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Multilevel and Latent Variable Modeling with Composite Links and Exploded Likelihoods

Published online by Cambridge University Press:  01 January 2025

Sophia Rabe-Hesketh  and
Anders Skrondal
Sophia Rabe-Hesketh*
Affiliation:
University of California at Berkeley and University of London
Anders Skrondal
Affiliation:
London School of Economics and Norwegian Institute of Public Health
*
Requests for reprints should be sent to Sophia Rabe-Hesketh, 3659 Tolman Hall, Graduate School of Education, University of California, Berkeley, CA 94720-1670, USA. E-mail: sophiarh@berkeley.edu
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Abstract

Composite links and exploded likelihoods are powerful yet simple tools for specifying a wide range of latent variable models. Applications considered include survival or duration models, models for rankings, small area estimation with census information, models for ordinal responses, item response models with guessing, randomized response models, unfolding models, latent class models with random effects, multilevel latent class models, models with log-normal latent variables, and zero-inflated Poisson models with random effects. Some of the ideas are illustrated by estimating an unfolding model for attitudes to female work participation.

Keywords

composite linkexploded likelihoodunfoldingmultilevel modelgeneralized linear mixed modellatent variable modelitem response modelfactor modelfrailtyzero-inflated Poisson modelgllamm

Information

Type
Original Paper
Information
Psychometrika , Volume 72 , Issue 2 , June 2007 , pp. 123 - 140
Copyright
Copyright © 2007 The Psychometric Society

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Footnotes

We wish to thank The Research Council of Norway for a grant supporting our collaboration.

References

Allison, P.D. (1982). Discrete time methods for the analysis of event histories. In Leinhardt, S. (Ed.), Sociological methodology 1982. San Francisco: Jossey-Bass.Google Scholar
Allison, P.D. (1987). Introducing a disturbance into logit and probit regression models. Sociological Methods & Research, 15, 355–374.CrossRefGoogle Scholar
Allison, P.D., Christakis, N.A. (1994). Logit models for sets of ranked items. In Marsden, P.V. (Eds.), Sociological methodology 1994 (pp. 199–228). Oxford: Blackwell.Google Scholar
Andrich, D. (1989). A probabilistic item response theory model for unfolding preference data. Applied Psychological Measurement, 13, 193–216.CrossRefGoogle Scholar
Andrich, D. (1995). Hyperbolic cosine latent trait models for unfolding direct-responses and pairwise preferences. Applied Psychological Measurement, 19, 269–290.CrossRefGoogle Scholar
Andrich, D. (1996). A hyperbolic cosine latent trait model for unfolding polytomous responses: Reconciling Thurstone and Likert methodologies. British Journal of Mathematical and Statistical Psychology, 49, 347–365.CrossRefGoogle Scholar
Andrich, D., Luo, G. (1993). A hyperbolic cosine latent trait model for unfolding dichotomous single stimulus responses. Applied Psychological Measurement, 17, 253–276.CrossRefGoogle Scholar
Bartholomew, D.J., Knott, M. (1999). Latent variable models and factor analysis, London: Arnold.Google Scholar
Birnbaum, A. (1968). Test scores, sufficient statistics, and the information structures of tests. In Lord, F.M. & Novick, M.R. (Eds.), Statistical theories of mental test scores (pp. 425–435). Reading, MA: Addison-Wesley.Google Scholar
Böckenholt, U. (2001). Mixed-effects analyses of rank-ordered data. Psychometrika, 66, 45–62.CrossRefGoogle Scholar
Böckenholt, U., & van der Heijden, P.G.M. (2007). Item randomized-response models for measuring noncompliance: Risk-return perceptions, social influences, and self-protective responses. Psychometrika, 72, DOI: 10.1007/s11336-005-1495-y.CrossRefGoogle Scholar
Breslow, N.E., Clayton, D.G. (1993). Approximate inference in generalized linear mixed models. Journal of the American Statistical Association, 88, 9–25.CrossRefGoogle Scholar
Candy, S.G. (1997). Estimation in forest yield models using composite link functions with random effects. Biometrics, 53, 146–160.CrossRefGoogle Scholar
Chapman, R.G., Staelin, R. (1982). Exploiting rank ordered choice set data within the stochastic utility model. Journal of Marketing Research, 14, 288–301.CrossRefGoogle Scholar
Chen, Z., Kuo, L. (2001). A note on the estimation of the multinomial logit model with random effects. The American Statistician, 55, 89–95.CrossRefGoogle Scholar
Clayton, D.G. (1988). The analysis of event history data: A review of progress and outstanding problems. Statistics in Medicine, 7, 819–841.CrossRefGoogle Scholar
Clayton, D.G., Spiegelhalter, D., Dunn, G., Pickles, A. (1998). Analysis of longitudinal binary data from multiphase sampling. Journal of the Royal Statistical Society, Series B, 60, 71–77.CrossRefGoogle Scholar
Coombs, C.H. (1964). A theory of data, New York: Wiley.Google Scholar
Cox, C. (1984). Generalized linear models—The missing link. Journal of the Royal Statistical Society, Series C, 33, 18–24.Google Scholar
Cox, D.R. (1972). Regression models and life tables. Journal of the Royal Statistical Society, Series B, 34, 187–203.CrossRefGoogle Scholar
Davis, J.A., Smith, T.W., & Marsden, P.V. (2003). General social surveys, 1972–2002 [Cumulative File]. Ann Arbor, MI: ICPSR [Distributor].CrossRefGoogle Scholar
Dayton, C.M., MacReady, G.B. (1988). Concomitant variable latent class models. Journal of the American Statistical Association, 83, 173–178.CrossRefGoogle Scholar
De Boeck, P., & Wilson, M. (Eds.) (2004). Explanatory item response models: A generalized linear and nonlinear approach. New York: Springer.CrossRefGoogle Scholar
DeSarbo, W.S., Hoffman, D.L. (1986). Simple unweighted unfolding threshold models for the spatial representation of binary choice data. Applied Psychological Measurement, 10, 247–264.CrossRefGoogle Scholar
Eilers, P.H.C., Borgdorff, M.W. (2004). Modeling and correction of digit preference in tuberculin surveys. The International Journal of Tubercolosis and Lung Disease, 8, 232–239.Google ScholarPubMed
Engel, R. (1993). On the analysis of grouped extreme value data with GLIM. Journal of the Royal Statistical Society, Series C, 42, 633–640.Google Scholar
Espeland, M.A., Hui, S.L. (1987). A general approach to analyzing epidemiologic data that contain misclassification errors. Biometrics, 43, 1001–1012.CrossRefGoogle ScholarPubMed
Fielding, A. (2003). Ordered category responses and random effects in multilevel and other complex structures. In Reise, S.P. & Duan, N. (Eds.), Multilevel modeling. Methodological advances, issues, and applications (pp. 181–208). Mahwah, NJ: Erlbaum.Google Scholar
Finney, D.J. (1971). Probit analysis, Cambridge: Cambridge University Press.Google Scholar
Fox, J.P. (2005). Randomized item response theory models. Journal of Educational and Behavioral Statistics, 30, 1–24.CrossRefGoogle Scholar
Fox, J.P., Glas, C.A.W. (2001). Bayesian estimation of a multilevel IRT model using Gibbs sampling. Psychometrika, 66, 271–288.CrossRefGoogle Scholar
Goldstein, H. (2003). Multilevel statistical models (3rd ed.). London: Arnold.Google Scholar
Goldstein, H., McDonald, R.P. (1988). A general model for the analysis of multilevel data. Psychometrika, 53, 455–467.CrossRefGoogle Scholar
Haberman, S.J. (1979). Analysis of qualitative data: New developments (Vol. 2). New York: Academic Press.Google Scholar
Hall, D.B. (2000). Zero-inflated Poisson and binomial regression with random effects: A case study. Biometrics, 56, 1030–1039.CrossRefGoogle ScholarPubMed
Heckman, J.J., Singer, B. (1984). A method of minimizing the impact of distributional assumptions in econometric models for duration data. Econometrica, 52, 271–320.CrossRefGoogle Scholar
Heisterkamp, S.H., van Houwelingen, J.C., Downs, A.M. (1999). Empirical Bayesian estimators for a Poisson process propagated in time. Biometrical Journal, 41, 385–400.3.0.CO;2-Z>CrossRefGoogle Scholar
Hoijtink, H. (1990). A latent trait model for dichotomous choice data. Psychometrika, 55, 641–656.CrossRefGoogle Scholar
Holford, T.R. (1976). Life tables with concomitant variables. Biometrics, 32, 587–597.CrossRefGoogle Scholar
Jansen, J. (1992). Statistical analysis of threshold data from experiments with nested errors. Computational Statistics & Data Analysis, 13, 319–330.CrossRefGoogle Scholar
Johnson, M.S. (2006). Nonparametric estimation of item and respondent locations from unfolding-type items. Psychometrika, 71, 257–279.CrossRefGoogle ScholarPubMed
Klein, A., Moosbrugger, H. (2000). Maximum likelihood estimation of latent interaction effects with the LMS method. Psychometrika, 65, 457–474.CrossRefGoogle Scholar
Läärä, E., Matthews, J.N.S. (1985). The equivalence of two models for ordinal data. Biometrika, 72, 206–207.CrossRefGoogle Scholar
Laird, N.M. (1978). Nonparametric maximum likelihood estimation of a mixing distribution. Journal of the American Statistical Association, 73, 805–811.CrossRefGoogle Scholar
Lambert, D. (1992). Zero-inflated Poisson-regression with an application to defects in manufacturing. Technometrics, 34, 1–14.CrossRefGoogle Scholar
Lee, S.-Y., Song, X.-Y. (2004). Maximum likelihood analysis of a general latent variable model with hierarchically mixed data. Biometrics, 60, 624–636.CrossRefGoogle ScholarPubMed
Luce, R.D. (1959). Individual choice behavior, New York: Wiley.Google Scholar
Magder, S.M., Hughes, J.P. (1997). Logistic regression when the outcome is measured with uncertainty. American Journal of Epidemiology, 146, 195–203.CrossRefGoogle ScholarPubMed
McCullagh, P., & Nelder, J.A. (1989). Generalized linear models (2nd ed.). London: Chapman & Hall.CrossRefGoogle Scholar
Muthén, B.O. (1989). Latent variable modeling in heterogeneous populations. Psychometrika, 54, 557–585.CrossRefGoogle Scholar
Muthén, B.O. (2002). Beyond SEM: General latent variable modeling. Behaviormetrika, 29, 81–117.CrossRefGoogle Scholar
Muthén, B.O., Masyn, K. (2005). Dicrete-time survival mixture analysis. Journal of Educational and Behavioral Statistics, 30, 27–58.CrossRefGoogle Scholar
Neuhaus, J.M. (1999). Bias and efficiency loss due to misclassified responses in binary regression. Biometrika, 86, 843–855.CrossRefGoogle Scholar
Noël, Y. (1999). Recovering unimodal latent patterns of change by unfolding analysis: Application to smoking cessation. Psychological Methods, 4, 173–191.CrossRefGoogle Scholar
Palmgren, J. (1981). The Fisher information matrix for log linear models arguing conditionally on observed explanatory variables. Biometrika, 68, 563–566.Google Scholar
Plackett, R.L. (1975). The analysis of permutations. Journal of the Royal Statistical Society, Series C, 24, 193–202.Google Scholar
Qu, Y., Tan, M., Kutner, M.H. (1996). Random effects models in latent class analysis for evaluating accuracy of diagnostic tests. Biometrics, 52, 797–810.CrossRefGoogle ScholarPubMed
Rabe-Hesketh, S., Skrondal, A. (2005). Multilevel and longitudinal modeling using Stata, College Station, TX: Stata Press.Google Scholar
Rabe-Hesketh, S., Skrondal, A. (2006). Multilevel modeling of complex survey data. Journal of the Royal Statistical Society, Series A, 116, 805–827.CrossRefGoogle Scholar
Rabe-Hesketh, S., Pickles, A., Skrondal, A. (2003). Correcting for covariate measurement error in logistic regression using nonparametric maximum likelihood estimation. Statistical Modelling, 3, 215–232.CrossRefGoogle Scholar
Rabe-Hesketh, S., Skrondal, A., Pickles, A. (2004a). Generalized multilevel structural equation modeling. Psychometrika, 69, 167–190.CrossRefGoogle Scholar
Rabe-Hesketh, S., Skrondal, A., & Pickles, A. (2004b). GLLAMM manual. Technical report 160. U.C. Berkeley Division of Biostatistics Working Paper Series. Downloadable from http://www.bepress.com/ucbbiostat/paper160/.Google Scholar
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.CrossRefGoogle Scholar
Rabe-Hesketh, S., Yang, S., Pickles, A. (2001). Multilevel models for censored and latent responses. Statistical Methods in Medical Research, 10, 409–427.CrossRefGoogle ScholarPubMed
Rijmen, F., Tuerlinckx, F., De Boeck, P., Kuppens, P. (2003). A nonlinear mixed model framework for item response theory. Psychological Methods, 8, 185–205.CrossRefGoogle ScholarPubMed
Rindskopf, D. (1992). A general approach to categorical data analysis with missing data, using generalized linear models with composite links. Psychometrika, 57, 29–42.CrossRefGoogle Scholar
Roberts, J.S., Laughlin, J.E. (1996). A unidimensional item response model for unfolding responses from a graded disagree–agree response scale. Applied Psychological Measurement, 20, 231–255.CrossRefGoogle Scholar
Samejima, F. (1969). Estimation of latent trait ability using a response pattern of graded scores. Psychometric Monograph, Vol. 17. Bowling Green, OH: The Psychometric Society.Google Scholar
Skrondal, A., Rabe-Hesketh, S. (2003). Multilevel logistic regression for polytomous data and rankings. Psychometrika, 68, 267–287.CrossRefGoogle Scholar
Skrondal, A., & Rabe-Hesketh, S. (2004a). Generalised linear latent and mixed models with composite links and exploded likelihoods. In Biggeri, A., Dreassi, E., Lagazio, C., & Marchi, M. (Eds.), 19th International workshop on statistical modeling (pp. 27–29). Florence: Firenze University Press.Google Scholar
Skrondal, A., & Rabe-Hesketh, S. (2004b). Generalized latent variable modeling: Multilevel, longitudinal, and structural equation models. Boca Raton, FL: Chapman & Hall/CRC.CrossRefGoogle Scholar
Takane, Y., de Leeuw, J. (1987). On the relationship between item response theory and factor analysis of discretized variables. Psychometrika, 52, 393–408.CrossRefGoogle Scholar
Terza, J.V. (1985). Ordinal probit: A generalization. Communications in Statistics, Theory and Methods, 14, 1–11.Google Scholar
Thissen, D., Steinberg, L. (1986). A taxonomy of item response models. Psychometrika, 51, 567–577.CrossRefGoogle Scholar
Thompson, R., Baker, R.J. (1981). Composite link functions in generalized linear models. Journal of the Royal Statistical Society, Series C, 30, 125–131.Google Scholar
Thurstone, L.L. (1928). Attitudes can be measured. American Journal of Sociology, 33, 529–554.CrossRefGoogle Scholar
Tranmer, M., Pickles, A., Fieldhouse, E., Elliot, M., Dale, A., Brown, M., Martin, D., Steel, D., Gardiner, C. (2005). The case for small area microdata. Journal of the Royal Statistical Society, Series A, 168, 29–39.CrossRefGoogle Scholar
Verhelst, N.D., Verstralen, H.H.F.M. (1993). A stochastic unfolding model derived from the partial credit model. Kwantitative Methoden, 42, 73–92.Google Scholar
Vermunt, J.K. 1997. Log-linear models for event histories. Thousand Oaks, CA: Sage.Google Scholar
Vermunt, J.K. (2001). The use of restricted latent class models for defining and testing nonparametric item response theory models. Applied Psychological Measurement, 25, 283–294.CrossRefGoogle Scholar
Vermunt, J.K. (2003). Multilevel latent class models. In Stolzenberg, R.M. (Ed.), Sociological methodology 2003 (Vol. 33, pp. 213–239). Oxford: Blackwell.Google Scholar
Vermunt, J.K. (in press). Latent class and finite mixture models for multilevel data sets. Statistical Methods in Medical Research.Google Scholar
Warner, S.L. (1965). Randomized response: A survey technique for eliminating evasive answer bias. Journal of the American Statistical Association, 60, 63–69.CrossRefGoogle ScholarPubMed
Whitehead, J. (1980). Fitting Cox’s regression model to survival data using GLIM. Journal of the Royal Statistical Society, Series C, 29, 269–275.Google Scholar