Two-Way ANOVA

Calculator

1. Load Your Data

Need to transform your data?

2. Select Columns & Options

Select Factor A:

Select Factor B:

Select Dependent Variable:

Significance Level:

Exclude Outliers

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

1# Create the data
2library(tidyverse)
3
4data <- tibble(
5  Diet = factor(rep(c("Low-fat", "High-fat"), each = 6)),
6  Exercise = factor(rep(c("Yes", "No"), each = 3, times = 2)),
7  WeightLoss = c(8, 10, 9, 6, 7, 8, 5, 7, 6, 3, 4, 5)
8)
9
10# Perform two-way ANOVA
11model <- aov(WeightLoss ~ Diet * Exercise, data = data)
12
13# Get summary
14summary(model)
15
16
17#------ Manual calculations ------#
18# Compute the grand mean
19grand_mean <- data |>
20  summarize(grand_mean = mean(WeightLoss)) |>
21  pull(grand_mean)
22
23# Compute SS Total
24ss_total <- data |>
25  summarize(ss_total = sum((WeightLoss - grand_mean)^2)) |>
26  pull(ss_total)
27
28# Compute SS for Diet
29ss_diet <- data |>
30  group_by(Diet) |>
31  summarize(group_mean = mean(WeightLoss), n = n()) |>
32  ungroup() |>
33  summarize(ss_diet = sum((group_mean - grand_mean)^2 * n)) |>
34  pull(ss_diet)
35
36# Compute SS for Exercise
37ss_exercise <- data |>
38  group_by(Exercise) |>
39  summarize(group_mean = mean(WeightLoss), n = n()) |>
40  ungroup() |>
41  summarize(ss_exercise = sum((group_mean - grand_mean)^2 * n)) |>
42  pull(ss_exercise)
43
44# Compute SS Interaction
45ss_interaction <- data |>
46  group_by(Diet, Exercise) |>
47  mutate(group_mean = mean(WeightLoss)) |>
48  ungroup() |>
49  group_by(Diet) |>
50  mutate(diet_mean = mean(WeightLoss)) |>
51  ungroup() |>
52  group_by(Exercise) |>
53  mutate(exercise_mean = mean(WeightLoss)) |>
54  ungroup() |>
55  mutate(interaction_term = (group_mean - diet_mean - exercise_mean + grand_mean)^2) |>
56  summarize(ss_interaction = sum(interaction_term)) |>
57  pull(ss_interaction)
58
59ss_error <- ss_total - ss_diet - ss_exercise - ss_interaction
60
61print(str_glue("SS total: {ss_total}"))
62print(str_glue("SS diet: {ss_diet}"))
63print(str_glue("SS exercise: {ss_exercise}"))
64print(str_glue("SS interaction: {ss_interaction}"))
65print(str_glue("SS error: {ss_error}"))
66
67ms_diet = ss_diet / (2 - 1)
68ms_exercise = ss_exercise / (2 - 1)
69ms_error = ss_error / (2 * 2 * (3 - 1))
70
71f_diet = ms_diet / ms_error
72f_exercise = ms_exercise / ms_error
73print(str_glue("F Diet: {f_diet}"))
74print(str_glue("F Exercise: {f_exercise}"))

Python

1import pandas as pd
2import numpy as np
3from scipy import stats
4
5# Create the data
6data = pd.DataFrame({
7    'Diet': pd.Categorical(np.repeat(['Low-fat', 'High-fat'], 6)),
8    'Exercise': pd.Categorical(np.tile(np.repeat(['Yes', 'No'], 3), 2)),
9    'WeightLoss': [8, 10, 9, 6, 7, 8, 5, 7, 6, 3, 4, 5]
10})
11
12# Using statsmodels for ANOVA
13import statsmodels.api as sm
14from statsmodels.stats.anova import anova_lm
15
16# Fit the model using statsmodels
17model = sm.OLS.from_formula('WeightLoss ~ Diet + Exercise + Diet:Exercise', data=data)
18fit = model.fit()
19anova_table = anova_lm(fit, typ=2)
20print("ANOVA results from statsmodels:")
21print(anova_table)
22
23#------ Manual calculations ------#
24grand_mean = data['WeightLoss'].mean()
25ss_total = np.sum((data['WeightLoss'] - grand_mean) ** 2)
26
27# Compute SS for Diet
28diet_means = data.groupby('Diet', observed=True)['WeightLoss'].agg(['mean', 'size']).reset_index()
29ss_diet = np.sum((diet_means['mean'] - grand_mean) ** 2 * diet_means['size'])
30
31# Compute SS for Exercise
32exercise_means = data.groupby('Exercise', observed=True)['WeightLoss'].agg(['mean', 'size']).reset_index()
33ss_exercise = np.sum((exercise_means['mean'] - grand_mean) ** 2 * exercise_means['size'])
34
35# Compute SS Interaction
36cell_means = data.groupby(['Diet', 'Exercise'], observed=True)['WeightLoss'].mean().reset_index()
37cell_means = cell_means.merge(
38    data.groupby('Diet', observed=True)['WeightLoss'].mean().reset_index().rename(columns={'WeightLoss': 'diet_mean'}),
39    on='Diet'
40)
41cell_means = cell_means.merge(
42    data.groupby('Exercise', observed=True)['WeightLoss'].mean().reset_index().rename(columns={'WeightLoss': 'exercise_mean'}),
43    on='Exercise'
44)
45
46cell_means['interaction_term'] = (
47    (cell_means['WeightLoss'] - cell_means['diet_mean'] - 
48     cell_means['exercise_mean'] + grand_mean) ** 2
49)
50ss_interaction = cell_means['interaction_term'].sum()
51
52ss_error = ss_total - ss_diet - ss_exercise - ss_interaction
53
54# Calculate Mean Squares
55df_diet = len(data['Diet'].unique()) - 1
56df_exercise = len(data['Exercise'].unique()) - 1
57df_interaction = df_diet * df_exercise
58df_error = len(data) - (df_diet + 1) * (df_exercise + 1)
59
60ms_diet = ss_diet / df_diet
61ms_exercise = ss_exercise / df_exercise
62ms_error = ss_error / df_error
63
64# Calculate F statistics and p-values
65f_diet = ms_diet / ms_error
66f_exercise = ms_exercise / ms_error
67p_diet = 1 - stats.f.cdf(f_diet, df_diet, df_error)
68p_exercise = 1 - stats.f.cdf(f_exercise, df_exercise, df_error)
69
70# Create ANOVA table
71anova_manual = pd.DataFrame({
72    'df': [df_diet, df_exercise, df_interaction, df_error],
73    'sum_sq': [ss_diet, ss_exercise, ss_interaction, ss_error],
74    'mean_sq': [ms_diet, ms_exercise, ss_interaction/df_interaction, ms_error],
75    'F': [f_diet, f_exercise, (ss_interaction/df_interaction)/ms_error, np.nan],
76    'PR(>F)': [p_diet, p_exercise, 
77             1 - stats.f.cdf((ss_interaction/df_interaction)/ms_error, df_interaction, df_error), 
78             np.nan]
79}, index=['Diet', 'Exercise', 'Diet:Exercise', 'Residuals'])
80
81print("ANOVA Table:")
82print(anova_manual.round(4))

Alternative Tests

When assumptions are violated:

Aligned Rank Transform ANOVA: For non-normal data
Scheirer-Ray-Hare Test: Non-parametric alternative

Related Calculators

Help us improve

Found an error or have a suggestion? Let us know!

Two-Way ANOVA

Calculator

1. Load Your Data

2. Select Columns & Options

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

Alternative Tests

Related Calculators

One-Way ANOVA Calculator

Independent T Test Calculator

Repeated Measures ANOVA Calculator

ANCOVA Calculator

Help us improve