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Mixed ANOVA (Between-Within)

Mixed ANOVA (Between-Within)

Definition

Core Statement

Mixed ANOVA combines between-subjects factors (different groups) and within-subjects factors (repeated measures on the same subjects). It is used when you have both independent groups and repeated measurements.

Purpose

  1. Test effects of both between-subjects and within-subjects factors.
  2. Test interactions between the two types of factors.
  3. Example: Compare treatment groups (between) across multiple time points (within).

When to Use

Use Mixed ANOVA When...

  • You have at least one between-subjects factor (e.g., Treatment Group: Control vs Experimental).
  • You have at least one within-subjects factor (e.g., Time: Pre, Mid, Post).
  • You want to test if groups change differently over time (Group × Time interaction).

Theoretical Background

Example Design

Factor Type Levels
Group Between-Subjects Control, Treatment
Time Within-Subjects Pre, Mid, Post

The Model

Yijk=μ+αi+πj(i)+βk+(αβ)ik+εijk
Term Meaning
αi Main effect of between factor (Group)
πj(i) Subject nested within Group
βk Main effect of within factor (Time)
(αβ)ik Interaction (Group × Time)

Three Effects Tested

Test Question
Main Effect: Group Do groups differ overall?
Main Effect: Time Does everyone change over time?
Interaction: Group × Time Do groups change differently over time?
The Interaction is Often Key

In longitudinal treatment studies, the Group × Time interaction answers: "Does the treatment group improve more than the control group over time?"

Assumptions

Limitations

Pitfalls

  1. Sphericity violations: Common in the within-subjects factor. Apply Greenhouse-Geisser correction.
  2. Missing data: Mixed ANOVA requires complete data. Use LMM for flexibility.
  3. Complex interpretation: Three F-tests (2 main effects + 1 interaction) require careful interpretation.

Python Implementation

import pandas as pd
from statsmodels.stats.anova import AnovaRM

# Example: Pain Reduction Study
# Between: Group (Control, Treatment)
# Within: Time (Pre, Post)

data = pd.DataFrame({
    'Subject': [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6],
    'Group': ['Control', 'Control', 'Control', 'Control', 'Control', 'Control',
              'Treatment', 'Treatment', 'Treatment', 'Treatment', 'Treatment', 'Treatment'],
    'Time': ['Pre', 'Post'] * 6,
    'Pain': [8, 7, 7, 6, 9, 8, 8, 4, 7, 3, 9, 5]
})

# For Mixed ANOVA in Python, use pingouin
import pingouin as pg

mixed_anova = pg.mixed_anova(dv='Pain', within='Time', between='Group', 
                              subject='Subject', data=data)
print(mixed_anova)

R Implementation

library(ez)

# Example Data
df <- data.frame(
  Subject = factor(rep(1:6, each = 2)),
  Group = factor(rep(c('Control', 'Treatment'), c(6, 6))),
  Time = factor(rep(c('Pre', 'Post'), 6)),
  Pain = c(8, 7, 7, 6, 9, 8, 8, 4, 7, 3, 9, 5)
)

# Mixed ANOVA
result <- ezANOVA(
  data = df,
  dv = Pain,
  wid = Subject,
  within = Time,
  between = Group,
  detailed = TRUE
)

print(result)

# Check Sphericity (for within factor)
# If violated, use Greenhouse-Geisser correction

Interpretation Guide

Output Interpretation
Main Effect Group: F=5.2, p=0.04 Treatment group differs from control overall.
Main Effect Time: F=18.3, p<0.001 Pain decreases over time for all subjects.
Interaction Group×Time: F=8.1, p=0.01 Treatment group improves MORE than control over time. This is the key finding.
No Interaction: p>0.05 Both groups change similarly over time.