Quick Answer: One Way MANOVA Result

The worked One Way MANOVA example compares four studytime groups across three related grade outcomes: G1, G2 and G3. The mean profile chart shows a clear group pattern. Studytime group 1 is lowest across all three outcomes, group 2 is higher, and groups 3 and 4 are in the highest range.

The follow-up univariate ANOVA chart reports statistically significant group effects for every dependent variable: G1 p = 3.467e-10, G2 p = 4.017e-09 and G3 p = 5.706e-10. The effect-size chart reports partial eta squared values of 0.070 for G1, 0.063 for G2 and 0.069 for G3.

What Is One Way MANOVA?

One Way MANOVA, or one-way multivariate analysis of variance, extends One Way ANOVA to multiple dependent variables. Instead of testing one outcome at a time, MANOVA tests whether groups differ on a combined multivariate outcome profile.

In this example, studytime is the single categorical factor, and G1, G2 and G3 are the dependent variables. These outcomes are conceptually related because they represent grade performance across periods. The correlation matrix confirms this relationship with strong correlations among the three grade outcomes.

The chart set shows a consistent pattern: students in higher studytime groups tend to have higher G1, G2 and G3 values. The follow-up ANOVA p-values are significant for all three outcomes, and the effect-size values are similar across G1, G2 and G3.

One Way MANOVA connects naturally with Factorial ANOVA, Fixed Effects ANOVA, F Distribution, ANOVA Effect Size, Eta Squared, Cohen’s F Formula, ANOVA Assumptions, ANOVA in SPSS, ANOVA in R and ANOVA in Python.

One Way MANOVA Formula

A one-way MANOVA model uses one grouping factor and multiple dependent variables. The model can be written as a multivariate group effect plus multivariate residual error.

The key difference from One Way ANOVA is that the outcome is a vector rather than a single variable. The model tests whether the studytime groups differ on the combined pattern of G1, G2 and G3.

Follow-up Univariate ANOVA Layer

After the multivariate test, follow-up univariate ANOVA results explain which dependent variables show group differences. In this output, the follow-up p-values for G1, G2 and G3 are all far below alpha = .05.

Effect Size Layer

The effect-size chart reports partial eta squared of 0.070 for G1, 0.063 for G2 and 0.069 for G3. These values show that studytime has a similar practical contribution across the three grade outcomes.

One Way MANOVA Hypotheses

One Way MANOVA tests whether the groups have equal mean vectors. A mean vector contains the means of all dependent variables together.

Dataset and MANOVA Variables Used

The worked example uses student performance data. The grouping factor is studytime, with four levels. The dependent variables are G1, G2 and G3. These outcomes are appropriate for a MANOVA demonstration because they are related grade measures rather than unrelated outcomes.

Approximate Mean Profile by Group

The correlation matrix supports the MANOVA setup. G1 and G2 correlate about 0.86, G1 and G3 correlate about 0.83, and G2 and G3 correlate about 0.92. This confirms that the dependent variables are strongly connected and should be interpreted as a related grade profile.

For background, review Descriptive Statistics, Correlation, Mean Median and Mode, Standard Deviation, Effect Size, Confidence Interval and P Value.

SPSS Output Interpretation for One Way MANOVA

The SPSS output PDF should be read in three stages. First, read the multivariate tests table to confirm the overall MANOVA decision. Second, read the tests of between-subjects effects for G1, G2 and G3. Third, read descriptive statistics, effect sizes and diagnostic plots to understand the practical direction and assumptions.

SPSS MANOVA Reading Order

Follow-up Results from the Chart Output

The follow-up results show that the group difference is not limited to one grade outcome. The studytime pattern appears across G1, G2 and G3. The effect sizes are close to each other, so the practical impact is consistent across the grade sequence.

Python Chart-by-Chart Interpretation

The Python chart sequence explains One Way MANOVA through mean profiles, grouped means, follow-up p-values, effect sizes, dependent-variable correlations, boxplots and residual Q-Q plots.

Python Chart 1: MANOVA Mean Profile by Group

This profile chart shows how each studytime group moves across G1, G2 and G3. Group 1 is the lowest line across all three grade outcomes. Group 2 is higher than group 1, while groups 3 and 4 remain in the highest range.

The lines are not random or flat. The grade profile rises with studytime, especially from group 1 to groups 3 and 4. This chart gives the practical visual reason why a MANOVA framework is useful: the group difference appears across a combined set of related outcomes.

Python Chart 2: Grouped Mean Comparison

The grouped bar chart confirms the same mean pattern in a more direct comparison. For G1, G2 and G3, group 1 has the lowest bar and the higher studytime groups have taller bars.

The chart is especially useful because it separates the outcomes visually. It shows that the group difference is not restricted to the final grade G3. The pattern is already visible in G1 and G2, then continues into G3.

Python Chart 3: Follow-up Univariate ANOVA p-values

This chart shows that all three follow-up ANOVA p-values are far below alpha = 0.05. The p-values are 3.467e-10 for G1, 4.017e-09 for G2 and 5.706e-10 for G3.

The chart supports a strong follow-up conclusion. After the multivariate group comparison, each dependent variable shows a significant studytime group effect. This means the grade profile difference is supported across G1, G2 and G3.

Python Chart 4: Univariate Effect Size Summary

The effect-size chart shows that the studytime effect has a similar size across the three grade outcomes. Partial eta squared is 0.070 for G1, 0.063 for G2 and 0.069 for G3.

This means the group effect is not only statistically significant; it also has a consistent practical size across the grade sequence. G1 has the largest effect-size value by a small margin, while G2 has the smallest but still meaningful value.

Python Chart 5: Dependent Variable Correlation Matrix

The correlation matrix shows strong positive relationships among the dependent variables. G1 and G2 correlate 0.86, G1 and G3 correlate 0.83, and G2 and G3 correlate 0.92.

This chart supports the use of MANOVA. The dependent variables are clearly related, but they are not identical. The strongest relationship is between G2 and G3, which makes sense because later grades are closely connected to final grade performance.

Python Chart 6A: G1 Distribution by Studytime

The G1 boxplot shows that group 1 is centered lowest, while groups 3 and 4 have higher medians and means. Group 2 is between the lower and higher groups.

There are low and high outlying values in some groups, but the overall distribution pattern supports the significant follow-up result for G1. The group difference is visible in both the mean marker and the box position.

Python Chart 6B: G2 Distribution by Studytime

The G2 boxplot confirms the same grade improvement pattern. Groups 3 and 4 are centered higher than groups 1 and 2, and group 1 remains the lowest distribution.

Low values appear in groups 1 and 2. These low observations explain why the Q-Q diagnostics later show lower-tail departures. The distribution still supports the significant G2 follow-up result.

Python Chart 6C: G3 Distribution by Studytime

The G3 boxplot shows that the final grade distribution follows the same group order. Group 1 is lowest, group 2 is higher, and groups 3 and 4 are centered in the highest range.

The lower outliers in groups 1 and 2 are visible again. This means final grades have a clear studytime pattern, but individual low-grade cases still remain important for assumption diagnostics.

Python Chart 7A: Residual Q-Q Plot for G1

The G1 residual Q-Q plot shows strong departure from the reference line in the lower tail and visible departure in the upper tail. The central points follow the general direction more closely than the extremes.

This means residual normality for G1 is approximate rather than perfect. The group result is still strong, but the diagnostics should be reported with caution.

Python Chart 7B: Residual Q-Q Plot for G2

The G2 Q-Q plot shows an even stronger lower-tail departure, with several points far below the reference line. The upper side also departs from the line, but the lower tail is the main issue.

This diagnostic pattern matches the low values visible in the G2 boxplot. The MANOVA and follow-up results should be interpreted with this residual shape in mind.

Python Chart 7C: Residual Q-Q Plot for G3

The G3 Q-Q plot shows clear lower-tail departure, with several residual points grouped far below the reference line. The middle of the distribution is closer to the line than the extreme lower tail.

This chart should be reported in the assumption section. The studytime effect on G3 is significant, but residual normality is not perfect because of low final-grade observations.

R Chart-by-Chart Validation

The R validation charts repeat the same One Way MANOVA workflow in a second software environment. The R charts confirm the mean profile, grouped mean comparison, follow-up p-values, effect sizes, dependent-variable correlations, boxplots and residual Q-Q diagnostics.

R Chart 1: MANOVA Mean Profile by Group

The R mean profile confirms the same group structure as the Python profile. Studytime group 1 is consistently lowest, and groups 3 and 4 remain highest across the grade sequence.

This validates the visual MANOVA conclusion. The group difference is visible as a grade profile, not only as one isolated final-grade comparison.

R Chart 2: Grouped Mean Comparison

The R grouped bar chart confirms that studytime group 1 has the lowest bars across G1, G2 and G3, while groups 3 and 4 have the highest bars.

The chart supports the same conclusion as the Python grouped mean comparison: the grade outcome profile improves across higher studytime groups.

R Chart 3: Follow-up Univariate ANOVA p-values

The R p-value chart confirms that all three follow-up ANOVA results are statistically significant. The p-values are far below the alpha = .05 reference line.

This confirms that G1, G2 and G3 each contribute to the studytime group difference rather than only one dependent variable driving the result.

R Chart 4: Univariate Effect Size Summary

The R effect-size chart confirms the same partial eta squared values: G1 is about 0.070, G2 is about 0.063 and G3 is about 0.069.

This validation is important because the practical effect-size interpretation is stable across software. The effect is similar across the three grade outcomes.

R Chart 5: Dependent Variable Correlation Matrix

The R correlation matrix confirms that G1, G2 and G3 are strongly related. The strongest relationship is between G2 and G3, while G1 is also strongly related to both later grade outcomes.

This supports the MANOVA design because the dependent variables form a connected academic performance profile. MANOVA is more appropriate than treating the outcomes as unrelated.

R Chart 6A: G1 Distribution by Studytime

The R G1 boxplot confirms that the lower studytime groups are centered lower than groups 3 and 4. The pattern matches the Python boxplot.

This validates the follow-up result for G1 and supports the direction shown in the mean profile.

R Chart 6B: G2 Distribution by Studytime

The R G2 boxplot shows the same upward group pattern. Groups 3 and 4 are centered higher than groups 1 and 2.

The low observations in the lower groups remain visible, which explains the residual lower-tail behavior in the Q-Q plot.

R Chart 6C: G3 Distribution by Studytime

The R G3 boxplot confirms the same final-grade pattern. Studytime group 1 is lowest, group 2 is higher, and groups 3 and 4 have the highest central values.

This supports the significant follow-up result for G3 and matches the earlier one-way ANOVA pattern for final grade.

R Chart 7A: Residual Q-Q Plot for G1

The R G1 residual Q-Q plot confirms that residual normality is not perfect. The lower tail departs from the reference line, while the center is closer to the expected direction.

This should be reported as diagnostic caution, not as a reason to ignore the group effect.

R Chart 7B: Residual Q-Q Plot for G2

The R G2 residual Q-Q plot shows strong lower-tail departure. This confirms the same diagnostic pattern as the Python chart.

The final report should state that G2 follow-up results are significant, but residual normality is approximate rather than ideal.

R Chart 7C: Residual Q-Q Plot for G3

The R G3 residual Q-Q plot confirms the lower-tail departure seen in the Python output. The lowest residual values sit far from the reference line.

This completes the diagnostic message: the studytime group differences are clear, but the residual assumptions are not perfect and should be reported transparently.

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

The same One Way MANOVA workflow can be reproduced in SPSS, R and Python. Excel can prepare mean tables, charts and correlation matrices, but SPSS, R or Python should be used for the formal MANOVA test.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for One Way MANOVA

SPSS Syntax for One Way MANOVA

* One Way MANOVA in SPSS.
* Factor: studytime.
* Dependent variables: G1 G2 G3.

TITLE "One Way MANOVA: G1 G2 G3 by Studytime".

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

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

Python Code for One Way MANOVA

import pandas as pd
import statsmodels.api as sm
from statsmodels.multivariate.manova import MANOVA
from statsmodels.formula.api import ols

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

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

df["studytime"] = df["studytime"].astype("category")
df_model = df.dropna(subset=["studytime", "G1", "G2", "G3"]).copy()

# One Way MANOVA
manova_model = MANOVA.from_formula("G1 + G2 + G3 ~ C(studytime)", data=df_model)
print(manova_model.mv_test())

# Follow-up univariate ANOVA tables
for outcome in ["G1", "G2", "G3"]:
    model = ols(f"{outcome} ~ C(studytime)", data=df_model).fit()
    table = sm.stats.anova_lm(model, typ=2)
    ss_effect = table.loc["C(studytime)", "sum_sq"]
    ss_error = table.loc["Residual", "sum_sq"]
    partial_eta = ss_effect / (ss_effect + ss_error)
    print(outcome)
    print(table)
    print("Partial eta squared:", partial_eta)

# Mean profile table
mean_profile = df_model.groupby("studytime")[["G1", "G2", "G3"]].mean()
print(mean_profile)

# Dependent variable correlation matrix
corr_matrix = df_model[["G1", "G2", "G3"]].corr()
print(corr_matrix)

R Code for One Way MANOVA

# One 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$studytime <- as.factor(df$studytime)

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

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

# Multivariate tests
summary(model, test = "Pillai")
summary(model, test = "Wilks")

# Follow-up univariate ANOVA
summary.aov(model)

# Mean profile
df_model %>%
  group_by(studytime) %>%
  summarise(
    G1_mean = mean(G1),
    G2_mean = mean(G2),
    G3_mean = mean(G3),
    .groups = "drop"
  )

# Correlation matrix
cor(df_model[, c("G1", "G2", "G3")])

# Boxplots and diagnostics can be created separately for each outcome.

Excel Notes for One Way MANOVA

Excel support workflow:

1. Keep columns:
   studytime, G1, G2, G3

2. Create PivotTable:
   Rows = studytime
   Values = average G1, average G2, average G3

3. Create grouped bar chart:
   X-axis = G1, G2, G3
   Bars = studytime groups

4. Create correlation matrix:
   =CORREL(G1_range, G2_range)
   =CORREL(G1_range, G3_range)
   =CORREL(G2_range, G3_range)

5. Create boxplots:
   Use Insert > Statistic Chart > Box and Whisker for each dependent variable.

6. Formal MANOVA:
   Use SPSS, R or Python for the actual multivariate test.

7. Reporting:
   Report the multivariate test first, then follow-up ANOVA p-values and effect sizes.

APA Reporting Wording

When reporting One Way MANOVA, start with the multivariate test from the SPSS or R output. Then report follow-up univariate ANOVA results, effect sizes, group mean direction and diagnostic notes.

Common Mistakes

When to Use One Way MANOVA

Use One Way MANOVA when there is one categorical grouping factor and two or more related numeric dependent variables. In this example, studytime is the factor, while G1, G2 and G3 are related grade outcomes.

One Way MANOVA should be compared with Fixed Effects ANOVA, Factorial ANOVA, Balanced ANOVA, Brown Forsythe ANOVA, ANOVA Assumptions, ANOVA Effect Size, Eta Squared, and Cohen’s F Formula.

Downloads and Resources for One Way MANOVA

Use these resources to reproduce the One Way MANOVA workflow. The 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.

Related Guides