Quick Answer: Three Way ANOVA Result

The worked Three Way ANOVA tests seven model effects: three main effects, three two-way interactions and one three-way interaction. The main effects are studytime, school and sex. The two-way interactions are studytime × school, studytime × sex and school × sex. The final interaction is studytime × school × sex.

The summary table shows that all three main effects are statistically significant. Studytime is significant, F = 9.176, p = 5.978e-06, partial η² = 0.04168. School is significant, F = 49.18, p = 6.001e-12, partial η² = 0.0721. Sex is significant, F = 7.417, p = 0.006639, partial η² = 0.01158.

The interaction tests are not significant. The studytime × school interaction has p = 0.6896, studytime × sex has p = 0.1157, school × sex has p = 0.8101, and the three-way interaction has p = 0.7414. Therefore, the correct interpretation focuses on the main effects, not on a complex three-factor interaction.

What Is Three Way ANOVA?

Three Way ANOVA is a factorial ANOVA test used when a researcher has one numeric dependent variable and three categorical independent variables. It extends One Way ANOVA and Factorial ANOVA by testing not only main effects but also interaction effects among three factors.

In this example, the dependent variable is G3 final grade. The three factors are studytime, school and sex. The model tests whether G3 differs by each factor separately and whether the effect of one factor depends on another factor.

The output shows that the three main effects are significant. Students with different studytime categories have different mean G3 scores. GP and MS schools have different mean G3 scores. Female and male groups also differ in mean G3. However, the interaction effects are not significant, so the model does not support a strong combined interaction among these factors.

This guide connects naturally with One Way ANOVA, Factorial ANOVA, Fixed Effects ANOVA, Balanced ANOVA, Brown Forsythe ANOVA, Nested ANOVA, ANOVA Effect Size, Eta Squared, Omega Squared, and F Distribution.

Three Way ANOVA Formula

A full three-way fixed-effects ANOVA model includes main effects, all two-way interactions and the three-way interaction. The model can be written as:

In this example, Y is G3 final grade, A is studytime, B is school, and C is sex. The model includes three main effects, three two-way interactions and one three-way interaction.

F Statistic Formula

Each effect has its own F statistic. A large F statistic means that the effect explains more variation than expected relative to the residual error. In this example, school has the largest F statistic among the main effects, followed by studytime and sex.

Partial Eta Squared Formula

Partial eta squared describes the practical strength of each effect after accounting for the model error term. In this output, school has the largest partial eta squared at 0.0721, studytime has 0.04168, and sex has 0.01158. The interaction effects are very small.

Three Way ANOVA Hypotheses

A Three Way ANOVA has more than one hypothesis. The researcher must test each main effect and each interaction separately. This is why a three-way ANOVA table contains several rows.

Dataset and Variables Used

The worked example uses the student performance dataset structure commonly used in Salar Cafe statistical tutorials. The dependent variable is G3 final grade. The three categorical factors are studytime, school and sex.

Main Effect Mean Pattern

For supporting concepts, review Descriptive Statistics, Mean Median and Mode, Standard Deviation, Variance, Confidence Interval, P Value and Null and Alternative Hypothesis.

SPSS Output Interpretation for Three Way ANOVA

The SPSS output for Three Way ANOVA should be read in a fixed order. First confirm the dependent variable and factors. Then review descriptive statistics and cell sizes. After that, read the tests of between-subject effects. Finally, check assumptions and residual diagnostics before writing the final interpretation.

SPSS Reading Order

SPSS Result Summary

Python Chart-by-Chart Interpretation

The Python chart sequence explains Three Way ANOVA through effect sizes, p-values, main effect means, interaction profiles, three-way cell means and residual diagnostics.

Python Chart 1: Three Way ANOVA Effect Sizes

The effect-size chart shows that school is the strongest practical effect with partial eta squared about 0.072. Studytime follows with about 0.042, and sex has a smaller effect of about 0.012.

The interaction effect sizes are very small. Studytime × sex is about 0.009, studytime × school is about 0.002, the three-way interaction is about 0.002, and school × sex is nearly zero. This supports a main-effects interpretation.

Python Chart 2: p-values by Effect

The p-value chart shows that studytime, school and sex are below alpha = .05. These are statistically significant main effects.

The interaction p-values are above alpha = .05. This means the two-way interactions and the three-way interaction are not statistically supported in this model.

Python Chart 3: Main Effect Mean Plot for Studytime

The studytime mean chart shows a clear upward pattern. Studytime group 1 has the lowest mean G3, group 2 is higher, and groups 3 and 4 are in the highest range.

This visual pattern explains why the studytime main effect is significant. The effect is not the largest in the model, but it is meaningful enough to report as a small main effect.

Python Chart 4: Main Effect Mean Plot for School

The school mean chart shows that GP has a higher mean G3 than MS. The separation is visually clear, and the confidence intervals are well apart.

This matches the summary table where school has the largest F statistic and the largest partial eta squared. School is the strongest practical factor in this Three Way ANOVA result.

Python Chart 5: Main Effect Mean Plot for Sex

The sex mean chart shows that the female group has a higher mean G3 than the male group. The difference is smaller than the school effect but still statistically significant.

This supports a careful interpretation: sex has a significant main effect, but its practical size is small compared with school and studytime.

Python Chart 6: Two-Way Interaction Profile

The interaction profile shows that GP is higher than MS across the studytime levels. Both school lines increase toward studytime level 3 and then slightly decline or flatten at level 4.

The lines are not perfectly parallel, but the formal studytime × school interaction is not significant. Therefore, the plot should be used descriptively, not as proof of an interaction.

Python Chart 7: Three-Way Cell Means

The three-way cell mean chart shows every factor combination. The highest cells are generally connected with higher studytime, GP school and female group combinations, while lower cells are connected with lower studytime and MS combinations.

This chart is useful for understanding the data structure, but it should not be over-interpreted as a significant three-way interaction. The formal three-way interaction p-value is 0.7414, so the visual cell differences are better explained by main effects.

Python Chart 8: Residuals vs Fitted Values

The residuals-versus-fitted chart shows vertical fitted-value bands because the model is based on categorical cell means. Most residuals are distributed around zero, but some negative residuals are large.

The diagnostic conclusion is balanced. The model captures the main mean patterns, but residual spread and lower-tail cases should be acknowledged when reporting model diagnostics.

Python Chart 9: Residual Q-Q Plot

The Q-Q plot shows visible departure from the reference line, especially in the lower tail. The middle residuals are closer to the expected pattern than the extreme lower residuals.

This means residual normality is approximate rather than perfect. The significant main effects can still be reported, but the diagnostics should be described honestly.

Python Chart 10: Three Way ANOVA Summary Table

The summary table gives the compact final result. The three main effects are significant, and all interaction effects are not significant.

This is the most important table for reporting because it contains the formal decision for every model effect. It supports a final interpretation focused on studytime, school and sex main effects.

R Chart-by-Chart Validation

The R charts validate the Python analysis in a second software workflow. The R outputs repeat the same effect-size pattern, p-value decisions, main effect means, interaction profile, cell means and residual diagnostics.

R Chart 1: Three Way ANOVA Effect Sizes

The R effect-size chart confirms that school has the strongest effect, followed by studytime and sex. The interaction effect sizes remain very small.

This validates the practical conclusion from Python. The model should be reported as a main-effects result rather than an interaction-driven result.

R Chart 2: p-values by Effect

The R p-value chart confirms that studytime, school and sex are statistically significant, while all interaction effects are not significant.

This software agreement strengthens the final decision because Python and R support the same interpretation.

R Chart 3: Main Effect Mean Plot for Factor A

The R studytime chart confirms the upward mean pattern from lower studytime to higher studytime categories.

This supports the conclusion that studytime has a significant small main effect on G3.

R Chart 4: Main Effect Mean Plot for Factor B

The R school chart confirms that GP has a higher mean G3 than MS.

This validates school as the largest practical effect in the model.

R Chart 5: Main Effect Mean Plot for Factor C

The R sex chart confirms that the female group has a higher mean G3 than the male group.

The practical effect is small, but the formal test supports reporting sex as a significant main effect.

R Chart 6: Two-Way Interaction Profile

The R interaction profile confirms the same descriptive pattern: GP stays higher than MS across studytime categories.

The interaction is not statistically significant, so the plot should be interpreted as descriptive support for main effects rather than evidence of a formal interaction.

R Chart 7: Three-Way Cell Means

The R cell-mean chart confirms that some cells have higher means than others, especially cells connected with GP school and higher studytime levels.

The three-way interaction remains non-significant, so the cell mean chart should be used for description and quality checking, not for claiming a complex three-way effect.

R Chart 8: Residuals vs Fitted Values

The R residual chart confirms the same fitted-value banding and residual spread seen in Python.

The diagnostic message remains the same: the model explains the main cell mean differences, but residual tail behavior should be mentioned.

R Chart 9: Residual Q-Q Plot

The R Q-Q plot confirms that residual normality is approximate rather than perfect. Lower-tail departure is visible.

This supports a transparent assumption statement in the final report.

R Chart 10: Three Way ANOVA Summary Table

The R summary table confirms the same final result as Python. Studytime, school and sex are significant; all interaction effects are not significant.

This agreement between Python and R makes the final interpretation stable and suitable for SPSS-style reporting.

SPSS, R, Python and Excel Workflows for Three Way ANOVA

The same Three Way ANOVA workflow can be reproduced in SPSS, R, Python and Excel. SPSS uses the General Linear Model univariate procedure. R can use aov() or lm(). Python can use statsmodels. Excel can support descriptive summaries and interaction charts, but SPSS, R or Python is better for the full factorial ANOVA table.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Three Way ANOVA

SPSS Syntax for Three Way ANOVA

* Three Way ANOVA in SPSS.
* Dependent variable: G3.
* Factors: studytime, school, sex.

TITLE "Three Way ANOVA: G3 by Studytime, School and Sex".

UNIANOVA G3 BY studytime school sex
  /METHOD=SSTYPE(3)
  /INTERCEPT=INCLUDE
  /PRINT=DESCRIPTIVE ETASQ HOMOGENEITY
  /PLOT=PROFILE(studytime*school)
  /EMMEANS=TABLES(studytime) COMPARE ADJ(BONFERRONI)
  /EMMEANS=TABLES(school)
  /EMMEANS=TABLES(sex)
  /CRITERIA=ALPHA(.05)
  /DESIGN=studytime school sex
          studytime*school studytime*sex school*sex
          studytime*school*sex.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="three_way_anova_spss_output.pdf".

Python Code for Three Way ANOVA

import pandas as pd
import statsmodels.api as sm
from statsmodels.formula.api import ols
from statsmodels.stats.anova import anova_lm

df = pd.read_csv("dataset.csv")

for col in ["G3", "studytime", "school", "sex"]:
    df[col] = df[col].astype("category") if col != "G3" else pd.to_numeric(df[col], errors="coerce")

data = df[["G3", "studytime", "school", "sex"]].dropna().copy()

model = ols("G3 ~ C(studytime) * C(school) * C(sex)", data=data).fit()

anova_table = anova_lm(model, typ=2)
print(anova_table)

# Partial eta squared
error_ss = anova_table.loc["Residual", "sum_sq"]
anova_table["partial_eta_sq"] = anova_table["sum_sq"] / (anova_table["sum_sq"] + error_ss)

print(anova_table)

# Estimated main effect means
print(data.groupby("studytime")["G3"].mean())
print(data.groupby("school")["G3"].mean())
print(data.groupby("sex")["G3"].mean())

# Cell means
cell_means = data.groupby(["studytime", "school", "sex"])["G3"].mean()
print(cell_means)

# Residual diagnostics
data["fitted"] = model.fittedvalues
data["residual"] = model.resid
print(data[["G3", "fitted", "residual"]].head())

R Code for Three Way ANOVA

# Three Way ANOVA in R

library(tidyverse)

df <- read.csv("dataset.csv")

df$G3 <- as.numeric(df$G3)
df$studytime <- as.factor(df$studytime)
df$school <- as.factor(df$school)
df$sex <- as.factor(df$sex)

data <- df %>%
  select(G3, studytime, school, sex) %>%
  drop_na()

model <- aov(G3 ~ studytime * school * sex, data = data)
summary(model)

# Main effect means
data %>% group_by(studytime) %>% summarise(mean_G3 = mean(G3), n = n(), .groups = "drop")
data %>% group_by(school) %>% summarise(mean_G3 = mean(G3), n = n(), .groups = "drop")
data %>% group_by(sex) %>% summarise(mean_G3 = mean(G3), n = n(), .groups = "drop")

# Three-way cell means
data %>%
  group_by(studytime, school, sex) %>%
  summarise(mean_G3 = mean(G3), n = n(), .groups = "drop")

# Optional effect sizes
# install.packages("effectsize")
# library(effectsize)
# eta_squared(model, partial = TRUE)

# Diagnostics
par(mfrow = c(1, 2))
plot(fitted(model), residuals(model),
     xlab = "Fitted values", ylab = "Residuals",
     main = "Residuals vs Fitted")
abline(h = 0, lty = 2)
qqnorm(residuals(model))
qqline(residuals(model))

Excel Notes for Three Way ANOVA

Excel support workflow:

1. Arrange the data:
   G3 | studytime | school | sex

2. Create main effect PivotTables:
   Rows = studytime, Values = Average of G3
   Rows = school, Values = Average of G3
   Rows = sex, Values = Average of G3

3. Create two-way cell means:
   Rows = studytime
   Columns = school
   Values = Average of G3

4. Create three-way cell means:
   Rows = studytime
   Columns = school and sex
   Values = Average of G3

5. Create charts:
   - effect-size chart from ANOVA table output
   - p-value chart from ANOVA table output
   - main effect bar charts
   - interaction profile chart
   - cell mean chart

6. Formal Three Way ANOVA:
   Use SPSS, R or Python for the correct factorial ANOVA table.
   Excel is useful for summaries and graphs but not ideal for the full model.

APA Reporting Wording

When reporting Three Way ANOVA, include the dependent variable, three factors, significant main effects, interaction results, effect sizes and assumption notes. Because the three-way interaction is non-significant, do not write the report as if the result is driven by a complex three-factor dependency.

Common Mistakes

When to Use Three Way ANOVA

Use Three Way ANOVA when the outcome is numeric and there are three categorical factors. It is common in education, psychology, agriculture, medicine, business experiments and social science research where the researcher needs to test several factors and their interactions in one model.

Three Way ANOVA should be compared with Factorial ANOVA, One Way ANOVA, Fixed Effects ANOVA, Nested ANOVA, Balanced ANOVA, Brown Forsythe ANOVA, ANOVA Effect Size, ANOVA in SPSS, ANOVA in R and ANOVA in Python.

Downloads and Resources for Three Way ANOVA

Use these resources to reproduce the Three Way ANOVA workflow. The Python report, R report and SPSS output PDF are included as verification files. Script and workbook placeholders can be replaced after the final downloadable files are uploaded to the WordPress Media Library.

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