Huynh-Feldt Correction: Complete Repeated-Measures ANOVA Guide with SPSS, R, Python and Excel 1

What Is the Huynh-Feldt Correction?

The Huynh-Feldt Correction is a degrees-of-freedom correction used in repeated-measures ANOVA when the assumption of sphericity is not met. Repeated-measures ANOVA is used when the same subjects are measured across three or more time points, conditions or related outcomes. In this example, the same students have three grade measurements: G1, G2 and G3.

The core issue is that repeated observations from the same person are correlated. Repeated-measures ANOVA expects those correlations and difference-score variances to follow a pattern called sphericity. If that assumption is violated, the standard ANOVA degrees of freedom can be too generous. A correction is then needed so that the p-value is more reliable.

The Huynh-Feldt method adjusts the numerator and denominator degrees of freedom using an epsilon value. It does not change the raw data. It does not change the group means. It changes the degrees of freedom used to calculate the corrected p-value.

If you are building a full data-analysis workflow, you may also need assumption tests such as the DAgostino Pearson Test for normality, the Cramer von Mises Test for distribution checking, the Brown-Forsythe Test for equality of variances, the Cochran C Test for largest-variance detection, the Hartley F Max Test for variance-ratio checking and the Goldfeld-Quandt Test for regression heteroscedasticity.

Huynh-Feldt Correction Formula

The Huynh-Feldt Correction adjusts repeated-measures ANOVA degrees of freedom by multiplying the original degrees of freedom by the Huynh-Feldt epsilon. The simplified correction logic is:

Corrected df = epsilon × original df

For the repeated-measures effect, the original degrees of freedom are based on the number of repeated levels. With three repeated grade measurements, the uncorrected effect df is 2. In this example, the Huynh-Feldt epsilon is approximately .855, so the corrected effect degrees of freedom become approximately 1.709.

Corrected effect df = 2 × .855 ≈ 1.709
Corrected error df  = 1296 × .855 ≈ 1107.581

The F statistic remains 36.287. The correction changes the degrees of freedom used to calculate the p-value. Since the corrected p-value remains below .001, the repeated-measures effect is still statistically significant.

Huynh-Feldt Correction Null Hypothesis and Alternative Hypothesis

The Huynh-Feldt Correction does not change the actual research hypothesis. It only adjusts the repeated-measures ANOVA test when sphericity is violated.

Because the Huynh-Feldt corrected p-value is less than .001, the null hypothesis is rejected. The repeated grade means are not equal.

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 are three related grade measurements from the same students, which makes repeated-measures ANOVA the correct design.

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

Verified Results in SPSS, R and Python

The analysis was reproduced in SPSS, R and Python. SPSS produced the main repeated-measures ANOVA output, Mauchly’s Test, epsilon values and corrected rows. R and Python were used to verify the same correction pattern and generate charts for publication.

Descriptive Statistics for G1, G2 and G3

Mauchly’s Test of Sphericity

Epsilon Values

Tests of Within-Subjects Effects

Pairwise Difference Variances for Sphericity Check

The pairwise difference variances are not equal. This explains why Mauchly’s Test is significant and why corrected repeated-measures rows are needed.

Huynh-Feldt Correction Charts and Interpretation

1. Greenhouse-Geisser and Huynh-Feldt Correction Comparison

This chart compares the strength of evidence across the sphericity-assumed result, Greenhouse-Geisser correction, Huynh-Feldt correction and lower-bound correction. All corrected rows remain far beyond the usual p = .05 threshold, meaning the repeated grade effect is stable. The result does not depend on choosing the most favorable row.

2. Degrees of Freedom Before and After Huynh-Feldt Correction

The degrees-of-freedom chart shows the actual job of the correction. The F statistic does not change, but the df values are reduced. Huynh-Feldt correction reduces the effect df from 2.000 to 1.709 and the error df from 1296.000 to 1107.581.

3. Epsilon Comparison

The epsilon comparison shows that Greenhouse-Geisser and Huynh-Feldt epsilon values are very close: .853 and .855. This means the two corrected rows will lead to almost the same conclusion. The lower-bound value of .500 is much more conservative.

4. G1, G2 and G3 Grade Distributions

The grade distribution boxplot shows that G3 has a slightly higher center and wider spread. Some low grade values are visible, but the overall distribution supports the descriptive result that final grades are higher on average than earlier grade measurements.

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

The mean profile is the clearest practical interpretation chart. The mean grade increases from 11.3991 to 11.5701 and then to 11.9060. The repeated-measures ANOVA confirms that this upward pattern is statistically significant.

6. Mauchly W and Huynh-Feldt Epsilon Summary

This chart helps separate two concepts that students often mix. Mauchly’s W is part of the sphericity test. Epsilon values are then used to correct the degrees of freedom after the assumption problem is found.

7. Mean Grade Changes Across Repeated Measures

The mean-change chart shows where the improvement occurs. The total increase from G1 to G3 is about .507 grade points. The G3 − G1 change is the largest because it captures the full movement from first grade measurement to final grade.

8. Pairwise Difference Variances for Sphericity Check

This chart explains the statistical reason for the correction. The difference variances are not equal: 2.188, 1.634 and 3.315. Since sphericity requires similar difference variances, this unequal pattern supports the need for corrected repeated-measures ANOVA rows.

9. Repeated-Measures ANOVA P-Value Comparison

The p-value chart shows that all correction methods remain significant. The result is not fragile. Even under the most conservative lower-bound correction, the repeated grade effect remains statistically significant.

10. Sample Student Grade Trajectories

The trajectory chart shows that individual students do not all follow the same pattern. Some improve, some remain stable and some decline. However, the average trajectory rises across G1, G2 and G3, which explains the significant repeated-measures effect.

Python Validation Charts for Huynh-Feldt Correction

The Python workflow produced matching validation charts. These are useful because they confirm that the statistical conclusion is not dependent on one software program only.

How to Run the Huynh-Feldt Correction in SPSS, R, Python and Excel

Huynh-Feldt Correction in SPSS

SPSS automatically reports the Huynh-Feldt row when repeated-measures ANOVA is run through General Linear Model. The key output tables are Mauchly’s Test of Sphericity and Tests of Within-Subjects Effects.

* Huynh-Feldt Correction repeated-measures ANOVA in SPSS.

GET DATA
 /TYPE=TXT
 /FILE='D:\low kda score priority basis posts\first post\Huynh Feldt Correction\Python\huynh_feldt_correction_python_clean_data.csv'
 /ENCODING='UTF8'
 /DELCASE=LINE
 /DELIMITERS=","
 /QUALIFIER='"'
 /ARRANGEMENT=DELIMITED
 /FIRSTCASE=2
 /VARIABLES=
 subject_id F8.0
 school A20
 sex A10
 age F8.0
 address A20
 famsize A20
 Pstatus A20
 Medu F8.0
 Fedu F8.0
 Mjob A30
 Fjob A30
 reason A30
 guardian A30
 traveltime F8.0
 studytime F8.0
 failures F8.0
 schoolsup A10
 famsup A10
 paid A10
 activities A10
 nursery A10
 higher A10
 internet A10
 romantic A10
 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 HuynhFeldtData WINDOW=FRONT.

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

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.

OUTPUT EXPORT
 /CONTENTS EXPORT=VISIBLE
 /PDF DOCUMENTFILE='D:\low kda score priority basis posts\first post\Huynh Feldt Correction\SPSS\Huynh-Feldt-Correction-SPSS-output.pdf'.

Huynh-Feldt Correction in R

In R, the workflow is to reshape the data from wide format to long format and then run repeated-measures ANOVA with sphericity corrections.

install.packages(c("tidyverse", "ez"))

library(tidyverse)
library(ez)

base_dir <- "D:/low kda score priority basis posts/first post/Huynh Feldt 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(subject_id = row_number()) %>%
  filter(!is.na(G1), !is.na(G2), !is.na(G3)) %>%
  select(subject_id, everything())

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

hf_result <- ezANOVA(
  data = long_data,
  dv = Grade,
  wid = subject_id,
  within = GradeTime,
  detailed = TRUE,
  type = 3
)

print(hf_result)

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)

diff_table <- clean_data %>%
  transmute(
    diff_G2_G1 = G2 - G1,
    diff_G3_G2 = G3 - G2,
    diff_G3_G1 = G3 - G1
  ) %>%
  summarise(
    var_G2_G1 = var(diff_G2_G1),
    var_G3_G2 = var(diff_G3_G2),
    var_G3_G1 = var(diff_G3_G1)
  )

print(diff_table)

Huynh-Feldt Correction in Python

In Python, packages such as pandas, NumPy, SciPy and Pingouin can be used for repeated-measures ANOVA and sphericity-related output. The code below gives a clean structure for practical use.

import pandas as pd
import numpy as np
import pingouin as pg

base_file = r"D:\low kda score priority basis posts\first post\Huynh Feldt Correction\student-por.csv"

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

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

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

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

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

print(anova)

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

print(sphericity)

student["diff_G2_G1"] = student["G2"] - student["G1"]
student["diff_G3_G2"] = student["G3"] - student["G2"]
student["diff_G3_G1"] = student["G3"] - student["G1"]

print(student[["diff_G2_G1", "diff_G3_G2", "diff_G3_G1"]].var())

Huynh-Feldt Correction in Excel

Excel is useful for checking means and pairwise difference variances, but SPSS, R or Python is better for final Mauchly’s Test, epsilon values and corrected ANOVA rows.

Variance of G2 - G1:
=VAR.S(E2:E650)

Variance of G3 - G2:
=VAR.S(F2:F650)

Variance of G3 - G1:
=VAR.S(G2:G650)

If the difference variances are not similar, sphericity may be violated and a correction such as Huynh-Feldt or Greenhouse-Geisser should be used in SPSS, R or Python.

How to Report the Huynh-Feldt Correction Result

A good report should mention Mauchly’s Test, the correction used, the corrected degrees of freedom, the F statistic, the p-value, effect size and mean pattern.

Related Data Analysis Guides

After learning the Huynh-Feldt Correction for repeated-measures ANOVA, continue with related statistical testing guides on Online Internet Cafe: Greenhouse-Geisser Correction for the more conservative sphericity correction, Goldfeld-Quandt Test for regression heteroscedasticity, Hartley F Max Test for variance-ratio checking, Brown-Forsythe Test for robust variance comparison, Cochran C Test for largest-variance detection, DAgostino Pearson Test for omnibus normality testing, and Cramer von Mises Test for distribution goodness-of-fit analysis.

When Should You Use Huynh-Feldt Correction?

Use Huynh-Feldt Correction when your design has three or more repeated conditions and the sphericity assumption is violated or doubtful. It is commonly used in education, psychology, medicine, repeated survey analysis, training evaluation and experimental designs with repeated measurements.

Common Mistakes

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

If Mauchly’s Test is significant, the sphericity-assumed row should not be the main reported row. Use a corrected row such as Huynh-Feldt or Greenhouse-Geisser.

2. Reporting p = .000

SPSS often displays very small p-values as .000. In final writing, report this as p < .001.

3. Thinking the correction changes the means

The correction does not change G1, G2 or G3 means. It changes the degrees of freedom used for the F-test.

4. Confusing Huynh-Feldt with Greenhouse-Geisser

Both are sphericity corrections, but Greenhouse-Geisser is usually more conservative. Huynh-Feldt is usually less conservative when epsilon is high.

5. Using it for independent groups

Huynh-Feldt Correction is for within-subject repeated-measures designs. It is not a general independent-group variance test.

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.

External References for Huynh-Feldt Correction and Sphericity

This post uses verified SPSS, R and Python outputs together with external statistical references and software documentation. These external resources help readers verify the background of repeated-measures ANOVA, Mauchly’s Test, sphericity and epsilon-based corrections.

• Mauchly’s Sphericity Test explanation
• Greenhouse-Geisser and Huynh-Feldt correction background
• UCI Machine Learning Repository: Student Performance dataset
• IBM SPSS Statistics documentation
• The Comprehensive R Archive Network
• Pingouin statistical package for Python

What Is the Huynh-Feldt Correction?

The Huynh-Feldt Correction is a degrees-of-freedom correction used in repeated-measures ANOVA when the assumption of sphericity is not met. Repeated-measures ANOVA is used when the same subjects are measured across three or more time points, conditions or related outcomes. In this example, the same students have three grade measurements: G1, G2 and G3.

The core issue is that repeated observations from the same person are correlated. Repeated-measures ANOVA expects those correlations and difference-score variances to follow a pattern called sphericity. If that assumption is violated, the standard ANOVA degrees of freedom can be too generous. A correction is then needed so that the p-value is more reliable.

The Huynh-Feldt method adjusts the numerator and denominator degrees of freedom using an epsilon value. It does not change the raw data. It does not change the group means. It changes the degrees of freedom used to calculate the corrected p-value.

If you are building a full data-analysis workflow, you may also need assumption tests such as the DAgostino Pearson Test for normality, the Cramer von Mises Test for distribution checking, the Brown-Forsythe Test for equality of variances, the Cochran C Test for largest-variance detection, the Hartley F Max Test for variance-ratio checking and the Goldfeld-Quandt Test for regression heteroscedasticity.

Huynh-Feldt Correction Formula

The Huynh-Feldt Correction adjusts repeated-measures ANOVA degrees of freedom by multiplying the original degrees of freedom by the Huynh-Feldt epsilon. The simplified correction logic is:

Corrected df = epsilon × original df

For the repeated-measures effect, the original degrees of freedom are based on the number of repeated levels. With three repeated grade measurements, the uncorrected effect df is 2. In this example, the Huynh-Feldt epsilon is approximately .855, so the corrected effect degrees of freedom become approximately 1.709.

Corrected effect df = 2 × .855 ≈ 1.709
Corrected error df  = 1296 × .855 ≈ 1107.581

The F statistic remains 36.287. The correction changes the degrees of freedom used to calculate the p-value. Since the corrected p-value remains below .001, the repeated-measures effect is still statistically significant.

Huynh-Feldt Correction Null Hypothesis and Alternative Hypothesis

The Huynh-Feldt Correction does not change the actual research hypothesis. It only adjusts the repeated-measures ANOVA test when sphericity is violated.

Because the Huynh-Feldt corrected p-value is less than .001, the null hypothesis is rejected. The repeated grade means are not equal.

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 are three related grade measurements from the same students, which makes repeated-measures ANOVA the correct design.

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

Verified Results in SPSS, R and Python

The analysis was reproduced in SPSS, R and Python. SPSS produced the main repeated-measures ANOVA output, Mauchly’s Test, epsilon values and corrected rows. R and Python were used to verify the same correction pattern and generate charts for publication.

Descriptive Statistics for G1, G2 and G3

Mauchly’s Test of Sphericity

Epsilon Values

Tests of Within-Subjects Effects

Pairwise Difference Variances for Sphericity Check

The pairwise difference variances are not equal. This explains why Mauchly’s Test is significant and why corrected repeated-measures rows are needed.

Huynh-Feldt Correction Charts and Interpretation

1. Greenhouse-Geisser and Huynh-Feldt Correction Comparison

This chart compares the strength of evidence across the sphericity-assumed result, Greenhouse-Geisser correction, Huynh-Feldt correction and lower-bound correction. All corrected rows remain far beyond the usual p = .05 threshold, meaning the repeated grade effect is stable. The result does not depend on choosing the most favorable row.

2. Degrees of Freedom Before and After Huynh-Feldt Correction

The degrees-of-freedom chart shows the actual job of the correction. The F statistic does not change, but the df values are reduced. Huynh-Feldt correction reduces the effect df from 2.000 to 1.709 and the error df from 1296.000 to 1107.581.

3. Epsilon Comparison

The epsilon comparison shows that Greenhouse-Geisser and Huynh-Feldt epsilon values are very close: .853 and .855. This means the two corrected rows will lead to almost the same conclusion. The lower-bound value of .500 is much more conservative.

4. G1, G2 and G3 Grade Distributions

The grade distribution boxplot shows that G3 has a slightly higher center and wider spread. Some low grade values are visible, but the overall distribution supports the descriptive result that final grades are higher on average than earlier grade measurements.

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

The mean profile is the clearest practical interpretation chart. The mean grade increases from 11.3991 to 11.5701 and then to 11.9060. The repeated-measures ANOVA confirms that this upward pattern is statistically significant.

6. Mauchly W and Huynh-Feldt Epsilon Summary

This chart helps separate two concepts that students often mix. Mauchly’s W is part of the sphericity test. Epsilon values are then used to correct the degrees of freedom after the assumption problem is found.

7. Mean Grade Changes Across Repeated Measures

The mean-change chart shows where the improvement occurs. The total increase from G1 to G3 is about .507 grade points. The G3 − G1 change is the largest because it captures the full movement from first grade measurement to final grade.

8. Pairwise Difference Variances for Sphericity Check

This chart explains the statistical reason for the correction. The difference variances are not equal: 2.188, 1.634 and 3.315. Since sphericity requires similar difference variances, this unequal pattern supports the need for corrected repeated-measures ANOVA rows.

9. Repeated-Measures ANOVA P-Value Comparison

The p-value chart shows that all correction methods remain significant. The result is not fragile. Even under the most conservative lower-bound correction, the repeated grade effect remains statistically significant.

10. Sample Student Grade Trajectories

The trajectory chart shows that individual students do not all follow the same pattern. Some improve, some remain stable and some decline. However, the average trajectory rises across G1, G2 and G3, which explains the significant repeated-measures effect.

Python Validation Charts for Huynh-Feldt Correction

The Python workflow produced matching validation charts. These are useful because they confirm that the statistical conclusion is not dependent on one software program only.

How to Run the Huynh-Feldt Correction in SPSS, R, Python and Excel

Huynh-Feldt Correction in SPSS

SPSS automatically reports the Huynh-Feldt row when repeated-measures ANOVA is run through General Linear Model. The key output tables are Mauchly’s Test of Sphericity and Tests of Within-Subjects Effects.

* Huynh-Feldt Correction repeated-measures ANOVA in SPSS.

GET DATA
 /TYPE=TXT
 /FILE='D:\low kda score priority basis posts\first post\Huynh Feldt Correction\Python\huynh_feldt_correction_python_clean_data.csv'
 /ENCODING='UTF8'
 /DELCASE=LINE
 /DELIMITERS=","
 /QUALIFIER='"'
 /ARRANGEMENT=DELIMITED
 /FIRSTCASE=2
 /VARIABLES=
 subject_id F8.0
 school A20
 sex A10
 age F8.0
 address A20
 famsize A20
 Pstatus A20
 Medu F8.0
 Fedu F8.0
 Mjob A30
 Fjob A30
 reason A30
 guardian A30
 traveltime F8.0
 studytime F8.0
 failures F8.0
 schoolsup A10
 famsup A10
 paid A10
 activities A10
 nursery A10
 higher A10
 internet A10
 romantic A10
 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 HuynhFeldtData WINDOW=FRONT.

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

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.

OUTPUT EXPORT
 /CONTENTS EXPORT=VISIBLE
 /PDF DOCUMENTFILE='D:\low kda score priority basis posts\first post\Huynh Feldt Correction\SPSS\Huynh-Feldt-Correction-SPSS-output.pdf'.

Huynh-Feldt Correction in R

In R, the workflow is to reshape the data from wide format to long format and then run repeated-measures ANOVA with sphericity corrections.

install.packages(c("tidyverse", "ez"))

library(tidyverse)
library(ez)

base_dir <- "D:/low kda score priority basis posts/first post/Huynh Feldt 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(subject_id = row_number()) %>%
  filter(!is.na(G1), !is.na(G2), !is.na(G3)) %>%
  select(subject_id, everything())

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

hf_result <- ezANOVA(
  data = long_data,
  dv = Grade,
  wid = subject_id,
  within = GradeTime,
  detailed = TRUE,
  type = 3
)

print(hf_result)

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)

diff_table <- clean_data %>%
  transmute(
    diff_G2_G1 = G2 - G1,
    diff_G3_G2 = G3 - G2,
    diff_G3_G1 = G3 - G1
  ) %>%
  summarise(
    var_G2_G1 = var(diff_G2_G1),
    var_G3_G2 = var(diff_G3_G2),
    var_G3_G1 = var(diff_G3_G1)
  )

print(diff_table)

Huynh-Feldt Correction in Python

In Python, packages such as pandas, NumPy, SciPy and Pingouin can be used for repeated-measures ANOVA and sphericity-related output. The code below gives a clean structure for practical use.

import pandas as pd
import numpy as np
import pingouin as pg

base_file = r"D:\low kda score priority basis posts\first post\Huynh Feldt Correction\student-por.csv"

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

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

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

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

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

print(anova)

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

print(sphericity)

student["diff_G2_G1"] = student["G2"] - student["G1"]
student["diff_G3_G2"] = student["G3"] - student["G2"]
student["diff_G3_G1"] = student["G3"] - student["G1"]

print(student[["diff_G2_G1", "diff_G3_G2", "diff_G3_G1"]].var())

Huynh-Feldt Correction in Excel

Excel is useful for checking means and pairwise difference variances, but SPSS, R or Python is better for final Mauchly’s Test, epsilon values and corrected ANOVA rows.

Variance of G2 - G1:
=VAR.S(E2:E650)

Variance of G3 - G2:
=VAR.S(F2:F650)

Variance of G3 - G1:
=VAR.S(G2:G650)

If the difference variances are not similar, sphericity may be violated and a correction such as Huynh-Feldt or Greenhouse-Geisser should be used in SPSS, R or Python.

How to Report the Huynh-Feldt Correction Result

A good report should mention Mauchly’s Test, the correction used, the corrected degrees of freedom, the F statistic, the p-value, effect size and mean pattern.

Related Data Analysis Guides

After learning the Huynh-Feldt Correction for repeated-measures ANOVA, continue with related statistical testing guides on Online Internet Cafe: Greenhouse-Geisser Correction for the more conservative sphericity correction, Goldfeld-Quandt Test for regression heteroscedasticity, Hartley F Max Test for variance-ratio checking, Brown-Forsythe Test for robust variance comparison, Cochran C Test for largest-variance detection, DAgostino Pearson Test for omnibus normality testing, and Cramer von Mises Test for distribution goodness-of-fit analysis.

When Should You Use Huynh-Feldt Correction?

Use Huynh-Feldt Correction when your design has three or more repeated conditions and the sphericity assumption is violated or doubtful. It is commonly used in education, psychology, medicine, repeated survey analysis, training evaluation and experimental designs with repeated measurements.

Common Mistakes

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

If Mauchly’s Test is significant, the sphericity-assumed row should not be the main reported row. Use a corrected row such as Huynh-Feldt or Greenhouse-Geisser.

2. Reporting p = .000

SPSS often displays very small p-values as .000. In final writing, report this as p < .001.

3. Thinking the correction changes the means

The correction does not change G1, G2 or G3 means. It changes the degrees of freedom used for the F-test.

4. Confusing Huynh-Feldt with Greenhouse-Geisser

Both are sphericity corrections, but Greenhouse-Geisser is usually more conservative. Huynh-Feldt is usually less conservative when epsilon is high.

5. Using it for independent groups

Huynh-Feldt Correction is for within-subject repeated-measures designs. It is not a general independent-group variance test.

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.

External References for Huynh-Feldt Correction and Sphericity

This post uses verified SPSS, R and Python outputs together with external statistical references and software documentation. These external resources help readers verify the background of repeated-measures ANOVA, Mauchly’s Test, sphericity and epsilon-based corrections.

• Mauchly’s Sphericity Test explanation
• Greenhouse-Geisser and Huynh-Feldt correction background
• UCI Machine Learning Repository: Student Performance dataset
• IBM SPSS Statistics documentation
• The Comprehensive R Archive Network
• Pingouin statistical package for Python