, Volume 69, Issue 3, pp 309–322 | Cite as

Sensitivity analysis for nonignorable missing responses with application to multivariate Random effect model

  • Ehsan Bahrami Samani
  • Mojtaba Ganjali


A joint model with random effects for longitudinal mixed ordinal and continuous responses, with potentially non-random missing values in both types of responses is proposed. The presented model simultaneously considers a multivariate probit regression model for the missing mechanisms, which provides the ability of examining the missing data assumptions, and a multivariate mixed model for the responses. Random effects are used to take into account the correlation between longitudinal responses of the same individual. A full likelihood-based approach that allows yielding maximum likelihood estimates of the model parameters is used. The joint modeling of responses with the possibility of missing values requires caution since the interpretation of the fitted model highly depends on the assumptions that are unexaminable in a fundamental sense. A sensitivity of the results to the assumptions is also investigated. To illustrate the application of such modeling the longitudinal data of PIAT (Peabody Individual Achievement Test) is analyzed.


Longitudinal studies Missing responses Mixed ordinal and continuous responses Random effect 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Agresti, A. (2002) Categorical data and analysis, Wiely.Google Scholar
  2. Anderson, T. W. and Goodman, L. A. (1957) Statistical inference about Markov chains, Ann. Math. Stat, 28, 89–110.MathSciNetMATHCrossRefGoogle Scholar
  3. Bahrami Samani, E. and Ganjali, M. (2008) A multivariate latent variable model for mixed continuous and ordinal responses, World Applied Sciences Journal, 3(2), 294–299.Google Scholar
  4. Bahrami Samani, E., Ganjali, M. and Eftekhari Mahabadi, S. (2010) A latent variable model for mixed continuous and ordinal responses with nonignorable missing responses, Sankhya, 72-B, 38–57.Google Scholar
  5. Baker, P. C., Keck, C. K., Mott, F. L. and Quinlan, S. V. (1993) NLSY child handbook: guide to the 1986–1990 national longitudinal survey of youth child data, Columbus, OH: Center for Human Resource Research.Google Scholar
  6. Berridge, D. M. and Dos Santos, D. M. (1996) Fitting a random effects model to ordinal recurrent events using existing software, J. Statist. Comput. Simul., 55, 73–86.MATHCrossRefGoogle Scholar
  7. Catalano, P. and Ryan, L. M. (1992) Bivariate latent variable models for clustered discrete and continuous outcoms, Journal of the American Statistical Association, 50(3), 1078–1095.Google Scholar
  8. Cook, R. D. (1986) Assessment of Local Influence (with discussion), Journal. Royal Statist. Soc., Ser.B.,48, 133–169.MATHGoogle Scholar
  9. Cox, D. R and Wermuth, N. (1992) Response models for mixed binary and quantitative variables, Biometrika, 79(3), 441–461.MathSciNetMATHCrossRefGoogle Scholar
  10. Diggle, P. J., Heagerty, P., Liang, K. Y. and Zeger, S. L. (2002) Analysis of longitudinal data, Oxford: University Press.Google Scholar
  11. Diggle, P. J. and Kenward, M. G. (1994) Informative drop-out in longitudinal data analysis, Journal of Applied Statistics, 43, 49–93.MATHCrossRefGoogle Scholar
  12. Fitzmaurice, G. M. and Laird, N. M. (1995) Regression models for bivariate discrete and continuous outcome with clustering, Journal of the American Statistical Association, 90, 845–852.MathSciNetMATHCrossRefGoogle Scholar
  13. Fitzmaurice, G. M. and Laird, N. M. (1997) Regression models for mix discrete and continuous responses with potentially missing values, Biometrics, 53, 110–122.MATHCrossRefGoogle Scholar
  14. Ganjali, M. (2003) A model for mixed continuous and discrete responses with possibilty of missing responses, Journal of Sciences, Islamic Republic of Iran, 14(1), 53–60.MathSciNetGoogle Scholar
  15. Ganjali, M. and Shafie, K. (2006) A transition model for an ordered cluster of mixed continuous and discrete responses with non-monotone missingness, Journal of Applied Statistical sciences; Volume 15 Issue 3.Google Scholar
  16. Harvile, D. A. and Mee, R. W. (1984) A mixed model procedure for analyzing ordered categorical data, Biometrics, 40, 393–408.MathSciNetCrossRefGoogle Scholar
  17. Heckman, J. J. D. (1978) Endogenous variable in a simutaneous Equation system, Econometrica, 46(6), 931–959.MathSciNetMATHCrossRefGoogle Scholar
  18. Liang, K. Y., Zeger, S. L. and Qaqish, B. F. (1992) Multivariate regression analyzes for categorical data (with discussion), J. Roy. Statist. Soc. B, 54, 3–40.MathSciNetMATHGoogle Scholar
  19. Little, R. J. and Schluchter M. (1987) Maximum likelihood estimation for mixed continuous and categorical data with missing values, Biometrika, 72, 497–512.MathSciNetCrossRefGoogle Scholar
  20. Little, R. J. and Rubin, D. (2002) Statistical analysis with missing data, Second edition, New york, Wiley.MATHGoogle Scholar
  21. Kaciroti, N. A., Raghunathan, T. E., Schork, M. A., Clark, N. M. and Gong, M. (2006) A Bayesian Approach for Clustered Longitudinal Ordinal Outcome With Nonignorable Missing Data: Evaluation of an Asthma Education Program, Journal of the American Statistical Association, 474, 435–446.MathSciNetCrossRefGoogle Scholar
  22. McCullagh, P. (1980) Regression models for ordinal data (with discussion), J. Roy. Statist. Soc. B, 42, 109–142.MathSciNetMATHGoogle Scholar
  23. Mcculloch, C. (2007) Joint modelling of mixed outcome type using latend variables, statistical methods in Medical Research, 1–27.Google Scholar
  24. Rubin, D. B. (1976) Inference and missing data, Biometrica, 82, 669–710.Google Scholar
  25. Olkin L. and Tate R. F. (1961) Multivariate correlation models with mixed discrete and continuous variables, Annals of Mathematical Statistics, 32, 448–456.MathSciNetMATHCrossRefGoogle Scholar
  26. Sengul, T. K., Stoffer, D. S. and Day N. L. (2007) A residuals-based transition model for longitudinal analysis with estimation in the presence of missing data, Stat. Med., 26, 3330–3341.MathSciNetCrossRefGoogle Scholar
  27. Stiratelli, R., Laird, N. and Ware, J. H. (1984) Random-effect models for serial observations with binary response, Biometrics, 40, 961–971.CrossRefGoogle Scholar
  28. Ten Have, T. R., Kunselman, A., Pulkstenis, E. and Landis, J. R. (1998a) Mixed effects logistic regression models for longitudinal binary response data with informative drop-out, Bimetrics, 54, 367–383.MATHGoogle Scholar
  29. Tutz, G. (2005) Modeling of repeated ordered measurements by isotonic sequential regression, Statist. Mod, 5, 269–287.MathSciNetCrossRefGoogle Scholar
  30. Verbeke, G. and Molenberghs, G. (1997) Linear mixed models in practice: A SAS Oriented Approach, Springer.Google Scholar
  31. Yang, Y., Kang, J., Mao, K. and Zhang, J. (2007) Regression models for mixed Poisson and continuous longitudinal data, Statistics in Medicine, 26, 3782–3800.MathSciNetCrossRefGoogle Scholar

Copyright information

© Sapienza Università di Roma 2011

Authors and Affiliations

  1. 1.Department of Statistics Faculty of Mathematical ScienceShahid Beheshti UniversityTehranIran
  2. 2.Department of Statistics Faculty of Mathematical ScienceShahid Beheshti UniversityTehranIran

Personalised recommendations