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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. Click here to populate the sample data for a quick example.

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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

Yijkl=μ+Ai+Bj+Ck+(AB)ij+(AC)ik+(BC)jk+(ABC)ijk+ϵijklY_{ijkl} = \mu + A_i + B_j + C_k + (AB)_{ij} + (AC)_{ik} + (BC)_{jk} + (ABC)_{ijk} + \epsilon_{ijkl}

Where:

  • YijklY_{ijkl} = dependent variable value for the llth observation in the group i,j,ki,j,k
  • μ\mu = overall mean
  • Ai,Bj,CkA_i, B_j, C_k = main effects of factors A,B,CA, B, C
  • (AB)ij,(AC)ik,(BC)jk(AB)_{ij}, (AC)_{ik}, (BC)_{jk} = two-way interaction effects
  • (ABC)ijk(ABC)_{ijk} = three-way interaction
  • ϵijkl\epsilon_{ijkl} = error (residual) term

Test statistic for each factor:

F=MSFactorMSErrorF = \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 MethodGenderGradeTest Scores
LectureMale9th98
LectureMale9th84
LectureMale10th98
GroupWorkMale9th72
GroupWorkMale10th72
...

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

Step 2: State Hypotheses

Main Effects:

  • H0AH_0^A: No effect of teaching method
  • H0BH_0^B: No effect of gender
  • H0CH_0^C: No effect of grade level

Interactions:

  • H0ABH_0^{AB}: No method × gender interaction
  • H0ACH_0^{AC}: No method × grade interaction
  • H0BCH_0^{BC}: No gender × grade interaction
  • H0ABCH_0^{ABC}: No three-way interaction
Step 3: ANOVA Results
SourceSSdfMSFp-value
Method68234.220.4030.671
Gender1801180.272.1230.152
Grade38138.400.4520.504
Method:Gender66232.920.3880.681
Method:Grade2182108.951.2830.286
Gender:Grade414.270.0500.824
Method:Gender:Grade137268.520.8070.452
Residuals40764884.91
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

R
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

Related Calculators

One-Way ANOVA Calculator

Two-Way ANOVA Calculator

Independent T-Test Calculator

Repeated Measures ANOVA Calculator