Mixed and mixture regression models for continuous bounded responses using the beta distribution

Journal of Educational and Behavioral Statistics

Volumn 37, Issue 1, 2012, Pages 82-113

  Verkuilen, Jay ^a   Smithson, Michael ^b  

^a CITY UNIVERSITY OF NEW YORK   (United States)

^b AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

beta distribution; general linear model; mixed model; mixture model

Indexed keywords

EID: 84856587257     PISSN: 10769986     EISSN: None     Source Type: Journal    
DOI: 10.3102/1076998610396895     Document Type: Review

References (54)

1. Aitchison J.The statistical analysis of compositional data. London, England: The Blackburn Press; 2003:.
2. Alicke M. D.,Govorun O., The better-than-average effect. In,The self in social judgment. Alicke M. D.Dunning D. A.Krueger J. I., ed. New York, NY: Psychology Press, 85-108; 2005:.
3. Atkinson A. C.Fast very robust methods for the detection of multiple outliers.Journal of the American Statistical Association. 1994;89:1329-1339.
4. Barndorff-Nielsen O. E.,Jorgensen B.Some parametric models on the simplex.Journal of Multivariate Analysis. 1991;39:106-116.
5. Brehm J.,Gates S.Donut shops and speed traps: Evaluating models of supervision on police behavior.American Journal of Political Science. 1993;37:555-581.
6. Browne W. J.,Draper D.A comparison of Bayesian and likelihood-based methods for fitting multilevel models.Bayesian Analysis. 2006;1:473-514.
7. Browne W. J.,Subramanian S. V.,Jones K.,Goldstein H.Variance partitioning in multilevel logistic models that exhibit over-dispersion.Journal of the Royal Statistical Society, Series A. 2005;168:599-613.
8. Buckley J.Estimation of models with beta-distributed dependent variables: A replication and extension of Paolino (2001).Political Analysis. 2002;11:1-12.
9. Carroll R. J.Variances are not always nuisance parameters.Biometrics. 2003;59:211-220.
10. Chen M.-H.,Ibrahim J. G.,Shao Q.-M.,Weiss R. E.Prior elicitation for model selection and estimation in generalized linear mixed models.Journal of Statistical Planning and Inference. 2003;111:57-76.
11. Congdon P.Applied Bayesian modelling. Chichester, England: Wiley; 2003:.
12. Cox C.Nonlinear quasi-likelihood models: Applications to continuous proportions.Computational Statistics and Data Analysis. 1996;21:449-461.
13. Crowder M. J.Beta-binomial ANOVA for proportions.Applied Statistics. 1978;27:34-37.
14. Dolnicar S.,Grun B.Cross-cultural differences in survey response patterns.International Marketing Review. 2007;24:127-143.
15. Epsinheira P. L.,Ferrari S. L. P.,Cribari-Neto F.On beta regression residuals.Journal of Applied Statistics. 2008a;35:407-419.
16. Epsinheira P. L.,Ferrari S. L. P.,Cribari-Neto F.Influence diagnostics in beta regression.Computational Statistics and Data Analysis. 2008b;52:4417-4431.
17. Fahrmeir L.,Tutz G.Multivariate statistical modelling based on generalized linear models. New York, NY: Springer; 2001:.
18. Ferrari S. L. P.,Cribari-Neto F.Beta regression for modeling rates and proportions.Journal of Applied Statistics. 2004;31:799-815.
19. Fox C. R.,Rottenstreich Y.Partition priming in judgment under uncertainty.Psychological Science. 2003;14:195-200.
20. Gelman A.Prior distributions for variance parameters in hierarchical models.Bayesian Analysis. 2006;1:516-534.
21. Gelman A.,Rubin D. B.Inference from iterative simulation using multiple sequences (with discussion).Statistical Science. 1992;7:457-511.
22. Goldstein H.,Browne W. J.,Rasbash J.Partitioning variation in multilevel models.Understanding Statistics. 2002;1:223-231.
23. Gupta, A. K. & Nadarajah, S.(Ed.). (2004) Handbook of beta distribution and its applications. Gupta A. K.Nadarajah S., ed. New York, NY: Marcel Dekker; 2004:.
24. Gurr M.Partition dependence: Investigating the principle of insufficient reason, uncertainty and dispositional predictors. Canberra, Australia: The Australian National University; 2009:.
25. Johnson T. R.On the use of heterogeneous thresholds ordinal regression models to account for individual differences in response style.Psychometrika. 2003;68:563-583.
26. Kieschnick R.,McCullogh B. D.Regression analysis of variates observed on (0,1): Percentages, proportions, and fractions.Statistical Modelling. 2003;3:193-213.
27. Kumaraswamy P.A generalized probability density function for double-bounded random processes.Journal of Hydrology. 1980;46:79-88.
28. Longford N. T.Simulation-based diagnostics in random-coefficient models.Journal of the Royal Statistical Society, Series A. 2001;164:259-273.
29. McAuliffe J. D.,Blei D. M.,Jordan M. I.Nonparametric empirical Bayes for the Dirichlet process mixture model.Statistics and Computing. 2006;16:5-14.
30. McCullagh P.,Nelder J. A.Generalized linear models. 1989). Generalized linear models2nd ed.London, England: Chapman and Hall.London, England: Chapman and Hall; 1989:.
31. McCulloch C. E.,Searle S. R.,Neuhaus J. M.Generalized, linear, and mixed models. 2008). Generalized, linear, and mixed models2nd ed.New York, NY: Wiley.New York, NY: Wiley; 2008:.
32. Merkle E.,Smithson M.,Verkuilen J., Hierarchical Bayesian beta models for subjective probabilities. In, Paper presented at the Joint Annual Convention for the Society of Mathematical Psychology and the European Mathematical Psychology Group; 2009Amsterdam; 2009.
33. Noël Y.When extreme responses are substantial: A generalized beta response model of behavior change., Paper presented at the Joint Annual Convention for the Society of Mathematical Psychology and the European Mathematical Psychology Group; 2008Amsterdam, the Netherlands; 2008.
34. Noël Y.,Dauvier B.A beta item response model for continuous bounded responses.Applied Psychological Measurement. 2007;31:47-73.
35. Ntzoufras I.Bayesian modeling using WinBUGS. Hoboken, NJ: Wiley; 2009:.
36. Ospina R.,Cribari-Neto F.,Vasconcellos K. L. P.Improved point and interval estimates for a beta regression model.Computational Statistics and Data Analysis. 2006;51:960-981.
37. Paolino P.Maximum likelihood estimation of models with beta-distributed dependent variables.Political Analysis. 2001;9:325-346.
38. Papke L.,Wooldridge J.Econometric methods for fractional response variables with an application to 401(K) plan participation rates.Journal of Applied Econometrics. 1996;11:619-632.
39. Pawitan Y.In all likelihood: Statistical modeling and inference using likelihood. Oxford, UK: Clarendon Press; 2001:.
40. Pinheiro J. C.,Chao E. C.Efficient Laplacian and adaptive Gaussian quadrature algorithms for multilevel generalized linear models.Journal of Computational and Graphical Statistics. 2006;15:58-81.
41. Qiu Z.,Song P. X. K.,Tan M.Simplex mixed-effects models for longitudinal proportional data.Scandinavian Journal of Statistics. 2008;35:577-596.
42. Roy M.,Liersch M.Most people are above average, sometimes. Elizabethtown, PA: Elizabethtown College; 2009:.
43. Cary, NC, U.S.A.: SAS Institute, Inc; 2006:.
44. Cary, NC, U.S.A.: SAS Institute, Inc; 2008:.
45. See K. E.,Fox C. R.,Rottenstreich Y. S.Between ignorance and truth: Partition dependence and learning in judgment under uncertainty.Journal of Experimental Psychology: Learning, Memory and Cognition. 2006;32:1385-1402.
46. Smithson M. J.,Merkle E.,Verkuilen J., Beta regression finite mixture models of polarization and priming., Paper presented at the Joint Annual Convention for the Society of Mathematical Psychology and the European Mathematical Psychology Group; 2009Amsterdam, the Netherlands; 2009.
47. Smithson M. J.,Segale C.Partition priming in judgments of imprecise probabilities.Journal of Statistical Theory and Practice. 2009;3:169-182.
48. Smithson M. J.,Verkuilen J.A better lemon squeezer? Maximum likelihood regression with beta-distributed dependent variables.Psychological Methods. 2006;11:54-71.
49. Smyth G. K.Generalized linear models with varying dispersion.Journal of the Royal Statistical Society, Series B. 1989;51:47-60.
50. Spiegelhalter D.,Thomas A.,Best N.,Lunn D.England: Medical Research Council, Biostatistics Unit; 2004:.
51. Steiger J. H.Driving fast in reverse: The relationship between software development, theory and education in structural equation modeling.Journal of the American Statistical Association. 2001;96:331-338.
52. Tadikamalla P. R.,Johnson N. L.Systems of frequency curves generated by transformation of logistic variables.Biometrika. 1982;69:461-465.
53. Vasconcellos K. L. P.,Cribari-Neto F.Improved maximum likelihood estimation in a new class of beta regression models.Brazilian Journal of Probability and Statistics. 2005;19:13-31.
54. Weiss R. E.,Cho M.Bayesian marginal influence assessment.Journal of Statistical Planning and Inference. 1998;71:163-177.