References
- Algina, J., & Swaminathan, H. (2011). Centering in two-level nested designs. In J. Hox & J. K. Roberts (Eds.), Handbook of advanced multilevel analysis (pp. 285–312). Taylor and Francis. https://www.routledgehandbooks.com/doi/10.4324/9780203848852.ch15
- Asparouhov, T., & Muthén, B. (2019). Latent variable centering of predictors and mediators in multilevel and time-series models. Structural Equation Modeling: A Multidisciplinary Journal, 26(1), 119–142. https://doi.org/https://doi.org/10.1080/10705511.2018.1511375
- Aydin, B., Algina, J., & Leite, W. L. (2019). Comparison of model- and design-based approaches to detect the treatment effect and covariate by treatment interactions in three-level models for multisite cluster-randomized trials. Behavior Research Methods, 51(1), 243–257. https://doi.org/https://doi.org/10.3758/s13428-018-1080-1
- Aydin, B., Leite, W. L., & Algina, J. (2016). The effects of including observed means or latent means as covariates in multilevel models for cluster randomized trials. Educational and Psychological Measurement, 76(5), 803–823. https://doi.org/https://doi.org/10.1177/0013164415618705
- Bradley, J. V. (1978). Robustness?. British Journal of Mathematical and Statistical Psychology, 31(2), 144–152. https://doi.org/https://doi.org/10.1111/j.2044-8317.1978.tb00581.x
- Brincks, A. M., Enders, C. K., Llabre, M. M., Bulotsky-Shearer, R. J., Prado, G., & Feaster, D. J. (2017). Centering predictor variables in three-level contextual models. Multivariate Behavioral Research, 52(2), 149–163. https://doi.org/https://doi.org/10.1080/00273171.2016.1256753
- Cornfield, J. (1978). Randomization by group: A formal analysis. American Journal of Epidemiology, 108(2), 100–102. https://doi.org/https://doi.org/10.1093/oxfordjournals.aje.a112592
- Croon, M. A., & Van Veldhoven, M. J. P. M. (2007). Predicting group-level outcome variables from variables measured at the individual level: A latent variable multilevel model. Psychological Methods, 12(1), 45–57. https://doi.org/https://doi.org/10.1037/1082-989X.12.1.45
- Daunic, A. P., Corbett, N., Smith, S. W., Crews, E., Poling, D. V., & Worth, M. (2018, April 13–17). Social-emotional learning foundations for K–1 students at risk for emotional and behavioral disorder: First-year findings. Paper presentation Annual Meeting of the American Educational Research Association.
- Enders, C. K., & Tofighi, D. (2007). Centering predictor variables in cross-sectional multilevel models: A new look at an old issue. Psychological Methods, 12(2), 121–138. https://doi.org/https://doi.org/10.1037/1082-989X.12.2.121
- Gardiner, J. C., Luo, Z., & Roman, L. A. (2009). Fixed effects, random effects and GEE: What are the differences? Statistics in Medicine, 28(2), 221–239. https://doi.org/https://doi.org/10.1002/sim.3478
- Gioia, G. A., Isquith, P. K., Guy, S. C., & Kenworthy, L. (2000). Behavior Rating Inventory of Executive Function professional manual. Psychological Assessment Resources Inc.
- Hedges, L. V., & Borenstein, M. (2014). Conditional optimal design in three- and four-level experiments. Journal of Educational and Behavioral Statistics, 39(4), 257–281. https://doi.org/https://doi.org/10.3102/1076998614534897
- Hedges, L. V., & Hedberg, E. C. (2013). Intraclass correlations and covariate outcome correlations for planning two- and three-level cluster-randomized experiments in education. Evaluation Review, 37(6), 445–489. https://doi.org/https://doi.org/10.1177/0193841X14529126
- Hoffman, L., & Rovine, M. J. (2007). Multilevel models for the experimental psychologist: Foundations and illustrative examples. Behavior Research Methods, 39(1), 101–117. https://doi.org/https://doi.org/10.3758/bf03192848
- Huang, F. L. (2018). Using cluster bootstrapping to analyze nested data with a few clusters. Educational and Psychological Measurement, 78(2), 297–318. https://doi.org/https://doi.org/10.1177/0013164416678980
- Jia, Y., & Konold, T. (2019). Moving to the next level: Doubly latent multilevel mediation models with a school climate illustration. The Journal of Experimental Education, 1–19. https://doi.org/https://doi.org/10.1080/00220973.2019.1675136
- Kelcey, B., Spybrook, J., Phelps, G., Jones, N., & Zhang, J. (2017). Designing large-scale multisite and cluster-randomized studies of professional development. The Journal of Experimental Education, 85(3), 389–410. https://doi.org/https://doi.org/10.1080/00220973.2016.1220911
- Kish, L. (1965). Survey sampling. Wiley.
- Konstantopoulos, S. (2011). Optimal sampling of units in three-level cluster randomized designs: An ANCOVA framework. Educational and Psychological Measurement, 71(5), 798–813. https://doi.org/https://doi.org/10.1177/0013164410397186
- Kreft, I. G., de Leeuw, J., & Aiken, L. S. (1995). The effect of different forms of centering in hierarchical linear models. Multivariate Behavioral Research, 30(1), 1–21. https://doi.org/https://doi.org/10.1207/s15327906mbr3001_1
- Littell, R. C., Milliken, G. A., Stroup, W. W., Wolfinger, R. D., & Schabenberger, O. (2006). SAS for mixed models (2nd ed.). SAS Institute, Inc.
- Lüdtke, O., Marsh, H. W., Robitzsch, A., Trautwein, U., Asparouhov, T., & Muthén, B. (2008). The multilevel latent covariate model: A new, more reliable approach to group-level effects in contextual studies. Psychological Methods, 13(3), 203–229. https://doi.org/https://doi.org/10.1037/a0012869
- Marsh, H. W., Lüdtke, O., Robitzsch, A., Trautwein, U., Asparouhov, T., Muthén, B., & Nagengast, B. (2009). Doubly-latent models of school contextual effects: Integrating multilevel and structural equation approaches to control measurement and sampling error. Multivariate Behavioral Research, 44(6), 764–802. https://doi.org/https://doi.org/10.1080/00273170903333665
- Mathieu, J. E., Aguinis, H., Culpepper, S. A., & Chen, G. (2012). Understanding and estimating the power to detect cross-level interaction effects in multilevel modeling. The Journal of Applied Psychology, 97(5), 951–966. https://doi.org/https://doi.org/10.1037/a0028380
- Mcneish, D. (2017). Small sample methods for multilevel modeling: A colloquial elucidation of REML and the Kenward-Roger correction. Multivariate Behavioral Research, 52(5), 661–670. https://doi.org/https://doi.org/10.1080/00273171.2017.1344538
- Mcneish, D. M. (2014). Modeling sparsely clustered data: Design-based, model-based, and single-level methods. Psychological Methods, 19(4), 552–563. https://doi.org/https://doi.org/10.1037/met0000024
- Mcneish, D., Stapleton, L. M., & Silverman, R. D. (2017). On the unnecessary ubiquity of hierarchical linear modeling. Psychological Methods, 22(1), 114–140. https://doi.org/https://doi.org/10.1037/met0000078
- Mcneish, D., & Wentzel, K. R. (2017, Mar–Apr). Accommodating small sample sizes in three-level models when the third level is incidental. Multivariate Behavioral Research, 52(2), 200–215. https://doi.org/https://doi.org/10.1080/00273171.2016.1262236
- Muthén, L., & Muthén, B. (2018). Mplus software (version 8). Muthén & Muthén.
- Pinheiro, J., Bates, D., DebRoy, S., Sarkar D., & R Core Team. (2018). nlme: Linear and nonlinear mixed effects models. https://CRAN.R-project.org/package=nlme.
- Pituch, K. A., Murphy, D. L., & Tate, R. L. (2009). Three-level models for indirect effects in school- and class-randomized experiments in education. The Journal of Experimental Education, 78(1), 60–95. https://doi.org/https://doi.org/10.1080/00220970903224685
- Preacher, K. J., & Sterba, S. K. (2019). Aptitude-by-treatment interactions in research on educational interventions. Exceptional Children, 85(2), 248–264. https://doi.org/https://doi.org/10.1177/0014402918802803
- Preacher, K. J., Zhang, Z., & Zyphur, M. J. (2016). Multilevel structural equation models for assessing moderation within and across levels of analysis. Psychological Methods, 21(2), 189–205. https://doi.org/https://doi.org/10.1037/met0000052
- Preacher, K. J., Zyphur, M. J., & Zhang, Z. (2010). A general multilevel SEM framework for assessing multilevel mediation. Psychological Methods, 15(3), 209–233. https://doi.org/https://doi.org/10.1037/a0020141
- R Core Team. (2018). R: A language and environment for statistical computing. https://www.R-project.org/
- Rao, P. S., & Heckler, C. E. (1997). The three-fold nested random effects model. Journal of Statistical Planning and Inference, 64(2), 341–352. https://doi.org/https://doi.org/10.1016/S0378-3758(97)00004-9
- Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (Vol. 1, 2nd ed.). Sage Publications.
- Robinson, W. S. (1950). Ecological correlations and the behavior of individuals. American Sociological Review, 15(3), 351. https://doi.org/https://doi.org/10.2307/2087176
- Ryu, E. (2015). The role of centering for interaction of level 1 variables in multilevel structural equation models. Structural Equation Modeling: A Multidisciplinary Journal, 22(4), 617–630. https://doi.org/https://doi.org/10.1080/10705511.2014.936491
- Shin, Y., & Raudenbush, S. W. (2010). A latent cluster-mean approach to the contextual effects model with missing data. Journal of Educational and Behavioral Statistics, 35(1), 26–53. https://doi.org/https://doi.org/10.3102/1076998609345252
- Snijders, T. A. B., & Bosker, R. J. (2012). Multilevel analysis: An introduction to basic and advanced multilevel modeling. SAGE.
- Spybrook, J., Bloom, H., Congdon, R., Hill, C., Martinez, A., Raudenbush, S. (2011). Optimal design for longitudinal and multilevel research: Documentation for the “Optimal Design” software, https://wtgrantfoundation.org/resource/optimal-design-with-empirical-information-od
- Sterba, S. K. (2009). Alternative model-based and design-based frameworks for inference from samples to populations: From polarization to integration. Multivariate Behavioral Research, 44(6), 711–740. https://doi.org/https://doi.org/10.1080/00273170903333574
- Yang, M., & Yuan, K.-H. (2016). Robust methods for moderation analysis with a two-level regression model. Multivariate Behavioral Research, 51(6), 757–771. https://doi.org/https://doi.org/10.1080/00273171.2016.1235965