We propose a marginalized joint-modeling approach for marginal inference on the association between longitudinal responses and covariates when longitudinal measurements are subject to informative dropouts. The proposed model is motivated by the idea of linking longitudinal responses and dropout times by latent variables while focusing on marginal inferences. We develop a simple inference procedure based on a series of estimating equations, and the resulting estimators are consistent and asymptotically normal with a sandwich-type covariance matrix ready to be estimated by the usual plug-in rule. The performance of our approach is evaluated through simulations and illustrated with a renal disease data application. 1. Introduction Longitudinal studies often encounter data attrition because subjects drop out before the designated study end. Both statistical analysis and practical interpretation of longitudinal data can be complicated by dropouts. For example, in the Modification of Diet in Renal Disease (MDRD) study [1, 2], one main interest was to investigate the efficacy of interventions of blood pressure control and diet modification on patients with impaired renal functions. The primary outcome was glomerular filtration rate (GFR), which measured filtering capacity of kidneys, and was repeatedly measured over the study period. However, some patients could leave the study prematurely for kidney transplant or dialysis, which precluded further GFR measurements. This resulted in a dropout mechanism that could relate to patients’ kidney function and correlate with their GFR values. Other patients were followed to the end of the study or dropped out due to independent reasons. Thus, statistical analysis of longitudinal GFR needs to take into consideration the presence of mixed types of informative and independent dropouts. Many statistical models and inference approaches have been proposed to accommodate the nonignorable missingness into modeling longitudinal data (see reviews [3–8]). According to the target of inference and the interpretation of model parameters, existing methods can be classified into three categories: subject-specific inference, event-conditioning inference, and marginal inference. First, a widely used modeling strategy for longitudinal data with informative dropouts is to specify their joint distribution via shared or correlated latent variables. Under such model assumptions, the longitudinal parameters have a conditional, subject-specific interpretation (e.g., [9–11]). But the interpretation of longitudinal parameters usually changes
References
[1]
G. J. Beck, R. L. Berg, C. H. Coggins, et al., “Design and statistical issues of the modification of diet in renal disease trial. The modification of diet in renal disease study group,” Controlled Clinical Ttrials, vol. 12, pp. 566–586, 19991.
[2]
S. Klahr, A. S. Levey, G. J. Beck et al., “The effects of dietary protein restriction and blood-pressure control on the progression of chronic renal disease,” The New England Journal of Medicine, vol. 330, no. 13, pp. 877–884, 1994.
[3]
J. W. Hogan and N. M. Laird, “Mixture models for the joint distribution of repeated measures and event times,” Statistics in Medicine, vol. 16, no. 1–3, pp. 239–257, 1997.
[4]
J. W. Hogan and N. M. Laird, “Model-based approaches to analysing incomplete longitudinal and failure time data,” Statistics in Medicine, vol. 16, no. 1–3, pp. 259–272, 1997.
[5]
J. W. Hogan, J. Roy, and C. Korkontzelou, “Tutorial in biostatistics. Handling drop-out in longitudinal studies,” Statistics in Medicine, vol. 23, no. 9, pp. 1455–1497, 2004.
[6]
J. G. Ibrahim and G. Molenberghs, “Missing data methods in longitudinal studies: a review,” Test, vol. 18, no. 1, pp. 1–43, 2009.
[7]
R. J. A. Little and D. B. Rubin, Statistical Analysis with Missing Data, John Wiley & Sons, New York, NY, USA, 1987.
[8]
A. A. Tsiatis and M. Davidian, “Joint modeling of longitudinal and time-to-event data: an overview,” Statistica Sinica, vol. 14, no. 3, pp. 809–834, 2004.
[9]
V. De Gruttola and X. M. Tu, “Modelling progression of CD4-lymphocyte count and its relationship to survival time,” Biometrics, vol. 50, no. 4, pp. 1003–1014, 1994.
[10]
M. D. Schluchter, “Methods for the analysis of informatively censored longitudinal data,” Statistics in Medicine, vol. 11, no. 14-15, pp. 1861–1870, 1992.
[11]
M. C. Wu and R. J. Carroll, “Estimation and comparison of changes in the presence of informative right censoring by modeling the censoring process,” Biometrics, vol. 44, no. 1, pp. 175–188, 1988.
[12]
R. J. A. Little, “Pattern-mixture models for multivariate incomplete data,” Journal of the American Statistical Association, vol. 88, pp. 125–134, 1993.
[13]
B. F. Kurland and P. J. Heagerty, “Directly parameterized regression conditioning on being alive: analysis of longitudinal data truncated by deaths,” Biostatistics, vol. 6, no. 2, pp. 241–258, 2005.
[14]
C. Dufouil, C. Brayne, and D. Clayton, “Analysis of longitudinal studies with death and drop-out: a case study,” Statistics in Medicine, vol. 23, no. 14, pp. 2215–2226, 2004.
[15]
D. B. Rubin, “Inference and missing data,” Biometrika, vol. 63, no. 3, pp. 581–592, 1976.
[16]
K. Y. Liang and S. L. Zeger, “Longitudinal data analysis using generalized linear models,” Biometrika, vol. 73, no. 1, pp. 13–22, 1986.
[17]
J. M. Robins, A. Rotnitzky, and L. P. Zhao, “Analysis of semiparametric regression-models for repeated outcomes in the presence of missing data,” Journal of the American Statistical Association, vol. 90, pp. 106–121, 1995.
[18]
G. Y. Yi and R. J. Cook, “Marginal methods for incomplete longitudinal data arising in clusters,” Journal of the American Statistical Association, vol. 97, no. 460, pp. 1071–1080, 2002.
[19]
J. J. Heckman, “Sample selection bias as a specification error,” Econometrica, vol. 47, pp. 153–162, 1979.
[20]
G. M. Fitzmaurice, G. Molenberghs, and S. R. Lipsitz, “Regression-models for longitudinal binary responses with informative drop-outs,” Journal of the Royal Statistical Society Series B, vol. 57, pp. 691–704, 1995.
[21]
M. G. Kenward, “Selection models for repeated measurements with non-random dropout: an illustration of sensitivity,” Statistics in Medicine, vol. 17, no. 23, pp. 2723–2732, 1998.
[22]
G. Molenberghs, M. G. Kenward, and E. Lesaffre, “The analysis of longitudinal ordinal data with nonrandom drop-out,” Biometrika, vol. 84, no. 1, pp. 33–44, 1997.
[23]
B. F. Kurland and P. J. Heagerty, “Marginalized transition models for longitudinal binary data with ignorable and non-ignorable drop-out,” Statistics in Medicine, vol. 23, no. 17, pp. 2673–2695, 2004.
[24]
K. J. Wilkins and G. M. Fitzmaurice, “A marginalized pattern-mixture model for longitudinal binary data when nonresponse depends on unobserved responses,” Biostatistics, vol. 8, no. 2, pp. 297–305, 2007.
[25]
D. R. Cox, “Regression models and life-tables,” Journal of the Royal Statistical Society Series B, vol. 34, pp. 187–220, 1972.
[26]
S. Bennett, “Analysis of survival data by the proportional odds model,” Statistics in Medicine, vol. 2, no. 2, pp. 273–277, 1983.
[27]
K. Chen, Z. Jin, and Z. Ying, “Semiparametric analysis of transformation models with censored data,” Biometrika, vol. 89, no. 3, pp. 659–668, 2002.
[28]
S. C. Cheng, L. J. Wei, and Z. Ying, “Analysis of transformation models with censored data,” Biometrika, vol. 82, no. 4, pp. 835–845, 1995.
[29]
D. Clayton and J. Cuzick, “Multivariate generalizations of the proportional hazards model,” Journal of the Royal Statistical Society Series A, vol. 148, pp. 82–117, 1985.
[30]
P. J. Heagerty, “Marginally specified logistic-normal models for longitudinal binary data,” Biometrics, vol. 55, no. 3, pp. 688–698, 1999.
[31]
P. J. Heagerty and S. L. Zeger, “Marginalized multilevel models and likelihood inference,” Statistical Science, vol. 15, no. 1, pp. 1–26, 2000.
[32]
W. H. Press, Numerical Recipes in C : The Art of Scientific Computing, Cambridge University Press, New York, NY, USA, 2nd edition, 1992.
[33]
M. D. Schluchter, T. Greene, and G. J. Beck, “Analysis of change in the presence of informative censoring: application to a longitudinal clinical trial of progressive renal disease,” Statistics in Medicine, vol. 20, no. 7, pp. 989–1007, 2001.
[34]
P. K. Andersen, O. Borgan, R. D. Gill, and N. Keiding, Statistical Models Based on Counting Processes, Springer Series in Statistics, Springer, New York, NY, USA, 1993.
[35]
R. Tsonaka, G. Verbeke, and E. Lesaffre, “A semi-parametric shared parameter model to handle nonmonotone nonignorable missingness,” Biometrics, vol. 65, no. 1, pp. 81–87, 2009.
