Abstract
We investigated 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. Effects of the level of inter-correlations between measurement occasions on Type I error rates were considered for the first time. Two populations with non-violation of the sphericity assumption, one with uncorrelated measurement occasions and one with moderately correlated measurement occasions, were generated. One population with violation of the sphericity assumption combines uncorrelated with highly correlated measurement occasions. A second population with violation of the sphericity assumption combines moderately correlated and highly correlated measurement occasions. From these four populations without any between-group effect or within-subject effect 5,000 random samples 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. With respect to rANOVA, 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. 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 proportionality of bias and number of measurement occasions should be considered when MLM-UN is used. The good news is that this proportionality can be compensated by means of large sample sizes. Accordingly, MLM-UN can be recommended even for small sample sizes for about three measurement occasions and for large sample sizes for about nine measurement occasions.
Highlights
Multilevel linear models (MLM) have been discussed as an alternative to repeated measures analysis of variance and, sometimes, researchers have even been urged to use MLM instead of rANOVA (Boisgontier and Cheval, 2016)
For the nonsphericity condition and three measurement occasions, the Type I error rates were close to the expectation for all methods except rANOVA and MLM with compound-symmetry (MLM-CS)
Note that here and in the following the results for rANOVA and MLM-CS were so similar that the respective lines completely overlap
Summary
Multilevel linear models (MLM) have been discussed as an alternative to repeated measures analysis of variance (rANOVA; Gueorguieva and Krystal, 2004; Arnau et al, 2010; Goedert et al, 2013) and, sometimes, researchers have even been urged to use MLM instead of rANOVA (Boisgontier and Cheval, 2016). Different labels are used to denote models encompassing fixed and random effects, covariance pattern models, and regression models that are based on more than one data level, where the levels are typically defined by the measurement occasions nested within individuals These models are referred to as hierarchical linear models (Bryk and Raudenbush, 1992; Raudenbush and Bryk, 2002), as (general) mixed linear models (McLean et al, 1991; Arnau et al, 2010), mixed effects models (Gueorguieva and Krystal, 2004), multilevel linear models (Hox, 2002), or as multilevel models (Maas and Hox, 2005; Lucas, 2014). MLM has substantial advantages when compared to rANOVA
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