Estimation and prediction of a generalized mixed-effects model with t-process for longitudinal correlated binary data

Abstract

We propose a generalized mixed-effects model based on t-process for longitudinal correlated binary data. The correlations among repeated binary outcomes are defined by a latent t-process, which provides a new framework on modeling nonlinear random- effects. The covariance kernel of the process can adaptively capture the subject-specific variations while the heavy-tails of the t-process enable robust inferences. We develop an efficient estimation procedure based on Monte Carlo EM algorithm and a prediction approach through conditional inference. Numerical studies indicate that the estimation and prediction based on the proposed model is robust against outliers compared with Gaussian model. We use the renal anemia and meteorological data as illustrative examples.

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References

  1. Breslow N, Clayton D (1993) Approximate inference in generalized linear mixed models. J Am Stat Assoc 88(421):9–25

    MATH  Google Scholar 

  2. Cao C, Shi JQ, Lee Y (2018) Robust functional regression model for marginal mean and subject-specific inferences. Stat Methods Med Res 27(11):3236–3254

    MathSciNet  Article  Google Scholar 

  3. Cheng L, Ramchandran S, Vatanen T et al (2019) An additive Gaussian process regression model for interpretable non-parametric analysis of longitudinal data. Nat Commun 10:1798

    Article  Google Scholar 

  4. Guo L, Jiang Z, Ding M, Chen W, Li L (2019) Downscaling and projection of summer rainfall in Eastern China using a nonhomogeneous hidden Markov model. Int J Climatol 39(3):1319–1330

    Article  Google Scholar 

  5. Hartmann M, Vanhatalo J (2019) Laplace approximation and natural gradient for Gaussian process regression with heteroscedastic student-\(t\) model. Stat Comput 29:753–773

    MathSciNet  Article  Google Scholar 

  6. Ho HJ, Lin T, Chen H et al (2012) Some results on the truncated multivariate \(t\) distribution. J Stat Plan Inference 142(1):25–40

    MathSciNet  Article  Google Scholar 

  7. Lee Y, Nelder JA, Pawitan Y (2017) Generalized Linear Models with Random-Effects, Unified Analysis via H-likelihood, 2nd edn. Chapman and Hall, London

    Google Scholar 

  8. Liu J, Dey D (2008) Skew random effects in multilevel binomial models: an alternative to nonparametric approach. Stat Model 8(3):221–241

    MathSciNet  Article  Google Scholar 

  9. McCulloch CE (1994) Maximum likelihood variance components estimation for binary data. J Am Stat Assoc 89:330–335

    Article  Google Scholar 

  10. McCulloch CE, Searle SR (2001) Generalized, linear, and mixed models. Wiley, New York

    Google Scholar 

  11. Prates MO, Costa DR, Lachos VH (2014) Generalized linear mixed models for correlated binary data with T-link. Stat Comput 24(6):1111–1123

    MathSciNet  Article  Google Scholar 

  12. Rasmussen CE, Williams CKI (2006) Gaussian processes for machine learning. The MIT Press, Cambridge

    Google Scholar 

  13. Santos CC, Loschi RH (2017a) Maximum likelihood estimation and parameter interpretation in elliptical mixed logistic regression. Test 26(1):209–230

    MathSciNet  Article  Google Scholar 

  14. Santos CC, Loschi RH (2017b) EM-type algorithms for heavy-tailed logistic mixed models. J Stat Comput Simul 87(15):2940–2961

    MathSciNet  Article  Google Scholar 

  15. Shi JQ, Choi T (2011) Gaussian process regression analysis for functional data. Chapman and Hall, London

    Google Scholar 

  16. Shi JQ, Wang B, Will EJ et al (2012) Mixed-effects Gaussian process functional regression models with application to dose-response curve prediction. Stat Med 31(26):3165–3177

    MathSciNet  Article  Google Scholar 

  17. Sofro A, Shi JQ, Cao C (2020) Regression analysis for multivariate process data of counts using convolved Gaussian processes. J Stat Plan Inference 206:57–74

    MathSciNet  Article  Google Scholar 

  18. Tan M, Tian GL, Fang HB (2007) An efficient MCEM algorithm for fitting generalized linear mixed models for correlated binary data. J Stat Comput Simul 77(11):929–943

    MathSciNet  Article  Google Scholar 

  19. Tolman C, Richardson D, Bartlett C et al (2005) Structured conversion from thrice weekly to weekly erythropoietic regimens using a computerized decision-support system: a randomized clinical study. J Am Soc Nephrol 16(5):1463–1470

    Article  Google Scholar 

  20. Wang B, Shi JQ (2014) Generalized Gaussian process regression model for non-Gaussian functional data. J Am Stat Assoc 109(507):1123–1133

    MathSciNet  Article  Google Scholar 

  21. Wang Z, Shi JQ, Lee Y (2017) Extended T-process regression models. J Stat Plan Inference 189:38–60

    MathSciNet  Article  Google Scholar 

  22. West RM, Harris K, Gilthorpe MS et al (2007) Functional data analysis applied to a randomized controlled clinical trial in hemodialysis patients describes the variability of patient responses in the control of renal anemia. J Am Soc Nephrol 18(8):2371–2376

    Article  Google Scholar 

  23. Will EJ, Richardson D, Tolman C et al (2007) Development and exploitation of a clinical decision support system for the management of renal anaemia. Nephrol Dial Transp 22(Suppl 4):iv31–iv36

    Google Scholar 

Download references

Acknowledgements

This work is supported by the National Social Science Foundation of China (Grant No. 16ZDA047), the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20191394), the National Natural Science Foundation of China (Grant No. 61672291) and the Key Research Project of Jiangsu Province of China (Grant No. 2019A005).

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Correspondence to Jian Qing Shi.

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Cao, C., He, M., Shi, J.Q. et al. Estimation and prediction of a generalized mixed-effects model with t-process for longitudinal correlated binary data. Comput Stat (2021). https://doi.org/10.1007/s00180-020-01057-0

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Keywords

  • Functional data
  • Heavy-tailed process
  • Prediction
  • Random-effects
  • Robustness