Graphical Models for Clustered Binary and Continuous Responses

  • Martin T. Wells
  • Dabao Zhang


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.


Conditional Expectation Stochastic Version Latent Continuous Variable SAEM Algorithm Directed Graphical Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Amemiya T (1985) Advanced econometrics. Harvard University Press, CambridgeGoogle Scholar
  2. 2.
    Berger J (2000) Bayesian analysis: A look at today and thoughts of tomorrow. J Am Stat Assoc 95:1269–1276MATHCrossRefGoogle Scholar
  3. 3.
    Bollen K (1989) Structural equations with latent variables. Wiley, New YorkMATHGoogle Scholar
  4. 4.
    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 359–374Google Scholar
  5. 5.
    Cappé O, Robert C (2000) Markov chain monte carlo: 10 years and still running! J Am Stat Assoc 95:1282–1286MATHCrossRefGoogle Scholar
  6. 6.
    Catalano P, Ryan L (1992) Bivariate latent variable models for clustered discrete and continuous outcomes. J Am Stat Assoc 87:651–658CrossRefGoogle Scholar
  7. 7.
    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 617–623Google Scholar
  8. 8.
    Celeux G, Diebolt J (1985) The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem. Comput Stat Q 2:73–82Google Scholar
  9. 9.
    Chan J, Kuk A (1997) Maximum likelihood estimation for probit-linear mixed models with correlated random effects. Biometrics 53:86–97MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Daniels M, Pourahmadi M (2002) Bayesian analysis of covariance matrices and dynamic models for longitudinal data. Biometrika 89:553–566MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Delyon B, Lavielle M, Moulines E (1999) Convergence of a stochastic approximation version of the EM algorithm. Ann Stat 27:94–128MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    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 39:1–38MATHMathSciNetGoogle Scholar
  13. 13.
    Dunson D (2000) Bayesian latent variable models for clustered mixed outcomes. J R Stat Soc Series B 62:355–366CrossRefMathSciNetGoogle Scholar
  14. 14.
    Dunson D, Chen Z, Harry J (2003) A bayesian approach for joint modeling of cluster size and subunit-specific outcomes. Biometrics 59:521–530CrossRefMathSciNetGoogle Scholar
  15. 15.
    Edwards D (1990) Hierarchical interaction models (with discussion). J R Stat Soc Series B 52:3–20MATHGoogle Scholar
  16. 16.
    Edwards D, Lauritzen S (2001) The TM algorithm for maximising a conditional likelihood function. Biometrika 88:961–972MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gueorguieva R, Agresti A (2001) A correlated probit model for joint modelling of clustered binary and continuous responses. J Am Stat Assoc 96:1102–1112MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Gueorguieva R, Sanacora G (2006) Joint analysis of repeatedly observed continuous and ordinal measures of disease severity. Stat Med 25:1307–1322CrossRefMathSciNetGoogle Scholar
  19. 19.
    Haavelmo T (1943) The statistical implications of a system of simultaneous equations. Econometrica 11:1–12MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Heckman J, MaCurdy T (1980) A life cycle model of female labour supply. Rev Econ Stud 47:47–74CrossRefGoogle Scholar
  21. 21.
    Koster J (1996) Markov properties of non-recursive causal models. Ann Stat 24:2148–2177MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Lauritzen S, Wermuth N (1989) Graphical models for associations between variables, some of which are qualitative and some quantitative. Ann Stat 17:31–57MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Lin X, Ryan L, Sammel M, Zhang D, Padungtod C, Xu X (2000) A scaled linear mixed model for multiple outcomes. Biometrics 56:593–601MATHCrossRefGoogle Scholar
  24. 24.
    Louis T (1982) Finding the observed information matrix when using the EM algorithm. J R Stat Soc Series B 44:226–233MATHMathSciNetGoogle Scholar
  25. 25.
    McLachlan GJ, Krishnan T (1997) The EM algorithm and extensions. Wiley, New YorkMATHGoogle Scholar
  26. 26.
    Miglioretti D (2003) Latent transition regression for mixed outcomes. Biometrics 59:710–720CrossRefMathSciNetGoogle Scholar
  27. 27.
    Price C, Kimmel C, Tyl R, Marr M (1985) The developmental toxicity of ethylene glycol in rats and mice. Toxicol Appl Pharmacol 81:113–127CrossRefGoogle Scholar
  28. 28.
    Rochon J (1996) Analyzing bivariate repeated measures for discrete and continuous outcome variables. Biometrics 52:740–750MATHCrossRefGoogle Scholar
  29. 29.
    Roy J, Lin X, Ryan L (2003) Scaled marginal models for multiple continuous outcomes. Biostatistics 4:371–383MATHCrossRefGoogle Scholar
  30. 30.
    Sammel M, Lin X, Ryan L (1999) Multivariate linear mixed models for multiple outcomes. Stat Med 18:2479–2492CrossRefGoogle Scholar
  31. 31.
    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 85:699–704CrossRefGoogle Scholar
  32. 32.
    Zellner A (1962) An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. J Am Stat Assoc 57:348–368MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of Statistical ScienceCornell UniversityIthacaUSA

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