The CHull procedure for selecting among multilevel component solutions

Recently, Timmerman [1] proposed a class of multilevel component models for the analysis of two-level multivariate data. These models consist of a separate component model for each level in the data. Specifically, the between-differences are captured by a between-component model and the within-differences by a within-component model. Within the class of multilevel component models a number of variants can be distinguished. These variants differ with respect to the within-component model, in that different sets of restrictions are imposed on the within-component loadings and on the variances and correlations of the within-component scores. The following question then may be raised: given a specific two-level data set, which of the multilevel component model variants should be selected, and with how many between- and within-components? We address this question by proposing a model selection procedure that builds on the CHull heuristic of Ceulemans and Kiers [2,3]. The results of an extensive simulation study show that the proposed CHull heuristic succeeds very well in assessing the number of between- and within-components. Tracing the underlying multilevel component model variant is more difficult: Whereas differences in within-loading matrices and differences in variances are very easy to detect, the precise correlational structure of the within-components is much harder to capture.