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Random Effects Models for Longitudinal Data

  • Geert VerbekeEmail author
  • Geert Molenberghs
  • Dimitris Rizopoulos
Chapter

Abstract

Mixed models have become very popular for the analysis of longitudinal data, partly because they are flexible and widely applicable, partly also because many commercially available software packages offer procedures to fit them. They assume that measurements from a single subject share a set of latent, unobserved, random effects which are used to generate an association structure between the repeated measurements. In this chapter, we give an overview of frequently used mixed models for continuous as well as discrete longitudinal data, with emphasis on model formulation and parameter interpretation. The fact that the latent structures generate associations implies that mixed models are also extremely convenient for the joint analysis of longitudinal data with other outcomes such as dropout time or some time-to-event outcome, or for the analysis of multiple longitudinally measured outcomes. All models will be extensively illustrated with the analysis of real data.

Keywords

Linear Mixed Model Random Effect Model Generalize Linear Mixed Model Generalize Estimate Equation American Statistical Association 
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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References

  1. Aerts, M., Geys, H., Molenberghs, G., & Ryan, L. (2002). Topics in modelling of clustered data. London: Chapman & Hall.zbMATHCrossRefGoogle Scholar
  2. Afifi, A., & Elashoff, R. (1966). Missing observations in multivariate statistics I: Review of the literature. Journal of the American Statistical Association, 61, 595-604.MathSciNetCrossRefGoogle Scholar
  3. Alonso, A., Geys, H., Molenberghs, G., & Vangeneugden, T. (2003). Validation of surrogate markers in multiple randomized clinical trials with repeated measurements. Biometrical Journal, 45, 931-945.MathSciNetCrossRefGoogle Scholar
  4. Alonso, A., & Molenberghs, G. (2007). Surrogate marker evaluation from an information theory perspective. Biometrics, 63, 180-186.zbMATHMathSciNetCrossRefGoogle Scholar
  5. Alonso, A., Molenberghs, G., Geys, H., & Buyse, M. (2005). A unifying approach for surrogate marker validation based on Prentice’s criteria. Statistics in Medicine, 25, 205-211.MathSciNetCrossRefGoogle Scholar
  6. Alonso, A., Molenberghs, G., Burzykowski, T., Renard, D., Geys, H., Shkedy, Z., Tibaldi, F., Abrahantes, J., & Buyse, M. (2004). Prentice’s approach and the meta analytic paradigm: a reflection on the role of statistics in the evaluation of surrogate endpoints. Biometrics, 60, 724-728.zbMATHMathSciNetCrossRefGoogle Scholar
  7. Altham, P. M. E. (1978). Two generalizations of the binomial distribution. Applied Statistics, 27, 162-167.zbMATHMathSciNetCrossRefGoogle Scholar
  8. Andersen, P., Borgan, O., Gill, R., & Keiding, N. (1993). Statistical models based on counting processes. New York: Springer.zbMATHGoogle Scholar
  9. Arnold, B. C., & Strauss, D. (1991). Pseudolikelihood estimation: some examples. Sankhya: The Indian Journal of Statistics - Series B, 53, 233-243.zbMATHMathSciNetGoogle Scholar
  10. Bahadur, R. R. (1961). A representation of the joint distribution of responses to n dichotomous items. In H. Solomon (Ed.), Studies in item analysis and prediction, Stanford Mathematical Studies in the Social Sciences VI. Stanford, CA: Stanford University Press.Google Scholar
  11. Beunckens, C., Sotto, C., & Molenberghs, G. (2007). A simulation study comparing weighted estimating equations with multiple imputation based estimating equations for longitudinal binary data. Computational Statistics and Data Analysis, 52, 1533-1548.MathSciNetCrossRefGoogle Scholar
  12. Brant, L. J., & Fozard, J. L. (1990). Age changes in pure-tone hearing thresholds in a longitudinal study of normal human aging. Journal of the Acoustical Society of America, 88, 813-820.CrossRefGoogle Scholar
  13. Breslow, N. E., & Clayton, D. G. (1993). Approximate inference in generalized linear mixed models. Journal of the American Statistical Association, 88, 9-25.zbMATHCrossRefGoogle Scholar
  14. Brown, E., & Ibrahim, J. (2003). A Bayesian semiparametric joint hierarchical model for longitudinal and survival data. Biometrics, 59, 221-228.MathSciNetCrossRefGoogle Scholar
  15. Brown, E., Ibrahim, J., & DeGruttola, V. (2005). A flexible B-spline model for multiple longitudinal biomarkers and survival. Biometrics, 61, 64-73.zbMATHMathSciNetCrossRefGoogle Scholar
  16. Burzykowski, T., Molenberghs, G., & Buyse, M. (2004). The validation of surrogate endpoints using data from randomized clinical trials: a case-study in advanced colorectal cancer. Journal of the Royal Statistical Society, Series A, 167, 103-124.MathSciNetGoogle Scholar
  17. Burzykowski, T., Molenberghs, G., & Buyse, M. (2005). The evaluation of surrogate endpoints. New York: Springer.zbMATHCrossRefGoogle Scholar
  18. Burzykowski, T., Molenberghs, G., Buyse, M., Geys, H., & Renard, D. (2001). Validation of surrogate endpoints in multiple randomized clinical trials with failure time end points. Applied Statistics, 50, 405-422.zbMATHMathSciNetGoogle Scholar
  19. Buyse, M., & Molenberghs, G. (1998). The validation of surrogate endpoints in randomized experiments. Biometrics, 54, 1014-1029.zbMATHCrossRefGoogle Scholar
  20. Buyse, M., Molenberghs, G., Burzykowski, T., Renard, D., & Geys, H. (2000). The validation of surrogate endpoints in meta-analyses of randomized experiments. Biostatistics, 1, 49-67.zbMATHCrossRefGoogle Scholar
  21. Cardiac Arrhythmia Suppression Trial (CAST) Investigators (1989). Preliminary report: effect of encainide and flecainide on mortality in a randomized trial of arrhythmia suppression after myocardial infraction. New England Journal of Medicine, 321, 406-412.CrossRefGoogle Scholar
  22. Catalano, P. J. (1997). Bivariate modelling of clustered continuous and ordered categorical outcomes. Statistics in Medicine, 16, 883-900.CrossRefGoogle Scholar
  23. Catalano, P. J., & Ryan, L. M. (1992). Bivariate latent variable models for clustered discrete and continuous outcomes. Journal of the American Statistical Association, 87, 651-658.CrossRefGoogle Scholar
  24. Chakraborty, H., Helms, R. W., Sen, P. K., & Cohen, M. S. (2003). Estimating correlation by using a general linear mixed model: Evaluation of the relationship between the concentration of HIV-1 RNA in blood and semen. Statistics in Medicine, 22, 1457-1464.CrossRefGoogle Scholar
  25. Chi, Y.-Y., & Ibrahim, J. (2006). Joint models for multivariate longitudinal and multivariate survival data. Biometrics, 62, 432-445.zbMATHMathSciNetCrossRefGoogle Scholar
  26. Clayton, D. G. (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65, 141-151.zbMATHMathSciNetCrossRefGoogle Scholar
  27. Cover, T., & Tomas, J. (1991). Elements of information theory. New York: Wiley.zbMATHCrossRefGoogle Scholar
  28. Cox, N. R. (1974). Estimation of the correlation between a continuous and a discrete variable. Biometrics, 30, 171-178.zbMATHMathSciNetCrossRefGoogle Scholar
  29. Cox, D. R., & Wermuth, N. (1992). Response models for mixed binary and quantitative variables. Biometrika, 79, 441-461.zbMATHMathSciNetCrossRefGoogle Scholar
  30. Cox, D. R., & Wermuth, N. (1994a). A note on the quadratic exponential binary distribution. Biometrika, 81, 403-408.zbMATHMathSciNetCrossRefGoogle Scholar
  31. Cox, D. R., & Wermuth, N. (1994b). Multivariate dependencies: Models, analysis and interpretation. London: Chapman & Hall.Google Scholar
  32. Dale, J. R. (1986). Global cross ratio models for bivariate, discrete, ordered responses. Biometrics, 42, 909-917.CrossRefGoogle Scholar
  33. Daniels, M. J., & Hughes, M. D. (1997). Meta-analysis for the evaluation of potential surrogate markers. Statistics in Medicine, 16, 1515-1527.CrossRefGoogle Scholar
  34. De Backer, M., De Keyser, P., De Vroey, C., & Lesaffre, E. (1996). A 12-week treatment for dermatophyte toe onychomycosis: terbinafine 250mg/day vs. itraconazole 200mg/day–a double-blind comparative trial. British Journal of Dermatology, 134, 16-17.Google Scholar
  35. DeGruttola, V., & Tu, X. (1994). Modeling progression of CD-4 lymphocyte count and its relationship to survival time. Biometrics, 50, 1003-1014.CrossRefGoogle Scholar
  36. Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society, Series B, 39, 1-38.zbMATHMathSciNetGoogle Scholar
  37. Diggle, P. J., Heagerty, P., Liang, K.-Y., & Zeger, S. L. (2002). Analysis of longitudinal data. New York: Oxford University Press.Google Scholar
  38. Ding, J., & Wang, J.-L. (2008). Modeling longitudinal data with nonparametric multiplicative random effects jointly with survival data. Biometrics, 64, 546-556.zbMATHMathSciNetCrossRefGoogle Scholar
  39. Dobson, A., & Henderson, R. (2003). Diagnostics for joint longitudinal and dropout time modeling. Biometrics, 59, 741-751.zbMATHMathSciNetCrossRefGoogle Scholar
  40. Efron, B. (1986). Double exponential families and their use in generalized linear regression. Journal of the American Statistical Association, 81, 709-721.zbMATHMathSciNetCrossRefGoogle Scholar
  41. Elashoff, R., Li, G., & Li, N. (2008). A joint model for longitudinal measurements and survival data in the presence of multiple failure types. Biometrics, 64, 762-771.zbMATHCrossRefMathSciNetGoogle Scholar
  42. Fahrmeir, L., & Tutz, G. (2001). Multivariate statistical modelling based on generalized linear models. Heidelberg: Springer.zbMATHGoogle Scholar
  43. Faucett, C., Schenker, N., & Elashoff, R. (1998). Analysis of censored survival data with intermittently observed time-dependent binary covariates. Journal of the American Statistical Association, 93, 427-437.zbMATHCrossRefGoogle Scholar
  44. Ferentz, A. E. (2002). Integrating pharmacogenomics into drug development. Pharmacogenomics, 3, 453-467.CrossRefGoogle Scholar
  45. Fieuws, S., & Verbeke, G. (2004). Joint modelling of multivariate longitudinal profiles: Pitfalls of the random-effects approach. Statistics in Medicine, 23, 3093-3104.CrossRefGoogle Scholar
  46. Fieuws, S., & Verbeke, G. (2006). Pairwise fitting of mixed models for the joint modelling of multivariate longitudinal profiles. Biometrics, 62, 424-431.zbMATHMathSciNetCrossRefGoogle Scholar
  47. Fieuws, S., Verbeke, G., Boen, F., & Delecluse, C. (2006). High-dimensional multivariate mixed models for binary questionnaire data. Applied Statistics, 55, 1-12.MathSciNetGoogle Scholar
  48. Fitzmaurice, G. M., & Laird, N. M. (1995). Regression models for a bivariate discrete and continuous outcome with clustering. Journal of the American Statistical Association, 90, 845-852.zbMATHMathSciNetCrossRefGoogle Scholar
  49. Fitzmaurice, G. M., Laird, N. M., & Ware, J. H. (2004). Applied longitudinal analysis. New York: John Wiley & Sons.zbMATHGoogle Scholar
  50. Fleming, T. R., & DeMets, D. L. (1996). Surrogate endpoints in clinical trials: are we being misled? Annals of Internal Medicine, 125, 605-613.Google Scholar
  51. Folk, V. G., & Green, B. F. (1989). Adaptive estimation when the unidimensionality assumption of IRT is violated. Applied Psychological Measurement, 13, 373-389.CrossRefGoogle Scholar
  52. Follmann, D., & Wu, M. (1995). An approximate generalized linear model with random effects for informative missing data. Biometrics, 51, 151-168.zbMATHMathSciNetCrossRefGoogle Scholar
  53. Forster, J. J., & Smith, P. W. (1998). Model-based inference for categorical survey data subject to non-ignorable non-response. Journal of the Royal Statistical Society, Series B, 60, 57-70.zbMATHMathSciNetCrossRefGoogle Scholar
  54. Freedman, L. S., Graubard, B. I., & Schatzkin, A. (1992). Statistical validation of intermediate endpoints for chronic diseases. Statistics in Medicine, 11, 167-178.CrossRefGoogle Scholar
  55. Gail, M. H., Pfeiffer, R., van Houwelingen, H. C., & Carroll, R. J. (2000). On meta-analytic assessment of surrogate outcomes. Biostatistics, 1, 231-246.zbMATHCrossRefGoogle Scholar
  56. Galecki, A. (1994). General class of covariance structures for two or more repeated factors in longitudinal data analysis. Communications in Statistics: Theory and Methods, 23, 3105-3119.zbMATHCrossRefGoogle Scholar
  57. Genest, C., & McKay, J. (1986). The joy of copulas: bivariate distributions with uniform marginals. American Statistician, 40, 280-283.MathSciNetCrossRefGoogle Scholar
  58. Geys, H., Molenberghs, G., & Ryan, L. M. (1997). Pseudo-likelihood inference for clustered binary data. Communications in Statistics: Theory and Methods, 26, 2743-2767.zbMATHMathSciNetCrossRefGoogle Scholar
  59. Geys, H., Molenberghs, G., & Ryan, L. (1999). Pseudolikelihood modeling of multivariate outcomes in developmental toxicology. Journal of the American Statistical Association, 94, 734-745.CrossRefGoogle Scholar
  60. Gueorguieva, R. (2001). A multivariate generalized linear mixed model for joint modelling of clustered outcomes in the exponential family. Statistical Modelling, 1, 177-193.zbMATHCrossRefGoogle Scholar
  61. Goldstein, H. (1979). The design and analysis of longitudinal studies. London: Academic Press.zbMATHGoogle Scholar
  62. Hartley, H. O., & Hocking, R. (1971). The analysis of incomplete data. Biometrics, 27, 7783-7808.CrossRefGoogle Scholar
  63. Harville, D. A. (1974). Bayesian inference for variance components using only error contrasts. Biometrika, 61, 383-385.zbMATHMathSciNetCrossRefGoogle Scholar
  64. Harville, D. A. (1976). Extension of the Gauss-Markov theorem to include the estimation of random effects. The Annals of Statistics, 4, 384-395.zbMATHMathSciNetCrossRefGoogle Scholar
  65. Harville, D. A. (1977). Maximum likelihood approaches to variance component estimation and to related problems. Journal of the American Statistical Association, 72, 320-340.zbMATHMathSciNetCrossRefGoogle Scholar
  66. Hedeker, D., & Gibbons, R. D. (1994). A random-effects ordinal regression model for multilevel analysis. Biometrics, 50, 933-944.zbMATHCrossRefGoogle Scholar
  67. Hedeker, D., & Gibbons, R. D. (1996). MIXOR: A computer program for mixed-effects ordinal regression analysis. Computer Methods and Programs in Biomedicine, 49, 157-176.CrossRefGoogle Scholar
  68. Henderson, R., Diggle, P., & Dobson, A. (2000). Joint modelling of longitudinal measurements and event time data. Biostatistics, 1, 465-480.zbMATHCrossRefGoogle Scholar
  69. Henderson, C. R., Kempthorne, O., Searle, S. R., & Von Krosig, C. N. (1959). Estimation of environmental and genetic trends from records subject to culling. Biometrics, 15, 192-218.zbMATHCrossRefGoogle Scholar
  70. Hougaard, P. (1986). Survival models for heterogeneous populations derived from stable distributions. Biometrika, 73, 387-396.zbMATHMathSciNetCrossRefGoogle Scholar
  71. Hsieh, F., Tseng, Y.-K., & Wang, J.-L. (2006). Joint modeling of survival and longitudinal data: likelihood approach revisited. Biometrics, 62, 1037-1043.zbMATHMathSciNetCrossRefGoogle Scholar
  72. Jennrich, R. I., & Schluchter, M. D. (1986). Unbalanced repeated measures models with structured covariance matrices. Biometrics, 42, 805-820.zbMATHMathSciNetCrossRefGoogle Scholar
  73. Kenward, M. G., & Molenberghs, G. (1998). Likelihood based frequentist inference when data are missing at random. Statistical Science, 12, 236-247.MathSciNetGoogle Scholar
  74. Krzanowski, W. J. (1988). Principles of multivariate analysis. Oxford: Clarendon Press.zbMATHGoogle Scholar
  75. Lagakos, S. W., & Hoth, D. F. (1992). Surrogate markers in AIDS: Where are we? Where are we going? Annals of Internal Medicine, 116, 599-601.Google Scholar
  76. Laird, N. M., & Ware, J. H. (1982). Random effects models for longitudinal data. Biometrics, 38, 963-974.zbMATHCrossRefGoogle Scholar
  77. Lang, J. B., & Agresti, A. (1994). Simultaneously modeling joint and marginal distributions of multivariate categorical responses. Journal of the American Statistical Association, 89, 625-632.zbMATHCrossRefGoogle Scholar
  78. Lange, K. (2004). Optimization. New York: Springer.zbMATHGoogle Scholar
  79. Lesko, L. J., & Atkinson, A. J. (2001). Use of biomarkers and surrogate endpoints in drug development and regulatory decision making: criteria, validation, strategies. Annual Review of Pharmacological Toxicology, 41, 347-366.CrossRefGoogle Scholar
  80. Liang, K.-Y., & Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73, 13-22.zbMATHMathSciNetCrossRefGoogle Scholar
  81. Liang, K.-Y., Zeger, S.L., & Qaqish, B. (1992). Multivariate regression analyses for categorical data. Journal of the Royal Statistical Society, Series B, 54, 3-40.zbMATHMathSciNetGoogle Scholar
  82. Lin, H., Turnbull, B., McCulloch, C., & Slate, E. (2002). Latent class models for joint analysis of longitudinal biomarker and event process data: Application to longitudinal prostate-specific antigen readings and prostate cancer. Journal of the American Statistical Association, 97, 53-65.zbMATHMathSciNetCrossRefGoogle Scholar
  83. Lipsitz, S. R., Laird, N. M., & Harrington, D. P. (1991). Generalized estimating equations for correlated binary data: using the odds ratio as a measure of association. Biometrika, 78, 153-160.MathSciNetCrossRefGoogle Scholar
  84. Little, R. J. A., & Rubin, D. B. (2002). Statistical analysis with missing data. New York: John Wiley & Sons.zbMATHGoogle Scholar
  85. Little, R. J. A., & Schluchter, M. D. (1985). Maximum likelihood estimation for mixed continuous and categorical data with missing values. Biometrika, 72, 497-512.zbMATHMathSciNetCrossRefGoogle Scholar
  86. Liu, L. C., & Hedeker, D. (2006). A mixed-effects regression model for longitudinal multivariate ordinal data. Biometrics, 62, 261-268.zbMATHMathSciNetCrossRefGoogle Scholar
  87. MacCallum, R., Kim, C., Malarkey, W., & Kiecolt-Glaser, J. (1997). Studying multivariate change using multilevel models and latent curve models. Multivariate Behavioral Research, 32, 215-253.CrossRefGoogle Scholar
  88. Mancl, L. A., & Leroux, B. G. (1996). Efficiency of regression estimates for clustered data. Biometrics, 52, 500-511.zbMATHCrossRefGoogle Scholar
  89. McCullagh, P., & Nelder, J. A. (1989). Generalized linear models. London: Chapman & Hall/CRC.zbMATHGoogle Scholar
  90. Michiels, B., Molenberghs, G., Bijnens, L., Vangeneugden, T., & Thijs, H. (2002). Selection models and pattern-mixture models to analyze longitudinal quality of life data subject to dropout. Statistics in Medicine, 21, 1023-1041.CrossRefGoogle Scholar
  91. Molenberghs, G., Burzykowski, T., Alonso, A., Assam, P., Tilahun, A., & Buyse, M. (2008). The meta-analytic framework for the evaluation of surrogate endpoints in clinical trials. Journal of Statistical Planning and Inference, 138, 432-449.zbMATHMathSciNetCrossRefGoogle Scholar
  92. Molenberghs, G., Burzykowski, T., Alonso, A., Assam, P., Tilahun, A., & Buyse, M. (2009). A unified framework for the evaluation of surrogate endpoints in clinical trials. Statistical Methods in Medical Research, 00, 000-000.Google Scholar
  93. Molenberghs, G., Geys, H., & Buyse, M. (2001). Evaluation of surrogate end-points in randomized experiments with mixed discrete and continuous outcomes. Statistics in Medicine, 20, 3023-3038.CrossRefGoogle Scholar
  94. Molenberghs, G., & Kenward, M. G. (2007). Missing data in clinical studies. Chichester: John Wiley & Sons.CrossRefGoogle Scholar
  95. Molenberghs, G., & Lesaffre, E. (1994). Marginal modelling of correlated ordinal data using a multivariate Plackett distribution. Journal of the American Statistical Association, 89, 633-644.zbMATHCrossRefGoogle Scholar
  96. Molenberghs, G., & Lesaffre, E. (1999). Marginal modelling of multivariate categorical data. Statistics in Medicine, 18, 2237-2255.CrossRefGoogle Scholar
  97. Molenberghs, G., & Verbeke, G. (2005). Models for discrete longitudinal data. New York: Springer.zbMATHGoogle Scholar
  98. Morrell, C. H., & Brant, L. J. (1991). Modelling hearing thresholds in the elderly. Statistics in Medicine, 10, 1453-1464.CrossRefGoogle Scholar
  99. Neuhaus, J. M., Kalbfleisch, J. D., & Hauck, W. W. (1991). A comparison of cluster-specific and population-averaged approaches for analyzing correlated binary data. International Statistical Review, 59, 25-30.CrossRefGoogle Scholar
  100. Ochi, Y., & Prentice, R. L. (1984). Likelihood inference in a correlated probit regression model. Biometrika, 71, 531-543.zbMATHMathSciNetCrossRefGoogle Scholar
  101. Olkin, I., & Tate, R. F. (1961). Multivariate correlation models with mixed discrete and continuous variables. Annals of Mathematical Statistics, 32, 448-465 (with correction in 36, 343-344).Google Scholar
  102. Oort, F. J. (2001). Three-mode models for multivariate longitudinal data. British Journal of Mathematical and Statistical Psychology, 54, 49-78.CrossRefGoogle Scholar
  103. Pearson, J. D., Morrell, C. H., Gordon-Salant, S., Brant, L. J., Metter, E. J., Klein, L. L., & Fozard, J. L. (1995). Gender differences in a longitudinal study of age-associated hearing loss. Journal of the Acoustical Society of America, 97, 1196-1205.CrossRefGoogle Scholar
  104. Pharmacological Therapy for Macular Degeneration Study Group (1997). Interferon α-IIA is ineffective for patients with choroidal neovascularization secondary to age-related macular degeneration. Results of a prospective randomized placebo-controlled clinical trial. Archives of Ophthalmology, 115, 865-872.Google Scholar
  105. Pinheiro, J. C., & Bates, D. M. (2000). Mixed effects models in S and S-Plus. New York: Springer.zbMATHGoogle Scholar
  106. Prentice, R. L., & Zhao, L. P. (1991). Estimating equations for parameters in means and covariances of multivariate discrete and continuous responses. Biometrics, 47, 825-839.zbMATHMathSciNetCrossRefGoogle Scholar
  107. Potthoff, R. F., & Roy, S. N. (1964). A generalized multivariate analysis of variance model useful especially for growth curve problems. Biometrika, 51, 313-326.zbMATHMathSciNetGoogle Scholar
  108. Prentice, R. (1982). Covariate measurement errors and parameter estimates in a failure time regression model. Biometrika, 69, 331-342.zbMATHMathSciNetCrossRefGoogle Scholar
  109. Prentice, R. L. (1988). Correlated binary regression with covariates specific to each binary observation. Biometrics, 44, 1033-1048.zbMATHMathSciNetCrossRefGoogle Scholar
  110. Prentice, R. L. (1989). Surrogate endpoints in clinical trials: definitions and operational criteria. Statistics in Medicine, 8, 431-440.CrossRefGoogle Scholar
  111. Proust-Lima, C., Joly, P., Dartigues, J. F., & Jacqmin-Gadda, H. (2009). Joint modelling of multivariate longitudinal outcomes and a time-to-event: a nonlinear latent class approach. Computational Statistics and Data Analysis, 53, 1142-1154.zbMATHCrossRefGoogle Scholar
  112. Raab, G. M., & Donnelly, C. A. (1999). Information on sexual behaviour when some data are missing. Applied Statistics, 48, 117-133.Google Scholar
  113. Regan, M. M., & Catalano, P. J. (1999a). Likelihood models for clustered binary and continuous outcomes: Application to developmental toxicology. Biometrics, 55, 760-768.zbMATHCrossRefGoogle Scholar
  114. Regan, M. M., & Catalano, P. J. (1999b). Bivariate dose-response modeling and risk estimation in developmental toxicology. Journal of Agricultural, Biological and Environmental Statistics, 4, 217-237.MathSciNetCrossRefGoogle Scholar
  115. Regan, M. M., & Catalano, P. J. (2000). Regression models for mixed discrete and continuous outcomes with clustering. Risk Analysis, 20, 363-376.CrossRefGoogle Scholar
  116. Regan, M. M., & Catalano, P. J. (2002). Combined continuous and discrete outcomes. In M. Aerts, H. Geys, G. Molenberghs, & L. Ryan (Eds.), Topics in modelling of clustered data. London: Chapman & Hall.Google Scholar
  117. Renard, D., Geys, H., Molenberghs, G., Burzykowski, T., & Buyse, M. (2002). Validation of surrogate endpoints in multiple randomized clinical trials with discrete outcomes. Biometrical Journal, 44, 1-15.MathSciNetCrossRefGoogle Scholar
  118. Rizopoulos, D., Verbeke, G., & Molenberghs, G. (2009a). Multiple-imputation-based residuals and diagnostic plots for joint models of longitudinal and survival outcomes. Biometrics, to appear. doi: 10.1111/j.1541-0420.2009.01273.xGoogle Scholar
  119. Rizopoulos, D., Verbeke, G., & Lesaffre, E. (2009b). Fully exponential Laplace approximation for the joint modelling of survival and longitudinal data. Journal of the Royal Statistical Society, Series B, 71, 637-654.CrossRefGoogle Scholar
  120. Rizopoulos, D., Verbeke, G., & Molenberghs, G. (2008). Shared parameter models under random effects misspecification. Biometrika, 95, 63-74.zbMATHMathSciNetCrossRefGoogle Scholar
  121. Robins, J. M., Rotnitzky, A., & Scharfstein, D. O. (1998). Semiparametric regression for repeated outcomes with non-ignorable non-response. Journal of the American Statistical Association, 93, 1321-1339.zbMATHMathSciNetCrossRefGoogle Scholar
  122. Robins, J. M., Rotnitzky, A., & Zhao, L. P. (1995). Analysis of semiparametric regression models for repeated outcomes in the presence of missing data. Journal of the American Statistical Association, 90, 106-121.zbMATHMathSciNetCrossRefGoogle Scholar
  123. Roy, J., & Lin, X. (2000). Latent variable models for longitudinal data with multiple continuous outcomes. Biometrics, 56, 1047-1054.zbMATHMathSciNetCrossRefGoogle Scholar
  124. Rubin, D. B. (1987). Multiple imputation for nonresponse in surveys. New York: John Wiley & Sons.CrossRefGoogle Scholar
  125. Rubin, D. B., Stern, H. S., & Vehovar, V. (1995). Handling “don’t know” survey responses: The case of the Slovenian plebiscite. Journal of the American Statistical Association, 90, 822-828.CrossRefGoogle Scholar
  126. Sammel, M. D., Ryan, L. M., & Legler, J. M. (1997). Latent variable models for mixed discrete and continuous outcomes. Journal of the Royal Statistical Society, Series B, 59, 667-678.zbMATHCrossRefGoogle Scholar
  127. Schafer J. L. (1997). Analysis of incomplete multivariate data. London: Chapman & Hall.zbMATHGoogle Scholar
  128. Schafer, J. L. (2003). Multiple imputation in multivariate problems when the imputation and analysis models differ. Statistica Neerlandica, 57, 19-35.MathSciNetCrossRefGoogle Scholar
  129. Schatzkin, A., & Gail, M. (2002). The promise and peril of surrogate end points in cancer research. Nature Reviews Cancer, 2, 19-27.CrossRefGoogle Scholar
  130. Schemper, M., & Stare, J. (1996). Explained variation in survival analysis. Statistics in Medicine, 15, 1999-2012.CrossRefGoogle Scholar
  131. Self, S., & Pawitan, Y. (1992). Modeling a marker of disease progression and onset of disease. In N.P. Jewell, K. Dietz, & V.T. Farewell (Eds.), AIDS epidemiology: Methodological issues. Boston: Birkhauser.Google Scholar
  132. Shah, A., Laird, N., & Schoenfeld, D. (1997). A random-effects model for multiple characteristics with possibly missing data. Journal of the American Statistical Association, 92, 775-779.zbMATHMathSciNetCrossRefGoogle Scholar
  133. Shannon, C. (1948). A mathematical theory of communication. Bell System Technical Journal, 27, 379-423 and 623-656.Google Scholar
  134. Shock, N. W., Greullich, R. C., Andres, R., Arenberg, D., Costa, P. T., Lakatta, E. G., & Tobin, J. D. (1984). Normal human aging: The Baltimore Longitudinal Study of Aging. National Institutes of Health publication 84-2450.Google Scholar
  135. Shih, J. H., & Louis, T. A. (1995). Inferences on association parameter in copula models for bivariate survival data. Biometrics, 51, 1384-1399.zbMATHMathSciNetCrossRefGoogle Scholar
  136. Sivo, S. A. (2001). Multiple indicator stationary time series models. Structural Equation Modeling, 8, 599-612.MathSciNetCrossRefGoogle Scholar
  137. Song, X., Davidian, M., & Tsiatis, A. (2002). A semiparameteric likelihood approach to joint modeling of longitudinal and time-to-event data. Biometrics, 58, 742-753.MathSciNetCrossRefGoogle Scholar
  138. Tate, R. F. (1954). Correlation between a discrete and a continuous variable. Annals of Mathematical Statistics, 25, 603-607.zbMATHMathSciNetCrossRefGoogle Scholar
  139. Tate, R.F. (1955). The theory of correlation between two continuous variables when one is dichotomized. Biometrika, 42, 205-216.zbMATHMathSciNetGoogle Scholar
  140. Thijs, H., Molenberghs, G., Michiels, B., Verbeke, G., & Curran, D. (2002). Strategies to fit pattern-mixture models. Biostatistics, 3, 245-265.zbMATHCrossRefGoogle Scholar
  141. Therneau, T., & Grambsch, P. (2000). Modeling survival data: Extending the Cox Model. New York: Springer.zbMATHGoogle Scholar
  142. Thiébaut, R., Jacqmin-Gadda, H., Chêne, G., Leport, C., & Commenges, D. (2002). Bivariate linear mixed models using SAS PROC MIXED. Computer Methods and Programs in Biomedicine, 69, 249-256.CrossRefGoogle Scholar
  143. Thum, Y. M. (1997). Hierarchical linear models for multivariate outcomes. Journal of Educational and Behavioral Statistics, 22, 77-108.Google Scholar
  144. Tibaldi, F. S, Cortiñas Abrahantes, J., Molenberghs, G., Renard, D., Burzykowski, T., Buyse, M., Parmar, M., Stijnen, T., & Wolfinger, R. (2003). Simplified hierarchical linear models for the evaluation of surrogate endpoints. Journal of Statistical Computation and Simulation, 73, 643-658.zbMATHMathSciNetCrossRefGoogle Scholar
  145. Tseng, Y.-K., Hsieh, F., & Wang, J.-L. (2005). Joint modelling of accelerated failure time and longitudinal data. Biometrika, 92, 587-603.zbMATHMathSciNetCrossRefGoogle Scholar
  146. Tsiatis, A., & Davidian, M. (2001). A semiparametric estimator for the proportional hazards model with longitudinal covariates measured with error. Biometrika, 88, 447-458.zbMATHMathSciNetCrossRefGoogle Scholar
  147. Tsiatis, A., & Davidian, M. (2004). Joint modeling of longitudinal and time-to-event data: An overview. Statistica Sinica, 14, 809-834.zbMATHMathSciNetGoogle Scholar
  148. Tsiatis, A., DeGruttola, V., & Wulfsohn, M. (1995). Modeling the relationship of survival to longitudinal data measured with error: applications to survival and CD4 counts in patients with AIDS. Journal of the American Statistical Association, 90, 27-37.zbMATHCrossRefGoogle Scholar
  149. Van der Laan, M. J., & Robins, J. M. (2002). Unified methods for censored longitudinal data and causality. New York: Springer.Google Scholar
  150. Verbeke, G., Lesaffre, E., & Spiessens, B. (2001). The practical use of different strategies to handle dropout in longitudinal studies. Drug Information Journal, 35, 419-434.Google Scholar
  151. Verbeke, G., & Molenberghs, G. (2000). Linear mixed models for longitudinal data. New York: Springer.zbMATHGoogle Scholar
  152. Verbeke, G., Molenberghs, G., Thijs, H., Lesaffre, E., & Kenward, M. G. (2001). Sensitivity analysis for non-random dropout: A local influence approach. Biometrics, 57, 7-14.MathSciNetCrossRefGoogle Scholar
  153. Wang, Y., & Taylor, J. (2001). Jointly modeling longitudinal and event time data with application to acquired immunodeficiency syndrome. Journal of the American Statistical Association, 96, 895-905.zbMATHMathSciNetCrossRefGoogle Scholar
  154. Wolfinger, R. D. (1998). Towards practical application of generalized linear mixed models. In B. Marx & H. Friedl (Eds.), Proceedings of the 13th International Workshop on Statistical Modeling (pp. 388-395). New Orleans, Louisiana, USA.Google Scholar
  155. Wolfinger, R., & O’Connell, M. (1993). Generalized linear mixed models: a pseudo-likelihood approach. Journal of Statistical Computation and Simulation, 48, 233-243.zbMATHCrossRefGoogle Scholar
  156. Wu, M., & Carroll, R. (1988). Estimation and comparison of changes in the presence of informative right censoring by modeling the censoring process. Biometrics, 44, 175-188.zbMATHMathSciNetCrossRefGoogle Scholar
  157. Wulfsohn, M., & Tsiatis, A. (1997). A joint model for survival and longitudinal data measured with error. Biometrics, 53, 330-339.zbMATHMathSciNetCrossRefGoogle Scholar
  158. Xu, J., & Zeger, S. (2001). Joint analysis of longitudinal data comprising repeated measures and times to events. Applied Statistics, 50, 375-387.zbMATHMathSciNetGoogle Scholar
  159. Yu, M., Law, N., Taylor, J., & Sandler, H. (2004). Joint longitudinal-survival-cure models and their application to prostate cancer. Statistica Sinica, 14, 835-832.zbMATHMathSciNetGoogle Scholar
  160. Zhao, L. P., Prentice, R. L., & Self, S. G. (1992). Multivariate mean parameter estimation by using a partly exponential model. Journal of the Royal Statistical Society B, 54, 805-811.Google Scholar

Copyright information

© Springer Berlin Heidelberg 2010

Authors and Affiliations

  • Geert Verbeke
    • 1
    Email author
  • Geert Molenberghs
    • 2
  • Dimitris Rizopoulos
    • 3
  1. 1.I-BioStatKatholieke Universiteit LeuvenLeuvenBelgium
  2. 2.I-BioStatUniversiteit HasseltDiepenbeekBelgium
  3. 3.Department of BiostatisticsErasmus University Medical CenterRotterdamThe Netherlands

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