Latent-variable models for longitudinal data with bivariate ordinal outcomes
We use the concept of latent variables to derive the joint distribution of bivariate ordinal outcomes, and then extend the model to allow for longitudinal data. Specifically, we relate the observed ordinal outcomes using threshold values to a bivariate latent variable, which is then modelled as a li...
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| Published in | Statistics in medicine Vol. 26; no. 5; pp. 1034 - 1054 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Chichester, UK
John Wiley & Sons, Ltd
28.02.2007
Wiley Subscription Services, Inc |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0277-6715 1097-0258 |
| DOI | 10.1002/sim.2599 |
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| Abstract | We use the concept of latent variables to derive the joint distribution of bivariate ordinal outcomes, and then extend the model to allow for longitudinal data. Specifically, we relate the observed ordinal outcomes using threshold values to a bivariate latent variable, which is then modelled as a linear mixed model. Random effects terms are used to tie all together repeated observations from the same subject. The cross‐sectional association between the two outcomes is modelled through the correlation coefficient of the bivariate latent variable, conditional on random effects. Assuming conditional independence given random effects, the marginal likelihood, under the missing data at random assumption, is approximated using an adaptive Gaussian quadrature for numerical integration. The model provides fixed effects parameters that are subject‐specific, but retain the population‐averaged interpretation when properly scaled. This is particularly well suited for the situation in which population comparisons and individual level contrasts are of equal importance. Data from a psychiatric trial, the Fluvoxamine (an antidepressant drug) study, are used to illustrate the methodology. Copyright © 2006 John Wiley & Sons, Ltd. |
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| AbstractList | We use the concept of latent variables to derive the joint distribution of bivariate ordinal outcomes, and then extend the model to allow for longitudinal data. Specifically, we relate the observed ordinal outcomes using threshold values to a bivariate latent variable, which is then modelled as a linear mixed model. Random effects terms are used to tie all together repeated observations from the same subject. The cross-sectional association between the two outcomes is modelled through the correlation coefficient of the bivariate latent variable, conditional on random effects. Assuming conditional independence given random effects, the marginal likelihood, under the missing data at random assumption, is approximated using an adaptive Gaussian quadrature for numerical integration. The model provides fixed effects parameters that are subject-specific, but retain the population-averaged interpretation when properly scaled. This is particularly well suited for the situation in which population comparisons and individual level contrasts are of equal importance. Data from a psychiatric trial, the Fluvoxamine (an antidepressant drug) study, are used to illustrate the methodology. [PUBLICATION ABSTRACT] We use the concept of latent variables to derive the joint distribution of bivariate ordinal outcomes, and then extend the model to allow for longitudinal data. Specifically, we relate the observed ordinal outcomes using threshold values to a bivariate latent variable, which is then modelled as a linear mixed model. Random effects terms are used to tie all together repeated observations from the same subject. The cross-sectional association between the two outcomes is modelled through the correlation coefficient of the bivariate latent variable, conditional on random effects. Assuming conditional independence given random effects, the marginal likelihood, under the missing data at random assumption, is approximated using an adaptive Gaussian quadrature for numerical integration. The model provides fixed effects parameters that are subject-specific, but retain the population-averaged interpretation when properly scaled. This is particularly well suited for the situation in which population comparisons and individual level contrasts are of equal importance. Data from a psychiatric trial, the Fluvoxamine (an antidepressant drug) study, are used to illustrate the methodology. We use the concept of latent variables to derive the joint distribution of bivariate ordinal outcomes, and then extend the model to allow for longitudinal data. Specifically, we relate the observed ordinal outcomes using threshold values to a bivariate latent variable, which is then modelled as a linear mixed model. Random effects terms are used to tie all together repeated observations from the same subject. The cross‐sectional association between the two outcomes is modelled through the correlation coefficient of the bivariate latent variable, conditional on random effects. Assuming conditional independence given random effects, the marginal likelihood, under the missing data at random assumption, is approximated using an adaptive Gaussian quadrature for numerical integration. The model provides fixed effects parameters that are subject‐specific, but retain the population‐averaged interpretation when properly scaled. This is particularly well suited for the situation in which population comparisons and individual level contrasts are of equal importance. Data from a psychiatric trial, the Fluvoxamine (an antidepressant drug) study, are used to illustrate the methodology. Copyright © 2006 John Wiley & Sons, Ltd. We use the concept of latent variables to derive the joint distribution of bivariate ordinal outcomes, and then extend the model to allow for longitudinal data. Specifically, we relate the observed ordinal outcomes using threshold values to a bivariate latent variable, which is then modelled as a linear mixed model. Random effects terms are used to tie all together repeated observations from the same subject. The cross-sectional association between the two outcomes is modelled through the correlation coefficient of the bivariate latent variable, conditional on random effects. Assuming conditional independence given random effects, the marginal likelihood, under the missing data at random assumption, is approximated using an adaptive Gaussian quadrature for numerical integration. The model provides fixed effects parameters that are subject-specific, but retain the population-averaged interpretation when properly scaled. This is particularly well suited for the situation in which population comparisons and individual level contrasts are of equal importance. Data from a psychiatric trial, the Fluvoxamine (an antidepressant drug) study, are used to illustrate the methodology.We use the concept of latent variables to derive the joint distribution of bivariate ordinal outcomes, and then extend the model to allow for longitudinal data. Specifically, we relate the observed ordinal outcomes using threshold values to a bivariate latent variable, which is then modelled as a linear mixed model. Random effects terms are used to tie all together repeated observations from the same subject. The cross-sectional association between the two outcomes is modelled through the correlation coefficient of the bivariate latent variable, conditional on random effects. Assuming conditional independence given random effects, the marginal likelihood, under the missing data at random assumption, is approximated using an adaptive Gaussian quadrature for numerical integration. The model provides fixed effects parameters that are subject-specific, but retain the population-averaged interpretation when properly scaled. This is particularly well suited for the situation in which population comparisons and individual level contrasts are of equal importance. Data from a psychiatric trial, the Fluvoxamine (an antidepressant drug) study, are used to illustrate the methodology. |
| Author | Kim, KyungMann Todem, David Lesaffre, Emmanuel |
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| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/16832841$$D View this record in MEDLINE/PubMed |
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| Cites_doi | 10.1080/01621459.1995.10476493 10.1007/978-1-4419-0318-1 10.1002/sim.4780100907 10.1093/oso/9780198524847.001.0001 10.1080/01621459.1965.10480807 10.2307/2529107 10.1002/1521-4036(200011)42:7<807::AID-BIMJ807>3.0.CO;2-3 10.2307/2530704 10.2307/2531734 10.1007/BF02293959 10.1097/00004850-199112003-00001 10.2307/2531158 10.1002/sim.4780131106 10.1080/01621459.1997.10473598 10.1080/01621459.1994.10476788 10.1093/biomet/63.3.581 10.1080/01621459.1994.10476474 10.1016/0169-2607(96)01720-8 10.1214/aoms/1177728725 10.1093/biomet/71.3.531 10.1080/01621459.1992.10475282 10.1080/01621459.1992.10475264 10.1080/01621459.1995.10476583 10.1093/biomet/75.1.57 10.1111/j.0006-341X.2003.00093.x 10.1111/j.0006-341X.1999.00085.x 10.1111/j.2517-6161.1995.tb02046.x 10.1002/(SICI)1097-0258(19960615)15:11<1123::AID-SIM228>3.0.CO;2-L 10.1002/sim.4780141207 10.2307/2685902 10.1002/(SICI)1097-0258(19960730)15:14<1507::AID-SIM316>3.0.CO;2-Z 10.2307/2533433 10.1093/biomet/84.1.33 10.2307/2286403 10.1093/biomet/73.1.13 |
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Molenberghs G, Kenward M, Lesaffre E. The analysis of longitudinal ordinal data with non-random dropout. Biometrika 1997; 84:33-44. Dale JR. Global cross-ratio models for bivariate, discrete, ordered responses. Biometrics 1986; 42:907-917. Ekholm A, Jokinen J, McDonald JW, Smith PWF. Joint regression and association modelling of longitudinal ordinal data. Biometrics 2003; 59:795-803. Molenberghs G, Lesaffre E. Marginal modeling of correlated ordinal data using a multivariate plackett distribution. Journal of the American Statistical Association 1994; 89:633-644. Lesaffre E, Molenberghs G, Dewulf L. Effects of dropouts in a longitudinal study: an application of a repeated ordinal model. Statistics in Medicine 1996; 15:1123-1141. Rubin DB. Inference and missing data. Biometrika 1976; 63:581-590. Chernoff H. On the distribution of the likelihood ratio. Annals of the Mathematical Statistics 1954; 25:573-578. Gibbons RD, Bock RD. Trend in correlated proportions. 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| References_xml | – reference: Lipsitz SR, Kim K, Zhao L. Analysis of repeated categorical data using generalized estimating equations. Statistics in Medicine 1994; 13:1149-1163. – reference: Hedeker D, Gibbons RD. A random-effects ordinal regression model for multilevel analysis. Biometrics 1994; 50:933-944. – reference: Ten Have T, Morabia A. Mixed effects models with bivariate and univariate association parameters for longitudinal bivariate binary response data. Biometrics 1999; 55:85-93. – reference: Robins J, Rotnitzky A, Zhao L-P. Analysis of semi-parametric regression models for repeated outcomes in the presence of missing data. Journal of the American Statistical Association 1985; 90:106-121. – reference: Dale JR. Global cross-ratio models for bivariate, discrete, ordered responses. Biometrics 1986; 42:907-917. – reference: Liang KY, Zeger SL. Longitudinal data analysis using generalized linear models. Biometrika 1986; 73:13-22. – reference: Muthén B. A structural probit model with latent variables. Journal of the American Statistical Association 1979; 74:807-811. – reference: Morrell CH, Pearson JD, Brant LJ. Linear transformations of linear mixed effects models. The American Statistician 1997; 51:338-343. – reference: Williamson JM, Kim K. A global odds ratio regression model for bivariate ordered categorical data from opthalmologic studies. Statistics in Medicine 1996; 15:1507-1518. – reference: O'Brien PC. Procedures for comparing multiples endpoints. Biometrics 1984; 40:1079-1087. – reference: McCullogh CE. Maximum likelihood variance components estimation for binary data. Journal of the American Statistical Association 1994; 89:330-335. – reference: Lesaffre E, Todem D, Verbeke G. Flexible modelling of the covariance matrix in a linear random effects model. Biometrical Journal 2000; 42:807-822. – reference: Ashford JR, Sowden RR. Multivariate probit analysis. Biometrics 1970; 26:535-546. – reference: Legler JM, Ryan LM. Latent variable model for teratogenesis using multiple binary outcomes. Journal of the American Statistical Association 1997; 92:13-20. – reference: Kim K. A bivariate cumulative probit regression model for ordered categorical data. Statistics in Medicine 1995; 14:1341-1352. – reference: Zeger SL, Liang K-Y, Albert PA. Models for longitudinal data: a generalized estimating equation approach. Biometrics 1988; 44:1049-1060. – reference: Ochi Y, Prentice RL. Likelihood inference in a correlated probit regression model. Biometrika 1984; 71:531-543. – reference: Hedeker D, Gibbons RD. A computer program for mixed-effects ordinal probit and logistic regression analysis. Computer Methods and Programs in Biomedicine 1996; 49:157-176. – reference: Ekholm A, Jokinen J, McDonald JW, Smith PWF. Joint regression and association modelling of longitudinal ordinal data. Biometrics 2003; 59:795-803. – reference: Diggle PJ, Heagerty PJ, Liang K-Y, Zeger SL. The Analysis of Longitudinal Data (2nd edn). 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| SubjectTerms | adaptive Gaussian quadrature Antidepressants Antidepressive Agents, Second-Generation - therapeutic use bivariate ordinal outcome Fluvoxamine - therapeutic use Humans latent variable Longitudinal Studies longitudinal/ clustered outcomes Medical statistics Models, Statistical Normal Distribution Outcome Assessment (Health Care) - statistics & numerical data Psychiatry random effects Random variables Safety |
| Title | Latent-variable models for longitudinal data with bivariate ordinal outcomes |
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