# Violation of the sphericity assumption and its effect on Type-I error rates in repeated measures ANOVA and multi-level linear models (MLM)

## TL;DR

This study investigates how violations of the sphericity assumption affect Type I error rates in repeated measures ANOVA and multi-level linear models, considering up to nine measurement occasions through extensive simulation.

## Contribution

It provides new insights into the impact of sphericity violations on error rates across different models and sample sizes, especially with many measurement occasions.

## Key findings

- MLM-UN shows significant bias with small samples and many measurement occasions.
- Greenhouse-Geisser correction is slightly conservative under certain conditions.
- Huynh-Feldt correction is recommended when sphericity is violated with small samples.

## Abstract

This study aims to investigate the effects of violations of the sphericity assumption on Type I error rates for different methodical approaches of repeated measures analysis using a simulation approach. In contrast to previous simulation studies on this topic, up to nine measurement occasions were considered. Therefore, two populations representing the conditions of a violation vs. a non-violation of the sphericity assumption without any between-group effect or within-subject effect were created and 5,000 random samples of each population were drawn. Finally, the mean Type I error rates for Multilevel linear models (MLM) with an unstructured covariance matrix (MLM-UN), MLM with compound-symmetry (MLM-CS) and for repeated measures analysis of variance (rANOVA) models (without correction, with Greenhouse-Geisser-correction, and Huynh-Feldt-correction) were computed. To examine the effect of both the sample size and the number of measurement occasions, sample sizes of n = 20, 40, 60, 80, and 100 were considered as well as measurement occasions of m = 3, 6 and 9. For MLM-UN, the results illustrate a massive progressive bias for small sample sizes (n =20) and m = 6 or more measurement occasions. This effect could not be found in previous simulation studies with a smaller number of measurement occasions. The mean Type I error rates for rANOVA with Greenhouse-Geisser-correction demonstrate a small conservative bias if sphericity was not violated, sample sizes were small (n = 20), and m = 6 or more measurement occasions were conducted. The results plead for a use of rANOVA with Huynh-Feldt-correction, especially when the sphericity assumption is violated, the sample size is rather small and the number of measurement occasions is large. MLM-UN may be used when the sphericity assumption is violated and when sample sizes are large.

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Source: https://tomesphere.com/paper/1704.07638