Two-Way ANOVA

Created:October 15, 2024

Last Updated:August 18, 2025

This calculator helps you examine how two different factors (independent variables) affect an outcome (dependent variable), both individually and in combination. Unlike simpler tests that only look at one factor at a time, Two-Way ANOVA reveals whether your factors work together in unexpected ways.

What You'll Get:

Main Effects: How each factor individually affects your outcome
Interaction Effects: Whether factors work together in surprising ways
Effect Sizes: Not just statistical significance, but practical importance
Visual Analysis: Interactive plots showing factor relationships
APA-Ready Report: Publication-quality results you can copy directly

Ready to explore your data? Start with our sample dataset to see how it works, or upload your own data to begin your analysis.

Calculator

1. Load Your Data

2. Select Columns & Options

Select Factor A:

Select Factor B:

Select Dependent Variable:

Significance Level:

Related Calculators

One-Way ANOVA Calculator

Independent T Test Calculator

Repeated Measures ANOVA Calculator

ANCOVA Calculator

Learn More

Two-Way ANOVA

Definition

Two-Way ANOVA examines the influence of two categorical independent variables on one continuous dependent variable. It tests for main effects of each factor and their interaction effect. It helps determine if there are significant differences between group means in a dataset.

Factors: The independent categorical variables.
Levels: The groups or categories within each factor.
Interaction: Determines whether the effect of one factor depends on the level of the other factor.

Formulas

Total Sum of Squares Decomposition:

SS_{Total} = SS_{FactorA} + SS_{FactorB} + SS_{Interaction} + SS_{Error}

$SS_{Total} = \sum_{i=1}^{a} \sum_{j=1}^{b} (X_{ij} - \bar{X})^2$ where $\bar{X}$ is the grand mean

$SS_{FactorA} = b \sum_{i=1}^{a} (\bar{X}_{i.} - \bar{X})^2$ where $\bar{X}_{i.}$ is the mean of level $i$ of Factor A, and $b$ is the number of levels in Factor B.

$SS_{FactorB} = a \sum_{j=1}^{b} (\bar{X}_{.j} - \bar{X})^2$ where $\bar{X}_{.j}$ is the mean of level $j$ of Factor B, and $a$ is the number of levels in Factor A.

$SS_{Interaction} = \sum_{i=1}^{a} \sum_{j=1}^{b} (\bar{X}_{ij} - \bar{X}_{i.} - \bar{X}_{.j} + \bar{X})^2$

Where:

$SS_{FactorA}$ = Sum of Squares for Factor A, $df = a - 1$ where $a$ is the number of levels in Factor A
$SS_{FactorB}$ = Sum of Squares for Factor B, $df = b - 1$ where $b$ is the number of levels in Factor B
$SS_{Interaction}$ = Sum of Squares for interaction with $df = (a - 1)(b - 1)$
$SS_{Error}$ = Residual Sum of Squares, $df = N - a*b$

Mean Square:

MS = \frac{SS}{df}

F-Statistic for each factor:

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

F-statistics are calculated separately for each factor and interaction effect

Key Assumptions

Independence: Observations must be independent

Normality: Residuals should be normally distributed

Homoscedasticity: Equal variances across groups

Practical Example

Step 1: State the Data

Weight loss study examining effects of diet and exercise:

Raw Data:

Diet	Exercise	Weight Loss (pounds)
Low-fat	Yes	8, 10, 9
Low-fat	No	6, 7, 8
High-fat	Yes	5, 7, 6
High-fat	No	3, 4, 5

Summary Statistics:

Diet	Exercise	Mean	N
Low-fat	Yes	9.00	3
Low-fat	No	7.00	3
High-fat	Yes	6.00	3
High-fat	No	4.00	3

Step 2: State Hypotheses

Main Effects:

Diet: $H_0: \alpha_1 = \alpha_2 = 0$
Exercise: $H_0: \beta_1 = \beta_2 = 0$

Interaction:

$H_0: (\alpha\beta)_{ij} = 0$ for all $i,j$

Step 3: Calculate Test Statistics

Source	df	SS	MS	F	p-value
Diet	1	27.0	27.0	27.0	0.000826
Exercise	1	12.0	12.0	12.0	0.008516
Diet:Exercise	1	0.0	0.0	0.0	1.0000
Residuals	8	8	1

Step 4: Draw Conclusions

Significant main effect of Diet (p = 0.000826)
Significant main effect of Exercise (p = 0.008516)
No significant interaction effect (p = 1.0000)
Diet and Exercise appear to have a significant effect on weight loss at $\alpha = 0.05$
The interaction between Diet and Exercise are not statistically significant

Effect Size

Partial Eta-squared:

\eta^2_p = \frac{SS_{Factor}}{SS_{Factor} + SS_{Error}}

For the example above,

Diet: $\eta^2_p = \frac{27}{27+8} = 0.77$ (large effect)
Exercise: $\eta^2_p = \frac{12}{12+8} = 0.60$ (large effect)
Interaction: $\eta^2_p = 0.00$ (no effect)

Code Examples

# Create the data
library(tidyverse)

data <- tibble(
  Diet = factor(rep(c("Low-fat", "High-fat"), each = 6)),
  Exercise = factor(rep(c("Yes", "No"), each = 3, times = 2)),
  WeightLoss = c(8, 10, 9, 6, 7, 8, 5, 7, 6, 3, 4, 5)
)

# Perform two-way ANOVA
model <- aov(WeightLoss ~ Diet * Exercise, data = data)

# Get summary
summary(model)


#------ Manual calculations ------#
# Compute the grand mean
grand_mean <- data |>
  summarize(grand_mean = mean(WeightLoss)) |>
  pull(grand_mean)

# Compute SS Total
ss_total <- data |>
  summarize(ss_total = sum((WeightLoss - grand_mean)^2)) |>
  pull(ss_total)

# Compute SS for Diet
ss_diet <- data |>
  group_by(Diet) |>
  summarize(group_mean = mean(WeightLoss), n = n()) |>
  ungroup() |>
  summarize(ss_diet = sum((group_mean - grand_mean)^2 * n)) |>
  pull(ss_diet)

# Compute SS for Exercise
ss_exercise <- data |>
  group_by(Exercise) |>
  summarize(group_mean = mean(WeightLoss), n = n()) |>
  ungroup() |>
  summarize(ss_exercise = sum((group_mean - grand_mean)^2 * n)) |>
  pull(ss_exercise)

# Compute SS Interaction
ss_interaction <- data |>
  group_by(Diet, Exercise) |>
  mutate(group_mean = mean(WeightLoss)) |>
  ungroup() |>
  group_by(Diet) |>
  mutate(diet_mean = mean(WeightLoss)) |>
  ungroup() |>
  group_by(Exercise) |>
  mutate(exercise_mean = mean(WeightLoss)) |>
  ungroup() |>
  mutate(interaction_term = (group_mean - diet_mean - exercise_mean + grand_mean)^2) |>
  summarize(ss_interaction = sum(interaction_term)) |>
  pull(ss_interaction)

ss_error <- ss_total - ss_diet - ss_exercise - ss_interaction

print(str_glue("SS total: {ss_total}"))
print(str_glue("SS diet: {ss_diet}"))
print(str_glue("SS exercise: {ss_exercise}"))
print(str_glue("SS interaction: {ss_interaction}"))
print(str_glue("SS error: {ss_error}"))

ms_diet = ss_diet / (2 - 1)
ms_exercise = ss_exercise / (2 - 1)
ms_error = ss_error / (2 * 2 * (3 - 1))

f_diet = ms_diet / ms_error
f_exercise = ms_exercise / ms_error
print(str_glue("F Diet: {f_diet}"))
print(str_glue("F Exercise: {f_exercise}"))

Python

import pandas as pd
import numpy as np
from scipy import stats

# Create the data
data = pd.DataFrame({
    'Diet': pd.Categorical(np.repeat(['Low-fat', 'High-fat'], 6)),
    'Exercise': pd.Categorical(np.tile(np.repeat(['Yes', 'No'], 3), 2)),
    'WeightLoss': [8, 10, 9, 6, 7, 8, 5, 7, 6, 3, 4, 5]
})

# Using statsmodels for ANOVA
import statsmodels.api as sm
from statsmodels.stats.anova import anova_lm

# Fit the model using statsmodels
model = sm.OLS.from_formula('WeightLoss ~ Diet + Exercise + Diet:Exercise', data=data)
fit = model.fit()
anova_table = anova_lm(fit, typ=2)
print("ANOVA results from statsmodels:")
print(anova_table)

#------ Manual calculations ------#
grand_mean = data['WeightLoss'].mean()
ss_total = np.sum((data['WeightLoss'] - grand_mean) ** 2)

# Compute SS for Diet
diet_means = data.groupby('Diet', observed=True)['WeightLoss'].agg(['mean', 'size']).reset_index()
ss_diet = np.sum((diet_means['mean'] - grand_mean) ** 2 * diet_means['size'])

# Compute SS for Exercise
exercise_means = data.groupby('Exercise', observed=True)['WeightLoss'].agg(['mean', 'size']).reset_index()
ss_exercise = np.sum((exercise_means['mean'] - grand_mean) ** 2 * exercise_means['size'])

# Compute SS Interaction
cell_means = data.groupby(['Diet', 'Exercise'], observed=True)['WeightLoss'].mean().reset_index()
cell_means = cell_means.merge(
    data.groupby('Diet', observed=True)['WeightLoss'].mean().reset_index().rename(columns={'WeightLoss': 'diet_mean'}),
    on='Diet'
)
cell_means = cell_means.merge(
    data.groupby('Exercise', observed=True)['WeightLoss'].mean().reset_index().rename(columns={'WeightLoss': 'exercise_mean'}),
    on='Exercise'
)

cell_means['interaction_term'] = (
    (cell_means['WeightLoss'] - cell_means['diet_mean'] - 
     cell_means['exercise_mean'] + grand_mean) ** 2
)
ss_interaction = cell_means['interaction_term'].sum()

ss_error = ss_total - ss_diet - ss_exercise - ss_interaction

# Calculate Mean Squares
df_diet = len(data['Diet'].unique()) - 1
df_exercise = len(data['Exercise'].unique()) - 1
df_interaction = df_diet * df_exercise
df_error = len(data) - (df_diet + 1) * (df_exercise + 1)

ms_diet = ss_diet / df_diet
ms_exercise = ss_exercise / df_exercise
ms_error = ss_error / df_error

# Calculate F statistics and p-values
f_diet = ms_diet / ms_error
f_exercise = ms_exercise / ms_error
p_diet = 1 - stats.f.cdf(f_diet, df_diet, df_error)
p_exercise = 1 - stats.f.cdf(f_exercise, df_exercise, df_error)

# Create ANOVA table
anova_manual = pd.DataFrame({
    'df': [df_diet, df_exercise, df_interaction, df_error],
    'sum_sq': [ss_diet, ss_exercise, ss_interaction, ss_error],
    'mean_sq': [ms_diet, ms_exercise, ss_interaction/df_interaction, ms_error],
    'F': [f_diet, f_exercise, (ss_interaction/df_interaction)/ms_error, np.nan],
    'PR(>F)': [p_diet, p_exercise, 
             1 - stats.f.cdf((ss_interaction/df_interaction)/ms_error, df_interaction, df_error), 
             np.nan]
}, index=['Diet', 'Exercise', 'Diet:Exercise', 'Residuals'])

print("ANOVA Table:")
print(anova_manual.round(4))

Two-Way ANOVA

What You'll Get:

Calculator

1. Load Your Data

2. Select Columns & Options

Related Calculators

One-Way ANOVA Calculator

Independent T Test Calculator

Repeated Measures ANOVA Calculator

ANCOVA Calculator

Learn More

Two-Way ANOVA

Definition

Formulas

Key Assumptions

Practical Example

Step 1: State the Data

Raw Data:

Summary Statistics:

Step 2: State Hypotheses

Step 3: Calculate Test Statistics

Step 4: Draw Conclusions

Effect Size

Code Examples

Verification

View Verification Details