Create example data
data = pd.DataFrame({
    'Method': np.repeat(['Lecture', 'GroupWork', 'Mixed'], 20),
    'Gender': np.tile(np.repeat(['Male', 'Female'], 10), 3),
    'Grade': np.tile(np.repeat(['9th', '10th'], 5), 6),
    'Score': np.random.randint(70, 100, 60)
})

# Fit the model
model = sm.OLS.from_formula('Score ~ Method * Gender * Grade', data=data).fit()

# Get ANOVA table
anova_table = anova_lm(model, typ=3)
print(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

Verification

Teaching Method

Gender

Grade

Test Scores

Lecture

Male

9th

Lecture

Male

9th

Lecture

Male

10th

GroupWork

Male

9th

GroupWork

Male

10th

...

Source

p-value

Method

34.22

0.403

0.671

Gender

180

180.27

2.123

0.152

Grade

38.40

0.452

0.504

Method:Gender

32.92

0.388

0.681

Method:Grade

218

108.95

1.283

0.286

Gender:Grade

4.27

0.050

0.824

Method:Gender:Grade

137

68.52

0.807

0.452

Residuals

4076

84.91

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

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

Teaching Method

Gender

Grade

Test Scores

Lecture

Male

9th

Lecture

Male

9th

Lecture

Male

10th

GroupWork

Male

9th

GroupWork

Male

10th

...

Source

p-value

Method

34.22

0.403

0.671

Gender

180

180.27

2.123

0.152

Grade

38.40

0.452

0.504

Method:Gender

32.92

0.388

0.681

Method:Grade

218

108.95

1.283

0.286

Gender:Grade

4.27

0.050

0.824

Method:Gender:Grade

137

68.52

0.807

0.452

Residuals

4076

84.91

Three-way ANOVA

Important data requirements for three-way ANOVA:

Calculator

1. Load Your Data

2. Select Columns & Options

Related Calculators

One-Way ANOVA Calculator

Two-Way ANOVA Calculator

Independent T-Test Calculator

Repeated Measures ANOVA Calculator

Learn More

Three-Way ANOVA

Definition

Model

Dataset Requirements

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

Verification

View Verification Details

Three-way ANOVA

Important data requirements for three-way ANOVA:

Calculator

1. Load Your Data

2. Select Columns & Options

Related Calculators

One-Way ANOVA Calculator

Two-Way ANOVA Calculator

Independent T-Test Calculator

Repeated Measures ANOVA Calculator

Learn More

Three-Way ANOVA

Definition

Model

Dataset Requirements

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

Verification

View Verification Details

Three-way ANOVA

Important data requirements for three-way ANOVA:

Calculator

1. Load Your Data

2. Select Columns & Options

Related Calculators

One-Way ANOVA Calculator

Two-Way ANOVA Calculator

Independent T-Test Calculator

Repeated Measures ANOVA Calculator

Learn More

Three-Way ANOVA

Definition

Model

Dataset Requirements

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

Verification

View Verification Details

Three-way ANOVA

Important data requirements for three-way ANOVA:

Calculator

1. Load Your Data

2. Select Columns & Options