Quick Answer: GLM ANOVA Result

The worked GLM ANOVA example shows that the model has a strong global result. The global F distribution chart reports observed F = 21.126, critical F = 2.228, df1 = 5 and df2 = 643. The observed F statistic is far to the right of the critical F value, so the combined fixed factors explain a statistically significant part of G3 variation.

The fixed-factor F statistic chart shows that school has the largest factor evidence, with an F statistic just above 50. Studytime is the next important source, with an F statistic around 9. Sex is smaller, with an F statistic around 7. The partial eta squared chart shows the same ranking: school has the largest partial eta squared near 0.073, studytime is near 0.041, and sex is near 0.011.

What Is GLM ANOVA?

GLM ANOVA is an ANOVA fitted inside the General Linear Model framework. Ordinary one-way ANOVA compares one categorical factor. GLM ANOVA can include more than one fixed factor, interaction terms, covariates and planned model terms, depending on the design.

In this article, the GLM uses G3 final grade as the dependent variable and uses studytime, school and sex as fixed factors. The purpose is to test whether these fixed factors explain differences in mean G3 after accounting for the other factors in the same model.

The supplied charts show that the global model is strong. The global F statistic is 21.126 and the critical F value is 2.228. The factor-level chart shows that school has the strongest F statistic, studytime is also important, and sex has the smallest partial eta squared among the three fixed factors.

GLM 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.

GLM ANOVA Formula

A GLM ANOVA model expresses the numeric outcome as a linear combination of fixed factors and residual error.

For the supplied chart set, the outcome is G3. The fixed factors are studytime, school and sex.

The exact model may be expanded when interactions or covariates are included, but the supplied charts focus on fixed-factor evidence for studytime, school and sex.

Factor-Level F Statistic

Each fixed factor gets an F statistic. In the supplied fixed-factor F chart, school is the largest, studytime is second and sex is smallest. A larger F statistic means stronger evidence that the factor explains outcome variation after the other terms are included.

Global F Statistic

The global model chart reports observed F = 21.126 and critical F = 2.228. This means the full set of fixed factors explains more variation than expected from residual error alone.

Partial Eta Squared

Partial eta squared measures the practical size of each fixed factor while accounting for error variation. The supplied chart shows school as the largest effect-size source, studytime as the next source and sex as the smallest source.

Fixed Factor Hypotheses

GLM ANOVA tests a separate hypothesis for each fixed factor, and it can also test a global model hypothesis. The global hypothesis asks whether the model explains meaningful outcome variation. The fixed-factor hypotheses ask whether each factor contributes after the other model terms are considered.

Dataset and Variables Used

The worked example uses a student performance dataset. The dependent variable is G3 final grade. The fixed factors are studytime, school and sex. The cell means heatmap shows mean G3 for school-by-studytime combinations.

For supporting concepts, review Descriptive Statistics, Mean Median and Mode, Standard Deviation, Variance, Standard Error, Confidence Interval, Effect Size and Cohen’s F Formula.

Python Chart-by-Chart Interpretation

The Python charts show the first GLM ANOVA workflow. They explain adjusted group means, school-by-studytime cell means, fixed-factor F statistics, global F decision, partial eta squared, residual diagnostics and observed-versus-fitted model fit.

Python Chart 1: Group Means with 95% Confidence Intervals

This chart shows the observed mean G3 score for each studytime group. Studytime group 1 is lowest, group 2 is higher, group 3 is highest and group 4 remains close to the high range.

The interval for group 4 is visibly wider than the intervals for the first three studytime groups. That wider interval means the estimate for group 4 is less precise, but the mean still stays around the higher G3 range.

This chart gives the practical direction behind the studytime part of the GLM ANOVA. The studytime means are not flat, so studytime contributes to the fixed-factor evidence in the model.

Python Chart 2: Cell Means Heatmap

This heatmap compares mean G3 across school and studytime combinations. The GP row is consistently higher than the MS row across the studytime levels. The brightest cell appears for GP at studytime level 3, while the darkest cell appears for MS at studytime level 1.

The heatmap explains why school becomes the strongest fixed factor in the GLM ANOVA. Across the studytime columns, GP cells generally stay in the higher mean range, while MS cells stay lower.

This chart is important because GLM ANOVA can interpret more than one fixed factor in the same model. The heatmap shows the combined school-by-studytime pattern before the F statistic chart summarizes factor evidence.

Python Chart 3: F Statistics by Fixed Factor

This chart compares the F statistic for each fixed factor after controlling for the other factors. School has the largest F statistic, just above 50. Studytime is much smaller but still visible around 9. Sex is the smallest of the three, around 7.

The chart shows that school is the strongest factor in this GLM ANOVA. Studytime still contributes evidence, but the school effect is visibly larger. Sex contributes the least factor evidence among the three fixed factors shown.

This chart should be reported carefully. It does not mean the variables have the same degrees of freedom or the same practical interpretation. It shows the relative strength of factor evidence in the fitted GLM ANOVA.

Python Chart 4: Global F Distribution

This chart shows the global F distribution for the GLM ANOVA model. The subtitle reports right-tail p-value = 0, df1 = 5 and df2 = 643. The legend shows observed F = 21.126 and critical F = 2.228.

The observed F line is far to the right of the critical F line. This means the full fixed-factor model explains G3 variation far beyond the rejection threshold.

This chart gives the main model-level decision. The GLM ANOVA model is statistically significant as a whole, even before interpreting each fixed factor separately.

Python Chart 5: Partial Eta Squared by Fixed Factor

This chart compares the practical effect size of each fixed factor. School has the largest partial eta squared, near 0.073. Studytime is lower, near 0.041. Sex is much smaller, near 0.011.

The effect-size ranking matches the F statistic chart. School is the strongest practical source, studytime is the next source, and sex is the smallest source in the fitted model.

This chart is essential because significance and practical size are not identical. The GLM ANOVA is statistically strong, but partial eta squared shows how much practical contribution each fixed factor has.

Python Chart 6: Residuals vs Fitted Values

This chart shows residuals against fitted values. The points form vertical bands because GLM ANOVA produces fitted values for combinations of categorical factor levels. Most residuals are scattered around the zero line.

The residual range extends roughly from strong negative values below -10 to positive values above 7. Several negative residuals appear far below zero, showing cases where observed G3 was much lower than the model-fitted value.

This chart supports a balanced diagnostic conclusion. The model is statistically significant, but individual prediction errors remain visible. The residual pattern should be reported as part of the model-checking section.

Python Chart 7: Residual Q-Q Plot

The Q-Q plot compares ordered residuals with a normal reference line. The middle part follows the general upward trend, but the lower tail departs strongly from the line.

The largest departure occurs in the negative residual tail, where several values are far below the reference pattern. The upper tail is closer to the line than the lower tail, although it also shows some spread.

The diagnostic conclusion is that residual normality is not perfect. The GLM ANOVA result remains strong, but the final report should mention lower-tail residual departures.

Python Chart 8: Observed vs Fitted Values

This chart compares observed G3 scores with fitted values. Fitted values are concentrated in a narrower range, roughly from about 9.5 to 14, while observed scores spread much more widely from very low scores to high scores.

The diagonal reference line marks perfect fit. Many points are not on the diagonal, showing that the model captures group-level mean patterns rather than perfectly predicting each individual student’s final grade.

This chart gives the best model-fit context. The fixed factors explain meaningful group differences, but individual G3 outcomes still contain substantial unexplained variation.

R Chart-by-Chart Validation

The R validation charts repeat the GLM ANOVA workflow in a second software environment. They confirm the group mean pattern, school-by-studytime cell means, factor-level F statistic ranking, global F decision, partial eta squared pattern and residual diagnostic behavior.

R Chart 1: Group Means with 95% Confidence Intervals

This R chart confirms the same studytime mean pattern. Mean G3 is lowest in studytime group 1, higher in group 2, highest in group 3 and still high in group 4.

The confidence interval pattern also matches the Python chart, with the fourth group showing wider uncertainty. The direction of the studytime pattern remains stable across software.

In reporting, this chart validates the studytime part of the GLM ANOVA result.

R Chart 2: Cell Means by School and Studytime

This R heatmap confirms the school-by-studytime pattern. GP cells are generally in the higher mean range, while MS cells are lower across the studytime levels.

The strongest high-value area remains in the GP row around studytime level 3. The lowest area remains in the MS row at studytime level 1.

The heatmap validates the reason school becomes the largest fixed factor in the F statistic and partial eta squared charts.

R Chart 3: F Statistics by Fixed Factor

This R chart confirms the same fixed-factor ranking. School has the highest F statistic, studytime has the next strongest factor evidence, and sex has the smallest F statistic among the fixed factors.

The high school F statistic shows that school contributes a large part of the model-level evidence. Studytime remains meaningful, while sex is weaker in this model.

This chart should be used as a validation of the Python factor-level interpretation.

R Chart 4: Global F Distribution

This R chart confirms the global model decision. The observed global F statistic is far to the right of the critical F value.

The model-level result is therefore statistically significant. The fixed factors together explain more G3 variation than expected from residual error alone.

In the final report, this chart supports the statement that the GLM ANOVA model is significant as a whole.

R Chart 5: Partial Eta Squared by Fixed Factor

This R chart confirms the same practical effect-size ranking. School has the largest partial eta squared, studytime is second and sex is smallest.

The effect-size pattern supports a practical interpretation rather than only a significance interpretation. School contributes the most practical explanatory share in the fitted GLM ANOVA.

This chart should be used with the F statistics chart because both show the same ordering of factor importance.

R Chart 6: Residuals vs Fitted Values

The R residual chart confirms the same fitted-value banding and residual spread. Most residuals cluster around zero, but several cases fall far below zero.

The negative residuals show students whose observed G3 was much lower than their fitted value. This pattern appears in both Python and R diagnostics.

The chart supports a careful conclusion: the model is statistically significant, but residual diagnostics should still be reported.

R Chart 7: Residual Q-Q Plot

The R Q-Q plot confirms the same lower-tail departure from the normal reference line. The central part of the residual distribution is closer to the line than the negative tail.

This means residual normality is approximate, not perfect. The diagnostic pattern should be mentioned in the assumptions section.

The Q-Q plot does not cancel the GLM ANOVA result. It adds context about lower-tail residual behavior.

R Chart 8: Observed vs Fitted Values

This R chart confirms that fitted values are more compressed than observed G3 scores. The model predicts group-level fitted means, while observed values vary widely across students.

Points away from the diagonal show individual prediction error. This is expected in ANOVA-style GLM because the model explains group patterns rather than every individual grade perfectly.

The chart supports a balanced fit statement: the model is significant and useful for group effects, but it does not fully explain individual performance differences.

SPSS Output and Report PDFs

The supplied report files provide downloadable support for the GLM ANOVA workflow. The Python report, R report and SPSS output PDF should be linked as verification resources for this article.

Download GLM ANOVA Python Report PDF

Download GLM ANOVA R Report PDF

Download GLM ANOVA SPSS Output PDF

Output Items to Read

SPSS, R, Python and Excel Workflows for GLM ANOVA

The same GLM ANOVA workflow can be reproduced in SPSS, R, Python and Excel. The main steps are to define the dependent variable, enter the fixed factors, fit the general linear model, inspect factor-level F statistics, report partial eta squared and check residual diagnostics.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for GLM ANOVA

SPSS Syntax for GLM ANOVA

* GLM ANOVA in SPSS.
* Dependent variable: G3.
* Fixed factors: studytime, school, sex.

TITLE "GLM ANOVA: G3 by Studytime, School and Sex".

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

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

Python Code for GLM 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["school"] = df["school"].astype("category")
df["sex"] = df["sex"].astype("category")

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

# GLM ANOVA with fixed factors
model = ols("G3 ~ C(studytime) + C(school) + C(sex)", data=df_model).fit()
anova_table = sm.stats.anova_lm(model, typ=2)

print(anova_table)

# Cell means for school by studytime
cell_means = (
    df_model
    .groupby(["school", "studytime"])["G3"]
    .mean()
    .reset_index()
)
print(cell_means)

# Partial eta squared by factor
ss_error = anova_table.loc["Residual", "sum_sq"]
partial_eta = {}

for source in anova_table.index:
    if source != "Residual":
        ss_factor = anova_table.loc[source, "sum_sq"]
        partial_eta[source] = ss_factor / (ss_factor + ss_error)

print("Partial eta squared:")
print(partial_eta)

# Global model F test
ss_model = anova_table.loc[anova_table.index != "Residual", "sum_sq"].sum()
df_model_terms = anova_table.loc[anova_table.index != "Residual", "df"].sum()

ss_resid = anova_table.loc["Residual", "sum_sq"]
df_resid = anova_table.loc["Residual", "df"]

ms_model = ss_model / df_model_terms
ms_error = ss_resid / df_resid

f_global = ms_model / ms_error
p_global = stats.f.sf(f_global, df_model_terms, df_resid)
f_critical = stats.f.ppf(0.95, df_model_terms, df_resid)

print("Global F:", f_global)
print("Global p:", p_global)
print("Critical F:", f_critical)

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

print(df_model[["G3", "fitted", "residual"]].head())

R Code for GLM ANOVA

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

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

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

# GLM ANOVA with fixed factors
model <- aov(G3 ~ studytime + school + sex, data = df_model)

summary(model)

# Cell means
aggregate(G3 ~ school + studytime, data = df_model, mean)

# Fitted values and residuals
df_model$fitted <- fitted(model)
df_model$residuals <- residuals(model)

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

ss_error <- anova_table["Residuals", "Sum Sq"]

partial_eta_squared <- data.frame(
  factor = rownames(anova_table)[rownames(anova_table) != "Residuals"],
  partial_eta_squared = NA
)

for(i in seq_len(nrow(partial_eta_squared))){
  factor_name <- partial_eta_squared$factor[i]
  ss_factor <- anova_table[factor_name, "Sum Sq"]
  partial_eta_squared$partial_eta_squared[i] <- ss_factor / (ss_factor + ss_error)
}

partial_eta_squared

# Diagnostic plots
plot(model)

Excel Notes for GLM ANOVA

Excel can summarize GLM ANOVA data, but full multi-factor GLM is easier in SPSS, R or Python.

Useful Excel steps:
1. Create PivotTable for mean G3 by studytime.
2. Create PivotTable for mean G3 by school and studytime.
3. Use Data Analysis ToolPak for simpler ANOVA layouts.
4. Use SPSS, R or Python for full GLM ANOVA with multiple fixed factors.
5. Calculate partial eta squared when SS factor and SS error are available:

Partial eta squared:
=SS_Factor/(SS_Factor+SS_Error)

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

Global model interpretation:
Observed F > Critical F means the GLM model is statistically significant.

APA Reporting Wording

When reporting GLM ANOVA, include the dependent variable, fixed factors, global model result, factor-level findings and effect sizes. Also mention residual diagnostics when the Q-Q plot or residual plot shows visible departures.

Common Mistakes

When to Use GLM ANOVA

Use GLM ANOVA when the dependent variable is numeric and the model includes one or more fixed categorical factors. GLM ANOVA is especially useful when the analysis needs adjusted factor effects, multiple fixed factors, estimated marginal means or effect sizes from one model.

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 GLM ANOVA

Use these resources to reproduce the GLM ANOVA workflow. The Python report, 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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