Quick Answer: Greenhouse-Geisser Correction Result

The verified repeated-measures ANOVA used three repeated grade measurements: G1, G2, and G3. The within-subjects factor was Time with three levels. The means increased from G1 = 11.40 to G2 = 11.57 and G3 = 11.91. Because the same students were measured at all three grade points, the correct design is repeated-measures, not three separate independent groups.

The sphericity assumption was checked using Mauchly’s Test of Sphericity. SPSS reported Mauchly’s W = .827, χ²(2) = 122.814, and p < .001. Since the p-value is below .05, sphericity is violated. Therefore, the Greenhouse-Geisser corrected row should be used instead of the sphericity-assumed row.

Hypothesis-style decision: The repeated-measures Time effect tests whether the repeated grade means are equal. The null hypothesis is H₀: μ_G1 = μ_G2 = μ_G3. The alternative hypothesis is H₁: at least one Time mean is different. After Greenhouse-Geisser correction, the Time effect remains significant, F(1.705, 1104.961) = 36.287, p < .001, partial η² = .053. Therefore, reject the null hypothesis and conclude that the repeated grade means changed significantly across G1, G2 and G3.

What Is Greenhouse-Geisser Correction?

Greenhouse-Geisser Correction is a repeated-measures ANOVA correction used when the sphericity assumption is violated. Sphericity means that the variances of all pairwise difference scores among repeated-measure levels are equal. When sphericity is violated, the usual sphericity-assumed F test can be too liberal. The Greenhouse-Geisser Correction protects the analysis by reducing the degrees of freedom using an epsilon value.

In this guide, the repeated measurements are G1, G2, and G3. These represent three repeated grade measurements for the same students. Because there are three repeated levels, the sphericity assumption applies. SPSS reports Mauchly’s Test of Sphericity, which is significant in this example. Therefore, the correct repeated-measures ANOVA interpretation should use the Greenhouse-Geisser corrected row.

The Greenhouse-Geisser epsilon in the verified SPSS output is .853. An epsilon value closer to 1 means the sphericity violation is not extremely severe, while a lower epsilon means stronger correction. Here, the correction is meaningful but not extreme. The corrected Time effect remains significant with p < .001. When writing the final report, use the corrected F test together with effect size and confidence interval interpretation where appropriate.

Greenhouse-Geisser Formula and Epsilon Logic

The Greenhouse-Geisser Correction adjusts the degrees of freedom by multiplying them by the Greenhouse-Geisser epsilon value. The F statistic itself remains the same in SPSS, but the degrees of freedom and p-value are recalculated using the corrected degrees of freedom.

For the Time effect in this output, the original numerator degrees of freedom were 2. The Greenhouse-Geisser epsilon was .853. Therefore, the corrected numerator degrees of freedom became approximately:

The original error degrees of freedom were 1296. The corrected error degrees of freedom became approximately:

The corrected repeated-measures Time effect was then reported as:

Null and Alternative Hypotheses

The Greenhouse-Geisser Correction is used after a sphericity decision, but the repeated-measures ANOVA still tests the Time effect. For this guide, there are two useful hypothesis sections: one for sphericity and one for the corrected repeated-measures Time effect.

Sphericity Assumption Hypotheses

Corrected Repeated-Measures Time Effect Hypotheses

Dataset and Repeated Measures Used

The worked example uses three repeated grade variables from the same 649 students: G1, G2, and G3. SPSS defines a within-subjects factor named Time with three levels. Because the same students have all three grade measurements, this is a repeated-measures design.

Before interpreting the repeated-measures model, it is useful to review descriptive summaries such as Descriptive Statistics, Frequency Distribution, Five Number Summary, Histogram Interpretation, Box Plot Interpretation, and Coefficient of Variation. These guides help explain the distribution and spread of G1, G2 and G3 before the repeated-measures ANOVA is reported.

Verified SPSS Output Interpretation

The SPSS output provides repeated-measures descriptives, Mauchly’s Test of Sphericity, epsilon values, uncorrected and corrected within-subjects effects, contrast tests, group context by sex, and homogeneity checks. The key reporting result is the Greenhouse-Geisser corrected Time effect.

SPSS Repeated-Measures Descriptive Statistics

SPSS Mauchly’s Test and Epsilon Values

Mauchly’s Test is significant, so the sphericity assumption is violated. Because the Greenhouse-Geisser epsilon is .853, the degrees of freedom are reduced from the sphericity-assumed values to the corrected values. If you need the full assumption explanation, read Mauchly’s Test of Sphericity before reporting the corrected row.

SPSS Tests of Within-Subjects Effects

SPSS Within-Subjects Contrasts

Group Context by Sex

Greenhouse-Geisser Group Model

If the repeated-measures analysis is later combined with categorical summaries, Cross Tabulation can help summarize group membership. For distribution-shape checks before modeling, use Q-Q Plot Normality Check, P-P Plot Normality Check, Kolmogorov-Smirnov Test, Lilliefors Test, D’Agostino-Pearson Test, Cramer-von Mises Test, and Ryan-Joiner Test.

Python Chart-by-Chart Interpretation

The Python charts visualize the Greenhouse-Geisser Correction workflow. They show the repeated-measures mean profile, individual spaghetti profiles, covariance heatmap, epsilon values, corrected p-values, pairwise difference boxplots, and group mean profiles.

Python Chart 1: Repeated-Measures Mean Profile

This chart shows the repeated-measures mean pattern. The average score increases from G1 = 11.40 to G2 = 11.57 and then to G3 = 11.91. This rising profile explains why the repeated-measures Time effect is statistically significant.

The chart shows the substantive mean trend, while the Greenhouse-Geisser Correction handles the assumption problem. Because Mauchly’s Test was significant, the mean profile should be interpreted with the corrected ANOVA row rather than only the sphericity-assumed row. For publication, pair the chart conclusion with Effect Size and Confidence Interval interpretation.

Python Chart 2: Individual Profiles Spaghetti Plot

The spaghetti plot shows that individual students do not all follow exactly the same path across G1, G2 and G3. Some profiles rise, some stay similar, and some may decline. This individual variability is important because repeated-measures ANOVA depends on within-person change patterns.

The Greenhouse-Geisser Correction is relevant because repeated-measures data include dependence across time. The spaghetti plot makes that repeated structure visible and supports the need to treat the observations as related, not independent.

Python Chart 3: Repeated-Measure Covariance Heatmap

The covariance heatmap shows how G1, G2 and G3 are related. Repeated grades are expected to be strongly associated because they come from the same students and measure related academic performance. The correction is not based only on high correlations. It is needed when the sphericity structure is not adequate.

The heatmap helps explain the repeated-measures dependence, while Mauchly’s Test and epsilon values determine whether correction is needed. If your project is a regression-diagnostics project instead of repeated measures, a covariance or residual workflow may connect better with Ramsey RESET Test or Goldfeld-Quandt Test.

Python Chart 4: Epsilon Values

This chart shows the correction values. The Greenhouse-Geisser epsilon is .853, the Huynh-Feldt epsilon is about .855, and the lower-bound epsilon is .500. The Greenhouse-Geisser value is the main correction used in this guide.

Because ε = .853 is close to 1, the correction is moderate rather than severe. The corrected degrees of freedom are lower than the original degrees of freedom, but not dramatically lower. This explains why the corrected p-value remains significant.

Python Chart 5: Corrected p-values

This chart compares the sphericity-assumed result with the Greenhouse-Geisser, Huynh-Feldt and lower-bound corrected results. In the verified output, the Time effect remains p < .001 under each correction method.

This is the main reporting conclusion: the correction changes the degrees of freedom but not the final decision. The grade means differ significantly across G1, G2 and G3 even after correcting for sphericity violation.

Python Chart 6: Pairwise Difference Boxplots

The pairwise difference boxplots help explain why the sphericity assumption matters. The differences between G2 and G1, G3 and G2, and G3 and G1 do not have identical spread. Unequal difference-score variances are the practical reason Mauchly’s Test can become significant.

This chart is useful for teaching because it connects the abstract correction to visible data behavior. The Greenhouse-Geisser Correction adjusts the ANOVA test after those difference-score patterns show that sphericity is not fully satisfied. If the analysis were instead about independent group variance equality, compare the logic with Levene Test, Brown-Forsythe Test, and Cochran C Test.

Python Chart 7: Group Mean Profiles

This chart compares repeated-measures mean profiles for female and male students. Female means are higher at each grade level, while both groups show an upward pattern from G1 to G3. Female means are 11.64, 11.82, and 12.25; male means are 11.06, 11.21, and 11.41.

SPSS shows that the corrected Time effect remains significant when sex is included, but the corrected Time × sex interaction is not significant at p = .071. This means the overall repeated profile changes, but the shape of change is not clearly different between female and male students after correction.

R Chart-by-Chart Validation

The R charts validate the same Greenhouse-Geisser Correction workflow using a separate software environment. They confirm the repeated-measures mean pattern, individual profile variability, covariance structure, epsilon correction values, corrected p-value decision, pairwise difference behavior, and group mean profile interpretation.

R Chart 1: Repeated-Measures Mean Profile

The R mean profile confirms the Python and SPSS pattern. Mean scores increase across the three repeated grade measures. This supports the significant corrected Time effect. It also confirms that the result is not software-specific.

R Chart 2: Individual Profiles Spaghetti Plot

The R spaghetti plot validates the repeated-measures structure by showing within-student grade profiles. It confirms that the data should be analyzed as repeated observations rather than independent groups.

R Chart 3: Repeated-Measure Covariance Heatmap

The R covariance heatmap confirms the repeated-measure dependence among G1, G2 and G3. The grade measures are strongly related, which is expected in a within-subjects design.

R Chart 4: Epsilon Values

The R epsilon chart confirms the correction values. Greenhouse-Geisser epsilon is about .853, meaning corrected degrees of freedom should be used after the sphericity violation.

R Chart 5: Corrected p-values

The R corrected p-value chart validates that the Time effect remains significant after correction. This confirms the SPSS interpretation and supports the final reporting conclusion.

R Chart 6: Pairwise Difference Boxplots

The R pairwise difference boxplots validate the sphericity context. Difference-score distributions are not identical, which supports the need to use the corrected repeated-measures result.

R Chart 7: Group Mean Profiles

The R group profile chart confirms the same pattern as Python and SPSS. Female students have higher mean scores at each grade level, while both groups show a rising profile across G1, G2 and G3.

SPSS, R, Python and Excel Workflows for Greenhouse-Geisser Correction

Greenhouse-Geisser Correction is usually applied by repeated-measures ANOVA software. SPSS reports it automatically when running repeated-measures GLM. R can report it through repeated-measures ANOVA packages. Python can calculate repeated-measures ANOVA and corrections through packages such as pingouin. Excel does not provide a direct Greenhouse-Geisser ANOVA row, but it can support interpretation by calculating difference scores, covariance matrices, mean profiles, and pairwise changes.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

For complete applied reporting, the workflow should not stop at the corrected p-value. Use Descriptive Statistics for the means and spread, Effect Size for magnitude, Confidence Interval for estimation, and Clinical Trial Data Analysis Using R if your repeated-measures structure comes from a medical or experimental study design.

Code Blocks for Greenhouse-Geisser Correction

SPSS Syntax for Greenhouse-Geisser Correction

* Greenhouse-Geisser Correction through repeated-measures GLM.
* Repeated measures: G1 G2 G3.

SET PRINTBACK=OFF MPRINT=OFF.

TITLE "Greenhouse-Geisser Correction for Repeated-Measures ANOVA".

GLM G1 G2 G3
  /WSFACTOR=Time 3 Polynomial
  /METHOD=SSTYPE(3)
  /PRINT=DESCRIPTIVE ETASQ
  /CRITERIA=ALPHA(.05)
  /WSDESIGN=Time.

* Repeated-measures GLM with sex as a between-subjects factor.
GLM G1 G2 G3 BY sex
  /WSFACTOR=Time 3 Polynomial
  /METHOD=SSTYPE(3)
  /PRINT=DESCRIPTIVE ETASQ HOMOGENEITY
  /PLOT=PROFILE(Time*sex)
  /CRITERIA=ALPHA(.05)
  /WSDESIGN=Time
  /DESIGN=sex.

* Pairwise difference diagnostics for correction context.
COMPUTE diff_G2_G1 = G2 - G1.
COMPUTE diff_G3_G2 = G3 - G2.
COMPUTE diff_G3_G1 = G3 - G1.
EXECUTE.

DESCRIPTIVES VARIABLES=diff_G2_G1 diff_G3_G2 diff_G3_G1
  /STATISTICS=MEAN STDDEV VARIANCE MIN MAX.

CORRELATIONS
  /VARIABLES=G1 G2 G3 diff_G2_G1 diff_G3_G2 diff_G3_G1
  /PRINT=TWOTAIL NOSIG
  /MISSING=PAIRWISE.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Greenhouse-Geisser-Correction-SPSS-Output.pdf".

Python Code for Greenhouse-Geisser Correction

import pandas as pd
import numpy as np

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

measures = ["G1", "G2", "G3"]

wide = df[measures + ["sex"]].copy()
for col in measures:
    wide[col] = pd.to_numeric(wide[col], errors="coerce")

wide = wide.dropna(subset=measures).copy()
wide["id"] = np.arange(1, len(wide) + 1)

# Difference score diagnostics
wide["diff_G2_G1"] = wide["G2"] - wide["G1"]
wide["diff_G3_G2"] = wide["G3"] - wide["G2"]
wide["diff_G3_G1"] = wide["G3"] - wide["G1"]

difference_table = wide[["diff_G2_G1", "diff_G3_G2", "diff_G3_G1"]].agg(
    ["mean", "std", "var", "min", "max"]
).T

print(difference_table)

# Long format for repeated-measures ANOVA packages
long_df = wide.melt(
    id_vars=["id", "sex"],
    value_vars=measures,
    var_name="Time",
    value_name="Score"
)

print(long_df.head())

# Optional pingouin workflow:
# pip install pingouin
# import pingouin as pg
# rm = pg.rm_anova(data=long_df, dv="Score", within="Time", subject="id", detailed=True)
# print(rm)
# sph = pg.sphericity(data=wide[measures])
# print(sph)

# Mean profile
mean_profile = long_df.groupby("Time")["Score"].agg(["mean", "std", "count"]).reset_index()
print(mean_profile)

# Group mean profile
group_profile = long_df.groupby(["sex", "Time"])["Score"].agg(["mean", "std", "count"]).reset_index()
print(group_profile)

R Code for Greenhouse-Geisser Correction

# Greenhouse-Geisser Correction in R

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

measures <- c("G1", "G2", "G3")
df[measures] <- lapply(df[measures], as.numeric)
df$id <- seq_len(nrow(df))

df$diff_G2_G1 <- df$G2 - df$G1
df$diff_G3_G2 <- df$G3 - df$G2
df$diff_G3_G1 <- df$G3 - df$G1

difference_table <- data.frame(
  difference = c("G2 - G1", "G3 - G2", "G3 - G1"),
  mean = c(mean(df$diff_G2_G1, na.rm = TRUE),
           mean(df$diff_G3_G2, na.rm = TRUE),
           mean(df$diff_G3_G1, na.rm = TRUE)),
  sd = c(sd(df$diff_G2_G1, na.rm = TRUE),
         sd(df$diff_G3_G2, na.rm = TRUE),
         sd(df$diff_G3_G1, na.rm = TRUE)),
  variance = c(var(df$diff_G2_G1, na.rm = TRUE),
               var(df$diff_G3_G2, na.rm = TRUE),
               var(df$diff_G3_G1, na.rm = TRUE))
)

print(difference_table)

library(tidyr)
library(dplyr)

long_df <- df %>%
  select(id, sex, G1, G2, G3) %>%
  pivot_longer(cols = c(G1, G2, G3),
               names_to = "Time",
               values_to = "Score") %>%
  filter(!is.na(Score))

mean_profile <- long_df %>%
  group_by(Time) %>%
  summarise(
    n = n(),
    mean = mean(Score),
    sd = sd(Score),
    .groups = "drop"
  )

print(mean_profile)

# Optional corrected repeated-measures ANOVA:
# install.packages("rstatix")
# library(rstatix)
# result <- anova_test(data = long_df, dv = Score, wid = id, within = Time)
# print(result)
# get_anova_table(result, correction = "GG")

Excel Support Formulas for Greenhouse-Geisser Context

Assume:
G1 is in column A
G2 is in column B
G3 is in column C

Mean of G1:
=AVERAGE(A2:A650)

Mean of G2:
=AVERAGE(B2:B650)

Mean of G3:
=AVERAGE(C2:C650)

Difference G2 minus G1:
=B2-A2

Difference G3 minus G2:
=C2-B2

Difference G3 minus G1:
=C2-A2

Variance of G2 minus G1:
=VAR.S(D2:D650)

Variance of G3 minus G2:
=VAR.S(E2:E650)

Variance of G3 minus G1:
=VAR.S(F2:F650)

Covariance between G1 and G2:
=COVARIANCE.S(A2:A650,B2:B650)

Covariance between G2 and G3:
=COVARIANCE.S(B2:B650,C2:C650)

Important:
Excel can support the interpretation by calculating means, differences and covariance values.
Excel does not provide a simple built-in Greenhouse-Geisser corrected repeated-measures ANOVA table.
Use SPSS, R or Python for formal epsilon corrections and corrected p-values.

APA Reporting Wording for Greenhouse-Geisser Correction

When reporting Greenhouse-Geisser Correction, include the sphericity decision, epsilon value, corrected degrees of freedom, F statistic, p-value, effect size, and mean profile. This makes the report transparent and prevents the common mistake of reporting the sphericity-assumed row after sphericity has been rejected.

APA-Style Sphericity and Correction Report

APA-Style Corrected Time Effect Report

APA-Style Group Context Report

For a stronger reporting section, pair this APA wording with guides on Effect Size, Confidence Interval, and Central Limit Theorem. If your paper also includes normality diagnostics, connect the repeated-measures report with Q-Q Plot Normality Check and P-P Plot Normality Check.

Common Mistakes in Greenhouse-Geisser Correction Interpretation

When to Use Greenhouse-Geisser Correction

Use Greenhouse-Geisser Correction when a repeated-measures ANOVA has a within-subjects factor with three or more levels and the sphericity assumption is violated. It is especially useful in SPSS repeated-measures GLM because SPSS automatically provides Greenhouse-Geisser, Huynh-Feldt and lower-bound rows.

For a complete assumption and reporting workflow, use Mauchly’s Test of Sphericity, Descriptive Statistics, Effect Size, Confidence Interval, Q-Q Plot Normality Check, P-P Plot Normality Check, and Kolmogorov-Smirnov Test.

Downloads and Resources for Greenhouse-Geisser Correction

The resources below include the SPSS output PDF, Python charts, and R validation charts used in this guide.

Related Statistical Guides

Use these related guides to connect Greenhouse-Geisser Correction with repeated-measures assumptions, sphericity, variance checks, normality checks, effect size, transformation decisions, descriptive statistics and APA reporting.