Introduction

Repeated Measures Analysis of Variance (Repeated Measures ANOVA) is one of the most widely used statistical methods, as it allows researchers to study differences across levels of a factor while using the same observation units. In other words, the same participants are measured repeatedly under different conditions or at different points in time. However, this method relies on certain assumptions in order for its results to be valid. The most important and, at the same time, the most “sensitive” assumption is that of sphericity.

The Concept of Sphericity

Sphericity is defined as the condition in which the variances of the differences between all pairs of factor levels are equal. It essentially represents a type of “homogeneity of differences,” similar to the homogeneity of variances assumption in between-subjects ANOVA. When sphericity is met, the analysis can yield accurate estimates for the F statistic. On the other hand, a violation of this assumption leads to distortions, particularly in the form of overly “liberal” tests, thereby increasing the risk of a Type I Error—that is, the probability of incorrectly rejecting the null hypothesis.

Consequences of Violating Sphericity

Violating sphericity is a serious problem. Specifically, the F statistic, which is used to test for significant differences, is calculated based on degrees of freedom that assume sphericity. When this assumption does not hold, the critical F values obtained from tables are too low, which results in a higher likelihood of drawing incorrect conclusions. Essentially, the researcher may conclude that a statistically significant difference exists when, in reality, it does not.

Estimation of Epsilon (ε)

To quantify the degree of sphericity violation, a coefficient called epsilon (ε) is used. Epsilon can take values from its lower bound up to 1. When ε = 1, sphericity is perfectly met. The smaller the value of ε (<1), the greater the violation of sphericity. This estimation provides researchers with a measure of how severe the violation is and whether adjustments are needed.

Corrections for Sphericity Violations

Fortunately, statistical corrections have been developed to handle violations of sphericity. The most well-known are the Greenhouse-Geisser and Huynh-Feldt corrections. Both are based on the estimation of epsilon, but they differ in how they calculate it. Essentially, these corrections apply the estimated ε to the degrees of freedom of the F distribution. As a result, the adjusted critical F values become larger, which in turn increases the p-value. In this way, the likelihood of a Type I Error is reduced, making the test more conservative.

Practical Implications of the Corrections

The application of corrections can drastically alter the results of an analysis. In many cases, a difference that initially appears statistically significant (without correction) may turn out to be non-significant after the adjustment. This is crucial for the proper interpretation of data and for avoiding incorrect scientific conclusions. At the same time, the use of such corrections demonstrates the researcher’s responsibility, as they account for the assumptions of the method and adjust their conclusions accordingly.

Conclusions

Sphericity is a key assumption for the correct application of Repeated Measures ANOVA. Its violation leads to significant statistical distortions, primarily an increase in Type I Error rates. The epsilon (ε) coefficient offers a way to estimate the degree of violation, while the Greenhouse-Geisser and Huynh-Feldt corrections provide practical solutions to ensure the validity of results. For researchers, knowledge and proper application of these procedures is essential, as they ensure more reliable and valid conclusions.