Analyzing data with repeated measures is crucial in many research fields, allowing researchers to track changes within the same subjects over time or across different conditions. When you have two independent variables (factors) that are both measured repeatedly within the same subjects, a two-way repeated measures ANOVA is the appropriate statistical test. This guide will walk you through performing this analysis in R, interpreting the results, and addressing potential issues.
Understanding Two-Way Repeated Measures ANOVA
Before diving into the R code, let's clarify the concept. A two-way repeated measures ANOVA examines the effects of two within-subjects factors and their interaction on a dependent variable. "Repeated measures" means that the same subjects are measured multiple times, under different conditions defined by the two factors. For instance, you might measure reaction time (dependent variable) under different levels of caffeine (factor 1) and different levels of sleep deprivation (factor 2) in the same participants.
This analysis allows you to test three main hypotheses:
- Main effect of Factor 1: Does Factor 1 significantly influence the dependent variable, averaging across the levels of Factor 2?
- Main effect of Factor 2: Does Factor 2 significantly influence the dependent variable, averaging across the levels of Factor 1?
- Interaction effect: Does the effect of Factor 1 on the dependent variable depend on the level of Factor 2 (and vice versa)? This indicates a combined effect beyond the individual effects of the factors.
Performing Two-Way Repeated Measures ANOVA in R using the ez package
The ez package simplifies the process of running repeated measures ANOVAs in R. First, ensure you've installed and loaded the package:
if(!require(ez)){install.packages("ez")}
library(ez)
Next, let's create a sample dataset. Imagine we're studying the effects of two types of exercise (Factor 1: "Cardio" and "Strength") and two different training schedules (Factor 2: "Short" and "Long") on muscle strength (dependent variable):
data <- data.frame(
Subject = rep(1:10, each = 4),
Exercise = rep(rep(c("Cardio", "Strength"), each = 2), 10),
Schedule = rep(rep(c("Short", "Long"), 2), 10),
Strength = c(10, 12, 15, 18, 11, 13, 16, 19, 12, 14, 17, 20, 9, 11, 14, 16, 13, 15, 18, 21, 14, 16, 19, 22, 10, 12, 15, 17, 11, 13, 16, 18, 15, 17, 20, 23)
)
Now, we can perform the two-way repeated measures ANOVA:
model <- ezANOVA(
data = data,
dv = Strength,
wid = Subject,
within = .(Exercise, Schedule)
)
print(model)
This code will output a table containing the F-statistics, degrees of freedom, p-values, and effect sizes (partial eta-squared) for each main effect and the interaction effect.
Interpreting the Results
The output will show the significance of each effect. A p-value less than your chosen alpha level (typically 0.05) indicates a statistically significant effect. For instance, if the p-value for the "Exercise" main effect is less than 0.05, you would conclude that the type of exercise significantly affects muscle strength. Similarly, you'd interpret the significance of the "Schedule" main effect and the interaction effect. The partial eta-squared provides a measure of the effect size, indicating the proportion of variance in the dependent variable explained by each effect.
Addressing Potential Issues
-
Sphericity: Repeated measures ANOVAs assume sphericity, meaning that the variances of the differences between levels of a within-subjects factor are equal. If sphericity is violated (indicated by Mauchly's test of sphericity in the
ezANOVAoutput), you might need to correct the degrees of freedom using Greenhouse-Geisser or Huynh-Feldt corrections (often automatically applied byezANOVA). -
Missing Data: Missing data can bias the results. Consider using appropriate methods for handling missing data, such as multiple imputation.
Conclusion
The two-way repeated measures ANOVA is a powerful tool for analyzing data with repeated measures. The ez package in R provides a user-friendly way to perform this analysis. However, it's crucial to understand the assumptions of the test and interpret the results carefully, considering potential violations of assumptions and the impact of missing data. Remember always to carefully consider the research question and the nature of your data before choosing and interpreting any statistical test.
