Multilevel Modeling in the ‘Wide Format’ Approach with Discrete Data: A Solution for Small Cluster Sizes

Abstract

In multilevel data, units at level 1 are nested in clusters at level 2, which in turn may be nested in even larger clusters at level 3, and so on. For continuous data, several authors have shown how to model multilevel data in a ‘wide’ or ‘multivariate’ format approach. We provide a general framework to analyze random intercept multilevel SEM in the ‘wide format’ (WF) and extend this approach for discrete data. In a simulation study, we vary response scale (binary, four response options), covariate presence (no, between-level, within-level), design (balanced, unbalanced), model misspecification (present, not present), and the number of clusters (small, large) to determine accuracy and efficiency of the estimated model parameters. With a small number of observations in a cluster, results indicate that the WF approach is a preferable approach to estimate multilevel data with discrete response options.

Keywords:

• Discrete data
• multilevel
• structural equation modeling
• random intercepts

Acknowledgements

The authors thank Helma Koomen of the University of Amsterdam for making her data available for secondary analysis. We are also thankful to Karin Schermelleh-Engel and Wen Wei Loh for helpful comments on an earlier draft of this article.

Notes

¹ ICC (see Hox et al., 2017) = $\mathrm{V}\mathrm{a}\mathrm{r}$ between/($\mathrm{V}\mathrm{a}\mathrm{r}$ between + $\mathrm{V}\mathrm{a}\mathrm{r}$ within); = (${.5}^2\ast 1$)/$(({.5}^2\ast 1)$ + (${.5}^2\times 4$) + $1$ (residual variance in theta parameterization)) = 0.111.

² All R scripts (e.g., data generation scripts and scripts to analyze the data) are available at the Open Science Foundation (OSF), following the link https://osf.io/fa2gp/.

³ The syntax can be found at the OSF. The syntax used to fit the STRS data is quite similar to the syntax printed in Appendix D (see Appendix B for adding covariates).

Additional information

Funding

This publication is supported by Special Research Fund (BOF) open competition grant from Ghent University.