Quick Answer: Repeated Measures MANOVA Result

The worked Repeated Measures MANOVA compares the same students across three repeated outcomes: G1, G2 and G3. The mean profile increases from about 11.40 at G1 to about 11.57 at G2 and about 11.91 at G3.

The Python multivariate change-score test reports Hotelling T² = 59.8558, F = 29.8817, p = 3.86691e-13, and partial eta squared = 0.0846. Because the p-value is far below alpha = .05, the repeated-measure change vector is statistically significant.

The SPSS repeated-measures MANOVA output also supports a significant multivariate Time effect. The SPSS multivariate table reports Pillai’s Trace = .049, Wilks’ Lambda = .951, Hotelling’s Trace = .051 and Roy’s Largest Root = .051, each with F = 16.435, df = 2, 644, p < .001, and partial eta squared = .049. The Time × studytime interaction is not significant in the SPSS multivariate output, with Pillai p = .089.

What Is Repeated Measures MANOVA?

Repeated Measures MANOVA is a multivariate version of a within-subject repeated-measures analysis. It tests whether repeated outcomes measured on the same subjects change jointly across time points or conditions.

In this example, each student has three repeated grade outcomes: G1, G2 and G3. These outcomes are not independent because they belong to the same students. The repeated-measure correlation matrix confirms this relationship: G1–G2 = .86, G1–G3 = .83 and G2–G3 = .92.

The multivariate approach is useful because it can test repeated-measure change through a change-score vector, such as G2 − G1 and G3 − G1, rather than relying only on the univariate sphericity-based ANOVA table.

Repeated Measures MANOVA connects naturally with Mixed MANOVA, Mixed ANOVA, Mauchly’s Test of Sphericity, Greenhouse-Geisser Correction, Huynh-Feldt Correction, F Distribution, Partial Eta Squared, Eta Squared, Omega Squared, and ANOVA Effect Size.

Repeated Measures MANOVA Formula

A repeated-measures MANOVA can be written as a multivariate test of repeated-measure contrasts. With three repeated measurements, two independent change contrasts are enough to test whether the repeated profile is flat.

The null hypothesis says that the mean change-score vector equals zero. In this example, the average changes are positive, so the repeated grade profile rises over time.

Hotelling T² Formula

Here, d̄ is the vector of average repeated-measure changes, and S_d is the covariance matrix of the change scores. Hotelling T² tests whether the whole change vector is different from zero.

F Conversion Formula

In the Python output, T² = 59.8558, F = 29.8817, and p = 3.86691e-13. The result rejects the no-change multivariate null hypothesis.

Effect Size Formula

The Python summary table reports partial eta squared = 0.0846. This is a moderate repeated-measure multivariate effect in this example. SPSS also reports a significant multivariate Time effect with partial eta squared of .049 in the GLM repeated-measures output.

Repeated Measures MANOVA Hypotheses

The main hypothesis focuses on the repeated-measure change-score vector. The test asks whether the repeated profile is flat or whether the vector of repeated changes is different from zero.

Dataset and Repeated Measures Variables Used

The worked example uses student grade variables as repeated measurements. Each student contributes scores for G1, G2 and G3. The SPSS output uses 649 valid cases and also includes studytime as an optional between-subject factor in the GLM repeated-measures setup.

Repeated-Measure Correlation Summary

The strong repeated-measure correlations support the use of a within-subject multivariate approach. The scores are related enough to justify modeling them as a repeated grade profile rather than as unrelated independent outcomes.

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

SPSS Output Interpretation for Repeated Measures MANOVA

The SPSS output uses the General Linear Model repeated-measures framework. It defines Time as the within-subject factor with three levels: G1, G2 and G3. It also includes studytime as a between-subject factor, so the output reports both the multivariate Time effect and the Time × studytime interaction.

SPSS Reading Order

SPSS Descriptive Statistics

SPSS Multivariate Time Effect

SPSS Time × Studytime Interaction

The SPSS interpretation is clear. The repeated-measure Time effect is significant, which means the grade profile changes across G1, G2 and G3. The Time × studytime interaction is not significant, so the evidence does not show a strong difference in the repeated-change pattern across studytime groups.

Python Chart-by-Chart Interpretation

The Python chart sequence explains Repeated Measures MANOVA through mean profiles, individual trajectories, change-score distributions, repeated-measure correlations, studytime group profiles, Hotelling T², change-score confidence intervals and the final summary table.

Python Chart 1: Repeated-Measure Mean Profile

The mean profile chart shows that the repeated-measure average rises across the grade sequence. G1 is about 11.40, G2 is about 11.57, and G3 is about 11.91.

This chart gives the simplest visual explanation of the result. The profile is not flat. The average student score increases across the repeated measures, especially by the time G3 is reached.

Python Chart 2: Individual Repeated-Measure Trajectories

The individual trajectory plot shows that not every student follows the same path. Some individual lines rise, some are flat, and some fall between G1, G2 and G3.

The thick overall line still rises across the repeated outcomes. This means individual variation exists, but the average multivariate repeated-measure trend remains positive.

Python Chart 3: Change Score Boxplots

The change-score boxplots show that the distributions are centered around small positive changes. The G3 − G1 contrast has the clearest positive average, while G2 − G1 is smaller.

The chart also shows outlying positive and negative changes. This is important because repeated-measure change is not identical for every student. The MANOVA result tests the average change vector while still allowing individual differences around that vector.

Python Chart 4: Repeated-Measure Correlation Heatmap

The correlation heatmap shows strong positive relationships among the repeated outcomes. G1 and G2 correlate .86, G1 and G3 correlate .83, and G2 and G3 correlate .92.

This chart supports the multivariate repeated-measures setup. The repeated outcomes are strongly connected, especially G2 and G3, but they are not identical. That makes a joint repeated-measure test meaningful.

Python Chart 5: Repeated-Measure Profiles by Studytime Group

The group profile chart shows that studytime group 1 has the lowest grade profile, group 2 is higher, and groups 3 and 4 are in the highest range. Most groups show an upward movement toward G3.

The lines are not perfectly parallel, but the SPSS multivariate interaction test does not show a significant Time × studytime effect at alpha .05. The practical conclusion is that the grade profile rises overall, while studytime group differences mainly reflect different average grade levels.

Python Chart 6: Hotelling T² Summary for Time Effect

The Hotelling T² summary chart shows alpha = .05, p = 3.867e-13 and partial eta squared = 0.08456. The p-value is far below alpha.

This chart is the main Python decision output. It confirms that the repeated-measure change vector is statistically significant and that the time effect has a meaningful effect-size estimate.

Python Chart 7: Mean Change Scores with 95% Confidence Intervals

The mean change chart shows that G2 − G1 is about 0.17, G3 − G2 is about 0.34, and G3 − G1 is about 0.51. The largest mean change is the full G1 to G3 contrast.

The confidence intervals help explain the direction and uncertainty of each change. The overall pattern is positive, with the strongest practical change occurring between the first and final repeated measure.

Python Chart 8: Repeated Measures MANOVA Summary Table

The summary table reports Hotelling T² = 59.8558, F = 29.8817, p = 3.86691e-13 and partial η² = 0.0846.

This table is the best compact Python result for reporting. It gives the statistic, inferential decision and effect-size estimate in one place.

R Chart-by-Chart Validation

The R validation charts repeat the same Repeated Measures MANOVA workflow in a second software environment. They confirm the mean profile, individual trajectories, change-score distributions, correlation heatmap, group profiles, Hotelling result, change-score confidence intervals and summary table.

R Chart 1: Repeated-Measure Mean Profile

The R mean profile confirms the Python result. The average grade profile rises from G1 through G3.

This validation shows that the positive repeated-measure trend is stable across software workflows.

R Chart 2: Individual Repeated-Measure Trajectories

The R trajectory plot confirms that individual students vary in their repeated-score paths. Some trajectories move upward, while others move less consistently.

The average path remains upward, supporting the same multivariate time-change conclusion.

R Chart 3: Change Score Boxplots

The R boxplots confirm the same change-score structure. The average changes are positive, but individual change scores vary around the mean.

This supports the repeated-measures MANOVA logic because the test evaluates a multivariate change vector rather than assuming identical change for every subject.

R Chart 4: Repeated-Measure Correlation Heatmap

The R heatmap confirms strong repeated-measure correlations. G2 and G3 remain the most strongly related measures.

This validates the decision to treat the grade outcomes as a repeated multivariate profile.

R Chart 5: Repeated-Measure Profiles by Studytime Group

The R group profile chart confirms that studytime groups differ in grade level. Group 1 is lowest, group 2 is higher, and groups 3 and 4 are highest.

The practical pattern matches the Python chart. The repeated grade profile rises overall, while studytime group membership mainly separates students by average grade level.

R Chart 6: Hotelling T² Summary for Time Effect

The R Hotelling summary confirms that the p-value is far below alpha and the partial eta squared value is meaningful.

This validates the main multivariate time-effect result outside Python.

R Chart 7: Mean Change Scores with 95% Confidence Intervals

The R mean change chart confirms the same ordering: G3 − G1 is the largest change, G3 − G2 is next, and G2 − G1 is the smallest.

This supports the final interpretation that most of the average improvement is seen by the final repeated measurement.

R Chart 8: Repeated Measures MANOVA Summary Table

The R summary table confirms the same formal result as the Python table. The repeated-measure change vector is statistically significant.

This software validation strengthens the final report because Python, R and SPSS all support a significant repeated-measure time effect.

SPSS, R, Python and Excel Workflows for Repeated Measures MANOVA

The same Repeated Measures MANOVA workflow can be reproduced in SPSS, R, Python and Excel. Python and R can run the Hotelling T² change-vector test directly. SPSS can run the GLM repeated-measures multivariate tests. Excel can calculate means, change scores 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 Repeated Measures MANOVA

SPSS Syntax for Repeated Measures MANOVA

* Repeated Measures MANOVA in SPSS.
* Repeated measures: G1, G2, G3.
* Optional between-subject factor: studytime.

TITLE "Repeated Measures MANOVA: G1, G2 and G3".

GLM G1 G2 G3 BY studytime
  /WSFACTOR=Time 3 Polynomial
  /MEASURE=Grade
  /METHOD=SSTYPE(3)
  /PRINT=DESCRIPTIVE ETASQ HOMOGENEITY
  /PLOT=PROFILE(Time*studytime)
  /CRITERIA=ALPHA(.05)
  /WSDESIGN=Time
  /DESIGN=studytime.

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

Python Code for Repeated Measures MANOVA

import pandas as pd
import numpy as np
from scipy import stats

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

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

wide = df[["G1", "G2", "G3"]].dropna().copy()

# Repeated-measure change-score matrix
# Two independent contrasts for three repeated outcomes
D = np.column_stack([
    wide["G2"] - wide["G1"],
    wide["G3"] - wide["G1"]
])

n, p = D.shape

mean_vector = D.mean(axis=0)
cov_matrix = np.cov(D, rowvar=False)

# Hotelling T-squared
T2 = n * mean_vector.T @ np.linalg.inv(cov_matrix) @ mean_vector

# Convert T2 to F
F_value = ((n - p) / (p * (n - 1))) * T2
df1 = p
df2 = n - p
p_value = stats.f.sf(F_value, df1, df2)

partial_eta_squared = T2 / (T2 + n - 1)

print("Mean vector:", mean_vector)
print("Hotelling T2:", T2)
print("F:", F_value)
print("df1:", df1)
print("df2:", df2)
print("p-value:", p_value)
print("Partial eta squared:", partial_eta_squared)

# Repeated-measure means
print(wide[["G1", "G2", "G3"]].mean())

# Change score means
wide["G2_G1"] = wide["G2"] - wide["G1"]
wide["G3_G2"] = wide["G3"] - wide["G2"]
wide["G3_G1"] = wide["G3"] - wide["G1"]
print(wide[["G2_G1", "G3_G2", "G3_G1"]].mean())

# Repeated-measure correlation matrix
print(wide[["G1", "G2", "G3"]].corr())

R Code for Repeated Measures MANOVA

# Repeated Measures MANOVA in R using Hotelling T-squared

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)

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

D <- cbind(
  G2_G1 = wide$G2 - wide$G1,
  G3_G1 = wide$G3 - wide$G1
)

n <- nrow(D)
p <- ncol(D)

mean_vector <- colMeans(D)
cov_matrix <- cov(D)

T2 <- n * t(mean_vector) %*% solve(cov_matrix) %*% mean_vector
T2 <- as.numeric(T2)

F_value <- ((n - p) / (p * (n - 1))) * T2
df1 <- p
df2 <- n - p
p_value <- pf(F_value, df1, df2, lower.tail = FALSE)

partial_eta_squared <- T2 / (T2 + n - 1)

T2
F_value
df1
df2
p_value
partial_eta_squared

colMeans(wide[, c("G1", "G2", "G3")])
cor(wide[, c("G1", "G2", "G3")])

Excel Notes for Repeated Measures MANOVA

Excel support workflow:

1. Keep repeated measures on the same row:
   Subject_ID | G1 | G2 | G3

2. Calculate repeated-measure means:
   G1 mean = AVERAGE(G1_range)
   G2 mean = AVERAGE(G2_range)
   G3 mean = AVERAGE(G3_range)

3. Calculate change scores:
   G2_G1 = G2 - G1
   G3_G2 = G3 - G2
   G3_G1 = G3 - G1

4. Calculate mean changes:
   =AVERAGE(change_score_range)

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

6. Create charts:
   - repeated-measure mean profile
   - change-score boxplots
   - correlation matrix
   - group profiles if a between-subject factor exists

7. Formal MANOVA:
   Use SPSS, Python or R for Hotelling T², Wilks, Pillai, Hotelling Trace or Roy's Root.

APA Reporting Wording

When reporting Repeated Measures MANOVA, include the repeated outcomes, the multivariate test statistic, F statistic, degrees of freedom, p-value, effect size, mean profile and change-score direction. When using SPSS, also state whether the Time × between-subject factor interaction was significant.

Common Mistakes

When to Use Repeated Measures MANOVA

Use Repeated Measures MANOVA when the same subjects have multiple related repeated outcomes and the research question focuses on whether the repeated outcome profile changes jointly. It is especially useful when a multivariate approach to repeated contrasts is preferred or when the analyst wants to avoid relying only on the sphericity-based univariate repeated-measures ANOVA layer.

Repeated Measures MANOVA should be compared with Mixed MANOVA, Mixed ANOVA, One Way MANOVA, One Way ANOVA, One Way ANCOVA, Fixed Effects ANOVA, Factorial ANOVA, ANOVA Assumptions, and ANOVA Effect Size.

Downloads and Resources for Repeated Measures MANOVA

Use these resources to reproduce the Repeated Measures 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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