Chapter 12 Problem Set – One-Way ANOVA

Where this problem set fits in the story

This Track C problem set is linked to Chapter 12: One-Way ANOVA in the Psychological Science & Statistics mini-book (Track B).

Track B Chapter 12 introduces the logic of one-way ANOVA: partitioning variance, interpreting the F-ratio, and understanding how post-hoc tests localize differences between means.
Track B Chapter 10 problem set showed how to work with two groups using independent-samples t-tests.
This Track C Chapter 12 problem set extends that workflow to three or more groups using one-way ANOVA.

The PyStatsV1 scripts here turn the conceptual ideas from Chapter 12 into a reusable analysis template that you can adapt for your own studies.

By working through this problem set and inspecting the solution scripts, you should be able to:

Recognize when a one-way ANOVA is appropriate (one categorical predictor, three or more groups, continuous outcome).
Simulate and analyze between-subjects designs with multiple levels of an independent variable (e.g., low, medium, high intervention).
Interpret the ANOVA table: sums of squares, degrees of freedom, mean squares, F-statistic, and p-value.
Inspect group means and effect sizes to distinguish statistical significance from practical significance.
Use PyStatsV1’s solution scripts as a template for running your own one-way ANOVAs.

From the project root (with your virtual environment activated), the Chapter 12 problem-set lab can be re-run with:

make psych-ch12-problems

Behind the scenes this target calls:

python -m scripts.psych_ch12_problem_set

This will:

Before looking at the code, think through these questions:

Why do we prefer ANOVA instead of running multiple t-tests when comparing three or more groups?
What does it mean, conceptually, to “partition” total variability into between-groups and within-groups components?
How does increasing the number of groups (but holding sample size fixed) affect the F-ratio and statistical power?
How would you explain to a collaborator the difference between a statistically significant F and a large, practically meaningful effect?

The script scripts.psych_ch12_problem_set contains three exercises that mirror common classroom and research scenarios.

Design: Three independent groups (control, low_dose, high_dose), equal sample sizes.
Outcome: Continuous score (e.g., stress reduction, performance, symptom change).
Effect: Moderate treatment effect – the means are separated enough that the omnibus F-test is usually significant.

Questions to consider:

What does the ANOVA table tell you about the overall effect of Group?
How large is the effect (e.g., partial eta-squared)?
Which pairwise differences would you expect to be significant in a post-hoc analysis?

Design: Same three groups as Exercise 1, but with smaller differences between group means.
Effect: Small effect size – the F-test may be non-significant or only weakly significant depending on the simulated sample.

Questions to consider:

Design: Three groups with unequal sample sizes (e.g., more participants in the control group than in one of the treatment groups).
Effect: Strong effect size – the F-test is clearly significant, and the pattern of means is obvious in the plot.

Questions to consider:

The solution code for these exercises lives in:

scripts/psych_ch12_problem_set.py

Key pieces to inspect in that module:

Data generation helpers that simulate one-way ANOVA designs with tunable effect sizes and sample sizes.
A `run_one_way_anova` function that wraps pingouin.anova to produce a tidy ANOVA table suitable for teaching.
A small `ProblemResult` dataclass that bundles the simulated data, ANOVA table, and metadata (exercise label, group means, effect size, etc.).
A main routine that writes all datasets and a summary CSV so you can see the three scenarios side-by-side.

To re-run all exercises and regenerate the outputs for this problem set, use:

make psych-ch12-problems

Then inspect:

data/synthetic/psych_ch12_exercise1.csv – Three-group design with a moderate treatment effect.
data/synthetic/psych_ch12_exercise2.csv – Three-group design with a small effect.
data/synthetic/psych_ch12_exercise3.csv – Three-group design with unequal n and a strong effect.
outputs/track_c/ch12_problem_set_results.csv – Summary ANOVA table (one row per exercise).
outputs/track_c/ch12_problem_set_group_means.png – Group means plot for the three exercises.

One-way ANOVA compares mean differences across three or more independent groups using a single F-test.
The F-ratio expresses the signal-to-noise logic of variance partitioning: how much variability is explained by group membership relative to residual variability within groups.
Sample size, group spacing (effect size), and error variance jointly determine whether an effect is likely to be detected as statistically significant.
PyStatsV1 solution scripts give you a reusable one-way ANOVA template: swap in your own dataset, re-run the analysis, and verify the results using the tests.