Quick Answer: Two Way ANOVA Result

The worked Two Way ANOVA tests three effects. The first main effect is studytime. The second main effect is school. The third effect is the studytime × school interaction.

The summary table reports a significant studytime effect, F = 12.43, p = 6.557e-08, partial η² = 0.05498. The school effect is also significant, F = 46.2, p = 2.445e-11, partial η² = 0.06723. The studytime × school interaction is not significant, F = 0.3636, p = 0.7793, partial η² = 0.001699.

What Is Two Way ANOVA?

Two Way ANOVA is a factorial ANOVA test used when one continuous dependent variable is compared across two categorical independent variables. It answers three questions in one model: whether factor A has a main effect, whether factor B has a main effect, and whether factor A and factor B interact.

In this example, G3 final grade is the continuous dependent variable. Studytime is the first factor, with four levels. School is the second factor, with GP and MS groups. The model asks whether mean G3 differs across studytime levels, whether mean G3 differs by school, and whether the studytime difference depends on school.

The result is a clear main-effects pattern. Studytime matters, school matters, but the combined studytime × school interaction does not matter statistically. This means the final report should explain the overall studytime difference and the overall school difference without claiming a meaningful interaction.

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

Two Way ANOVA Formula

A two-factor ANOVA model separates each outcome score into the grand mean, the first factor effect, the second factor effect, the interaction effect and random error.

In this formula, Y is the dependent variable, μ is the grand mean, A is the first factor, B is the second factor, AB is the interaction and e is the residual error. In this example, Y is G3, A is studytime and B is school.

F Statistic Formula

Each effect has its own F statistic. Studytime has an F statistic, school has an F statistic, and the studytime × school interaction has an F statistic. The p-value attached to each F statistic tells whether that model effect is statistically significant.

Partial Eta Squared Formula

Partial eta squared describes the practical size of each effect. In this output, school has the largest partial eta squared at 0.06723. Studytime is slightly smaller at 0.05498. The interaction is very small at 0.001699.

Two Way ANOVA Hypotheses

Two Way ANOVA has separate hypotheses for the first main effect, the second main effect and the interaction. The interaction is especially important because it tells whether the effect of one factor depends on the level of the other factor.

Dataset and Variables Used

The worked example uses student performance data. The dependent variable is G3 final grade. The two factors are studytime and school. Studytime has four groups, while school has two groups, GP and MS.

Cell Mean Pattern

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

Two Way ANOVA Assumptions

Two Way ANOVA assumes a numeric outcome, categorical factors, independent observations, approximate residual normality and reasonably similar residual variance across cells. These assumptions should be checked before relying on the final F tests.

For deeper assumption checks, use Levene Test, Bartlett’s Test, Q-Q Plot Normality Check, P-P Plot Normality Check, Shapiro-Wilk Test, Anderson-Darling Test, Studentized Residuals, Cook’s Distance and Outlier Detection.

SPSS Output Interpretation for Two Way ANOVA

The SPSS output for Two Way ANOVA should be read from the factorial model table. The dependent variable is G3, and the fixed factors are studytime and school. The important rows are the studytime main effect, the school main effect and the studytime × school interaction.

SPSS Reading Order

SPSS Result Summary

Python Chart-by-Chart Interpretation

The Python chart sequence explains the Two Way ANOVA result through interaction lines, cell means, p-values, effect sizes, boxplots, residual diagnostics and a summary table.

Python Chart 1: Two Way ANOVA Interaction Plot

The interaction plot shows that GP is higher than MS across every studytime level. Both school lines rise from studytime 1 toward studytime 3, then slightly decline or flatten at studytime 4.

The lines are not perfectly parallel, but the formal interaction test is not significant. This means the plot should be used to explain the main effects, not to claim a statistically supported interaction.

Python Chart 2: Two Way ANOVA Cell Means

The cell mean chart confirms that GP is higher than MS in each studytime category. The highest GP and MS means appear around studytime group 3, while the lowest means appear in studytime group 1.

This chart clearly supports the significant main effects. It also shows why the interaction is weak: the school difference remains in the same direction across studytime groups.

Python Chart 3: p-value Decision Summary

The p-value chart shows that studytime and school are far below alpha = .05. The interaction p-value is far above alpha at 0.7793.

This is the clearest decision chart. The model supports two significant main effects and one non-significant interaction.

Python Chart 4: Partial Eta Squared by Effect

The effect-size chart shows school as the largest practical effect, with partial eta squared around 0.067. Studytime is close behind at about 0.055.

The interaction effect is very small, about 0.002. This supports a main-effects interpretation and prevents overstatement of the interaction plot.

Python Chart 5: Distribution by Cell

The boxplots show that cell distributions vary in center and spread. GP cells generally sit higher than MS cells, and higher studytime groups generally have higher centers.

Some low outlying values are visible, including very low scores in several cells. These values explain why residual diagnostics should be reviewed before final reporting.

Python Chart 6: Residuals vs Fitted Values

The residual plot shows vertical fitted-value bands because the model uses categorical cell means. Most residuals are scattered around zero, but there are several strong negative residuals.

This diagnostic suggests that the model captures average cell differences, but some individual students have much lower G3 scores than their fitted cell means. These cases should be acknowledged as residual diagnostic context.

Python Chart 7: Residual Q-Q Plot

The Q-Q plot shows visible departure from the reference line, especially in the lower tail. The center of the distribution is closer to the expected pattern than the extremes.

This means residual normality is approximate rather than perfect. The large sample and clear main effects support interpretation, but the residual tail behavior should be reported transparently.

Python Chart 8: Two Way ANOVA Summary Table

The summary table gives the final decision in one place. Studytime and school are significant. The studytime × school interaction is not significant.

This table is the best source for final reporting because it includes the model effects, F values, p-values, effect sizes and decisions.

R Chart-by-Chart Validation

The R validation charts repeat the same Two Way ANOVA workflow in a second software environment. They confirm the interaction pattern, cell means, p-value decisions, effect sizes, distribution checks, residual diagnostics and final summary table.

R Chart 1: Two Way ANOVA Interaction Plot

The R interaction plot confirms the same pattern as Python. GP remains above MS across the studytime levels, and both schools show a broadly upward pattern toward studytime group 3.

The R plot supports the same interpretation: use the graph descriptively, but do not report a significant interaction because the formal interaction test is non-significant.

R Chart 2: Two Way ANOVA Cell Means

The R cell mean chart confirms that GP cell means are consistently higher than MS cell means.

This agreement between R and Python strengthens the main-effect interpretation for school and studytime.

R Chart 3: p-value Decision Summary

The R p-value chart confirms that studytime and school are statistically significant and the studytime × school interaction is not.

This validates the final decision structure across software tools.

R Chart 4: Partial Eta Squared by Effect

The R effect-size chart confirms that school has the largest practical effect, studytime has a meaningful smaller effect, and the interaction is very small.

This supports the same practical interpretation as Python: the article should discuss main effects, not an interaction story.

R Chart 5: Distribution by Cell

The R boxplots confirm the same distribution pattern as Python. Higher studytime and GP cells tend to have higher central values.

Some low outliers remain visible, so assumption and residual diagnostics should be included in the final report.

R Chart 6: Residuals vs Fitted Values

The R residual plot confirms the same fitted-value bands and residual spread seen in Python.

The model explains the group mean structure, but some individual cases deviate strongly from their cell means.

R Chart 7: Residual Q-Q Plot

The R Q-Q plot confirms that residual normality is approximate. Tail departures are visible.

This validates the diagnostic conclusion and supports transparent assumption reporting.

R Chart 8: Two Way ANOVA Summary Table

The R summary table confirms the final result. Studytime and school are significant, while the interaction is not significant.

This software agreement makes the final interpretation stable and suitable for SPSS, R, Python and Excel teaching.

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

The same Two 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 create 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 Two Way ANOVA

SPSS Syntax for Two Way ANOVA

* Two Way ANOVA in SPSS.
* Dependent variable: G3.
* Factors: studytime and school.

TITLE "Two Way ANOVA: G3 by Studytime and School".

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

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

Python Code for Two 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")

df["G3"] = pd.to_numeric(df["G3"], errors="coerce")
df["studytime"] = df["studytime"].astype("category")
df["school"] = df["school"].astype("category")

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

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

anova_table = anova_lm(model, typ=2)
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)

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

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

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

R Code for Two Way ANOVA

# Two 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)

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

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

# Cell means
data %>%
  group_by(studytime, school) %>%
  summarise(
    n = n(),
    mean_G3 = mean(G3),
    sd_G3 = sd(G3),
    .groups = "drop"
  )

# Optional Type III ANOVA
# install.packages("car")
# library(car)
# Anova(lm(G3 ~ studytime * school, data = data), type = 3)

# 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 Two Way ANOVA

Excel support workflow:

1. Arrange the data:
   G3 | studytime | school

2. Create a PivotTable:
   Rows = studytime
   Columns = school
   Values = average of G3

3. Create an interaction chart:
   X-axis = studytime
   Lines = school
   Y-axis = mean G3

4. Create a cell mean chart:
   Clustered bar chart by studytime and school.

5. Create cell boxplots:
   Separate G3 values by studytime and school cell.

6. Formal Two Way ANOVA:
   Use SPSS, R or Python for the complete factorial ANOVA table,
   p-values, interaction test and partial eta squared.

APA Reporting Wording

When reporting Two Way ANOVA, include the dependent variable, both factors, the two main effects, the interaction test, effect sizes and assumption notes. Because the interaction is non-significant, avoid reporting a complex interaction pattern as the main result.

Common Mistakes

When to Use Two Way ANOVA

Use Two Way ANOVA when you have one numeric outcome and two categorical factors. It is common in education, psychology, agriculture, medicine, business experiments and social science research where the researcher wants to test two factors and their interaction in one model.

Two 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 Two Way ANOVA

Use these resources to reproduce the Two 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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