Quick Answer: Two Way MANOVA Result

The worked Two Way MANOVA tests three multivariate effects. First, it tests whether the combined grade profile G1, G2 and G3 differs by the first factor. Second, it tests whether the same combined grade profile differs by the second factor. Third, it tests whether the two factors interact on the combined outcome profile.

The Python MANOVA summary uses the labels Factor A, Factor B and Factor A × Factor B. Pillai’s Trace p-values show that Factor A is significant, p = 0.0002264, and Factor B is significant, p = 0.0004081. The interaction is not significant, p = 0.9336. The SPSS output labels the design as school_id + studytime + school_id × studytime, and its Pillai’s Trace line also supports significant main effects and a non-significant interaction.

What Is Two Way MANOVA?

Two Way MANOVA is a multivariate extension of Factorial ANOVA. It is used when a researcher has more than one related dependent variable and two categorical independent variables. Instead of testing each outcome separately at the start, MANOVA tests whether the combined outcome vector differs across the factors.

In this example, the three dependent variables are G1, G2 and G3. These grade variables are strongly correlated, which makes MANOVA useful. The two grouping factors are school and studytime. The model tests whether the multivariate grade profile differs by school, differs by studytime, and whether school and studytime interact.

The output supports significant multivariate main effects. The cell mean profile and heatmap show that GP school combinations generally have higher grade profiles than MS school combinations, and higher studytime groups generally have higher grade means. The interaction is not significant, so the main message should not be written as a complex school-by-studytime interaction.

This guide connects naturally with One Way MANOVA, Mixed MANOVA, One Way ANOVA, Factorial ANOVA, Fixed Effects ANOVA, ANOVA Effect Size, Eta Squared, Omega Squared and F Distribution.

Two Way MANOVA Formula

A two-factor MANOVA model can be written as a vector model. Instead of one outcome score, the dependent variable is a vector of several related outcomes.

In this formula, Y is the dependent-variable vector. In this guide, Y contains G1, G2 and G3. The term A is the first grouping factor, B is the second grouping factor, AB is the two-way interaction and e is the multivariate residual vector.

Pillai’s Trace Decision

In MANOVA, H represents the hypothesis sums-of-squares-and-cross-products matrix, and E represents the error matrix. Pillai’s Trace measures how much multivariate variation is explained by the effect. It is often preferred when covariance and variance assumptions are under pressure.

Follow-up Partial Eta Squared Formula

After a significant MANOVA effect, follow-up ANOVA can show which dependent variables contribute to the multivariate result. In this example, follow-up plots show meaningful main-effect patterns for G1, G2 and G3, while the interaction effect sizes are very small.

Two Way MANOVA Hypotheses

Two Way MANOVA has separate multivariate hypotheses for each main effect and for the interaction. Each hypothesis is about the combined dependent-variable vector, not just one outcome.

Dataset and Variables Used

The worked example uses student performance data. The dependent variables are G1, G2 and G3. The grouping factors are school and studytime. The SPSS output reports 649 included cases for the MANOVA tables.

Important Cell Mean Pattern

The cell mean pattern explains the MANOVA result. GP cells generally sit above MS cells, and higher studytime groups generally sit above lower studytime groups. The profiles are not identical, but the formal multivariate interaction is not statistically significant.

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 MANOVA Assumptions

Two Way MANOVA has stronger assumptions than a simple One Way ANOVA. The dependent variables should be numeric and conceptually related. The grouping factors should be categorical. Observations should be independent. The dependent variables should have reasonable multivariate normality, and covariance matrices should be reasonably comparable across groups.

For deeper assumption checks, use Levene Test, Box Plot Interpretation, Q-Q Plot Normality Check, P-P Plot Normality Check, Shapiro-Wilk Test, Anderson-Darling Test, Outlier Detection and Studentized Residuals.

SPSS Output Interpretation for Two Way MANOVA

The SPSS output uses GLM Multivariate with G1, G2 and G3 as dependent variables. The factors are school and studytime, and the model includes school, studytime and school × studytime.

SPSS Reading Order

SPSS Multivariate Test Summary

SPSS Follow-up ANOVA Summary

Python Chart-by-Chart Interpretation

The Python chart sequence explains Two Way MANOVA through Pillai p-values, multivariate cell mean profiles, heatmaps, follow-up effect sizes, follow-up p-values, score scatter, dependent-variable correlations and a summary table.

Python Chart 1: Pillai p-values

The Pillai p-value chart is the primary decision chart. Factor A and Factor B are both below the alpha = .05 decision line, while the interaction is far above alpha.

This means the model supports significant multivariate main effects, but it does not support a significant two-factor multivariate interaction. The article should therefore explain the main effects first and keep the interaction interpretation conservative.

Python Chart 2: Multivariate Cell Mean Profile

The profile plot shows that GP cells generally sit above MS cells, and higher studytime groups generally have higher grade profiles. G3 is usually higher than G1 and G2 within many cells, showing an upward grade profile.

The profiles are not perfectly parallel, but the formal multivariate interaction is not significant. This is why the profile plot should be treated as descriptive evidence of main-effect patterns rather than proof of an interaction.

Python Chart 3: Cell Mean Heatmap

The heatmap makes the cell mean pattern easy to see. GP studytime 3 has the highest visible G3 mean, while MS studytime 1 has the lowest visible grade profile.

This supports the significant main effects. The heatmap also shows that G1, G2 and G3 move together, which supports using MANOVA instead of treating the outcomes as unrelated.

Python Chart 4: Follow-up ANOVA Partial Eta Squared

The follow-up effect-size chart shows that the two main effects have visible partial eta squared values across G1, G2 and G3. The interaction bars are very small for every outcome.

This supports the same final interpretation as the MANOVA table. The main effects explain meaningful variation in the grade outcomes, while the interaction has little practical importance.

Python Chart 5: Follow-up ANOVA p-values

The follow-up p-value chart shows that the main effects are statistically significant across the grade outcomes, while the interaction terms are not significant.

Follow-up ANOVA should not replace MANOVA. It should be used after the multivariate decision to explain which dependent variables carry the significant multivariate pattern.

Python Chart 6: Multivariate Score Scatter

The multivariate scatter plot reduces the three dependent variables into a visual score space. The groups overlap, but there is visible separation among some school and studytime combinations.

This chart is useful for understanding why the main effects are significant while the interaction is not. The groups are not completely separate, but their centers differ enough for the multivariate main effects to be detected.

Python Chart 7: Dependent Variable Correlation

The correlation matrix shows strong positive relationships among the three outcomes. G1 and G2 correlate about 0.86, G2 and G3 correlate about 0.92, and G1 and G3 correlate about 0.83.

This is strong support for MANOVA. When dependent variables are conceptually related and statistically correlated, a multivariate test is more appropriate than immediately running isolated ANOVAs.

Python Chart 8: Two Way MANOVA Summary Table

The summary table gives the final Python result in one place. Both main effects are significant, and the interaction is not significant.

This table is the best Python source for final reporting because it includes Pillai’s Trace, F statistics, p-values and decisions for the main MANOVA effects.

R Chart-by-Chart Validation

The R validation charts repeat the same workflow in a second software environment. They confirm the Pillai p-value pattern, multivariate cell mean profile, heatmap structure, follow-up ANOVA pattern, score scatter, dependent-variable correlation and final summary table.

R Chart 1: Pillai p-values

The R Pillai p-value chart confirms that both main effects are statistically significant and the interaction is not significant.

This agreement between Python and R strengthens the final decision and reduces the chance that the interpretation depends on one software environment.

R Chart 2: Multivariate Cell Mean Profile

The R profile plot confirms the same main pattern: GP cells generally have higher profiles than MS cells, and higher studytime groups generally have higher scores.

The interaction remains descriptive rather than inferential because the formal multivariate interaction p-value is not significant.

R Chart 3: Cell Mean Heatmap

The R heatmap confirms the same visible cell mean structure. Higher GP and higher studytime combinations are generally brighter and higher.

This validates the descriptive pattern behind the significant main effects.

R Chart 4: Follow-up ANOVA Partial Eta Squared

The R effect-size chart confirms that main effects have meaningful follow-up effect sizes across G1, G2 and G3, while interaction effect sizes are very small.

This supports a main-effects report and prevents overstating the interaction.

R Chart 5: Follow-up ANOVA p-values

The R p-value chart confirms that follow-up main effects are significant across the dependent variables, while interaction follow-up tests are not significant.

This software validation supports the same reporting decision as Python and SPSS.

R Chart 6: Multivariate Score Scatter

The R scatterplot confirms that the groups overlap but still show some separation in the combined outcome space.

This matches the main-effect conclusion: the groups are not completely distinct, but their multivariate centers differ enough to produce significant main effects.

R Chart 7: Dependent Variable Correlation

The R correlation matrix confirms that G1, G2 and G3 are strongly related. This is one of the key reasons MANOVA is useful for this example.

If the dependent variables were unrelated, separate ANOVA tests might be more defensible. Here, the strong correlations support a multivariate approach.

R Chart 8: Two Way MANOVA Summary Table

The R summary table confirms the final MANOVA decision. The two main effects are significant, and the interaction is not significant.

This agreement between Python, R and SPSS makes the final interpretation stable for a teaching post and for practical reporting.

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

The same Two Way MANOVA workflow can be reproduced in SPSS, R, Python and Excel. SPSS uses GLM Multivariate. R can use manova(). Python can use statsmodels.multivariate.manova.MANOVA. Excel can prepare cell means, profiles and heatmaps, but SPSS, R or Python is recommended for the formal multivariate test.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Two Way MANOVA

SPSS Syntax for Two Way MANOVA

* Two Way MANOVA in SPSS.
* Dependent variables: G1, G2, G3.
* Factors: school and studytime.

TITLE "Two Way MANOVA: G1 G2 G3 by School and Studytime".

GLM G1 G2 G3 BY school studytime
  /METHOD=SSTYPE(3)
  /INTERCEPT=INCLUDE
  /PRINT=DESCRIPTIVE ETASQ HOMOGENEITY
  /PLOT=PROFILE(school*studytime)
  /CRITERIA=ALPHA(.05)
  /DESIGN=school studytime school*studytime.

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

Python Code for Two Way MANOVA

import pandas as pd
from statsmodels.multivariate.manova import MANOVA
from statsmodels.formula.api import ols
from statsmodels.stats.anova import anova_lm

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

for col in ["G1", "G2", "G3"]:
    df[col] = pd.to_numeric(df[col], errors="coerce")

df["school"] = df["school"].astype("category")
df["studytime"] = df["studytime"].astype("category")

data = df[["G1", "G2", "G3", "school", "studytime"]].dropna().copy()

# Two Way MANOVA
mv_model = MANOVA.from_formula(
    "G1 + G2 + G3 ~ C(school) * C(studytime)",
    data=data
)

print(mv_model.mv_test())

# Correlation among dependent variables
print(data[["G1", "G2", "G3"]].corr())

# Follow-up ANOVA for each dependent variable
for outcome in ["G1", "G2", "G3"]:
    model = ols(f"{outcome} ~ C(school) * C(studytime)", data=data).fit()
    table = anova_lm(model, typ=2)
    error_ss = table.loc["Residual", "sum_sq"]
    table["partial_eta_sq"] = table["sum_sq"] / (table["sum_sq"] + error_ss)
    print("\nOutcome:", outcome)
    print(table)

# Cell means for profile plots and heatmaps
cell_means = data.groupby(["school", "studytime"])[["G1", "G2", "G3"]].mean()
print(cell_means)

R Code for Two Way MANOVA

# Two Way MANOVA in R

library(tidyverse)

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

df$G1 <- as.numeric(df$G1)
df$G2 <- as.numeric(df$G2)
df$G3 <- as.numeric(df$G3)
df$school <- as.factor(df$school)
df$studytime <- as.factor(df$studytime)

data <- df %>%
  select(G1, G2, G3, school, studytime) %>%
  drop_na()

# MANOVA model
model <- manova(cbind(G1, G2, G3) ~ school * studytime, data = data)

# Pillai's Trace is recommended when assumptions are not perfect
summary(model, test = "Pillai")

# Other MANOVA criteria if needed
summary(model, test = "Wilks")
summary(model, test = "Hotelling-Lawley")
summary(model, test = "Roy")

# Follow-up ANOVA
summary.aov(model)

# Dependent variable correlations
cor(data[, c("G1", "G2", "G3")])

# Cell means
data %>%
  group_by(school, studytime) %>%
  summarise(
    n = n(),
    G1_mean = mean(G1),
    G2_mean = mean(G2),
    G3_mean = mean(G3),
    .groups = "drop"
  )

Excel Notes for Two Way MANOVA

Excel support workflow:

1. Arrange the data:
   G1 | G2 | G3 | school | studytime

2. Create a cell mean PivotTable:
   Rows = school and studytime
   Values = average of G1, average of G2, average of G3

3. Create a profile plot:
   X-axis = G1, G2, G3
   Lines = school and studytime cell combinations

4. Create a heatmap:
   Use conditional formatting on the cell mean table.

5. Create dependent-variable correlations:
   =CORREL(G1_range, G2_range)
   =CORREL(G2_range, G3_range)
   =CORREL(G1_range, G3_range)

6. Formal Two Way MANOVA:
   Use SPSS, R or Python for Pillai's Trace, Wilks' Lambda,
   Hotelling's Trace, Roy's Root and follow-up ANOVA.

APA Reporting Wording

When reporting Two Way MANOVA, include the dependent variables, factors, multivariate test statistic, p-values, effect sizes and follow-up ANOVA interpretation. Because Box’s M and Levene diagnostics show assumption pressure, use Pillai’s Trace as the main decision line.

Common Mistakes

When to Use Two Way MANOVA

Use Two Way MANOVA when you have two categorical factors and two or more related numeric dependent variables. It is common in education, psychology, social science, medicine, business experiments and survey research where several outcomes belong to one conceptual family.

Two Way MANOVA should be compared with One Way MANOVA, Mixed MANOVA, Factorial ANOVA, One Way ANOVA, Mixed ANOVA, Fixed Effects ANOVA, Nested ANOVA, ANOVA Assumptions, ANOVA in SPSS, ANOVA in R and ANOVA in Python.

Downloads and Resources for Two Way MANOVA

Use these resources to reproduce the Two Way MANOVA 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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