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Abstract
In this article, the authors suggest a profile-likelihood approach for estimating complex models by maximum likelihood (ML) using standard software and minimal programming. The method works whenever setting some of the parameters of the model to known constants turns the model into a standard model. An important class of models that can be estimated this way is generalized linear mixed models with factor structures. Such models are useful in educational research, for example, for estimation of value-added teacher or school effects with persistence parameters and for analysis of large-scale assessment data using multilevel item response models with discrimination parameters. The authors describe the profile-likelihood approach, implement it in the R software, and apply the method to longitudinal data and binary item response data. Simulation studies and comparison with gllamm show that the profile-likelihood method performs well in both types of applications. The authors also briefly discuss other types of models that can be estimated using the profile-likelihood idea.
References
|
Adams, R., Wilson, M., Wu, M. (1997). Multilevel item response models: An approach to errors in variables regression. Journal of Educational and Behavioral Statistics, 22, 47–76. Google Scholar | SAGE Journals | |
|
Bates, D., Maechler, M. (2009). lme4: Linear mixed-effects models using S4 classes R package version 0.999375-31. Retrieved from http://CRAN.Rproject.org/package=lme4 Google Scholar | |
|
Berkhof, J., Snijders, T. A. B. (2001). Variance component testing in multilevel models. Journal of Educational and Behavioral Statistics, 26, 132–152. Google Scholar | SAGE Journals | |
|
Browne, W. J. (2009). MCMC estimation in MLwiN. Bristol, UK: Centre for Multilevel Modelling. Google Scholar | |
|
Browne, W. J., Draper, D. (2006). A comparison of Bayesian and likelihood methods for fitting multilevel models. Bayesian Analysis, 1, 473–514. Google Scholar | Crossref | |
|
Browne, W. J., Goldstein, H., Rasbash, J. (2001). Multiple membership multiple classification (MMMC) models. Statistical Modelling, 1, 103–124. Google Scholar | SAGE Journals | |
|
Byrd, R. H., Lu, P., Nocedal, J., Zhu, C. (1995). A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing, 16, 1190–1208. Google Scholar | Crossref | |
|
Cai, L., Yang, J. S., Hansen, M. P. (2011). Generalized full-information item bifactor analysis. Psychological Methods, 16, 221–248. Google Scholar | Crossref | Medline | |
|
Cho, S.-J., Rabe-Hesketh, S. (2011). Alternating imputation posterior estimation of models with crossed random effects. Computational Statistics and Data Analysis, 55, 12–25. Google Scholar | Crossref | |
|
De Boeck, P., Bakker, M., Zwitser, R., Nivard, M., Hofman, A., Tuerlinckx, F., Partchev, I. (2011). The estimation of item response models with the lmer function from the lme4 package in R. Journal of Statistical Software, 39, 1–28. Google Scholar | Crossref | |
|
De Boeck, P., Wilson, M. (2004). Explanatory item response models: A generalized and nonlinear approach. New York, NY: Springer. Google Scholar | Crossref | |
|
Doolaard, S. (1999). Schools in change or school in chain. Enschede, Netherlands: University of Twente. Google Scholar | |
|
Doran, H., Bates, D., Bliese, P., Dowling, M. (2007). Estimating the multilevel Rasch model: With the lme4 package. Journal of Statistical Software, 20, 1–18. Google Scholar | Crossref | |
|
Embretson, S. (1999). Generating items during testing: Psychometric issues and models. Psychometrika, 64, 407–433. Google Scholar | Crossref | |
|
Fisher, G. H. (1983). Logistic latent trait models with linear constraints. Psychometrika, 48, 3–26. Google Scholar | Crossref | |
|
Fox, J. P. (2007). Multilevel IRT modeling in practice with the package mlirt. Journal of Statistical Software, 20, 1–16. Google Scholar | Crossref | |
|
Fox, J. P., Glas, A. C. (2001). Bayesian estimation of a multilevel IRT model using Gibbs sampling. Psychometrika, 66, 271–288. Google Scholar | Crossref | |
|
Gibbons, R. D., Hedeker, D. (1992). Full-information item bi-factor analysis. Psychometrika, 57, 423–436. Google Scholar | Crossref | |
|
Goldstein, H. (1987). Multilevel covariance component models. Biometrika, 74, 430–431. Google Scholar | Crossref | |
|
Goldstein, H. (2003). Multilevel statistical models. 3rd ed. London, England: Arnold. Google Scholar | |
|
Goldstein, H., Bonnet, G., Rocher, T. (2007). Multilevel structural equation models for the analysis of comparative data on educational performance. Journal of Educational and Behavioral Statistics, 32, 252–286. Google Scholar | SAGE Journals | |
|
Goldstein, H., Browne, J. W. (2005). Multilevel factor analysis models for continuous and discrete data. In Maydeu-Olivares, A., McArdle, J. J. (Eds.),. In Contemporary psychometrics: A festschrift for Roderick P. McDonald (pp. 453–475). Mahwah, New Jersey:Lawrence Erlbaum Google Scholar | |
|
Goldstein, H., Burgess, S., McConnell, B. (2007). Modelling the effect of pupil mobility on school differences in educational achievement. Journal of the Royal Statistical Society Series A, 170, 841–954. Google Scholar | Crossref | |
|
Greenland, S. (1984). A counterexample to the test-based principle of setting confidence limits. American Journal of Epidemiology, 120, 4–7. Google Scholar | Crossref | Medline | |
|
Halperin, M. (1977). Estimability and estimation in case-referent studies. Letters to the Editor. American Journal of Epidemiology, 105, 496–498. Google Scholar | Crossref | Medline | |
|
Hill, P., Goldstein, H. (1998). Multilevel modeling of educational data with cross-classification and missing identification for units. Journal of Educational and Behavioral Statistics, 23, 117–128. Google Scholar | SAGE Journals | |
|
Jeon, M., Rijmen, F., Rabe-Hesketh, S. (in press). Modeling differential item functioning using a generalized multiple-group bifactor model. Journal of Educational and Behavioral Statistics. Google Scholar | |
|
Joe, H. (2008). Accuracy of Laplace approximation for discrete response mixed models. Computational Statistics and Data Analysis, 52, 5066–5074. Google Scholar | Crossref | |
|
Kamata, A. (2001). Item analysis by the hierarchical generalized linear model. Applied Psychological Measurement, 38, 79–93. Google Scholar | |
|
Li, D., Oranje, A., Jiang, Y. (2009). On the estimation of hierarchical latent regression models for large-scale assessments. Journal of Educational and Behavioral Statistics, 34, 433–463. Google Scholar | SAGE Journals | |
|
Lockwood, J., McCaffrey, D., Mariano, L., Setodji, C. (2007). Bayesian methods for scalable multivariate value-added assessment. Journal of Educational and Behavioral Statistics, 32, 125–150. Google Scholar | SAGE Journals | |
|
Maas, C. M., Snijders, T. A. B. (2003). The multilevel approach to repeated measures for complete and incomplete data. Quality & Quantity, 37, 71–89. Google Scholar | Crossref | |
|
Maier, K. S. (2001). A Rasch hierarchical measurement model. Journal of Educational and Behavioral Statistics, 26, 307–330. Google Scholar | SAGE Journals | |
|
Mariano, L., McCaffrey, D., Lockwood, J. (2010). A model for teacher effects from longitudinal data without assuming vertical scaling. Journal of Educational and Behavioral Statistics, 35, 253–279. Google Scholar | SAGE Journals | |
|
McCaffrey, D. F., Lockwood, J. R., Koretz, D., Louis, T. A., Hamilton, L. (2004). Models for value-added modeling of teacher effects. Journal of Educational and Behavioral Statistics, 29, 67–101. Google Scholar | SAGE Journals | |
|
Mellenbergh, J. G. (1994). Generalized linear item response theory. Psychological Bulletin, 115, 300–307. Google Scholar | Crossref | |
|
Meredith, W., Tisak, J. (1988). Latent curve analysis. Psychometrika, 55, 107–122. Google Scholar | Crossref | |
|
Miettinen, O. (1976). Estimability and estimation in case-referent studies. American Journal of Epidemiology, 103, 226–235. Google Scholar | Crossref | Medline | |
|
Muthén, L., Muthén, B. (2008). Mplus user’s guide. Los Angeles, CA: Muthen & Muthen. Google Scholar | |
|
Natarajan, R., Kass, R. E. (2000). Reference Bayesian methods for generalized linear mixed models. Journal of the American Statistical Association, 95, 227–237. Google Scholar | Crossref | |
|
Parke, R. W (1986). Pseudo maximum likelihood estimation: The asymptotic distribution. Annals of Statistics, 14, 355–357. Google Scholar | Crossref | |
|
Pawitan, Y. (2001). In all likelihood: Statistical modeling and inference using likelihood. New York, NY: Oxford. Google Scholar | |
|
Rabe-Hesketh, S., Skrondal, A., Pickles, A. (2004). Generalized multilevel structural equation modelling. Psychometrika, 69, 167–190. Google Scholar | Crossref | |
|
Rabe-Hesketh, S., Skrondal, A., Pickles, A. (2005). Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects. Journal of Econometrics, 128, 301–323. Google Scholar | Crossref | |
|
Raudenbush, S. W. (1993). A crossed random effects model for unbalanced data with applications in cross-sectional and longitudinal research. Journal of Educational Statistics, 18, 321–349. Google Scholar | SAGE Journals | |
|
Raudenbush, S. W., Bryk, A. (2002). Hierarchical linear models: Applications and data analysis methods. Thousand Oaks, CA: Sage. Google Scholar | |
|
Raudenbush, S. W., Rowan, B., Kang, S. J. (1991). A multilevel, multivariate model for studying school climate with estimation via the EM algorithm and application to U.S. high-school data. Journal of Educational Statistics, 16, 295–330. Google Scholar | SAGE Journals | |
|
Raudenbush, S. W., Sampson, R. (1999). Ecometrics: Toward a science of assessing ecological settings, with application to the systematic social observation of neighborhoods. Sociological Methodology, 29, 1–41. Google Scholar | SAGE Journals | |
|
Skrondal, A., Rabe-Hesketh, S. W. (2004). Generalized latent variable modeling: Multilevel, longitudinal, and structural equation models. Boca Raton, FL: Chapman & Hall/CRC. Google Scholar | Crossref | |
|
Skrondal, A., Rabe-Hesketh, S. W. (2007). Latent variable modelling: A survey. Scandinavian Journal of Statistics, 34, 712–745. Google Scholar | Crossref | |
|
Spiegelhalter, D. J., Thomas, A., Best, N. G., Gilks, W. R. (1996). BUGS 0.5 Bayesian analysis using Gibbs sampling. Manual (version ii). Cambridge, UK: MRC-Biostatistics Unit. Retrieved from http://www.mrc-bsu.cam.ac.uk/bugs/documentation/contents.shtml Google Scholar | |
|
StataCorp . (2009). Stata statistical software: Release 11. College Station, TX: Author. Google Scholar | |
|
Swaminathan, H., Rogers, H. J. (1990). Detecting differential item functioning using logistic regression procedures. Journal of Educational Measurement, 27, 361–370. Google Scholar | Crossref | |
|
Vermunt, J. (2007). Multilevel latent variable modeling: An application in education testing. Austrian Journal of Statistics, 38, 285–299. Google Scholar | |
|
Wolfinger, R. D. (1999). Fitting non-linear mixed models with the new NLMIXED procedure (Technical Report). Cary, NC: SAS Institute. Google Scholar | |
|
Zheng, X., Rabe-Hesketh, S. (2007). Estimating parameters of dichotomous and ordinal item response models with gllamm. The Stata Journal, 7, 313–333. Google Scholar | SAGE Journals |
