Quick Answer: Two Way ANCOVA Result

The worked Two Way ANCOVA compares adjusted G3 means for school and sex after controlling for G1. The covariate is extremely important in this model because earlier grade performance strongly predicts final grade.

The Python summary table reports that G1 is significant, F = 1200, p = 2.77e-149, partial η² = 0.6508. After this covariate adjustment, school remains significant, F = 5.136, p = 0.02377, partial η² = 0.007912. Sex is also significant, F = 4.93, p = 0.02674, partial η² = 0.007598. The school × sex interaction is not significant, F = 0.0006807, p = 0.9792.

The SPSS output confirms the same decision pattern. The Type III ANCOVA table reports G1 p < .001, school p = .027, sex p = .036, and school × sex p = .979. The corrected model explains a large amount of variation, with R² = .687 and adjusted R² = .685.

What Is Two Way ANCOVA?

Two Way ANCOVA is an analysis of covariance model with one numeric dependent variable, two categorical independent variables and at least one numeric covariate. It combines the group-comparison logic of Factorial ANOVA with the covariate-adjustment logic of ANCOVA.

In a standard One Way ANOVA, group means are compared directly. In Two Way ANCOVA, group means are compared after adjusting for a covariate. In this example, G1 is the covariate because earlier grade performance is strongly related to G3 final grade. The model asks whether school and sex still matter after G1 is controlled.

The result shows that G1 explains the largest share of G3 variation. However, school and sex still show statistically significant adjusted effects. The interaction between school and sex is almost zero and not significant, which means the school difference does not meaningfully depend on sex and the sex difference does not meaningfully depend on school.

This guide connects naturally with One Way ANCOVA, ANOVA Assumptions, ANOVA Effect Size, Eta Squared, Omega Squared, F Distribution, ANOVA in SPSS, ANOVA in R and ANOVA in Python.

Two Way ANCOVA Formula

A two-factor ANCOVA model includes the covariate, the two main effects, the two-way interaction and the residual error term. The model can be written as:

In this formula, Y is the dependent variable, A is the first factor, B is the second factor, AB is the interaction, X is the covariate, and β is the covariate slope. In this example, Y is G3, A is school, B is sex and X is G1.

F Statistic Formula

Each ANCOVA effect has its own F statistic. The covariate has an F test, each main effect has an F test, and the interaction has an F test. A significant F test means that the effect explains variation in G3 after the other model terms are considered.

Partial Eta Squared Formula

Partial eta squared shows the practical size of each adjusted effect. In this model, the covariate G1 has a very large effect, with partial eta squared around 0.651. School and sex are statistically significant but much smaller, with partial eta squared values around 0.008. The school × sex interaction is essentially zero.

Two Way ANCOVA Hypotheses

Two Way ANCOVA has separate hypotheses for the covariate, each main effect and the interaction. The interaction should be interpreted before over-explaining the main effects.

Dataset and Variables Used

The worked example uses student performance data. The dependent variable is G3 final grade. The covariate is G1. The two categorical factors are school and sex. The SPSS output includes 649 valid cases.

Raw Cell Means Before Covariate Adjustment

The raw cell means show a strong pattern before adjustment: GP students have higher G3 means than MS students, and female students have higher G3 means than male students. ANCOVA then checks whether these differences remain after controlling for G1.

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

Two Way ANCOVA Assumptions

Two Way ANCOVA has the usual ANOVA assumptions plus additional covariate assumptions. The most important extra idea is that the covariate should have a reasonably linear relationship with the outcome and the covariate slope should be comparable across groups.

For deeper assumption checks, use Levene Test, Bartlett’s Test, Q-Q Plot Normality Check, P-P Plot Normality Check, Shapiro-Wilk Test, Studentized Residuals, Cook’s Distance and Outlier Detection.

SPSS Output Interpretation for Two Way ANCOVA

The SPSS output uses UNIANOVA with G3 as the dependent variable, school and sex as fixed factors, and G1 as the covariate. The model is written as G3 = G1 + school + sex + school × sex.

SPSS Reading Order

SPSS Type III ANCOVA Table

SPSS Descriptive Pattern

The SPSS descriptive table shows that GP female students have the highest raw G3 mean at about 13.00, while MS male students have the lowest raw G3 mean at about 9.95. The total female mean is about 12.25, and the total male mean is about 11.41.

After controlling for G1, the adjusted means are closer together than the raw means, but the school and sex effects remain statistically significant. This is the central ANCOVA message: the group differences are smaller after adjustment, yet still detectable.

Python Chart-by-Chart Interpretation

The Python chart sequence explains Two Way ANCOVA through adjusted means, covariate relationship, p-value decisions, partial eta squared, raw cell means and residual diagnostics.

Python Chart 1: Adjusted Means by School and Sex

The adjusted mean chart shows GP higher than MS after covariate adjustment. Within both schools, female students have higher adjusted G3 means than male students.

The adjusted means are closer together than the raw means, which shows the effect of controlling for G1. The chart supports significant school and sex main effects but does not show a meaningful school × sex interaction.

Python Chart 2: Covariate Relationship Between G1 and G3

The covariate plot shows a strong positive relationship between G1 and G3. Students with higher G1 scores generally have higher G3 scores. This explains why G1 has the largest partial eta squared in the ANCOVA model.

The school trend lines both rise, but they should be reviewed as part of the homogeneity of regression slopes assumption. The main model adjusts G3 at the average G1 level, so the covariate relationship is central to the analysis.

Python Chart 3: ANCOVA Effect p-values

The p-value chart shows that school, sex and G1 fall below the alpha = .05 line. The school × sex interaction is far above alpha.

This chart gives the main decision visually. The covariate is significant, both main effects are significant, and the interaction should not be interpreted as meaningful.

Python Chart 4: Partial Eta Squared by Effect

The effect-size chart shows that G1 dominates the model with partial eta squared around 0.651. School and sex are much smaller, each around 0.008.

This means the final report should separate statistical significance from practical size. G1 is the major explanatory variable, while school and sex are significant but small adjusted effects.

Python Chart 5: Raw Cell Means Before Covariate Adjustment

The raw mean chart shows wider group separation before covariate adjustment. GP female students have the highest raw mean, and MS male students have the lowest raw mean.

When this chart is compared with the adjusted means chart, the influence of G1 becomes clear. Covariate adjustment narrows the differences, but it does not remove the school and sex effects completely.

Python Chart 6: Residuals vs Fitted Values

The residual plot shows most residuals scattered around zero across the fitted-value range. This supports the basic linear-model structure of the ANCOVA.

Some large negative residuals are visible, and one high positive residual appears on the left side. These points should be mentioned as diagnostic context, especially when writing a transparent assumptions section.

Python Chart 7: Residual Q-Q Plot

The Q-Q plot shows that the middle residuals are closer to the reference line than the tails. The lower tail departs strongly from the line.

This means residual normality is approximate rather than perfect. With a large sample, the ANCOVA results can still be reported, but the lower-tail departure should be described honestly.

Python Chart 8: ANCOVA Summary Table

The summary table confirms the final decision. G1, school and sex are significant, while school × sex is not significant.

This is the best Python table for final reporting because it gives the formal model effects, p-values, effect sizes and decisions in one place.

R Chart-by-Chart Validation

The R validation charts repeat the same ANCOVA workflow in a second software environment. The R images confirm the same decision pattern: G1 is the strongest effect, school and sex are significant adjusted effects, and the school × sex interaction is not significant.

R Chart 1: Adjusted Means by School and Sex

The R adjusted mean chart confirms the same pattern as Python. GP remains above MS, and female students remain above male students after controlling for G1.

This software agreement strengthens the adjusted main-effect interpretation.

R Chart 2: Covariate Relationship

The R scatterplot confirms the strong positive relationship between G1 and G3. Higher G1 scores generally correspond to higher G3 scores.

This validates why G1 must be included as a covariate instead of comparing raw group means only.

R Chart 3: ANCOVA Effect p-values

The R p-value chart confirms the same decision structure: G1, school and sex are significant, while the interaction is not significant.

This supports a final report focused on adjusted main effects rather than interaction explanation.

R Chart 4: Partial Eta Squared by Effect

The R effect-size chart confirms that G1 has the dominant practical effect. School and sex are much smaller.

This helps prevent overstatement. The group effects are statistically significant, but G1 explains most of the adjusted variation.

R Chart 5: Raw Cell Means Before Adjustment

The R raw mean chart confirms that raw group differences are larger before G1 adjustment.

This validates the ANCOVA logic: raw means are useful descriptively, but adjusted means are the correct focus of interpretation.

R Chart 6: Residuals vs Fitted Values

The R residual plot confirms the same diagnostic message as Python. Most residuals are centered around zero, while some lower-tail residuals remain visible.

The model is useful for adjusted mean comparison, but diagnostics should be reported with caution about extreme residuals.

R Chart 7: Residual Q-Q Plot

The R Q-Q plot confirms that residual normality is approximate. The lower tail departs from the reference line.

This supports a transparent diagnostic statement in the final report.

R Chart 8: ANCOVA Summary Table

The R summary table confirms the same final interpretation. G1 is strongly significant, school and sex are significant, and school × sex is not significant.

The R table may show small numerical differences from the Python table because of implementation and table-format choices, but the statistical decisions are the same.

SPSS, R, Python and Excel Workflows for Two Way ANCOVA

The same Two Way ANCOVA workflow can be reproduced in SPSS, R, Python and Excel. SPSS uses the General Linear Model univariate procedure. R and Python use linear-model syntax with categorical factors and a covariate. Excel can prepare adjusted summaries and visuals, but SPSS, R or Python is recommended for the full ANCOVA table.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Two Way ANCOVA

SPSS Syntax for Two Way ANCOVA

* Two Way ANCOVA in SPSS.
* Dependent variable: G3.
* Factors: school and sex.
* Covariate: G1.

TITLE "Two Way ANCOVA: G3 by School and Sex Controlling for G1".

UNIANOVA G3 BY school sex WITH G1
  /METHOD=SSTYPE(3)
  /INTERCEPT=INCLUDE
  /PRINT=DESCRIPTIVE ETASQ HOMOGENEITY PARAMETER
  /EMMEANS=TABLES(school*sex) WITH(G1=MEAN)
  /EMMEANS=TABLES(school) WITH(G1=MEAN) COMPARE ADJ(LSD)
  /EMMEANS=TABLES(sex) WITH(G1=MEAN) COMPARE ADJ(LSD)
  /CRITERIA=ALPHA(.05)
  /DESIGN=G1 school sex school*sex.

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

Python Code for Two Way ANCOVA

import pandas as pd
import statsmodels.api as sm
from statsmodels.formula.api import ols
from statsmodels.stats.anova import anova_lm

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

df["G3"] = pd.to_numeric(df["G3"], errors="coerce")
df["G1"] = pd.to_numeric(df["G1"], errors="coerce")
df["school"] = df["school"].astype("category")
df["sex"] = df["sex"].astype("category")

data = df[["G3", "G1", "school", "sex"]].dropna().copy()

model = ols("G3 ~ G1 + C(school) * C(sex)", data=data).fit()

anova_table = anova_lm(model, typ=2)
error_ss = anova_table.loc["Residual", "sum_sq"]
anova_table["partial_eta_sq"] = anova_table["sum_sq"] / (anova_table["sum_sq"] + error_ss)

print(anova_table)
print(model.summary())

# Adjusted means at average G1
mean_g1 = data["G1"].mean()
cells = pd.DataFrame(
    [(s, x, mean_g1) for s in data["school"].cat.categories for x in data["sex"].cat.categories],
    columns=["school", "sex", "G1"]
)
cells["adjusted_G3"] = model.predict(cells)
print(cells)

# Raw means for comparison
print(data.groupby(["school", "sex"])["G3"].mean())

# Residual diagnostics
data["fitted"] = model.fittedvalues
data["residual"] = model.resid
print(data[["G3", "G1", "school", "sex", "fitted", "residual"]].head())

R Code for Two Way ANCOVA

# Two Way ANCOVA in R

library(tidyverse)

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

df$G3 <- as.numeric(df$G3)
df$G1 <- as.numeric(df$G1)
df$school <- as.factor(df$school)
df$sex <- as.factor(df$sex)

data <- df %>%
  select(G3, G1, school, sex) %>%
  drop_na()

model <- lm(G3 ~ G1 + school * sex, data = data)

anova(model)
summary(model)

# Optional Type III ANOVA
# install.packages("car")
# library(car)
# Anova(model, type = 3)

# Adjusted means
# install.packages("emmeans")
# library(emmeans)
# emmeans(model, ~ school * sex)

# Raw means
data %>%
  group_by(school, sex) %>%
  summarise(
    n = n(),
    raw_mean_G3 = mean(G3),
    mean_G1 = mean(G1),
    .groups = "drop"
  )

# Diagnostics
par(mfrow = c(1, 2))
plot(fitted(model), residuals(model),
     xlab = "Fitted values", ylab = "Residuals",
     main = "Residuals vs Fitted")
abline(h = 0, lty = 2)
qqnorm(residuals(model))
qqline(residuals(model))

Excel Notes for Two Way ANCOVA

Excel support workflow:

1. Arrange the data:
   G3 | G1 | school | sex

2. Create raw cell means:
   PivotTable:
   Rows = school
   Columns = sex
   Values = average of G3

3. Calculate covariate mean:
   mean_G1 = AVERAGE(G1_range)

4. Fit a regression model:
   G3 = intercept + G1 + school dummy + sex dummy + school*sex dummy

5. Calculate adjusted cell means:
   Use the regression equation with G1 fixed at mean_G1.

6. Create charts:
   - raw cell means
   - adjusted means
   - G1 vs G3 scatterplot
   - residuals vs fitted
   - Q-Q plot if available

7. Formal ANCOVA:
   Use SPSS, R or Python for the official Type III ANCOVA table.

APA Reporting Wording

When reporting Two Way ANCOVA, include the outcome, two factors, covariate, adjusted mean interpretation, interaction decision, effect sizes and assumption notes. The school × sex interaction should not be interpreted as meaningful because its p-value is almost 1.

Common Mistakes

When to Use Two Way ANCOVA

Use Two Way ANCOVA when you have one numeric outcome, two categorical factors and at least one numeric covariate that should be statistically controlled. It is common in education, psychology, medicine, business testing and social science research.

Two Way ANCOVA should be compared with One Way ANCOVA, ANCOVA, Factorial ANOVA, Fixed Effects ANOVA, ANOVA Assumptions, ANOVA in SPSS, ANOVA in R, ANOVA in Python, F Distribution, and Statistical Power.

Downloads and Resources for Two Way ANCOVA

Use these resources to reproduce the Two Way ANCOVA 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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