Quick Answer: Mixed ANOVA Result

The worked Mixed ANOVA example shows two clear results and one weak interaction result. The between-subjects studytime effect is statistically significant, F = 16.979, p < .001, partial η² = .073. This means the four studytime groups differ in their average grade profile across G1, G2 and G3.

The within-subject Grade_Period effect is also statistically significant, F = 18.865, p < .001, partial η² = .028. This means the overall score changes across G1, G2 and G3. The interaction between Grade_Period and studytime is small and not statistically significant under the sphericity-assumed test, F = 1.731, p = .110, partial η² = .008. The profile lines are not identical, but the formal interaction result does not support a strong grade-period-by-studytime interaction.

What Is Mixed ANOVA?

Mixed ANOVA, also called mixed design ANOVA or split-plot ANOVA, combines a repeated-measures factor and a between-subjects factor. The repeated-measures factor is measured on the same participants more than once. The between-subjects factor divides participants into independent groups.

In this article, the same students have three grade measurements: G1, G2 and G3. That makes Grade_Period the within-subject factor. Students also belong to studytime groups 1, 2, 3 and 4. That makes studytime the between-subjects factor.

The key advantage of Mixed ANOVA is that it answers three questions at once. It tests whether grades change over repeated measurements, whether groups differ overall, and whether the repeated-measures pattern differs by group. This is why Mixed ANOVA is more informative than running separate one-sample t tests, several two-sample t tests, or a simple Factorial ANOVA that ignores the repeated nature of the data.

Mixed ANOVA is closely connected with Fixed Effects ANOVA, Factorial ANOVA, ANOVA in SPSS, ANOVA in Python, ANOVA in R, ANOVA Assumptions, Eta Squared, and Cohen’s F Formula.

Mixed ANOVA Formula

A Mixed ANOVA model includes a between-subjects effect, a within-subject effect, and their interaction.

For this worked example, the model becomes:

The studytime term compares the average grade profile across studytime groups. The Grade_Period term compares G1, G2 and G3 within the same students. The studytime × Grade_Period interaction tests whether the G1-to-G3 change differs across studytime groups.

Between-Subjects F Test

In this example, the between-subjects studytime effect is significant, F = 16.979, p < .001, partial η² = .073. Studytime groups differ in their overall average grade level.

Within-Subjects F Test

The within-subject Grade_Period effect is significant, F = 18.865, p < .001, partial η² = .028. Overall scores change across G1, G2 and G3.

Interaction F Test

The interaction effect is small, F = 1.731, p = .110, partial η² = .008. The studytime profile lines are not identical, but the formal interaction result does not support a strong difference in grade-period change by studytime group.

Mixed ANOVA Hypotheses

Mixed ANOVA has three main hypothesis tests. Each test answers a different research question, so all three should be reported separately.

Dataset and Variables Used

The worked example uses student performance data with 649 valid cases. The repeated outcome measurements are G1, G2 and G3. The between-subjects grouping variable is studytime, with group sizes of 212, 305, 97 and 35.

For background concepts, review Descriptive Statistics, Mean Median and Mode, Standard Deviation, Variance, Confidence Interval, Effect Size, F Distribution, and P Value.

Python Chart-by-Chart Interpretation

The Python chart sequence explains the Mixed ANOVA result using repeated-measures profile lines, within-subject means, cell means, F statistics, partial eta squared, within-subject F distribution, subject trajectories and difference-score distributions.

Python Chart 1: Profile Plot of Repeated Measures by Studytime Group

This profile plot shows the repeated grade trajectory for each studytime group. Studytime group 1 stays lowest across G1, G2 and G3, moving from about 10.50 to 10.84. Group 2 is higher, moving from about 11.54 to 12.09. Group 3 is high throughout and ends highest at G3, while group 4 begins high, dips slightly at G2 and rises again at G3.

The vertical separation between the lines explains the significant between-subjects studytime effect. Students in higher studytime groups have higher overall grade profiles than students in group 1.

The lines are not perfectly parallel, but the interaction F statistic is small and non-significant. The chart should therefore be interpreted as strong evidence of group-level differences and overall grade-period change, not as strong evidence that each studytime group changes differently over time.

Python Chart 2: Overall Within-Subject Means

This chart summarizes the repeated-measures pattern after averaging across studytime groups. The overall mean is about 11.40 for G1, about 11.57 for G2 and about 11.91 for G3.

The bar heights rise across the grade periods. This explains the significant within-subject Grade_Period result: scores are not constant across G1, G2 and G3.

This chart is the cleanest visual explanation of the repeated-measures main effect. It shows the overall grade-period increase without the extra group lines from the profile plot.

Python Chart 3: Cell Means Heatmap

The heatmap gives the group-by-period means in a compact visual form. Studytime group 1 has the lowest values across the row. Group 2 is higher than group 1. Groups 3 and 4 have the highest color intensity, especially at G3.

The exact pattern matches the SPSS descriptive table: group 1 has means of 10.5047, 10.7028 and 10.8443; group 2 has 11.5377, 11.6623 and 12.0918; group 3 has 12.4227, 12.7938 and 13.2268; and group 4 has 12.7714, 12.6286 and 13.0571.

This chart explains the result in practical terms. The significant studytime effect comes from consistent group separation, and the significant Grade_Period effect comes from the upward movement across the repeated measures.

Python Chart 4: Mixed ANOVA F Statistics

This chart compares the three main Mixed ANOVA F tests. The within-subject Grade_Period effect has the largest F statistic, about 36.41 in the Python chart. The between-subject studytime effect is also large, around 16.98. The interaction is much smaller, around 1.73.

The chart shows the strength ranking of the model effects. Grade_Period and studytime are the main signals. The interaction bar is much shorter, matching the non-significant interaction result in the SPSS table.

This chart is useful for explaining the statistical story quickly: grade scores change across repeated grade periods, studytime groups differ overall, and the group-by-time interaction is weak.

Python Chart 5: Partial Eta Squared for Mixed ANOVA Effects

This chart compares the practical effect size of each Mixed ANOVA source. The between-subject studytime effect is largest at about .073. The within-subject Grade_Period effect is smaller but still meaningful. The interaction effect is very small at about .008.

The effect-size chart helps prevent overclaiming. The studytime effect is the strongest practical effect, while the interaction is small.

The correct report should include partial eta squared because an F test alone only gives statistical evidence. Partial eta squared shows how much of the relevant explainable variation is associated with each effect.

Python Chart 6: Within-Subject F Distribution

This F distribution chart shows the within-subject Grade_Period decision. The observed F is about 36.410 in the chart, and the critical F is about 3.003. The right-tail p-value is shown as about 4.441e-16.

The observed F line is far to the right of the critical F line. This means the repeated-measures effect is far beyond the rejection threshold.

This chart gives the formal repeated-measures decision visually. The overall scores differ across G1, G2 and G3.

Python Chart 7: Subject Trajectory Spaghetti Plot

This spaghetti plot shows individual student trajectories across G1, G2 and G3. The thin lines represent sampled students, while the thicker line shows the overall mean trajectory.

The individual lines vary widely. Some students improve, some remain stable, and some decline. A few trajectories fall to very low values, including zero scores. The overall mean line still rises gradually from G1 to G3.

This chart explains why Mixed ANOVA is useful. It handles repeated measurements from the same people while summarizing the overall pattern across many individual trajectories.

Python Chart 8: Difference Score Distributions

This chart shows the distribution of change scores. The G3 – G1 and G3 – G2 distributions are concentrated near small positive values, which supports the overall increase shown in the within-subject means chart.

The chart also shows extreme negative values, especially for G3 – G1. Those negative differences represent students whose final grade was much lower than an earlier grade. The positive tail shows students who improved strongly.

This chart is a practical repeated-measures check. It shows that the average change is positive, but individual change is not uniform across all students.

R Chart-by-Chart Validation

The R validation charts repeat the same Mixed ANOVA workflow in a second software environment. They confirm the profile plot, within-subject means, cell means, F statistics, partial eta squared, within-subject F distribution, subject trajectories and difference-score pattern.

R Chart 1: Profile Plot of Repeated Measures by Studytime Group

The R profile plot confirms the same pattern as the Python chart. Studytime group 1 remains lowest, group 2 is higher, and groups 3 and 4 are in the higher score range.

The group 4 line dips slightly at G2 and rises again at G3, while group 3 rises more steadily. This creates some non-parallel line movement, but the formal interaction remains small and non-significant.

This validation chart supports the same interpretation: the studytime groups differ overall, and scores change across periods, but the interaction is not the main finding.

R Chart 2: Overall Within-Subject Means

The R bar chart confirms that the overall mean increases from G1 to G3. The values match the SPSS descriptives, with G1 around 11.40, G2 around 11.57 and G3 around 11.91.

This validates the repeated-measures conclusion. The Grade_Period effect is statistically significant because the repeated means do not remain equal.

The chart should be used to explain the within-subject effect in a simple way before discussing sphericity corrections.

R Chart 3: Cell Means Heatmap

The R heatmap confirms the same cell-mean structure. Studytime group 1 has the lowest row, group 2 is higher, and groups 3 and 4 have the highest values.

The highest G3 mean appears in studytime group 3, while group 4 remains close to that high range. This supports the studytime main effect.

This validation output is useful because it shows the full repeated-measures-by-group mean table in one visual.

R Chart 4: Mixed ANOVA F Statistics

The R F statistic chart labels the main values directly: studytime has F = 16.979, Grade_Period has F = 36.41 in the R chart workflow, and the interaction has F = 1.731.

The same ranking appears again. The repeated-measures effect and studytime effect are clearly larger than the interaction.

This confirms that the interaction should not be over-interpreted. The main results are the between-group studytime difference and the within-student grade-period change.

R Chart 5: Partial Eta Squared

The R effect-size chart confirms that studytime has the largest partial eta squared, Grade_Period has a smaller but meaningful partial eta squared, and the interaction is very small.

This effect-size pattern matches the SPSS output: studytime partial η² is .073, Grade_Period partial η² is .028 in the within-subjects table, and the interaction partial η² is .008.

The final report should state both statistical significance and effect size. This avoids treating every significant test as equally important.

R Chart 6: Within-Subject F Distribution

The R F distribution chart shows observed F = 36.41 and critical F = 3.003, with the observed F far to the right of the critical line.

The chart label reports the Grade_Period p-value as about 4.146e-16. This is far below .05 and supports the repeated-measures result.

This chart visually confirms the repeated-measures decision in the R workflow.

R Chart 7: Subject Trajectory Spaghetti Plot

The R trajectory plot confirms that individual student paths vary strongly. Some students improve, others decline, and many remain close to their starting range.

The overall mean trajectory still increases gradually across G1, G2 and G3. This validates the within-subject effect while showing that individual patterns are heterogeneous.

This chart is useful for explaining why repeated-measures models should respect within-person structure rather than treating all grade observations as independent.

R Chart 8: Difference Score Distributions

The R difference-score chart confirms that most changes are near small positive values, while a smaller number of students have large negative or positive changes.

The distribution supports the overall Grade_Period effect but also shows that repeated-measures change is not identical for every student.

This chart should be used as a practical diagnostic explanation alongside the formal F test and sphericity discussion.

SPSS Output and Report PDFs

The supplied report files support the Mixed ANOVA workflow. The Python report provides the first chart set, the R report validates the same chart sequence, and the SPSS output PDF provides the repeated-measures GLM tables used for formal reporting.

Download Mixed ANOVA Python Report PDF

Download Mixed ANOVA R Report PDF

Download Mixed ANOVA SPSS Output PDF

SPSS Output Items to Read

SPSS, R, Python and Excel Workflows for Mixed ANOVA

The same Mixed ANOVA workflow can be reproduced in SPSS, R and Python. Excel can prepare profile charts, cell means and difference scores, but formal Mixed ANOVA should be run in SPSS, R or Python because the design contains both repeated and between-subjects components.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Mixed ANOVA

SPSS Syntax for Mixed ANOVA

* Mixed ANOVA in SPSS.
* Repeated measures: G1, G2, G3.
* Within-subject factor: Grade_Period.
* Between-subject factor: studytime.

TITLE "Mixed ANOVA: G1 G2 G3 Repeated Measures by Studytime".

GLM G1 G2 G3 BY studytime
  /WSFACTOR=Grade_Period 3 Polynomial
  /METHOD=SSTYPE(3)
  /PLOT=PROFILE(Grade_Period*studytime)
  /PRINT=DESCRIPTIVE ETASQ HOMOGENEITY
  /CRITERIA=ALPHA(.05)
  /WSDESIGN=Grade_Period
  /DESIGN=studytime.

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

Python Code for Mixed ANOVA

import pandas as pd
import pingouin as pg

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

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

df["studytime"] = df["studytime"].astype("category")
df = df.dropna(subset=["G1", "G2", "G3", "studytime"]).copy()
df["subject_id"] = range(1, len(df) + 1)

long_df = df.melt(
    id_vars=["subject_id", "studytime"],
    value_vars=["G1", "G2", "G3"],
    var_name="Grade_Period",
    value_name="Score"
)

# Mixed ANOVA
mixed_result = pg.mixed_anova(
    data=long_df,
    dv="Score",
    within="Grade_Period",
    between="studytime",
    subject="subject_id"
)

print(mixed_result)

# Cell means
cell_means = (
    long_df
    .groupby(["studytime", "Grade_Period"])["Score"]
    .mean()
    .reset_index()
)

print(cell_means)

# Difference scores
df["G3_minus_G1"] = df["G3"] - df["G1"]
df["G3_minus_G2"] = df["G3"] - df["G2"]

print(df[["G3_minus_G1", "G3_minus_G2"]].describe())

R Code for Mixed ANOVA

library(tidyverse)
library(afex)

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

df$G1 <- as.numeric(df$G1)
df$G2 <- as.numeric(df$G2)
df$G3 <- as.numeric(df$G3)
df$studytime <- as.factor(df$studytime)
df$subject_id <- seq_len(nrow(df))

long_df <- df %>%
  select(subject_id, studytime, G1, G2, G3) %>%
  pivot_longer(
    cols = c(G1, G2, G3),
    names_to = "Grade_Period",
    values_to = "Score"
  ) %>%
  drop_na()

long_df$Grade_Period <- as.factor(long_df$Grade_Period)

# Mixed ANOVA
model <- aov_ez(
  id = "subject_id",
  dv = "Score",
  data = long_df,
  within = "Grade_Period",
  between = "studytime",
  type = 3
)

print(model)

# Cell means
long_df %>%
  group_by(studytime, Grade_Period) %>%
  summarise(
    n = n(),
    mean = mean(Score),
    sd = sd(Score),
    .groups = "drop"
  )

Excel Notes for Mixed ANOVA

Excel can support Mixed ANOVA reporting, but it should not be the main statistical engine.

Useful Excel steps:
1. Keep one row per student.
2. Columns: studytime, G1, G2, G3.
3. Create PivotTable:
   Rows = studytime
   Values = mean G1, mean G2, mean G3
4. Create profile plot from the PivotTable.
5. Calculate overall repeated-measures means:
   =AVERAGE(G1_range)
   =AVERAGE(G2_range)
   =AVERAGE(G3_range)
6. Calculate difference scores:
   =G3-G1
   =G3-G2
7. Run the formal Mixed ANOVA in SPSS, R or Python.
8. Report F, p value, partial eta squared and sphericity correction when needed.

APA Reporting Wording

When reporting Mixed ANOVA, state the within-subject factor, the between-subjects factor, the interaction result, sphericity decision and effect sizes. Report the interaction before overinterpreting separate main effects.

Common Mistakes

When to Use Mixed ANOVA

Use Mixed ANOVA when the same participants are measured more than once and those participants also belong to different independent groups. In this example, G1, G2 and G3 are repeated measurements, while studytime is a between-subjects grouping factor.

Mixed ANOVA is part of the broader ANOVA family. Compare it with Fixed Effects ANOVA, Factorial ANOVA, Balanced ANOVA, Brown Forsythe ANOVA, ANOVA in SPSS, ANOVA in R, ANOVA Effect Size, and ANOVA Assumptions.

Downloads and Resources for Mixed ANOVA

Use these resources to reproduce the Mixed 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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