Quick Answer: Repeated Measures ANOVA Result

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

The overall repeated-measures p-value is 4.637e-16, which is far below alpha = .05. The Bonferroni-adjusted paired comparisons are also significant: G2 − G1 p = 0.01002, G3 − G1 p = 1.035e-11, and G3 − G2 p = 1.408e-10. The effect-size chart reports partial eta squared = 0.053, while eta squared and omega squared are both about 0.005.

What Is Repeated Measures ANOVA?

Repeated Measures ANOVA is an ANOVA test for comparing three or more related means measured on the same subjects. It is used when each participant contributes a score at every repeated time point or condition.

In this example, each student has three grade measurements: G1, G2 and G3. Because these repeated scores belong to the same students, they are correlated. The correlation matrix confirms this: G1–G2 = 0.86, G1–G3 = 0.83 and G2–G3 = 0.92.

The test asks whether the average repeated-measure profile is flat or whether at least one repeated measurement has a different mean. Here, the mean profile is not flat. It rises from G1 to G2 and rises more strongly from G2 to G3.

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

Repeated Measures ANOVA Formula

A repeated-measures model separates the outcome into a subject component, a repeated-measure component and residual error. The subject component is important because every subject is measured more than once.

Here, Y_ij is the score for subject i at repeated measure j. The term μ is the grand mean, τ_j is the repeated-measure effect, s_i is the subject effect, and e_ij is residual error.

F Statistic Formula

The repeated-measures F statistic compares variation explained by repeated time or condition with the appropriate within-subject error term. Because the same subjects are measured repeatedly, the error term is not the same as an independent-groups ANOVA error term.

Effect Size Formula

The effect-size chart reports partial eta squared = 0.053. This means the repeated-measure factor explains a modest portion of the effect-plus-error variation. The same chart reports eta squared and omega squared near 0.005, which are more conservative practical-size summaries.

Repeated Measures ANOVA Hypotheses

The null hypothesis says that the repeated-measure means are equal. The alternative hypothesis says that at least one repeated mean is different.

Dataset and Repeated Measures Variables Used

The worked example uses student grade variables as repeated measurements. Each subject has scores for G1, G2 and G3. These repeated measures are not independent groups; they are repeated observations from the same subjects.

Pairwise Change Summary

The repeated-measure correlation matrix shows strong tracking across grades. G2 and G3 have the strongest correlation at 0.92, followed by G1 and G2 at 0.86 and G1 and G3 at 0.83. This confirms that students’ scores are strongly related across the repeated grade sequence.

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 Repeated Measures ANOVA

The SPSS output PDF should be read in a structured order. Start with the within-subject factor definition, then the descriptive statistics, then Mauchly’s test of sphericity, then the tests of within-subject effects, pairwise comparisons, effect size and residual diagnostics.

SPSS Reading Order

SPSS Result Summary

The SPSS interpretation should emphasize the repeated nature of the data. Because the same subjects contribute G1, G2 and G3, the analysis controls for stable subject-to-subject differences. The significant result means the average within-subject grade profile changes across repeated measurements.

Python Chart-by-Chart Interpretation

The Python chart sequence explains Repeated Measures ANOVA through mean profiles, individual subject profiles, repeated-measure distributions, pairwise changes, p-value decisions, effect sizes, repeated-measure correlations and residual diagnostics.

Python Chart 1: Repeated Measures Mean Profile

The mean profile chart shows a steady upward trend. The mean is about 11.40 at G1, about 11.57 at G2 and about 11.91 at G3.

The shaded confidence band shows uncertainty around the repeated-measure means. The line still moves upward across the sequence, which supports the conclusion that the repeated grade profile is not flat.

Python Chart 2: Individual Subject Profiles

The individual profile chart shows that students do not all follow exactly the same path. Some lines rise, some stay close to flat, and a few show strong changes across repeated measures.

The bold overall line still rises from G1 to G3. This means the average repeated-measure trend is upward even though individual students vary around that pattern.

Python Chart 3: Distribution by Repeated Measure

The boxplots show that the distributions shift upward from G1 and G2 toward G3. G3 has the highest central position, while G1 and G2 are slightly lower.

Low outlying values are visible, especially around zero for some grade measures. These low values help explain why the residual Q-Q plot later shows lower-tail departure.

Python Chart 4: Pairwise Mean Changes

The pairwise change chart shows a mean increase of 0.17 from G1 to G2, 0.51 from G1 to G3 and 0.34 from G2 to G3.

The largest change is from G1 to G3, which makes sense because it spans the full repeated-measure sequence. The G2 to G3 change is also clear, while G1 to G2 is smaller but still positive.

Python Chart 5: p-value Decision Summary

The p-value decision chart shows that the overall repeated effect has p = 4.637e-16. The paired comparisons are also below alpha = .05: G2 − G1 p = 0.01002, G3 − G1 p = 1.035e-11 and G3 − G2 p = 1.408e-10.

The result is not borderline. The repeated grade means differ overall, and every pairwise comparison is statistically significant after adjustment.

Python Chart 6: Effect Size Summary

The effect-size chart reports eta squared about 0.005, partial eta squared about 0.053 and omega squared about 0.005. Partial eta squared is the largest because it uses an effect-plus-error denominator.

The practical interpretation should be careful. The repeated-measures effect is statistically strong, but the effect sizes show that the average grade increase is modest rather than large.

Python Chart 7: Repeated Measure Correlation Matrix

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

The strongest relationship is between G2 and G3. This supports the repeated-measures design because the same students’ scores track strongly over time.

Python Chart 8: Residuals vs Fitted Values

The residuals-versus-fitted chart shows vertical bands because the model predicts repeated-measure means. Most residuals are distributed around zero, but some negative residuals fall below -10.

This plot shows that the model captures the mean pattern, but some individual observations are much lower than their fitted repeated-measure mean. The final report should include this diagnostic caution.

Python Chart 9: Residual Q-Q Plot

The residual Q-Q plot shows visible departure from the reference line, especially in the lower tail. The middle residuals follow the general direction more closely than the extreme lower residuals.

This means residual normality is approximate rather than perfect. The repeated-measures result is statistically clear, but the diagnostics should be described honestly.

R Chart-by-Chart Validation

The R validation charts repeat the same workflow in a second software environment. They confirm the repeated-measure mean profile, individual subject variation, distribution pattern, pairwise changes, p-value decision, effect-size summary, correlation matrix and residual diagnostics.

R Chart 1: Repeated Measures Mean Profile

The R mean profile confirms the Python result. The average grade profile increases from G1 to G2 and reaches its highest point at G3.

This software-to-software agreement strengthens the conclusion that the repeated-measure effect is a real feature of the data.

R Chart 2: Individual Subject Profiles

The R individual profile chart confirms that student-level paths vary. The overall average trend still moves upward, even though individual students follow different patterns.

This chart is important because repeated-measures data always contain both average change and individual change variation.

R Chart 3: Distribution by Repeated Measure

The R distribution chart confirms that G3 is centered higher than G1 and G2. The visible low values also match the Python distribution chart.

The distribution pattern supports the repeated-measures result and explains why residual diagnostics show lower-tail behavior.

R Chart 4: Pairwise Mean Changes

The R pairwise change chart confirms the same mean differences: G2 − G1 is about 0.17, G3 − G1 is about 0.51 and G3 − G2 is about 0.34.

The largest repeated-measure change is again G3 minus G1, confirming the practical direction of the result.

R Chart 5: p-value Decision Summary

The R p-value chart confirms that the overall repeated effect and all pairwise repeated comparisons are statistically significant at alpha = .05.

This validates the formal decision across workflows. The repeated means differ overall, and the post-analysis pairwise comparisons explain where the differences occur.

R Chart 6: Effect Size Summary

The R effect-size chart confirms that partial eta squared is about 0.053, while eta squared and omega squared are about 0.005.

This repeated validation supports a careful wording: the result is statistically significant, but the practical effect size is modest.

R Chart 7: Repeated Measure Correlation Matrix

The R correlation matrix confirms strong repeated-measure correlations. G2 and G3 have the strongest relationship, while G1 is also strongly related to both later grade measures.

This confirms that G1, G2 and G3 are related repeated measurements from the same subjects, which supports using repeated-measures ANOVA instead of independent-groups ANOVA.

R Chart 8: Residuals vs Fitted Values

The R residuals-versus-fitted chart confirms the same diagnostic pattern as Python. Most residuals are centered around zero, but there are lower negative residuals.

This diagnostic supports transparent reporting. The model shows a clear repeated-measure effect, but residual behavior is not perfectly ideal.

R Chart 9: Residual Q-Q Plot

The R Q-Q plot confirms the lower-tail departure visible in Python. The central residuals are closer to the reference direction than the extreme lower residuals.

The final diagnostic message is consistent across software: the repeated-measures result is strong, but residual normality should be described as approximate.

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

The same Repeated Measures ANOVA workflow can be reproduced in SPSS, R, Python and Excel. SPSS uses the GLM Repeated Measures menu. R can use repeated-measures ANOVA functions or long-format models. Python can use long-format repeated-measures tools. Excel can calculate supporting summaries and paired changes, but SPSS, R or Python is better for the formal repeated-measures test.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Repeated Measures ANOVA

SPSS Syntax for Repeated Measures ANOVA

* Repeated Measures ANOVA in SPSS.
* Repeated measures: G1, G2, G3.

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

GLM G1 G2 G3
  /WSFACTOR=GradeTime 3 Polynomial
  /METHOD=SSTYPE(3)
  /PRINT=DESCRIPTIVE ETASQ
  /EMMEANS=TABLES(GradeTime) COMPARE ADJ(BONFERRONI)
  /PLOT=PROFILE(GradeTime)
  /CRITERIA=ALPHA(.05)
  /WSDESIGN=GradeTime.

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

Python Code for Repeated Measures ANOVA

import pandas as pd
from statsmodels.stats.anova import AnovaRM
from scipy import stats
from itertools import combinations

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()
wide["subject_id"] = range(1, len(wide) + 1)

long = wide.melt(
    id_vars="subject_id",
    value_vars=["G1", "G2", "G3"],
    var_name="time",
    value_name="score"
)

# Repeated Measures ANOVA
rm_model = AnovaRM(
    data=long,
    depvar="score",
    subject="subject_id",
    within=["time"]
).fit()

print(rm_model)

# Means
means = long.groupby("time")["score"].mean()
print(means)

# Pairwise paired t tests with Bonferroni adjustment
pairs = list(combinations(["G1", "G2", "G3"], 2))
m = len(pairs)

for a, b in pairs:
    t_stat, p_value = stats.ttest_rel(wide[b], wide[a])
    p_adj = min(p_value * m, 1.0)
    mean_change = (wide[b] - wide[a]).mean()
    print(f"{b} - {a}: mean change = {mean_change:.3f}, Bonferroni p = {p_adj:.6g}")

# Correlation matrix
print(wide[["G1", "G2", "G3"]].corr())

# Residuals around repeated-measure means
long["fitted"] = long.groupby("time")["score"].transform("mean")
long["residual"] = long["score"] - long["fitted"]
print(long.head())

R Code for Repeated Measures ANOVA

# Repeated Measures ANOVA 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)

wide <- df %>%
  select(G1, G2, G3) %>%
  drop_na() %>%
  mutate(subject_id = row_number())

long <- wide %>%
  pivot_longer(
    cols = c(G1, G2, G3),
    names_to = "time",
    values_to = "score"
  )

# Repeated Measures ANOVA
model <- aov(score ~ time + Error(subject_id/time), data = long)
summary(model)

# Repeated-measure means
long %>%
  group_by(time) %>%
  summarise(
    n = n(),
    mean = mean(score),
    sd = sd(score),
    .groups = "drop"
  )

# Bonferroni-adjusted pairwise paired t tests
pairwise.t.test(
  x = long$score,
  g = long$time,
  paired = TRUE,
  p.adjust.method = "bonferroni"
)

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

# Residual diagnostics
long <- long %>%
  group_by(time) %>%
  mutate(
    fitted = mean(score),
    residual = score - fitted
  ) %>%
  ungroup()

qqnorm(long$residual)
qqline(long$residual)

Excel Formulas for Repeated Measures ANOVA Support

Step 1:
Keep repeated measures on the same row:
Subject ID | G1 | G2 | G3

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

Step 3:
Calculate pairwise changes:
G2 - G1 = G2_cell - G1_cell
G3 - G1 = G3_cell - G1_cell
G3 - G2 = G3_cell - G2_cell

Step 4:
Calculate average changes:
=AVERAGE(change_range)

Step 5:
Run paired t tests:
=T.TEST(G2_range, G1_range, 2, 1)
=T.TEST(G3_range, G1_range, 2, 1)
=T.TEST(G3_range, G2_range, 2, 1)

Step 6:
Apply Bonferroni adjustment:
Adjusted p = raw p * number_of_pairwise_tests

Step 7:
Create mean profile:
Insert line chart using G1, G2 and G3 means.

Step 8:
Formal Repeated Measures ANOVA:
Use SPSS, R or Python for the full within-subject ANOVA table, sphericity tests and corrected p-values.

APA Reporting Wording

When reporting Repeated Measures ANOVA, include the repeated factor, repeated measures, overall test result, effect size, pairwise comparison results and assumption note. If sphericity is violated, report the corrected Greenhouse-Geisser or Huynh-Feldt result from SPSS or R.

Common Mistakes

When to Use Repeated Measures ANOVA

Use Repeated Measures ANOVA when the same subjects are measured three or more times or under three or more related conditions. It is commonly used for time points, pre-mid-post testing, repeated clinical measurements, repeated exam scores, repeated lab measures and matched condition designs.

Repeated Measures ANOVA should be compared with One Way ANOVA, Mixed ANOVA, Mixed MANOVA, Fixed Effects ANOVA, Factorial ANOVA, ANOVA Assumptions, ANOVA Effect Size, F Distribution, and Cohen’s F Formula.

Downloads and Resources for Repeated Measures ANOVA

Use these resources to reproduce the Repeated Measures ANOVA 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.

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