Quick Answer: One Way ANCOVA Result

The worked One Way ANCOVA example shows that studytime groups differ in G3 after controlling for absences. The Python p-value decision chart reports C(studytime) p = 2.547e-09, which is far below alpha = .05. The covariate absences has p = 0.1203, which is above .05 in this model.

The raw group means were 10.84, 12.09, 13.23 and 13.06. After adjusting at the overall mean of absences, the adjusted means were 10.87, 12.09, 13.18 and 13.03. The adjusted pattern remains almost the same as the raw pattern, meaning the studytime group difference is not explained away by absences.

What Is One Way ANCOVA?

One Way ANCOVA, or one-way analysis of covariance, compares one categorical factor on a numeric dependent variable while controlling one or more covariates. It is useful when the group comparison should be adjusted for a continuous variable that may be related to the outcome.

In this example, studytime is the group factor, G3 is the outcome, and absences is the covariate. The goal is to test whether studytime groups still differ in G3 after the analysis accounts for differences in absences.

The raw means and adjusted means are very close in this output. Group 1 stays lowest, group 2 stays in the middle, and groups 3 and 4 stay highest. That means the covariate adjustment did not remove the main studytime pattern.

One Way ANCOVA is part of the broader ANCOVA and ANOVA family. It connects naturally with Fixed Effects ANOVA, Factorial ANOVA, ANOVA in Python, ANOVA in R, ANOVA in SPSS, ANOVA Assumptions, ANOVA Effect Size, Eta Squared, and Cohen’s F Formula.

One Way ANCOVA Formula

A one-way ANCOVA model includes a categorical group factor and a continuous covariate. The dependent variable is adjusted for the covariate before group means are interpreted.

For this worked example, the model becomes:

The studytime term tests whether adjusted G3 means differ across the four studytime groups. The absences term tests whether G3 is related to absences after studytime group differences are included.

Adjusted Mean Idea

The adjusted mean chart states that means are adjusted at the overall mean value of absences. This places all studytime groups on the same absences reference point before comparing G3.

Partial Eta Squared

The effect size chart reports partial eta squared of 0.064 for studytime and 0.004 for absences. This means the adjusted studytime effect is much larger than the adjusted absences effect in this model.

One Way ANCOVA Hypotheses

One Way ANCOVA tests the adjusted group effect and the covariate effect. The group hypothesis is the main research question, while the covariate result explains whether the adjustment variable contributes significantly to the outcome.

Dataset and Variables Used

The worked example uses student performance data. The dependent variable is G3 final grade. The categorical factor is studytime, with four groups. The covariate is absences, which is included to adjust the G3 group comparison.

The adjusted means are extremely close to the raw means. That means the covariate adjustment slightly changes the group values but does not change the overall ranking. The strongest difference remains between studytime group 1 and the higher studytime groups.

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 One Way ANCOVA through raw group means, adjusted group means, covariate scatter, p-value decisions, partial eta squared, group distributions, residuals versus fitted values and residual Q-Q diagnostics.

Python Chart 1: Raw Group Means

This chart shows the unadjusted mean G3 scores before controlling for absences. Studytime group 1 has a raw mean of 10.84, group 2 has 12.09, group 3 has 13.23, and group 4 has 13.06.

The raw pattern is clear. Group 1 is lowest, group 2 is higher, and groups 3 and 4 are in the highest score range. This is the group difference visible before covariate adjustment.

The chart provides the baseline comparison. The adjusted mean chart should be read next because ANCOVA interpretation depends on the adjusted comparison, not only the raw means.

Python Chart 2: Adjusted Means with 95% Confidence Intervals

This chart shows the ANCOVA-adjusted group means at the overall mean value of absences. The adjusted means are 10.87, 12.09, 13.18 and 13.03 for studytime groups 1 through 4.

The adjusted pattern remains very close to the raw pattern. Group 1 remains lowest, group 2 remains higher, and groups 3 and 4 remain highest. This means absences adjustment does not explain away the studytime difference.

The confidence interval for group 4 is wider than the other groups, so its adjusted mean is less precise. Even with that wider interval, group 4 stays in the high adjusted mean range.

Python Chart 3: Outcome by Covariate Scatter Plot

The scatter plot shows G3 on the vertical axis and absences on the horizontal axis. Most observations are clustered at lower absence values, especially between 0 and 10 absences. A smaller number of observations appear at higher absence values.

The overall dashed trend line slopes downward. This means higher absences are visually associated with lower G3 scores. However, the p-value chart shows absences p = 0.1203, so this covariate trend is not statistically significant at .05 in the displayed ANCOVA model.

The scatter chart is still useful because it explains why absences was included as a covariate. It shows the adjustment variable and the outcome together before reading the formal model results.

Python Chart 4: ANCOVA P-Value Decision Summary

This chart gives the formal ANCOVA decision. The studytime source is labelled C(studytime) p = 2.547e-09, which is far below alpha = .05. The absences source is labelled p = 0.1203, which is above alpha = .05.

The result means the adjusted studytime group effect is statistically significant. Absences is not statistically significant in this model, even though the scatter plot shows a downward visual trend.

This chart is the main decision chart for the post. It confirms that the adjusted group comparison remains significant after including the covariate.

Python Chart 5: Partial Eta Squared Summary

The effect-size chart shows that studytime has partial eta squared of 0.064, while absences has partial eta squared of 0.004. Studytime has the much larger adjusted contribution.

This effect-size pattern matches the p-value decision. Studytime is significant and has a meaningful adjusted effect size. Absences has a very small adjusted effect size in this ANCOVA model.

The chart supports reporting more than a p-value. It shows the practical size of the adjusted studytime effect and the small contribution of absences.

Python Chart 6: G3 Distribution by Studytime

The boxplot chart shows the distribution of G3 by studytime group. Groups 3 and 4 are centered higher than groups 1 and 2. Group 1 has the lowest center, while group 2 is between group 1 and the higher groups.

Low outlying values are visible in groups 1 and 2, including values near zero. These values help explain why the residual diagnostics show strong lower-tail departures.

The chart supports the adjusted result because the group distributions are not identical. The ANCOVA model then tests the adjusted group difference after adding absences as a covariate.

Python Chart 7: Residuals vs Fitted Values

The residuals-versus-fitted chart shows fitted values between roughly 10 and 13.5. Most residuals are spread around the zero line, but several negative residuals extend far below zero, reaching around -12.

The vertical banding appears because the model uses group differences and the covariate structure to generate fitted values. The spread shows that individual G3 values still vary strongly around the fitted ANCOVA means.

This chart supports diagnostic caution. The studytime effect is strong and significant, but the model does not perfectly explain every individual score.

Python Chart 8: Residual Q-Q Plot

The residual Q-Q plot shows clear departure from the reference line. The central residuals follow the general pattern, but the lower tail falls away strongly.

The strongest lower-tail points match the negative residuals seen in the residuals-versus-fitted plot and the low observations in the boxplots. This means residual normality is approximate rather than perfect.

The final report should include this diagnostic issue. The adjusted studytime result remains interpretable, but assumptions should be reported honestly.

R Chart-by-Chart Validation

The R validation chart sequence repeats the same One Way ANCOVA workflow in a second software environment. It confirms the raw means, adjusted means, covariate pattern, p-value decision, partial eta squared, distribution context and residual diagnostics.

R Chart 1: Raw Group Means

The R raw mean chart confirms the same pattern as the Python chart. Studytime group 1 is lowest, group 2 is higher, and groups 3 and 4 are highest.

The raw group mean structure gives the first visual explanation of the result. The groups are separated before the covariate adjustment is applied.

This validation chart confirms that the raw mean pattern is stable across software workflows.

R Chart 2: Adjusted Means with 95% Confidence Intervals

The R adjusted mean chart confirms that the adjusted studytime means remain separated. Group 1 remains lowest, group 2 remains in the middle, and groups 3 and 4 remain highest.

The adjustment for absences changes the means only slightly. This matches the interpretation that the studytime effect remains after controlling for the covariate.

The R chart validates the adjusted mean interpretation used in the Python section.

R Chart 3: Covariate Scatter by Group

The R scatter chart confirms the visual covariate pattern. Most students have low absence counts, and the overall trend line slopes downward.

The formal result still shows that absences is not significant at alpha = .05 in this model. The chart therefore supports including absences as context but not reporting it as a significant covariate effect.

This validation chart helps explain the difference between visual trend and formal ANCOVA decision.

R Chart 4: ANCOVA P-Value Decision Summary

The R decision chart confirms the same statistical conclusion. Studytime is far below the alpha line, while absences is above the alpha line.

The R chart labels the studytime p-value as 5.446e-10. This is still far below .05 and supports the same decision as the Python p-value chart.

The validation result is the same: the adjusted studytime effect is significant, and the covariate is not significant at alpha = .05.

R Chart 5: Partial Eta Squared Summary

The R effect-size chart confirms the same effect-size pattern. Studytime has partial eta squared around 0.064, while absences has partial eta squared around 0.004.

The studytime adjusted effect is clearly larger than the absences effect. This matches the Python effect-size chart and the p-value summary.

The R validation supports reporting studytime as the main ANCOVA effect.

R Chart 6: G3 Distribution by Studytime

The R boxplot chart confirms that groups 3 and 4 sit higher than groups 1 and 2. Group 1 remains the lowest distribution.

Low outlying values are visible again. These values explain why residual diagnostics later show lower-tail departure.

This validation chart confirms the same distribution context from the Python workflow.

R Chart 7: Residuals vs Fitted Values

The R residual chart confirms the same residual pattern. Residuals are centered around zero overall, but strong negative residuals appear in the lower tail.

This means the ANCOVA fitted values explain the group pattern but do not fully capture all individual student outcomes.

The chart supports diagnostic caution in the final article.

R Chart 8: Residual Q-Q Plot

The R Q-Q plot confirms the same lower-tail departure as the Python Q-Q plot. The middle part of the residual distribution follows the expected direction more closely than the lower tail.

This diagnostic pattern should be reported. It does not erase the significant adjusted studytime result, but it prevents overstating perfect model assumptions.

The R validation chart confirms that the residual issue is part of the data pattern rather than a software artifact.

SPSS Output and Report PDFs

The supplied report files support the One Way ANCOVA workflow. The R report validates the chart sequence, and the SPSS output PDF provides menu-based output for reporting and verification.

Download One Way ANCOVA R Report PDF

Download One Way ANCOVA SPSS Output PDF

Output Items to Read

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

The same One Way ANCOVA workflow can be reproduced in SPSS, R, Python and Excel. The key requirement is a numeric outcome, a categorical factor, and a continuous covariate.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for One Way ANCOVA

SPSS Syntax for One Way ANCOVA

* One Way ANCOVA in SPSS.
* Outcome: G3.
* Factor: studytime.
* Covariate: absences.

TITLE "One Way ANCOVA: G3 by Studytime Controlling Absences".

UNIANOVA G3 BY studytime WITH absences
  /METHOD=SSTYPE(3)
  /INTERCEPT=INCLUDE
  /PRINT=DESCRIPTIVE ETASQ HOMOGENEITY PARAMETER
  /EMMEANS=TABLES(studytime) WITH(absences=MEAN) COMPARE ADJ(BONFERRONI)
  /CRITERIA=ALPHA(.05)
  /DESIGN=absences studytime.

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

Python Code for One Way ANCOVA

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["absences"] = pd.to_numeric(df["absences"], errors="coerce")
df["studytime"] = df["studytime"].astype("category")

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

# One Way ANCOVA model
model = ols("G3 ~ C(studytime) + absences", data=df_model).fit()

# ANCOVA table
ancova_table = sm.stats.anova_lm(model, typ=2)
print(ancova_table)

# Partial eta squared
ss_error = ancova_table.loc["Residual", "sum_sq"]
partial_eta = {}
for source in ["C(studytime)", "absences"]:
    ss_effect = ancova_table.loc[source, "sum_sq"]
    partial_eta[source] = ss_effect / (ss_effect + ss_error)

print(partial_eta)

# Adjusted means at overall mean absences
mean_absences = df_model["absences"].mean()
groups = sorted(df_model["studytime"].cat.categories)

adjusted_rows = []
for g in groups:
    new_data = pd.DataFrame({
        "studytime": pd.Categorical([g], categories=df_model["studytime"].cat.categories),
        "absences": [mean_absences]
    })
    adjusted_mean = model.predict(new_data)[0]
    adjusted_rows.append({"studytime": g, "adjusted_mean_G3": adjusted_mean})

adjusted_means = pd.DataFrame(adjusted_rows)
print(adjusted_means)

# Diagnostics
df_model["fitted"] = model.fittedvalues
df_model["residual"] = model.resid
print(df_model[["G3", "studytime", "absences", "fitted", "residual"]].head())

R Code for One Way ANCOVA

library(tidyverse)
library(car)
library(emmeans)

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

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

df_model <- df %>%
  select(G3, studytime, absences) %>%
  drop_na()

# One Way ANCOVA model
model <- lm(G3 ~ studytime + absences, data = df_model)

# Type III ANCOVA table
Anova(model, type = 3)

# Adjusted means
emmeans(model, ~ studytime)

# Pairwise adjusted comparisons
pairs(emmeans(model, ~ studytime), adjust = "bonferroni")

# Partial eta squared manually from Type II/III table if SS values are extracted
anova_model <- anova(model)
anova_model

# Diagnostics
plot(model)

Excel Notes for One Way ANCOVA

Excel support workflow for One Way ANCOVA:

1. Keep columns:
   G3, studytime, absences

2. Create dummy variables for studytime:
   studytime_2, studytime_3, studytime_4
   Keep studytime group 1 as the reference group.

3. Run regression:
   Dependent variable = G3
   Predictors = studytime_2, studytime_3, studytime_4, absences

4. Calculate adjusted means:
   Use the model equation.
   Set absences equal to the overall mean absences.
   Change group dummy values to represent each studytime group.

5. Interpret:
   Studytime group coefficients test adjusted group differences.
   Absences coefficient tests the covariate effect.
   Use SPSS, R or Python for formal ANCOVA tables and effect sizes.

APA Reporting Wording

When reporting One Way ANCOVA, state the outcome, factor, covariate, adjusted group result, covariate result, effect size and diagnostic context. The adjusted means should be reported because they are the key ANCOVA comparison.

Common Mistakes

When to Use One Way ANCOVA

Use One Way ANCOVA when you have one categorical factor, one numeric outcome and one continuous covariate that should be controlled before comparing group means. In this example, studytime groups are compared on G3 after adjusting for absences.

One Way ANCOVA should be compared with ANCOVA, Fixed Effects ANOVA, Factorial ANOVA, Balanced ANOVA, Brown Forsythe ANOVA, ANOVA in SPSS, ANOVA in R, ANOVA Effect Size, and T Test vs ANOVA.

Downloads and Resources for One Way ANCOVA

Use these resources to reproduce the One Way ANCOVA workflow. The 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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