Abstract
Graphical models for clustered data mixed with discrete and continuous responses are developed. Discrete responses are assumed to be regulated by some latent continuous variables and particular link functions are used to describe the regulatory mechanisms. Inferential procedures are constructed using the full-information maximum likelihood estimation and observed/empirical Fisher information matrices. Implementation is carried out by stochastic versions of the generalized EM algorithm. As an illustrative application, clustered data from a developmental toxicity study is re-investigated using the directed graphical model and the proposed algorithms. A new interesting directed association between two mixed outcomes reveals. The proposed methods also apply to cross-sectional data with discrete and continuous responses.
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References
Amemiya T (1985) Advanced econometrics. Harvard University Press, Cambridge
Berger J (2000) Bayesian analysis: A look at today and thoughts of tomorrow. J Am Stat Assoc 95:1269–1276
Bollen K (1989) Structural equations with latent variables. Wiley, New York
Broniatowski M, Celeux G, Diebolt J (1983) Reconnaissance de mélanges de densités par un algorithme d’apprentissage probabiliste. In: Diday E (ed) Data analysis and informatics, vol 3. Amsterdam, North Holland, pp 359–374
Cappé O, Robert C (2000) Markov chain monte carlo: 10 years and still running! J Am Stat Assoc 95:1282–1286
Catalano P, Ryan L (1992) Bivariate latent variable models for clustered discrete and continuous outcomes. J Am Stat Assoc 87:651–658
Celeux G, Diebolt J (1983) A probabilistic teacher algorithm for iterative maximum likelihood estimation. In: Bock H (ed) Classification and related methods of data analysis. Amsterdam, North Holland, pp 617–623
Celeux G, Diebolt J (1985) The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem. Comput Stat Q 2:73–82
Chan J, Kuk A (1997) Maximum likelihood estimation for probit-linear mixed models with correlated random effects. Biometrics 53:86–97
Daniels M, Pourahmadi M (2002) Bayesian analysis of covariance matrices and dynamic models for longitudinal data. Biometrika 89:553–566
Delyon B, Lavielle M, Moulines E (1999) Convergence of a stochastic approximation version of the EM algorithm. Ann Stat 27:94–128
Dempster A, Laird N, Rubin D (1977) Maximum likelihood estimation form incomplete data via the em algorithm (with discussion). J R Stat Soc Series B 39:1–38
Dunson D (2000) Bayesian latent variable models for clustered mixed outcomes. J R Stat Soc Series B 62:355–366
Dunson D, Chen Z, Harry J (2003) A bayesian approach for joint modeling of cluster size and subunit-specific outcomes. Biometrics 59:521–530
Edwards D (1990) Hierarchical interaction models (with discussion). J R Stat Soc Series B 52:3–20
Edwards D, Lauritzen S (2001) The TM algorithm for maximising a conditional likelihood function. Biometrika 88:961–972
Gueorguieva R, Agresti A (2001) A correlated probit model for joint modelling of clustered binary and continuous responses. J Am Stat Assoc 96:1102–1112
Gueorguieva R, Sanacora G (2006) Joint analysis of repeatedly observed continuous and ordinal measures of disease severity. Stat Med 25:1307–1322
Haavelmo T (1943) The statistical implications of a system of simultaneous equations. Econometrica 11:1–12
Heckman J, MaCurdy T (1980) A life cycle model of female labour supply. Rev Econ Stud 47:47–74
Koster J (1996) Markov properties of non-recursive causal models. Ann Stat 24:2148–2177
Lauritzen S, Wermuth N (1989) Graphical models for associations between variables, some of which are qualitative and some quantitative. Ann Stat 17:31–57
Lin X, Ryan L, Sammel M, Zhang D, Padungtod C, Xu X (2000) A scaled linear mixed model for multiple outcomes. Biometrics 56:593–601
Louis T (1982) Finding the observed information matrix when using the EM algorithm. J R Stat Soc Series B 44:226–233
McLachlan GJ, Krishnan T (1997) The EM algorithm and extensions. Wiley, New York
Miglioretti D (2003) Latent transition regression for mixed outcomes. Biometrics 59:710–720
Price C, Kimmel C, Tyl R, Marr M (1985) The developmental toxicity of ethylene glycol in rats and mice. Toxicol Appl Pharmacol 81:113–127
Rochon J (1996) Analyzing bivariate repeated measures for discrete and continuous outcome variables. Biometrics 52:740–750
Roy J, Lin X, Ryan L (2003) Scaled marginal models for multiple continuous outcomes. Biostatistics 4:371–383
Sammel M, Lin X, Ryan L (1999) Multivariate linear mixed models for multiple outcomes. Stat Med 18:2479–2492
Wei G, Tanner M (1999) A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithm. J Am Stat Assoc 85:699–704
Zellner A (1962) An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. J Am Stat Assoc 57:348–368
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Wells, M.T., Zhang, D. (2011). Graphical Models for Clustered Binary and Continuous Responses. In: Wells, M., SenGupta, A. (eds) Advances in Directional and Linear Statistics. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2628-9_19
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DOI: https://doi.org/10.1007/978-3-7908-2628-9_19
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