Sage Journals: Discover world-class research

Abstract

In this paper, a new discrete statistical model for ordered categorical data is proposed via fixed-point discretization of a beta latent variable. The resulting discretized beta distribution has a highly flexible shape and it can be either over-dispersed or under-dispersed with respect to the binomial distribution. It has only two parameters, which may therefore parsimoniously depend on covariates and on random effects, providing new tools for the analysis of structured, clustered or longitudinal ordinal data. Practical examples and advices are given and an application of the new model to subjective evaluations of a gastrointestinal disease is shown.

Keywords

Longitudinal data mixed effect model ordinal data ordinal regression beta distribution

Get full access to this article

View all access options for this article.

References

Agresti

. Analysis of ordinal categorical data 2010; Vol. 656, Hoboken, NJ: John Wiley & Sons.

Becker

. Encyclopedia of biostatistics 2005; Vol. 6, Chichester, UK: John Wiley, pp. 3869–3876.

Snell

. A scaling procedure for ordered categorical data. Biometrics 1964; 20: 592–607.

Bock

Jones

. The measurement and prediction of judgment and choice, San Francisco, CA: Holden-Day, 1968.

Cullagh

. Regression models for ordinal data (with discussion). J Royal Stat Soc Ser B 1980; 42: 109–142.

Goodman

. Simple models for the analysis of association in cross-classifications having ordered categories. J Am Stati Assoc 1979; 74: 537–552.

Muniz-Terrera

van den Hout

Rigby

, et al. Analysing cognitive test data: Distributions and non-parametric random effects. Stat Meth Med Res 2012; 25: 741–753.

Stasinopoulos

Rigby

. Generalized Additive Models for Location Scale and Shape (GAMLSS) in R. J Stat Software 2007; 23: 1–46.

D’Elia

Piccolo

. A mixture model for preferences data analysis. Computat Stat Data Analysis 2005; 49: 917–934.

10.

Iannario

. Hierarchical cub models for ordinal variables. Commun Stat-Theory Meth 2012; 41: 3110–3125.

11.

Lee

Daniels

. Marginalized models for longitudinal ordinal data with application to quality of life studies. Stat Med 2008; 27: 4359–4380.

12.

Crouchley

. A random-effects model for ordered categorical data. J Am Stat Assoc 1995; 90: 489–498.

13.

Thomas

Have

. A mixed effects model for multivariate ordinal response data including correlated discrete failure times with ordinal responses. Biometrics 1996; 52: 473–491.

14.

Breslow

Clayton

. Approximate inference in generalized linear mixed models. J Am Stat Assoc 1993; 88: 9–25.

15.

Engel

. A simple illustration of the failure of PQL, IRREML and APHL as approximate ML methods for mixed models for binary data. Biometric J 1998; 40: 141–154.

16.

Gajewski

Hart

Bergquist-Beringer

, et al. Inter-rater reliability of pressure ulcer staging: ordinal probit bayesian hierarchical model that allows for uncertain rater response. Stat Med 2007; 26: 4602–4618.

17.

Tan

Mascha

, et al. A bayesian hierarchical model for multi-level repeated ordinal data: analysis of oral practice examinations in a large anaesthesiology training programme. Stat Med 1999; 18: 1983–1992.

18.

Lunn

Wakefield

Racine-Poon

. Cumulative logit models for ordinal data: a case study involving allergic rhinitis severity scores. Stat Med 2001; 20: 2261–2285.

19.

Lipsitz

Kim

Zhao

. Analysis of repeated categorical data using generalized estimating equations. Stat Med 1994; 13: 1149–1163.

20.

Bonney

. Logistic regression for dependent binary observations. Biometrics 1987; 43: 951–973.

21.

Muenz

Rubinstein

. Markov models for covariate dependence of binary sequences. Biometrics 1985; 41: 91–101.

22.

Battaglia

Grassini

Navino

, et al. Water load test before and after ppi therapy in patients with gastro-oesophageal reflux disease. Digest Liver Dis 2007; 39: 1052–1056.

23.

Johnson

Albert

. Ordinal data modeling. Statistics for social science and public policy, New York, NY: Springer, 1999, pp. 258–258.

24.

Ferrari

Cribari-Neto

. Beta regression for modelling rates and proportions. J Appl Stat 2004; 31: 799–815.

25.

Johnson

Kotz

. Distributions in statistics, Wiley series in probability and mathematical statistics, Wiley, 1970. ISBN 9780471443605.

26.

Punzo

Discrete beta-type models. In: Hermann

L-J

Claus

(eds). Classification as a tool for research, Berlin, Heidelberg: Springer, 2010, pp. 253–261.

27.

Byrd

Nocedal

, et al. A limited memory algorithm for bound constrained optimization. SIAM J Scientific Comput 1995; 16: 1190–1208.

28.

Team SD. Rstan: the R interface to Stan, version 2.8.0, http://mc-stan.org/rstan.html (2015, accessed 5 July 2016).

29.

Laird

Ware

. Random-effects models for longitudinal data. Biometrics 1982; 38: 963–974.

30.

Vaida

. Proportional hazards model with random effects. Stat Med 2000; 19: 3309–3324.

31.

Stiratelli

Laird

Ware

. Random-effects models for serial observations with binary response. Biometrics 1984; 40: 961–971.

32.

Daniels

. A prior for the variance in hierarchical models. Canadian J Stat 1999; 27: 567–578.

33.

Chung

Gelman

Rabe-Hesketh

, et al. Weakly informative prior for point estimation of covariance matrices in hierarchical models. J Educ Behavior Stat 2015; 40: 136–157.

34.

Christensen RHB. Ordinal—regression models for ordinal data, 2015. R package version 2015.6-28, http://www.cran.r-project.org/package=ordinal/.

35.

Dennis

Jr Schnabel

. Numerical methods for unconstrained optimization and nonlinear equations 1996; Vol. 16 Englewood Cliffs, NJ: Siam.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

1.12 MB

A new parsimonious model for ordinal longitudinal data with application to subjective evaluations of a gastrointestinal disease

Abstract

Keywords

Get full access to this article

References

Supplementary Material