Greenhouse Geisser Correction: Formula, SPSS, R, Python and Excel Guide 1

What Is the Greenhouse Geisser Correction?

The Greenhouse Geisser Correction is used in repeated-measures ANOVA when the assumption of sphericity is not satisfied. Repeated-measures ANOVA compares the same subjects across three or more related conditions. In this example, the same students are measured on G1, G2 and G3, so the observations are not independent in the ordinary between-groups sense.

The important assumption is sphericity. Sphericity means that the variances of the paired differences between repeated measures are equal enough for the ordinary F-test to be trusted. If that assumption is violated, the standard repeated-measures ANOVA can become too liberal. In simple terms, it may report significance too easily because the degrees of freedom are too optimistic.

The Greenhouse-Geisser correction solves this problem by reducing the degrees of freedom using an epsilon value. It does not change the means. It does not change the raw grade scores. It changes the degrees of freedom used to calculate the corrected p-value.

If you are building a complete data-analysis workflow, you may also find these related guides useful: DAgostino Pearson Test for skewness-kurtosis normality checking, Cramer von Mises Test for distribution goodness-of-fit, Cochran C Test for testing the largest variance, and Brown Forsythe Test for robust equality-of-variance analysis.

Greenhouse Geisser Correction Formula

The core idea is simple. The ordinary repeated-measures ANOVA degrees of freedom are multiplied by the Greenhouse-Geisser epsilon value:

Corrected df = ε × original df

Here, ε is the epsilon correction factor. If epsilon is close to 1, the correction is mild. If epsilon is much smaller, the correction is stronger. With three repeated measures, the lowest possible epsilon is 0.500, also called the lower-bound correction.

Formula applied to this example

In this dataset, the original effect degrees of freedom are 2, and the original error degrees of freedom are 1296. The Greenhouse-Geisser epsilon is 0.853. Therefore, the corrected effect df becomes about 1.705, and the corrected error df becomes about 1104.961.

Effect df: 2 × 0.853 ≈ 1.705
Error df: 1296 × 0.853 ≈ 1104.961

Greenhouse Geisser Correction Null Hypothesis and Alternative Hypothesis

The Greenhouse-Geisser row belongs to the repeated-measures ANOVA test. The hypothesis is still about whether the repeated-measure means are equal.

Because the corrected p-value is less than .001, the null hypothesis is rejected. The data show a statistically significant change in mean grades across the three repeated grade measures.

Dataset and Repeated-Measures Model Used

This worked example uses the student-por.csv student performance dataset. The repeated outcome variables are G1, G2 and G3. These represent grade measurements for the same students, so they are suitable for a repeated-measures ANOVA example.

External dataset source: UCI Machine Learning Repository: Student Performance dataset.

Verified Results in SPSS, R and Python

The analysis was reproduced with SPSS output and supporting R/Python-style workflows. The key result is that the ordinary repeated-measures ANOVA is significant, Mauchly’s test is also significant, and therefore the corrected Greenhouse-Geisser row should be reported.

Descriptive Statistics for G1, G2 and G3

Pairwise Difference Variances for Sphericity Check

The paired-difference variances are not equal. The variance for G3 − G1 is 3.315, while the variance for G3 − G2 is only 1.634. This is the practical reason why the sphericity assumption is not satisfied.

Mauchly’s Test of Sphericity

Epsilon Values

Tests of Within-Subjects Effects

The F statistic is the same across correction rows because the correction changes the degrees of freedom, not the basic mean-square ratio. The important change is in the degrees of freedom and the corrected p-value. In this example, all rows remain significant, so the conclusion is stable.

Greenhouse Geisser Correction Charts and Interpretation

1. Degrees of Freedom Before and After Sphericity Correction

This chart shows the main purpose of the Greenhouse Geisser Correction. The sphericity-assumed row uses the full original degrees of freedom. The Greenhouse-Geisser row reduces both the effect df and the error df. This reduction protects the repeated-measures ANOVA from becoming too liberal when sphericity is violated.

2. Sphericity Correction Epsilon Comparison

The epsilon chart shows that the Greenhouse-Geisser value is about 0.853. This is below 1, so some correction is needed, but it is not close to the lower-bound value of 0.500. The violation is real, but the correction is not extremely severe.

3. G1, G2 and G3 Grade Distributions

The grade distribution chart shows that G3 has a slightly higher central tendency than G1 and G2. It also has a wider spread. This supports the descriptive pattern shown in the means: student grades move upward on average from G1 to G3, but individual student patterns still vary.

4. Mean G1, G2 and G3 Grades with Standard Error

The mean profile is one of the easiest charts for readers to understand. The mean rises from 11.3991 to 11.5701 and then to 11.9060. The increase is not huge, but because the same 649 students are measured repeatedly, the repeated-measures ANOVA detects a statistically significant mean change.

5. Mauchly W and Sphericity Epsilon Summary

This chart is useful because many learners confuse Mauchly’s W with epsilon. Mauchly’s W is part of the test of sphericity. Epsilon values are used after the violation is found. In this example, Mauchly’s W confirms the violation, and the Greenhouse-Geisser epsilon provides the corrected degrees of freedom.

6. Mean Grade Changes Across Repeated Measures

The total mean change from G1 to G3 is about 0.5069 grade points. The change from G1 to G2 is smaller, while the change from G2 to G3 is larger. This supports the conclusion that the final-grade measure is higher on average than the earlier grade measures.

7. Pairwise Difference Variances for Sphericity Check

This is the most important assumption chart. Sphericity requires the paired-difference variances to be roughly equal. Here they are not equal: 2.188, 3.315 and 1.634. This unequal pattern explains why Mauchly’s test is significant and why the Greenhouse-Geisser correction is needed.

8. Repeated-Measures ANOVA P-Value Comparison

The p-value comparison shows that all correction methods remain statistically significant. This makes the result robust. Even after the most conservative lower-bound correction, the repeated-measures grade effect is still significant.

9. Sample Student Grade Trajectories

The trajectory chart shows that students do not all follow the same pattern. Some improve, some decline, and some stay stable. The blue mean trajectory rises gradually, which is the pattern tested by the repeated-measures ANOVA.

How to Run Greenhouse Geisser Correction in SPSS, R, Python and Excel

Greenhouse Geisser Correction in SPSS

SPSS automatically provides Greenhouse-Geisser, Huynh-Feldt and lower-bound rows when you run a repeated-measures GLM with three or more repeated measures. The important tables are Mauchly’s Test of Sphericity and Tests of Within-Subjects Effects.

GET DATA
 /TYPE=TXT
 /FILE="D:\low kda score priority basis posts\first post\Greenhouse Geisser Correction\student_por_greenhouse_geisser_clean.csv"
 /ENCODING='UTF8'
 /DELCASE=LINE
 /DELIMITERS=","
 /QUALIFIER='"'
 /ARRANGEMENT=DELIMITED
 /FIRSTCASE=2
 /VARIABLES=
 case_id F8.0
 school A12
 sex A8
 age F8.0
 address A8
 famsize A8
 Pstatus A8
 Medu F8.0
 Fedu F8.0
 Mjob A20
 Fjob A20
 reason A20
 guardian A20
 traveltime F8.0
 studytime F8.0
 failures F8.0
 schoolsup A8
 famsup A8
 paid A8
 activities A8
 nursery A8
 higher A8
 internet A8
 romantic A8
 famrel F8.0
 freetime F8.0
 goout F8.0
 Dalc F8.0
 Walc F8.0
 health F8.0
 absences F8.0
 G1 F8.0
 G2 F8.0
 G3 F8.0.
CACHE.
EXECUTE.

DATASET NAME GreenhouseGeisserClean WINDOW=FRONT.

VARIABLE LABELS
 case_id "Case ID for repeated-measures analysis"
 G1 "First period grade"
 G2 "Second period grade"
 G3 "Final grade"
 studytime "Study time group"
 failures "Number of past class failures"
 absences "School absences".

FORMATS G1 G2 G3 (F8.2).
EXECUTE.

FREQUENCIES VARIABLES=G1 G2 G3
 /STATISTICS=MEAN MEDIAN STDDEV MINIMUM MAXIMUM
 /ORDER=ANALYSIS.

COMPUTE diff_G2_G1 = G2 - G1.
COMPUTE diff_G3_G1 = G3 - G1.
COMPUTE diff_G3_G2 = G3 - G2.
EXECUTE.

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

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

OUTPUT EXPORT
 /CONTENTS EXPORT=ALL
 /PDF DOCUMENTFILE="D:\low kda score priority basis posts\first post\Greenhouse Geisser Correction\SPSS\Greenhouse-Geisser-Correction-SPSS-output.pdf"
 EMBEDBOOKMARKS=YES
 EMBEDFONTS=YES.

Greenhouse Geisser Correction in R

In R, repeated-measures ANOVA can be run with packages such as rstatix, afex or ez. The script below reshapes the data from wide format to long format and reports the corrected repeated-measures result.

install.packages(c("tidyverse", "rstatix", "afex"))

library(tidyverse)
library(rstatix)
library(afex)

base_dir <- "D:/low kda score priority basis posts/first post/Greenhouse Geisser Correction"
data_file <- file.path(base_dir, "student-por.csv")
out_dir <- file.path(base_dir, "R")
dir.create(out_dir, showWarnings = FALSE, recursive = TRUE)

student <- read.csv(data_file, sep = ";", stringsAsFactors = FALSE)

clean_data <- student %>%
  mutate(case_id = row_number()) %>%
  filter(!is.na(G1), !is.na(G2), !is.na(G3)) %>%
  select(case_id, everything())

write.csv(clean_data, file.path(out_dir, "greenhouse_geisser_clean_data_r.csv"), row.names = FALSE)

long_data <- clean_data %>%
  select(case_id, G1, G2, G3) %>%
  pivot_longer(cols = c(G1, G2, G3),
               names_to = "GradeTime",
               values_to = "Grade") %>%
  mutate(
    case_id = factor(case_id),
    GradeTime = factor(GradeTime, levels = c("G1", "G2", "G3"))
  )

desc_table <- long_data %>%
  group_by(GradeTime) %>%
  summarise(
    n = n(),
    mean = mean(Grade),
    sd = sd(Grade),
    se = sd / sqrt(n),
    .groups = "drop"
  )

print(desc_table)

anova_result <- anova_test(
  data = long_data,
  dv = Grade,
  wid = case_id,
  within = GradeTime,
  detailed = TRUE
)

print(get_anova_table(anova_result, correction = "GG"))
print(anova_result$Mauchly)
print(anova_result$`Sphericity Corrections`)

diff_table <- clean_data %>%
  transmute(
    diff_G2_G1 = G2 - G1,
    diff_G3_G1 = G3 - G1,
    diff_G3_G2 = G3 - G2
  ) %>%
  pivot_longer(everything(), names_to = "difference", values_to = "value") %>%
  group_by(difference) %>%
  summarise(
    mean_difference = mean(value),
    sd_difference = sd(value),
    variance_difference = var(value),
    .groups = "drop"
  )

print(diff_table)

write.csv(desc_table, file.path(out_dir, "greenhouse_geisser_descriptives_r.csv"), row.names = FALSE)
write.csv(diff_table, file.path(out_dir, "greenhouse_geisser_pairwise_difference_variances_r.csv"), row.names = FALSE)

Greenhouse Geisser Correction in Python

In Python, the workflow can be run with pandas and pingouin. The rm_anova() function provides repeated-measures ANOVA output, while sphericity() and epsilon() help verify the assumption and correction.

from pathlib import Path
import pandas as pd
import numpy as np
import pingouin as pg
import matplotlib.pyplot as plt

base_dir = Path(r"D:\low kda score priority basis posts\first post\Greenhouse Geisser Correction")
data_file = base_dir / "student-por.csv"
out_dir = base_dir / "Python"
out_dir.mkdir(parents=True, exist_ok=True)

student = pd.read_csv(data_file, sep=";")

clean = student.copy()
clean.insert(0, "case_id", np.arange(1, len(clean) + 1))
clean = clean.dropna(subset=["G1", "G2", "G3"]).copy()

clean_path = base_dir / "student_por_greenhouse_geisser_clean.csv"
clean.to_csv(clean_path, index=False)

long_df = clean[["case_id", "G1", "G2", "G3"]].melt(
    id_vars="case_id",
    value_vars=["G1", "G2", "G3"],
    var_name="GradeTime",
    value_name="Grade"
)

desc = long_df.groupby("GradeTime")["Grade"].agg(["count", "mean", "std"]).reset_index()
desc["se"] = desc["std"] / np.sqrt(desc["count"])
print(desc)

anova = pg.rm_anova(
    data=long_df,
    dv="Grade",
    within="GradeTime",
    subject="case_id",
    detailed=True,
    correction=True
)
print(anova)

sphericity = pg.sphericity(
    data=long_df,
    dv="Grade",
    within="GradeTime",
    subject="case_id"
)
print(sphericity)

epsilon = pg.epsilon(
    data=long_df,
    dv="Grade",
    within="GradeTime",
    subject="case_id"
)
print("Greenhouse-Geisser epsilon:", epsilon)

diffs = pd.DataFrame({
    "G2 - G1": clean["G2"] - clean["G1"],
    "G3 - G1": clean["G3"] - clean["G1"],
    "G3 - G2": clean["G3"] - clean["G2"]
})

diff_summary = diffs.agg(["mean", "std", "var"]).T.reset_index()
diff_summary.columns = ["difference", "mean_difference", "sd_difference", "variance_difference"]
print(diff_summary)

desc.to_csv(out_dir / "greenhouse_geisser_python_descriptives.csv", index=False)
diff_summary.to_csv(out_dir / "greenhouse_geisser_python_pairwise_difference_variances.csv", index=False)
anova.to_csv(out_dir / "greenhouse_geisser_python_anova.csv", index=False)

Greenhouse Geisser Correction in Excel

Excel does not automatically provide Mauchly’s test, Greenhouse-Geisser epsilon and corrected repeated-measures ANOVA rows in the same way as SPSS, R or Python. However, Excel can still be used to explain the assumption and manually inspect paired-difference variances.

Variance of G2 - G1 = VAR.S(E2:E650)
Variance of G3 - G1 = VAR.S(F2:F650)
Variance of G3 - G2 = VAR.S(G2:G650)

For this dataset, Excel should give paired-difference variances close to 2.188, 3.315 and 1.634. Since these variances are not equal, Excel supports the same practical conclusion: sphericity is not satisfied, so a correction is needed for the repeated-measures ANOVA.

How to Report the Greenhouse Geisser Correction Result

A good report should mention Mauchly’s test, the correction used, the corrected degrees of freedom, the F statistic, the p-value and the direction of the mean change. Avoid writing only “Greenhouse-Geisser was significant” because that does not explain what was corrected.

Related Data Analysis Guides

After learning the Greenhouse Geisser Correction for repeated-measures ANOVA, continue with related assumption-testing and model-checking guides on Online Internet Cafe: DAgostino Pearson Test for omnibus normality testing, Cramer von Mises Test for goodness-of-fit analysis, Cochran C Test for detecting the largest variance, and Brown Forsythe Test for robust variance comparison.

When Should You Use Greenhouse Geisser Correction?

Use this correction when your repeated-measures ANOVA has three or more related levels and Mauchly’s test indicates that sphericity is violated. It is especially common in psychology, education, medical trials, sports science, repeated survey analysis and before-after-follow-up designs.

Common Mistakes

1. Reporting the sphericity-assumed row after Mauchly’s test is significant

If Mauchly’s test is significant, the ordinary sphericity-assumed row should not be the main reported result. Report the Greenhouse-Geisser row or another justified correction row.

2. Thinking Greenhouse-Geisser changes the mean scores

The correction does not change G1, G2 or G3 means. It only changes the degrees of freedom and therefore the p-value calculation.

3. Using it when there are only two repeated measures

With only two related measurements, sphericity is automatically satisfied. Greenhouse-Geisser correction becomes relevant when there are three or more repeated conditions.

4. Ignoring pairwise difference variances

Mauchly’s test can look abstract. Pairwise difference variances make the assumption easier to understand. In this example, the difference variances are unequal, which explains the sphericity violation.

5. Confusing Greenhouse-Geisser and Huynh-Feldt

Both are sphericity corrections, but Greenhouse-Geisser is usually more conservative. Huynh-Feldt is often less conservative. In this dataset, both values are close, so both lead to the same conclusion.

Download SPSS Output and Verification Files

The SPSS PDF verifies the clean data import, descriptive statistics, Mauchly’s test, epsilon values, corrected within-subjects effects and repeated-measures ANOVA interpretation.

Sources and Method Notes

This post uses verified SPSS output, reproducible R/Python-style workflows and official/reference sources for the repeated-measures ANOVA and dataset context.

• UCI Machine Learning Repository: Student Performance dataset
• IBM SPSS Statistics Documentation
• The Comprehensive R Archive Network
• Pingouin statistical package for Python