Quick Answer: Factorial ANOVA Result

The worked Factorial ANOVA example shows that studytime is the only clearly significant factor in the supplied p-value charts. The studytime effect is far below the alpha line, while sex has p ≈ 0.07362 and the studytime × sex interaction has p ≈ 0.08323. Because both of those p values are above 0.05, sex and the interaction are not statistically significant at the 5% level in this chart set.

The interaction plot is still important because the two lines are not perfectly parallel. Female mean G3 increases steadily from studytime group 1 to group 4, while male mean G3 rises from group 1 to group 3 and then drops in group 4. The plotted interaction pattern is visible, but the p-value chart shows that it is not strong enough to be declared statistically significant at alpha = 0.05.

What Is Factorial ANOVA?

Factorial ANOVA is an ANOVA model that studies more than one categorical factor at the same time. A simple one-way ANOVA tests one factor. A factorial ANOVA tests two or more factors and also checks whether the effect of one factor depends on another factor.

In this example, the two factors are studytime and sex. The dependent variable is G3 final grade. The model tests whether mean G3 differs across studytime groups, whether mean G3 differs between female and male students, and whether the studytime pattern changes depending on sex.

The supplied charts show a clear studytime pattern. Female students rise from about 11.19 in studytime group 1 to about 14.19 in group 4. Male students rise from about 10.59 in group 1 to about 13.59 in group 3, then drop to about 11.36 in group 4. This produces a visible non-parallel pattern in the interaction plot, but the formal interaction p value is above .05.

Factorial 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.

Factorial ANOVA Formula

A two-way factorial ANOVA with two factors can be written as a model with two main effects and one interaction term.

In this example, Y is G3 final grade, A is studytime, B is sex, and A×B is the studytime × sex interaction. The error term represents variation in G3 not explained by the two factors and their interaction.

Factorial ANOVA for This Example

This model tests three ANOVA sources: studytime, sex, and studytime × sex. The p-value charts show that studytime is statistically significant, while sex and the interaction are above the .05 alpha line.

F Test Formula

Each source in the factorial ANOVA table has its own F statistic. The model compares the mean square for that source with the error mean square. A small p value means the source explains more variation than expected from error variation alone.

Effect Size Formula

The effect-size charts use partial eta squared to compare the relative practical size of the studytime effect, sex effect and interaction. The visual scale shows very small values, so the effect-size summary should be interpreted as a supporting diagnostic rather than a large-effect claim.

Main Effect and Interaction Hypotheses

Factorial ANOVA tests more than one hypothesis in the same model. The main effect of studytime, the main effect of sex and the studytime × sex interaction each have their own null and alternative hypothesis.

Dataset and Variables Used

The worked example uses a student performance dataset. The dependent variable is G3 final grade. The first factor is studytime, with four levels. The second factor is sex, with female and male groups. The model compares cell means for every studytime × sex combination.

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 factorial ANOVA workflow. They explain the interaction plot, studytime-sex cell means, main effects, p-value decision, partial eta squared summary and residual diagnostics.

Python Chart 1: Interaction Plot

This interaction plot compares female and male mean G3 across studytime groups. The female line rises steadily from group 1 through group 4. The male line rises from group 1 to group 3, then drops sharply at group 4.

The two lines are not parallel, which visually suggests that the relationship between studytime and G3 may differ by sex. The strongest visual difference appears at studytime group 4, where the female mean is high while the male mean is much lower.

The p-value chart must be used with this visual pattern. The interaction plot shows a possible interaction shape, but the formal interaction p value is about 0.08323, which is above the .05 threshold. The correct conclusion is visible non-parallel lines without a statistically significant interaction at alpha = .05.

Python Chart 2: Cell Means

This chart ranks the mean G3 values for each combination of studytime and sex. The lowest cell mean is 1 | M = 10.59, followed by 1 | F = 11.19. The highest cell mean is 4 | F = 14.19, followed by 3 | M = 13.59 and 3 | F = 13.12.

The cell means show why the interaction plot is not perfectly parallel. Female students continue upward into studytime group 4, while male students peak in studytime group 3 and then drop in group 4.

This chart gives the best practical explanation of the model. Factorial ANOVA is not only about marginal means; it also studies the individual combinations of the factors.

Python Chart 3: Main Effect of Studytime

This chart shows the overall mean G3 for each studytime level after averaging across sex. The means increase from group 1 to group 3 and remain high in group 4. The visible values are about 10.84 for group 1, 12.09 for group 2 and about 13.23 for group 3, with group 4 near the same high range.

The studytime pattern is the strongest result in the factorial ANOVA. The p-value summary places the studytime effect far below the alpha line, so studytime is statistically significant.

This chart should be used to explain the main finding. Higher studytime levels are associated with higher G3 means, even after the model includes sex and the interaction term.

Python Chart 4: Main Effect of Sex

This chart compares overall mean G3 between female and male students. The female mean is about 12.25, while the male mean is about 11.41. The chart therefore shows a higher average G3 for female students.

The p-value summary shows that this difference does not reach the .05 threshold. The sex p value is about 0.07362, which is above .05.

The correct interpretation is that the sample shows a visible female-male mean difference, but the factorial ANOVA does not declare the sex main effect statistically significant at the 5% level.

Python Chart 5: P-Value Decision Summary

This chart compares the p values for the factorial ANOVA sources with the alpha line at 0.05. The studytime p value is extremely small and is placed far to the left of the alpha line. The sex p value is about 0.07362, and the interaction p value is about 0.08323.

The chart gives the formal decision. Studytime is statistically significant because its p value is below .05. Sex and the studytime × sex interaction are not statistically significant because their p values are above .05.

This chart is the most important statistical decision summary. It prevents over-reading the interaction plot by showing that the visible non-parallel pattern is not significant at alpha .05.

Python Chart 6: Effect Size Summary

This chart summarizes the practical size of the factorial ANOVA effects with partial eta squared. The bars are shown on a wide partial eta squared scale, so the effects appear very small visually.

The effect-size chart should be read with the p-value chart. Studytime is statistically significant, while sex and the interaction do not cross the .05 threshold. The practical-size summary does not support a large-effect claim for any source.

The safest reporting approach is to state the significance decisions first, then report effect sizes as supporting information. In this example, the studytime effect is the meaningful source to discuss, while sex and the interaction are not significant at alpha .05.

Python Chart 7: Residuals vs Fitted Values

This residual plot shows residuals against fitted values from the factorial ANOVA model. The points appear in vertical groups because each fitted value corresponds to a studytime-sex cell mean. Most residuals are centered around the zero line.

Several large negative residuals appear below the main cloud, especially around fitted values near 11.2, 11.9 and 12.2. These cases represent students whose observed G3 scores were much lower than their fitted cell mean.

The diagnostic conclusion is that the model captures the group structure, but some low-score cases create a negative residual tail. These cases should be mentioned in the assumptions section rather than ignored.

Python Chart 8: Residual Q-Q Plot

The Q-Q plot compares the residual distribution with a normal reference line. The middle part of the residual distribution is closer to the reference line, but the lower tail moves far below the line.

The largest departures occur on the negative side, matching the residuals-versus-fitted chart. This shows that a small group of students scored far below the model’s fitted cell mean.

The Q-Q plot does not change the main studytime result, but it adds diagnostic context. The final report should state that the main effect of studytime is significant while residual diagnostics show lower-tail departures from normality.

R Chart-by-Chart Validation

The R validation charts repeat the same factorial ANOVA workflow in a second software environment. They confirm the interaction shape, cell means, main effects, p-value decisions, effect-size context and residual diagnostic pattern.

R Chart 1: Interaction Plot

This R chart confirms the non-parallel pattern between female and male students. The female line increases steadily across studytime groups, while the male line rises through group 3 and then falls in group 4.

The visible shape is important, but it must still be interpreted with the interaction p value. The supplied p-value chart reports the interaction above .05, so the visual interaction is not statistically significant at alpha .05.

This chart validates the Python interaction plot and supports the same conclusion: discuss the visible pattern, but do not claim a statistically significant interaction.

R Chart 2: Cell Means

This R chart confirms the cell mean ranking. The lowest mean appears for 1 | M = 10.59, while the highest mean appears for 4 | F = 14.19. The values for studytime group 3 are high for both sexes, with male students around 13.59 and female students around 13.12.

The chart makes the interaction pattern easier to understand. The female group reaches its highest value at studytime group 4, while the male group peaks at group 3 and then falls at group 4.

In reporting, this chart should be used after the interaction plot because it shows the exact cell combinations responsible for the plotted line pattern.

R Chart 3: Main Effect of Studytime

This R chart confirms the main effect of studytime. The mean G3 is about 10.84 at studytime group 1 and about 12.09 at group 2, then rises into the higher range for groups 3 and 4.

This pattern matches the p-value decision chart, where studytime is the significant source. The visible increase in means explains why the studytime main effect is the central result of the factorial ANOVA.

The final interpretation should emphasize studytime first. It is the factor that clearly crosses the statistical decision threshold in the supplied chart set.

R Chart 4: Main Effect of Sex

This R chart confirms that the female mean G3 is higher than the male mean G3. The female mean is about 12.25, and the male mean is about 11.41.

The p-value decision chart shows that this difference is not statistically significant at the .05 level because the sex p value is about 0.07362. The visible difference is therefore reported as a sample pattern, not as a statistically significant main effect.

This chart supports careful wording. Do not write that sex significantly affects G3 at .05; write that female students had a higher sample mean, but the main effect of sex did not reach the .05 significance level.

R Chart 5: P-Value Decision Summary

This R chart confirms the same decision pattern. Studytime is below the alpha threshold, while sex is about 0.07362 and the studytime × sex interaction is about 0.08323.

The chart includes the .05 alpha line, making the decision direct. Only effects to the left of that line are statistically significant at the 5% level.

The R validation result agrees with the Python workflow: studytime is significant, sex is not significant at .05, and the interaction is not significant at .05.

R Chart 6: Effect Size Summary

This R chart displays the partial eta squared context for the factorial ANOVA sources. The bars are visually tiny on the chart scale, so the effect sizes should not be presented as large.

The effect-size chart should be interpreted alongside the p-value chart. Studytime is the statistically supported effect, while sex and the interaction do not reach significance at alpha .05.

In final reporting, effect sizes should be used as supporting context. The strongest practical message remains the significant studytime effect and the non-significant sex and interaction tests.

R Chart 7: Residuals vs Fitted Values

The R residuals-versus-fitted chart confirms the same vertical-band pattern as the Python chart. The fitted values represent studytime-sex cell means, and most residuals are centered around zero.

Several negative residuals fall well below the main pattern. These cases represent students whose observed G3 scores were much lower than the fitted value for their cell.

The chart supports a balanced diagnostic statement. The factorial ANOVA model is useful for the studytime effect, but the residual diagnostics show lower-tail cases that should be mentioned.

R Chart 8: Residual Q-Q Plot

The R Q-Q plot confirms that the central residuals are closer to the reference line, while the lower tail departs from the line. This matches the Python Q-Q plot.

The lower-tail departure means some observations are much lower than the model expects from their studytime-sex cell. This should be reported as a diagnostic limitation or assumption note.

The Q-Q plot does not remove the significant studytime effect. It adds context: the model finds a clear studytime difference, while residual normality is not perfect because of low-tail cases.

SPSS Output and Report PDFs

The supplied report files provide downloadable support for the Factorial ANOVA workflow. The Python report, R report and SPSS output PDF should be linked in the resources section and used as verification files for the WordPress article.

Download Factorial ANOVA Python Report PDF

Download Factorial ANOVA R Report PDF

Download Factorial ANOVA SPSS Output PDF

Output Items to Read

SPSS, R, Python and Excel Workflows for Factorial ANOVA

The same Factorial ANOVA workflow can be reproduced in SPSS, R, Python and Excel. The key steps are to define the two categorical factors, fit a model with main effects and interaction, read the ANOVA table, inspect interaction and cell means, report effect sizes and check residual diagnostics.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Factorial ANOVA

SPSS Syntax for Factorial ANOVA

* Factorial ANOVA / Two-Way ANOVA in SPSS.
* Dependent variable: G3.
* Fixed factors: studytime and sex.

TITLE "Factorial ANOVA: G3 by Studytime and Sex".

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

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

Python Code for Factorial ANOVA

import pandas as pd
import numpy as np
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["sex"] = df["sex"].astype("category")

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

# Factorial ANOVA model with main effects and interaction
model = ols("G3 ~ C(studytime) * C(sex)", data=df_model).fit()

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

# Cell means
cell_means = (
    df_model
    .groupby(["studytime", "sex"])["G3"]
    .agg(["count", "mean", "std"])
    .reset_index()
)
print(cell_means)

# Main effects
studytime_means = df_model.groupby("studytime")["G3"].mean()
sex_means = df_model.groupby("sex")["G3"].mean()

print("Studytime means")
print(studytime_means)

print("Sex means")
print(sex_means)

# Partial eta squared
ss_error = anova_table.loc["Residual", "sum_sq"]

effect_sizes = {}
for source in anova_table.index:
    if source != "Residual":
        ss_effect = anova_table.loc[source, "sum_sq"]
        effect_sizes[source] = ss_effect / (ss_effect + ss_error)

print("Partial eta squared")
print(effect_sizes)

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

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

R Code for Factorial ANOVA

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

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

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

# Factorial ANOVA model
model <- aov(G3 ~ studytime * sex, data = df_model)

summary(model)

# Cell means
aggregate(G3 ~ studytime + sex, data = df_model, FUN = function(x) {
  c(n = length(x), mean = mean(x), sd = sd(x))
})

# Main effect means
aggregate(G3 ~ studytime, data = df_model, mean)
aggregate(G3 ~ sex, data = df_model, mean)

# Diagnostic values
df_model$fitted <- fitted(model)
df_model$residuals <- residuals(model)

# Residual diagnostics
plot(model)

# Optional interaction plot
interaction.plot(
  x.factor = df_model$studytime,
  trace.factor = df_model$sex,
  response = df_model$G3,
  fun = mean,
  xlab = "studytime",
  ylab = "Mean G3",
  trace.label = "sex"
)

Excel Formulas and Tool Steps for Factorial ANOVA

Excel Tool:
Data > Data Analysis > ANOVA: Two-Factor With Replication

Use this when:
- Factor 1 = studytime
- Factor 2 = sex
- Outcome = G3
- Each studytime-sex cell has repeated observations

Decision rule:
If p-value for rows < alpha, factor 1 is significant.
If p-value for columns < alpha, factor 2 is significant.
If p-value for interaction < alpha, interaction is significant.

Alpha:
0.05

Example interpretation from this output:
studytime = significant
sex = not significant at .05
studytime × sex = not significant at .05

Cell mean:
=AVERAGE(cell_range)

Cell standard deviation:
=STDEV.S(cell_range)

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

APA Reporting Wording

When reporting Factorial ANOVA, report the main effects and interaction separately. Do not only report the strongest effect. The reader needs to know whether each factor and their interaction were significant.

Common Mistakes

When to Use Factorial ANOVA

Use Factorial ANOVA when the dependent variable is numeric and the research question includes two or more categorical factors. It is especially useful when the analyst wants to test both main effects and interactions in the same model.

For related guides, see ANOVA in Python, ANOVA in R, ANOVA in SPSS, ANOVA Assumptions, ANOVA Effect Size, Balanced ANOVA, Brown Forsythe ANOVA, ANCOVA, and T Test vs ANOVA.

Downloads and Resources for Factorial ANOVA

Use these resources to reproduce the Factorial 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 files are uploaded to the WordPress Media Library.

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