Quick Answer: Fixed Effects ANOVA Result

The worked Fixed Effects ANOVA example compares mean G3 across four fixed studytime levels. The fixed factor mean chart shows that mean G3 is lowest at studytime level 1, higher at level 2, highest at level 3, and still high at level 4. The fixed effect estimate chart shows level 1 below the grand mean, while levels 2, 3 and 4 are above the grand mean.

The F statistic distribution chart reports observed F ≈ 15.88 and critical F ≈ 2.62. Since the observed F statistic is far to the right of the critical value, the fixed studytime factor has a statistically significant effect on G3. The effect-size chart reports eta squared = 0.0688, partial eta squared = 0.0688, omega squared = 0.0643, epsilon squared = 0.0644, and Cohen’s f = 0.2717.

What Is Fixed Effects ANOVA?

Fixed Effects ANOVA is an ANOVA model where the factor levels are deliberately selected and are the exact levels the researcher wants to compare. The levels are not treated as a random sample from a larger population of possible levels.

In this example, studytime has four fixed levels. The purpose is not to generalize to every possible studytime category. The purpose is to compare these four defined studytime groups and test whether their mean G3 final grades differ.

The fixed factor means show a clear pattern. Studytime level 1 has the lowest mean G3, level 2 is higher, level 3 is highest, and level 4 remains high. The fixed effect estimate chart converts that pattern into deviations from the grand mean: level 1 is negative, while levels 2, 3 and 4 are positive.

Fixed Effects ANOVA should be interpreted with ANOVA in Python, ANOVA in R, ANOVA in SPSS, ANOVA Assumptions, ANOVA Effect Size, Effect Size, Cohen’s F Formula, P Value, and Null and Alternative Hypothesis.

Fixed Effects ANOVA Formula

A one-way fixed effects ANOVA can be written as a model where the outcome is explained by the grand mean, the fixed factor effect and random error.

In this formula, Y is the observed G3 score, μ is the grand mean, α_i is the fixed effect for studytime level i, and ε is the residual error.

Fixed Effects ANOVA for This Example

The fixed effect estimates are interpreted as deviations from the grand mean. Level 1 is below the grand mean, level 2 is slightly above it, level 3 has the largest positive deviation, and level 4 also has a positive deviation.

F Statistic Formula

The F statistic compares variation explained by the fixed factor with the remaining error variation. In this output, the observed F statistic is about 15.88 and the critical F value is about 2.62, so the fixed factor effect is statistically significant.

Effect Size Formula

The effect size chart reports eta squared = 0.0688. This means the fixed studytime factor explains about 6.9% of total variation in G3.

Null and Alternative Hypothesis

For this fixed effects ANOVA, the null hypothesis says that all fixed studytime levels have the same mean G3 score. The alternative hypothesis says that at least one fixed studytime level has a different mean G3 score.

Dataset and Variables Used

The worked example uses a student performance dataset. The dependent variable is G3 final grade, and the fixed factor is studytime. The studytime levels are treated as fixed because they are the exact categories being compared in the analysis.

For supporting concepts, review Descriptive Statistics, Mean Median and Mode, Standard Deviation, Variance, Standard Error, Confidence Interval, Box Plot Interpretation, and Histogram Interpretation.

Python Chart-by-Chart Interpretation

The Python charts show the first fixed effects ANOVA workflow. They explain fixed factor means, fixed effect estimates, sum of squares decomposition, F statistic decision, residual distribution, Q-Q diagnostic pattern and effect-size summary.

Python Chart 1: Fixed Factor Means with 95% Confidence Intervals

This chart shows the mean G3 score at each fixed studytime level. Studytime level 1 has the lowest mean, level 2 is higher, level 3 has the highest mean, and level 4 remains close to the level 3 range.

The confidence interval for level 4 is wider than the intervals for the first three levels, which indicates less precision for that fixed level. Even with wider uncertainty, level 4 remains in the higher mean G3 range.

This chart gives the main practical direction of the fixed effects ANOVA. The fixed studytime levels do not form a flat mean pattern, which explains why the ANOVA F statistic is significant.

Python Chart 2: Fixed Effect Estimates

This chart shows each fixed factor level as a deviation from the grand mean. Level 1 has a negative fixed effect, meaning its mean G3 is below the overall average. Level 2 has a small positive fixed effect, while levels 3 and 4 have much larger positive effects.

The largest positive estimate is at level 3. Level 4 is also positive and close to level 3, but slightly lower. This tells the same story as the means chart in a more model-based way.

The fixed effect estimate chart is useful because it shows direction and magnitude relative to the grand mean. It clearly identifies level 1 as below average and levels 3 and 4 as above average.

Python Chart 3: Sum of Squares Decomposition

This chart separates the total variation into fixed factor variation and error variation. The fixed effect bar represents variation explained by studytime. The error bar represents variation that remains inside the fixed factor levels.

The error sum of squares is much larger than the fixed effect sum of squares. This means studytime explains a real part of G3 variation, but most of the variation remains among students within the same studytime levels.

This chart explains why the effect size is meaningful but not large. The fixed factor explains enough variation to produce a significant F statistic, but the error component remains the largest part of the total variation.

Python Chart 4: F Statistic Distribution Curve

This chart shows the F distribution used to test the fixed studytime effect. The critical F value is about 2.62, and the observed F statistic is about 15.88.

The observed F statistic is far to the right of the critical boundary. This means the fixed studytime factor explains much more variation than expected under the equal-means null hypothesis.

This chart gives the formal ANOVA decision. Since the observed F is far larger than the critical F value, the fixed factor effect is statistically significant.

Python Chart 5: Residual Histogram

The residual histogram shows most residuals clustered around the middle of the distribution, with many values between roughly -3 and +3. The dashed vertical line marks zero residual.

The histogram also shows a left-tail group of very negative residuals near -12. These cases represent students whose observed G3 scores are far lower than the fitted mean for their studytime level.

This chart supports a careful diagnostic statement. The fixed factor result is significant, but the residual distribution is not perfectly symmetric because of the lower-tail cases.

Python Chart 6: Residual Q-Q Plot

The Q-Q plot compares ordered residuals with theoretical normal quantiles. The central part of the plot follows an increasing pattern, but the lower tail falls well below the reference line.

The strongest departure appears among the most negative residuals. Several points are clustered near residual values around -12, which matches the left-tail pattern seen in the residual histogram.

The Q-Q plot does not remove the significant fixed studytime effect. It adds diagnostic context: the ANOVA result is statistically strong, while residual normality is not perfect because of lower-tail observations.

Python Chart 7: Effect Size Summary

This chart summarizes the practical size of the fixed studytime effect. Eta squared and partial eta squared are both 0.0688. Omega squared is 0.0643, epsilon squared is 0.0644, and Cohen’s f is 0.2717.

The adjusted effect sizes are slightly smaller than eta squared, which is expected because omega squared and epsilon squared are more conservative. The Cohen’s f value of 0.2717 places the effect near the medium range.

This chart should be used in the final reporting section. The fixed effect is statistically significant, and the effect-size summary shows that the practical effect is meaningful but not large.

R Chart-by-Chart Validation

The R validation charts repeat the same fixed effects ANOVA workflow in a second software environment. They confirm the same fixed factor means, fixed effect estimate pattern, sum-of-squares structure, F statistic decision, residual distribution, Q-Q diagnostic pattern and effect-size summary.

R Chart 1: Fixed Factor Means with 95% Confidence Intervals

This R chart validates the same fixed factor mean pattern. Studytime level 1 is lowest, level 2 is higher, level 3 is highest, and level 4 stays close to the high range.

The wider interval for level 4 appears again, showing that the fourth fixed level has less precise estimation than the larger groups. The overall direction remains stable.

In reporting, this chart confirms that the fixed factor conclusion is not a Python-only result. The same mean pattern appears in the R validation workflow.

R Chart 2: Fixed Effect Estimates

This R chart confirms the same fixed effect estimate pattern. Studytime level 1 is below the grand mean, while levels 2, 3 and 4 are above the grand mean.

Level 3 has the strongest positive estimate, and level 4 remains positive. Level 2 is only slightly above the grand mean.

The chart validates the interpretation that the fixed effect is driven mainly by the low performance at level 1 and the higher performance at levels 3 and 4.

R Chart 3: Sum of Squares Decomposition

This R chart confirms that the error sum of squares is much larger than the fixed effect sum of squares. The fixed factor explains a visible portion of total variation, but most variation remains inside the studytime levels.

The chart explains why eta squared is around 0.0688 rather than a large value. The fixed factor is statistically significant, but the error component is still dominant.

In reporting, this chart supports a balanced conclusion: the fixed factor effect is meaningful, but it does not explain most of the variation in G3.

R Chart 4: F Statistic Distribution Curve

The R F distribution chart validates the same significance decision. The observed F statistic is far beyond the critical F value.

The observed line is positioned deep in the right tail of the distribution. This confirms that the fixed studytime factor has a statistically significant effect on G3.

In the final article, this chart should be used as a second-software confirmation of the fixed effects ANOVA decision.

R Chart 5: Residual Histogram

This R histogram confirms that most residuals are concentrated near the center, while a smaller group of residuals appears far into the negative tail.

The negative tail matches the Python diagnostic chart. It shows that some observations are much lower than the fitted mean for their fixed studytime level.

The chart supports the same diagnostic note: the fixed factor effect is significant, but residual shape should be mentioned because the distribution is not perfectly normal.

R Chart 6: Residual Q-Q Plot

The R Q-Q plot confirms the same lower-tail departure from the normal reference line. The middle residuals follow a smoother pattern, but the most negative residuals are far below the line.

This pattern shows that a small set of low residuals affects the normality diagnostic. The Q-Q plot should be included in the assumptions discussion.

The final interpretation should not ignore this chart. The ANOVA result is significant, but residual diagnostics should be reported honestly.

R Chart 7: Effect Size Summary

This R chart confirms the same effect-size values. Eta squared and partial eta squared remain around 0.0688, while adjusted effect-size estimates are slightly lower.

Cohen’s f remains around 0.2717, which supports a medium practical effect interpretation. The fixed factor effect is not large, but it is meaningful.

In reporting, this validation chart confirms that the effect-size conclusion is stable across the Python and R workflows.

SPSS Output Interpretation for Fixed Effects ANOVA

The SPSS output PDF is included as the downloadable software output for the Fixed Effects ANOVA workflow. SPSS can run this model through One-Way ANOVA or GLM Univariate. GLM is useful when effect size, estimated marginal means and model output are needed together.

Download Fixed Effects ANOVA SPSS Output PDF

SPSS Output Items to Read

SPSS Reporting Summary

SPSS, R, Python and Excel Workflows for Fixed Effects ANOVA

The same Fixed Effects ANOVA workflow can be reproduced in SPSS, R, Python and Excel. The main steps are to define the categorical factor as fixed, compare factor means, fit the ANOVA model, inspect fixed effect estimates, review sum of squares, check residual diagnostics and report effect size.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Fixed Effects ANOVA

SPSS Syntax for Fixed Effects ANOVA

* Fixed Effects ANOVA in SPSS.
* Dependent variable: G3.
* Fixed factor: studytime.

TITLE "Fixed Effects ANOVA: G3 by Studytime".

UNIANOVA G3 BY studytime
  /METHOD=SSTYPE(3)
  /INTERCEPT=INCLUDE
  /PRINT=DESCRIPTIVE ETASQ HOMOGENEITY
  /EMMEANS=TABLES(studytime)
  /SAVE=PRED RESID
  /CRITERIA=ALPHA(.05)
  /DESIGN=studytime.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Fixed-Effects-ANOVA-SPSS-Output.pdf".

Python Code for Fixed Effects ANOVA

import pandas as pd
import numpy as np
import scipy.stats as stats
import statsmodels.api as sm
from statsmodels.formula.api import ols

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

df["G3"] = pd.to_numeric(df["G3"], errors="coerce")
df["studytime"] = df["studytime"].astype("category")

df_model = df.dropna(subset=["G3", "studytime"]).copy()

# Fixed effects ANOVA model
model = ols("G3 ~ C(studytime)", data=df_model).fit()
anova_table = sm.stats.anova_lm(model, typ=2)

print(anova_table)

# Group means and fixed effect estimates
grand_mean = df_model["G3"].mean()
group_means = df_model.groupby("studytime")["G3"].mean()
fixed_effects = group_means - grand_mean

print("Grand mean:", grand_mean)
print("Group means:")
print(group_means)
print("Fixed effect estimates:")
print(fixed_effects)

# Sum of squares and effect sizes
ss_fixed = anova_table.loc["C(studytime)", "sum_sq"]
df_fixed = anova_table.loc["C(studytime)", "df"]
ss_error = anova_table.loc["Residual", "sum_sq"]
df_error = anova_table.loc["Residual", "df"]

ss_total = ss_fixed + ss_error
ms_fixed = ss_fixed / df_fixed
ms_error = ss_error / df_error

f_observed = ms_fixed / ms_error
p_value = stats.f.sf(f_observed, df_fixed, df_error)
f_critical = stats.f.ppf(.95, df_fixed, df_error)

eta_squared = ss_fixed / ss_total
partial_eta_squared = ss_fixed / (ss_fixed + ss_error)
omega_squared = (ss_fixed - df_fixed * ms_error) / (ss_total + ms_error)
epsilon_squared = (ss_fixed - df_fixed * ms_error) / ss_total
cohens_f = np.sqrt(eta_squared / (1 - eta_squared))

print("Observed F:", f_observed)
print("Critical F:", f_critical)
print("p value:", p_value)
print("Eta squared:", eta_squared)
print("Partial eta squared:", partial_eta_squared)
print("Omega squared:", omega_squared)
print("Epsilon squared:", epsilon_squared)
print("Cohen's f:", cohens_f)

# Residuals for diagnostics
df_model["fitted"] = model.fittedvalues
df_model["residual"] = model.resid

R Code for Fixed Effects ANOVA

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

df$G3 <- as.numeric(df$G3)
df$studytime <- as.factor(df$studytime)

df_model <- na.omit(df[, c("G3", "studytime")])

# Fixed effects ANOVA model
model <- aov(G3 ~ studytime, data = df_model)
summary(model)

# Group means and fixed effect estimates
grand_mean <- mean(df_model$G3)
group_means <- aggregate(G3 ~ studytime, data = df_model, mean)
group_means$fixed_effect <- group_means$G3 - grand_mean

grand_mean
group_means

# ANOVA components
anova_table <- summary(model)[[1]]

ss_fixed <- anova_table["studytime", "Sum Sq"]
df_fixed <- anova_table["studytime", "Df"]
ss_error <- anova_table["Residuals", "Sum Sq"]
df_error <- anova_table["Residuals", "Df"]

ss_total <- ss_fixed + ss_error
ms_fixed <- ss_fixed / df_fixed
ms_error <- ss_error / df_error

f_observed <- ms_fixed / ms_error
p_value <- pf(f_observed, df_fixed, df_error, lower.tail = FALSE)
f_critical <- qf(.95, df_fixed, df_error)

eta_squared <- ss_fixed / ss_total
partial_eta_squared <- ss_fixed / (ss_fixed + ss_error)
omega_squared <- (ss_fixed - df_fixed * ms_error) / (ss_total + ms_error)
epsilon_squared <- (ss_fixed - df_fixed * ms_error) / ss_total
cohens_f <- sqrt(eta_squared / (1 - eta_squared))

data.frame(
  f_observed = f_observed,
  f_critical = f_critical,
  p_value = p_value,
  eta_squared = eta_squared,
  partial_eta_squared = partial_eta_squared,
  omega_squared = omega_squared,
  epsilon_squared = epsilon_squared,
  cohens_f = cohens_f
)

# Diagnostics
plot(model)

Excel Formulas for Fixed Effects ANOVA

Run ANOVA:
Data > Data Analysis > ANOVA: Single Factor

Grand mean:
=AVERAGE(all_G3_values)

Fixed factor level mean:
=AVERAGE(level_values)

Fixed effect estimate:
=LevelMean - GrandMean

F statistic:
=MS_Between / MS_Within

Critical F value:
=F.INV.RT(alpha, df_between, df_within)

P value:
=F.DIST.RT(F_observed, df_between, df_within)

Eta squared:
=SS_Between / SS_Total

Omega squared:
=(SS_Between - df_between * MS_Within) / (SS_Total + MS_Within)

Cohen's f:
=SQRT(EtaSquared / (1 - EtaSquared))

Decision:
=IF(p_value<0.05,"Reject H0","Fail to reject H0")

APA Reporting Wording

When reporting Fixed Effects ANOVA, state that the factor levels were treated as fixed, report the F statistic, degrees of freedom, p value if available, and effect size. Also explain which levels are above or below the grand mean.

Common Mistakes

When to Use Fixed Effects ANOVA

Use Fixed Effects ANOVA when the categorical factor levels are deliberately selected and are the exact levels you want to compare. In this article, the four studytime levels are fixed because the analysis is about those four categories.

For related guides, see ANOVA in Python, ANOVA in R, ANOVA in SPSS, ANOVA Assumptions, ANOVA Effect Size, Balanced ANOVA, Brown Forsythe ANOVA, Cohen’s F Formula, Effect Size, and T Test vs ANOVA.

Downloads and Resources for Fixed Effects ANOVA

Use these resources to reproduce the Fixed Effects ANOVA workflow. The SPSS output PDF is included as the software verification file, while the script and workbook placeholders can be replaced after the final downloadable files are uploaded to the WordPress Media Library.

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