Sphericity estimation bias for repeated measures designs in simulation studies

Abstract

In this study, we explored the accuracy of sphericity estimation and analyzed how the sphericity of covariance matrices may be affected when the latter are derived from simulated data. We analyzed the consequences that normal and nonnormal data generated from an unstructured population covariance matrix-with low (ε = .57) and high (ε = .75) sphericity-can have on the sphericity of the matrix that is fitted to these data. To this end, data were generated for four types of distributions (normal, slightly skewed, moderately skewed, and severely skewed or log-normal), four sample sizes (very small, small, medium, and large), and four values of the within-subjects factor (K = 4, 6, 8, and 10). Normal data were generated using the Cholesky decomposition of the correlation matrix, whereas the Vale-Maurelli method was used to generate nonnormal data. The results indicate the extent to which sphericity is altered by recalculating the covariance matrix on the basis of simulated data. We concluded that bias is greater with spherical covariance matrices, nonnormal distributions, and small sample sizes, and that it increases in line with the value of K. An interaction was also observed between sample size and K: With very small samples, the observed bias was greater as the value of K increased.

Non-normal Distributions Commonly Used in Health, Education, and Social Sciences: A Systematic Review

“…Monte Carlo simulation studies are commonly used to identify the robustness of statistical techniques under violation of underlying assumptions. In relation to continuous distributions, numerous simulation studies have analyzed the lognormal distribution ( Algina and Keselman, 1998 ; Keselman et al, 2000 ; Kowalchuk et al, 2004 ; Arnau et al, 2012 ; Oberfeld and Franque, 2013 ; Bono et al, 2016 , among others), and also the exponential distribution ( Lix et al, 2003 ; Arnau et al, 2012 ). Among discrete distributions, simulation studies have been conducted with binomial ( Wu and Wu, 2007 ; Fang and Louchin, 2013 ) and multinomial distributions ( Kuo-Chin, 2010 ; Bauer and Sterba, 2011 ; Jiang and Oleson, 2011 ).…”