Quick Answer: Nested ANOVA Result

The worked Nested ANOVA example shows a strong outer factor result and a significant nested factor result. The p-value decision chart reports outer factor p = 4.202e-14 and nested factor p = 1.478e-06. Both values are far below alpha = .05, so the outer school effect and the studytime-within-school nested effect are statistically significant.

The nested F distribution chart reports nested F = 6.397, df1 = 6, df2 = 641, and right-tail p-value = 1.478e-06. The observed F statistic is far into the right tail, so the nested studytime-within-school cells do not have equal mean G3 scores.

What Is Nested ANOVA?

Nested ANOVA is an analysis of variance design where one factor is contained inside another factor. The inner factor levels are interpreted within each level of the outer factor. This is different from a fully crossed Factorial ANOVA, where every level of one factor is crossed with every level of another factor in the same way.

In this worked example, school is the outer factor. Studytime cells are treated as nested inside each school, producing cells such as GP:1, GP:2, GP:3, GP:4, MS:1, MS:2, MS:3 and MS:4. The dependent variable is G3 final grade.

The outer factor chart shows that GP has a higher mean G3 than MS. The nested group means chart shows that studytime cells inside each school are not equal. The formal p-value summary confirms both conclusions: the outer school effect is significant, and the nested studytime-within-school effect is also significant.

Nested ANOVA is part of the wider ANOVA family. It should be compared with Fixed Effects ANOVA, Factorial ANOVA, Balanced ANOVA, Brown Forsythe ANOVA, ANOVA in Python, ANOVA in R, ANOVA in SPSS, and ANOVA Assumptions.

Nested ANOVA Formula

A nested ANOVA model separates the outer factor effect from the nested factor effect. The nested term is written inside parentheses to show that it belongs within the outer factor.

For this worked example, the model becomes:

The school term tests whether the outer school means differ. The studytime(school) term tests whether the studytime cells inside schools differ after the outer school structure is considered.

Outer Factor Test

The p-value decision chart reports outer p = 4.202e-14. This value is below .05, so the outer school means are not equal.

Nested Factor Test

The nested F distribution reports nested F = 6.397, df1 = 6, df2 = 641 and p = 1.478e-06. This shows that the nested studytime-within-school cells differ significantly.

Nested ANOVA Hypotheses

Nested ANOVA has a hypothesis for the outer factor and a hypothesis for the nested factor. The two hypotheses answer different questions and should not be mixed together.

Dataset and Variables Used

The worked example uses student performance data. The outcome variable is G3 final grade. The outer factor is school, with GP and MS as the two school groups. The nested factor is studytime inside school, producing eight nested cells.

The cell-size chart shows why Nested ANOVA interpretation should include sample-size context. GP:2 is the largest cell, while MS:4 is very small. Small cells can produce wider confidence intervals and less stable mean estimates.

For supporting concepts, review Descriptive Statistics, Mean Median and Mode, Standard Deviation, Standard Error, Confidence Interval, F Distribution, P Value, and Null and Alternative Hypothesis.

Python Chart-by-Chart Interpretation

The Python chart sequence explains the Nested ANOVA result through nested group means, nested cell distributions, outer school means, nested effect F distribution, p-value decision summary, residual diagnostics and cell-size context.

Python Chart 1: Nested Group Means with 95% Confidence Intervals

This chart shows the mean G3 value for each nested school-by-studytime cell. GP cells are generally higher than MS cells. GP:1 is the lowest GP cell, while GP:3 and GP:4 are in the highest GP range. MS:1 is the lowest MS cell, while MS:3 has the highest MS mean.

The confidence intervals are not equal across cells. MS:4 has the widest interval because it is the smallest cell. GP:4 also has a wider interval than the larger GP cells. The chart therefore shows both mean differences and uncertainty around those means.

This chart gives the practical direction behind the formal Nested ANOVA result. The nested cell means are not flat, and the formal nested p-value confirms that these differences are statistically significant.

Python Chart 2: Distribution by Nested Cell

The boxplot chart shows that the nested cells differ not only in their means but also in their spread. GP:3 and GP:4 are centered higher than GP:1. MS:1 and MS:2 are centered lower, while MS:3 and MS:4 shift upward.

Several cells include very low outlying values, including zero values in GP:1, MS:1 and MS:2. These low values help explain why residual diagnostics later show lower-tail departures.

This chart supports a careful interpretation of Nested ANOVA. The means differ significantly, but the distributions are not identical and the presence of low outliers should be described in the assumptions section.

Python Chart 3: Outer Factor Means

This chart compares the two outer school means. GP has a higher mean G3 than MS. The GP bar is around the 12.5 to 12.6 range, while the MS bar is around the 10.6 to 10.7 range.

The confidence interval around GP is narrower than the MS interval. This reflects stronger precision for the GP mean and more uncertainty for the MS mean.

This chart explains the outer p-value result. The p-value decision chart reports outer p = 4.202e-14, which confirms that the school-level difference is statistically significant.

Python Chart 4: Nested Effect on F Distribution

This chart gives the formal nested-effect test. The subtitle reports right-tail p-value = 1.478e-06, df1 = 6 and df2 = 641. The dashed vertical line marks nested F = 6.397.

The observed F statistic is far to the right of the main density region. This means the nested effect is much larger than expected under the equal nested-cell means null hypothesis.

The statistical decision is clear. The studytime cells nested within school differ significantly in mean G3.

Python Chart 5: P-Value Decision Summary

This decision chart places alpha = 0.05 next to the two tested p-values. The outer p-value is 4.202e-14, and the nested p-value is 1.478e-06.

Both p-values are essentially at the baseline compared with the alpha bar. This means both the outer factor and nested factor are statistically significant.

This chart should be used for the final decision paragraph. It confirms that school matters and that studytime cells inside school also matter.

Python Chart 6: Residuals vs Fitted Values

This chart shows residuals against fitted values from the nested model. The points form vertical bands because the fitted values come from nested cell means. Most residuals are spread around zero, but several negative residuals extend far below the center.

The lower-tail points show students whose observed G3 values were much lower than the nested cell fitted values. This pattern matches the low outlying points visible in the boxplot chart.

The diagnostic conclusion is balanced. The nested effects are statistically significant, but residual spread and lower-tail values should be reported as part of model checking.

Python Chart 7: Residual Q-Q Plot

The Q-Q plot shows visible departures from the normal reference line. The central residuals follow the general diagonal direction, but the lower tail bends away strongly.

The negative residual tail reaches far below the reference line. This means the residual distribution is not perfectly normal and contains strong low-end departures.

The final report should mention this diagnostic issue. The Nested ANOVA p-values are strong, but the residual normality assumption is approximate rather than perfect.

Python Chart 8: Cell Size and Standard Deviation

This chart shows that the nested cells are uneven in size. GP:2 is the largest cell with about 206 cases. GP:1 has about 119 cases. MS:2 and MS:1 are also large, while MS:4 is very small with about 8 cases.

The standard deviation line shows that spread differs across cells. MS:2 has one of the largest standard deviations, while GP:3 has one of the smaller spreads.

This chart is important for interpretation because significant nested differences should be read with cell-size and spread context. Very small cells can produce less stable mean estimates and wider intervals.

R Chart-by-Chart Validation

The R validation charts repeat the same Nested ANOVA workflow in a second software environment. They confirm the nested group mean pattern, cell boxplot structure, outer factor difference, nested F decision, p-value summary, residual diagnostics and cell-size context.

R Chart 1: Nested Group Means with 95% Confidence Intervals

The R chart confirms the same nested cell mean pattern. GP:1 is lower than the other GP cells, GP:3 and GP:4 are in the high range, MS:1 and MS:2 are lower, and MS:3 is the highest MS cell.

The confidence intervals again show that not all cells have the same precision. MS:4 is visibly the least stable cell because its interval is wide.

This validation chart supports the Python conclusion that the nested studytime-within-school cells differ meaningfully.

R Chart 2: Distribution by Nested Cell

The R boxplot chart confirms that the distributions differ across nested cells. Higher GP and MS studytime cells tend to sit above the lower cells, but spread varies by cell.

Low outlying values remain visible in several cells. These outliers are important because they help explain the lower-tail departure in the residual Q-Q plot.

The R chart validates the same practical interpretation: Nested ANOVA detects mean differences, while boxplots show spread and outlier context.

R Chart 3: Outer Factor Means

The R outer factor chart confirms that GP has a higher mean G3 than MS. The gap between the two school bars is clear.

The confidence interval around MS is wider than GP, showing more uncertainty in the MS mean. The mean difference remains large enough to support the very small outer p-value.

This validation chart confirms the outer school effect in the same direction as the Python output.

R Chart 4: Nested Effect on F Distribution

The R F distribution chart repeats the same formal nested-effect result. The subtitle reports right-tail p-value = 1.478e-06, df1 = 6 and df2 = 641.

The observed nested F line is far into the right tail. This confirms that the nested-cell differences are statistically significant.

This chart validates the nested factor decision from the Python workflow.

R Chart 5: P-Value Decision Summary

The R decision chart confirms that the outer and nested p-values are both far below alpha = .05. The outer p-value is 4.202e-14, and the nested p-value is 1.478e-06.

This chart confirms the same final decision: both the school effect and studytime-within-school effect are statistically significant.

It should be used as the compact statistical decision figure after the detailed mean and distribution charts.

R Chart 6: Residuals vs Fitted Values

The R residuals-versus-fitted chart confirms that residuals are centered around zero but include strong negative residuals. The vertical banding appears because fitted values come from the nested cell means.

The lower residual tail shows that some students had observed G3 scores far below their fitted nested-cell expectation.

This chart validates the diagnostic caution reported from the Python residual chart.

R Chart 7: Residual Q-Q Plot

The R Q-Q plot confirms the same lower-tail departure. The central points follow the general reference direction, while the lower tail departs strongly from the line.

This means residual normality is approximate rather than perfect. The result remains statistically strong, but the assumption check should be reported.

The chart supports a transparent final report that includes both significance and diagnostic limitations.

R Chart 8: Cell Size and Standard Deviation

The R cell-size chart confirms that the design is unbalanced. GP:2 is the largest cell, while MS:4 is the smallest cell.

The standard deviation line confirms that cell spread differs across nested cells. MS cells show higher spread in some levels than GP cells.

This validation chart supports the final recommendation to report cell size and spread together with the Nested ANOVA p-values.

SPSS Output and Report PDFs

The supplied report files support the Nested ANOVA workflow. The Python report provides the first chart set, the R report validates the chart sequence, and the SPSS output PDF provides the menu-based output for reporting and verification.

Download Nested ANOVA Python Report PDF

Download Nested ANOVA R Report PDF

Download Nested ANOVA SPSS Output PDF

Output Items to Read

SPSS, R, Python and Excel Workflows for Nested ANOVA

The same Nested ANOVA workflow can be reproduced in SPSS, R and Python. Excel can prepare nested means, cell sizes, standard deviations and charts, but the formal nested ANOVA model should be run in SPSS, R or Python.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Nested ANOVA

SPSS Syntax for Nested ANOVA

* Nested ANOVA in SPSS.
* Outcome: G3.
* Outer factor: school.
* Nested factor: studytime nested within school.

TITLE "Nested ANOVA: G3 by Studytime Nested within School".

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

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="nested_anova_spss_output.pdf".

Python Code for Nested ANOVA

import pandas as pd
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["school"] = df["school"].astype("category")
df["studytime"] = df["studytime"].astype("category")

df_model = df.dropna(subset=["G3", "school", "studytime"]).copy()
df_model["nested_cell"] = df_model["school"].astype(str) + ":" + df_model["studytime"].astype(str)

# Nested ANOVA model
# school is the outer factor
# studytime is treated as nested within school through the school:studytime term
model = ols("G3 ~ C(school) + C(school):C(studytime)", data=df_model).fit()

anova_table = sm.stats.anova_lm(model, typ=2)
print(anova_table)

# Cell summaries
cell_summary = (
    df_model
    .groupby(["school", "studytime", "nested_cell"])["G3"]
    .agg(["count", "mean", "std"])
    .reset_index()
)

print(cell_summary)

# Outer factor summaries
outer_summary = (
    df_model
    .groupby("school")["G3"]
    .agg(["count", "mean", "std"])
    .reset_index()
)

print(outer_summary)

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

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

R Code for Nested ANOVA

library(tidyverse)

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

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

df_model <- df %>%
  select(G3, school, studytime) %>%
  drop_na() %>%
  mutate(nested_cell = interaction(school, studytime, sep = ":"))

# Nested ANOVA
model <- aov(G3 ~ school + school:studytime, data = df_model)

summary(model)

# Cell summaries
df_model %>%
  group_by(school, studytime, nested_cell) %>%
  summarise(
    n = n(),
    mean = mean(G3),
    sd = sd(G3),
    .groups = "drop"
  )

# Outer factor summaries
df_model %>%
  group_by(school) %>%
  summarise(
    n = n(),
    mean = mean(G3),
    sd = sd(G3),
    .groups = "drop"
  )

# Residual diagnostics
plot(model)

Excel Notes for Nested ANOVA

Excel can support Nested ANOVA reporting, but use SPSS, R or Python for the final test.

Useful Excel steps:
1. Keep columns: G3, school, studytime.
2. Create nested cell label:
   =A2&":"&B2
   where A2 contains school and B2 contains studytime.
3. Create PivotTable:
   Rows = nested cell
   Values = mean G3, count G3, standard deviation G3
4. Create another PivotTable:
   Rows = school
   Values = mean G3
5. Create bar charts for nested means and outer means.
6. Use SPSS, R or Python for the nested F test and p-values.
7. Report outer p-value, nested p-value, F statistic, df, residual diagnostics and cell-size context.

APA Reporting Wording

When reporting Nested ANOVA, identify the outer factor, the nested factor, the dependent variable, the F statistic, degrees of freedom, p-values and diagnostic context. Also state that the nested factor is interpreted inside the outer factor.

Common Mistakes

When to Use Nested ANOVA

Use Nested ANOVA when lower-level groups are naturally contained within higher-level groups. Examples include students nested within classrooms, classrooms nested within schools, plots nested within farms, batches nested within factories, and samples nested within laboratories.

Nested ANOVA should be compared with Factorial ANOVA, Fixed Effects ANOVA, Balanced ANOVA, Brown Forsythe ANOVA, ANOVA in SPSS, ANOVA in R, ANOVA in Python, ANOVA Effect Size, and ANOVA Assumptions.

Downloads and Resources for Nested ANOVA

Use these resources to reproduce the Nested 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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