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A robust joint modeling approach for longitudinal data with informative dropouts

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Abstract

This article proposes a robust method for analysing longitudinal continuous responses with informative dropouts and potential outliers by using the multivariate t-distribution. We specify a dropout mechanism and a missing covariate distribution and incorporate them into the complete data log-likelihood. Unlike the existing approaches which mainly focus on the inference of regression mean and dropouts process, our approach aims to reveal the dynamics in the location function, marginal scale function and association by joint parsimonious modeling the location and dependence structure. A parametric fractional imputation algorithm is developed to speed up the computation associated with the EM algorithm for maximum likelihood estimation with missing data. The resulting estimators are shown to be consistent and asymptotically normally distributed. Data examples and simulations demonstrate the effectiveness of the proposed approach.

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References

  • Bang H, Robins JM (2005) Doubly robust estimation in missing data and causal inference models. Biometrics 61:962–73

    MathSciNet  MATH  Google Scholar 

  • Booth JG, Hobert JP (1999) Maximizing generalized linear models with an automated Monte Carlo EM algorithm. J R Stat Soc Ser B Stat Methodol 61:625–685

    MATH  Google Scholar 

  • Diggle PJ (2002) The analysis of longitudinal data. Oxford University Press, Oxford

    Google Scholar 

  • Diggle P, Kenward MG (1994) Informative drop-out in longitudinal data analysis. J R Stat Soc Ser B Stat Methodol 43:49–93

    MATH  Google Scholar 

  • Detke MJ, Wiltse CG, Mallinckrodt CH, McNamara RK, Demitrack MA, Bitter I (2004) Duloxetine in the acute and long-term treatment of major depressive disorder: a placebo and paroxetine controlled trial. Eur Neuropsychopharmacol 14:457–470

    Google Scholar 

  • Fan J, Wu Y (2008) Semiparametric estimation of covariance matrices for longitudinal data. J Am Stat Assoc 103:1520–1533

    MATH  Google Scholar 

  • Fan J, Huang T, Li R (2007) Analysis of longitudinal data with semiparametric estimation of covariance function. J Am Stat Assoc 102:632–641

    MATH  Google Scholar 

  • Fitzmaurice GM, Laird NM, Zahner GEP (1996) Multivariate logistic models for incomplete binary responses. J Am Stat Assoc 91:99–108

    MATH  Google Scholar 

  • Goldstein DJ, Lu Y, Detke MJ, Wiltse C, Mallinckrodt C, Demitrack MA (2004) Duloxetine in the treatment of depression: a double-blind placebo-controlled comparison with paroxetine. J Clin Psychopharmacol 24:389–399

    Google Scholar 

  • Hedeker D, Gibbons RD (1997) Application of random-effects pattern-mixture models for missing data in longitudinal studies. Psychol Methods 2:64–78

    Google Scholar 

  • Hogan JW, Roy J, Korkontzelou C (2004) Handling drop-out in longitudinal studies. Stat Med 23(9):1455–1497

    Google Scholar 

  • Ibrahim JG, Molenberghs G (2009) Missing data methods in longitudinal studies: a review. Test (Madrid) 18(1):1–43

    MathSciNet  MATH  Google Scholar 

  • Joseph G, Ibrahim HZ, Tang N (2008) Model selection criteria for missing-data problems using the EM algorithm. J Am Stat Assoc 103:1648–1658

    MathSciNet  MATH  Google Scholar 

  • Kim JK (2011) Parametric fractional imputation for missing data analysis. Biometrika 98:119–132

    MathSciNet  MATH  Google Scholar 

  • Kotz S, Nadarajah S (2004) Multivariate \(t\) distributions and their applications. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Lange KL (1989) Robust statistical modeling using the t-distribution. J Am Stat Assoc 84:881–896

    MathSciNet  Google Scholar 

  • Leng C, Zhang W, Pan J (2010) Semiparametric mean–covariance regression analysis for longitudinal data. J Am Stat Assoc 105:181–193

    MathSciNet  MATH  Google Scholar 

  • Liang KY, Zeger SL (1986) Longitudinal data analysis using generalized linear models. Biometrika 73:13–22

    MathSciNet  MATH  Google Scholar 

  • Lin TI, Wang WL (2011) Bayesian inference in joint modeling of location and scale parameters of the t distribution for longitudinal data. J Stat Plan Inference 141:1543–1553

    MATH  Google Scholar 

  • Little RJA (1995) Modeling the drop-out mechanism in repeated-measures studies. J Am Stat Assoc 90:1112–1121

    MathSciNet  MATH  Google Scholar 

  • Little RJ, Rubin DB (1987) Statistical analysis with missing data. Wiley, New York

    MATH  Google Scholar 

  • Louis TA (1982) Finding the observed information matrix when using the EM algorithm. J R Stat Soc Ser B Stat Methodol 44:226–233

    MathSciNet  MATH  Google Scholar 

  • Maadooliat M, Pourahmadi M, Huang JZ (2013) Robust estimation of the correlation matrix of longitudinal data. Stat Comput 23:17–28

    MathSciNet  MATH  Google Scholar 

  • Newey WK, Mcfadden D (1994) Large sample estimation and hypothesis testing. In: Engle R, McFadden D (eds) Handbook of econometrics. Elsevier, Berlin, pp 2111–2245

    Google Scholar 

  • Pan J, Mackenzie G (2003) Model selection for joint mean–covariance structures in longitudinal studies. Biometrika 90:239–244

    MathSciNet  MATH  Google Scholar 

  • Pauler DK, McCoy S, Moinpour C (2003) Pattern mixture models for longitudinal quality of life studies in advanced stage disease. Stat Methods Med Res 22:795–809

    Google Scholar 

  • Pourahmadi M (1999) Joint mean–covariance models with applications to longitudinal data: unconstrained parameterisation. Biometrika 86:677–690

    MathSciNet  MATH  Google Scholar 

  • Pourahmadi M (2000) Maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix. Biometrika 87:425–435

    MathSciNet  MATH  Google Scholar 

  • Pourahmadi M (2007) Cholesky decompositions and estimation of a covariance matrix: orthogonality of variance–correlation parameters. Biometrika 94:1006–1013

    MathSciNet  MATH  Google Scholar 

  • Roy J, Lin X (2003) Analysis of multivariate longitudinal outcomes with nonignorable dropouts and missing covariates: changes in methadone treatment practices. J Am Stat Assoc 97:40–52

    MathSciNet  MATH  Google Scholar 

  • Roy J, Lin X (2005) Missing covariates in longitudinal data with informative dropouts: bias analysis and inference. Biometrics 61:837–846

    MathSciNet  MATH  Google Scholar 

  • Rubin DB (1976) Inference and missing data. Biometrika 63:581–592

    MathSciNet  MATH  Google Scholar 

  • Seaman S, Copas A (2009) Doubly robust generalized estimating equations for longitudinal data. Stat Med 28(6):937–955

    MathSciNet  Google Scholar 

  • Stubbendick AL, Ibrahim JG (2003) Maximum likelihood methods for nonignorable missing responses and covariates in random effects models. Biometrics 59:1140–1150

    MathSciNet  MATH  Google Scholar 

  • Stubbendick AL, Ibrahim JG (2006) Likelihood-based inference with nonignorable missing responses and covariates in models for discrete longitudinal data. Stat Sin 16:1143–1167

    MathSciNet  MATH  Google Scholar 

  • Troxel AB, Ma G, Heitjan DF (2004) An index of local sensitivity to nonignorability. Stat Sin 14:1221–1237

    MathSciNet  MATH  Google Scholar 

  • Tukey JW (1949) One degree of freedom for non-additivity. Biometrics 5:232–242

    Google Scholar 

  • Vansteelandt S, Rotnitzky A, Robins J (2007) Estimation of regression models for the mean of repeated outcomes under nonignorable nonmonotone nonresponse. Biometrika 94:841–860

    MathSciNet  MATH  Google Scholar 

  • Verbeke G, Molenberghs G, Thijs H, Lesaffre E, Kenward MG (2001) Sensitivity analysis for nonrandom dropout: a local influence approach. Biometrics 57:7–14

    MathSciNet  MATH  Google Scholar 

  • Willoughby I, Stokes V, Poole J, White JEJ, Hodge SJ (2007) The potential of 44 native and non-native tree species for woodland creation on a range of contrasting sites in lowland Britain. Forestry 80(5):531–553

    Google Scholar 

  • Ye H, Pan J (2006) Modeling covariance structures in generalized estimating equations for longitudinal data. Biometrika 93:927–941

    MathSciNet  MATH  Google Scholar 

  • Zeger SL, Liang KY (1986) Longitudinal data analysis for discrete and continuous outcomes. Biometrics 42:121–130

    Google Scholar 

  • Zhang W, Leng C (2012) A moving average Cholesky factor model in covariance modeling for longitudinal data. Biometrika 99:141–150

    MathSciNet  MATH  Google Scholar 

  • Zhang W, Leng C, Tang CY (2015) A joint modelling approach for longitudinal studies. J R Stat Soc Ser B Stat Methodol 77:219–238

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank the Editor, and two referees for their constructive comments and suggestions that have greatly improved the paper.

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Authors and Affiliations

  1. Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, 230026, China

    Weiping Zhang, Feiyue Xie & Jiaxin Tan

Authors
  1. Weiping Zhang
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  2. Feiyue Xie
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  3. Jiaxin Tan
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Correspondence to Weiping Zhang.

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This work is supported by the National Key Research and Development Plan (No. 2016YFC0800100) and the NSFC of China (Nos. 11671374, 71771203, 71631006).

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Zhang, W., Xie, F. & Tan, J. A robust joint modeling approach for longitudinal data with informative dropouts. Comput Stat 35, 1759–1783 (2020). https://doi.org/10.1007/s00180-020-00972-6

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  • DOI: https://doi.org/10.1007/s00180-020-00972-6

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