The Repeated Measures ANOVA Calculator helps you analyze data where the same subjects are measured multiple times under different conditions or time points. It tests whether there are significant differences between related means across these repeated measurements. This test is widely used in fields like psychology, medicine, and education to analyze longitudinal studies, clinical trials with multiple follow-ups, and experimental designs with multiple treatments on the same subjects. Common applications include tracking changes in patient outcomes over time, measuring learning effects across multiple sessions, or comparing the effectiveness of different interventions on the same group of participants. Click here to populate the sample data for a quick example.
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Repeated Measures ANOVA
Definition
Repeated Measures ANOVA analyzes the differences between three or more related measurements on the same subjects over time or under different conditions. It accounts for the correlation between repeated measurements on the same subject. It is commonly applied in within-subjects experimental designs where the same subjects are exposed to different treatments or conditions.
Formulas
F-statistic:
Sum of Squares:
Key Assumptions
Practical Example using rstatix
Step 1: Prepare Data
We’ll organize the scores data in a long format, which is required by rstatix. Each score will be associated with its respective student and teaching method.
library(tidyverse)
library(rstatix)
scores_long <- tibble(
Student = rep(1:5, each = 3),
Method = rep(c("A", "B", "C"), times = 5),
Score = c(85, 90, 88, 78, 85, 82, 92, 95, 93, 88, 86, 89, 80, 84, 83)
)
Step 2: Perform Repeated Measures ANOVA
Using rstatixwe will compute repeated measures ANOVA and summarize the results. Here’s the code:
# Perform repeated measures ANOVA
res.aov <- scores_long |>
anova_test(dv = Score, wid = Student, within = Method)
# Get ANOVA table
res.aov %>%
get_anova_table()
Step 3: Interpret Results
The output provides the following key information:
- F-statistic:
- p-value:
Output:
# ANOVA Table (type III tests)
# Effect DFn DFd F p p<.05 ges
# 1 Method 2 8 4.925 0.04 * 0.094
- Effect: The within-subjects factor
- DFn: Degrees of freedom for the numerator
- DFd: Degrees of freedom for the denominator
- F: The F-statistic
- p: The p-value
- ges: The generalized eta-squared effect size
Step 4: Draw Conclusion
Since and , we reject . There is sufficient evidence to conclude that the teaching methods differ significantly.
Effect Size
Generalized Eta-squared ():
Interpretation guidelines:
- Small effect:
- Medium effect:
- Large effect:
For the example above, the generalized eta-squared effect size is , indicating a medium to large effect.
Code Examples
library(tidyverse)
library(rstatix)
# Original Data
scores <- tibble(
Student = 1:5,
Method_A = c(85, 78, 92, 88, 80),
Method_B = c(90, 85, 95, 86, 84),
Method_C = c(88, 82, 93, 89, 83)
)
# Convert to long format
scores_long <- scores |>
pivot_longer(
cols = starts_with("Method"),
names_to = "Method",
values_to = "Score"
)
res.aov <- scores_long %>%
anova_test(dv = Score, wid = Student, within = Method)
# Get ANOVA table
res.aov |>
get_anova_table()import pandas as pd
from statsmodels.stats.anova import AnovaRM
# Data
data = pd.DataFrame({
'Student': [1, 2, 3, 4, 5],
'Method_A': [85, 78, 92, 88, 80],
'Method_B': [90, 85, 95, 86, 84],
'Method_C': [88, 82, 93, 89, 83]
})
# Reshape to long format
data_long = pd.melt(data, id_vars=['Student'], var_name='Method', value_name='Score')
# Repeated Measures ANOVA
anova = AnovaRM(data_long, 'Score', 'Student', within=['Method'])
res = anova.fit()
print(res)Alternative Tests
Consider these alternatives:
- Friedman Test: Non-parametric alternative when assumptions are violated
- Mixed Models: When dealing with missing data or unequal time points