Abstract
Longitudinal studies with binary outcomes characterized by informative right censoring are commonly encountered in clinical, basic, behavioral, and health sciences. Approaches developed to analyze data with binary outcomes were mainly tailored to clustered or longitudinal data with missing completely at random or at random. Studies that focused on informative right censoring with binary outcomes are characterized by their imbedded computational complexity and difficulty of implementation. Here we present a new maximum likelihood-based approach with repeated binary measures modeled in a generalized linear mixed model as a function of time and other covariates. The longitudinal binary outcome and the censoring process determined by the number of times a subject is observed share latent random variables (random intercept and slope) where these subject-specific random effects are common to both models. A simulation study and sensitivity analysis were conducted to test the model under different assumptions and censoring settings. Our results showed accuracy of the estimates generated under this model when censoring was fully informative or partially informative with dependence on the slopes. A successful implementation was undertaken on a cohort of renal transplant patients with blood urea nitrogen as a binary outcome measured over time to indicate normal and abnormal kidney function until the emanation of graft rejection that eventuated in informative right censoring. In addition to its novelty and accuracy, an additional key feature and advantage of the proposed model is its viability of implementation on available analytical tools and widespread application on any other longitudinal dataset with informative censoring.
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References
Diggle P, Kenward MG (1994) Informative dropout in longitudinal data analysis. Appl Stat 43:49–94
Rubin DB (1976) Inference and missing data. Biometrika 63:581–590
De Gruttola V, Ming TuX (1994) Modelling progression of CD4-lymphocyte count and its relationship to survival time. Biometrics 50:1003–1014
Wu MC, Carroll RJ (1988) Estimation and comparison of changes in the presence of informative right censoring by modeling the censoring process. Biometrics 44:175–188
Guo W, Ratcliffe SJ, Ten Have TT (2004) A random pattern-mixture model for longitudinal data with dropouts. J Am Stat Assoc 99:929–937
Birmingham J, Rotnitzky A, Fitzmaurice GM (2003) Pattern-mixture and selection models for analyzing longitudinal data with monotone missing patterns. J Roy Stat Soc B 65:275–297
Little RJA (1995) Modeling the dropout mechanism in repeated measures studies. J Am Stat Assoc 90:1113–1121
Little RJA, Rubin DB (2002) Statistical analysis with missing data, 2nd edn. Wiley, New York
Rotnitzky A, Scharfstein DO, Su T, Robins JM (2001) Methods for conducting sensitivity analysis of trials with potentially nonignorable competing causes of censoring. Biometrics 57:103–113
Scharfstein DO, Robins JM, Eddings W, Rotnitzky A (2001) Inference in randomized studies with informative censoring and discrete time-to-event endpoints. Biometrics 57:404–413
Scharfstein DO, Robins JM (2002) Estimation of the failure time distribution in the presence of informative censoring. Biometrika 89:617–634
Roy J, Daniels M (2008) A general class of pattern-mixture models for nonignorable dropout with many possible dropout times. Biometrics 64:538–545
Lipsitz SR, Fitzmaurice GM, Sleeper L, Zhao LP (1995) Estimation methods for the joint distribution of repeated binary observations. Biometrics 51(2):562–570
Heagerty PJ (1999) Marginally specified logistic-normal models for longitudinal binary data. Biometrics 55:688–698
Parzen M, Ghosh S, Lipsitz SR, Sinha D, Fitzmaurice GM, Mallick BK, Ibrahim JG (2011) A generalized linear mixed model for longitudinal binary data with a marginal logit link function. Ann Appl Stat 5(1):449–467
Koch GG, Imrey PB, Reinfurt DW (1972) Linear model analysis of categorical data with incomplete response vectors. Biometrics 28:633–692
Lehnen RG, Koch GG (1974) Analyzing panel data with uncontrolled attrition. Public Opin Q 38:40–56
Lipsitz SR, Laird NM, Harrington DP (1994) Weighted least squares analysis of repeated categorical measurements with outcomes subject to nonresponse. Biometrics 50(1):11–24
Woolson RF, Clarke WR (1984) Analysis of categorical incomplete longitudinal data. J R Stat Soc Ser A 147:87–99
Liang KY, Zeger SL (1986) Longitudinal data analysis using generalized linear models. Biometrika 73:13–22
Fitzmaurice GM, Laird NM (1993) A likelihood-based method for analysing longitudinal binary responses. Biometrika 80:141–151
Fitzmaurice GM, Laird NM, Lipsitz SR (1994) Analyzing incomplete longitudinal binary responses: a likelihood-based approach. Biometrics 50(3):601–612
Ten Have TR, Pulkstenis EP, Kunselman AR, Landis RJ (1998) Mixed effects logistic regression models for longitudinal binary response data with informative drop-out. Biometrics 54(1):367–383
Pulkstenis ER, Ten Have TR, Landis RJ (1998) Model for the analysis of binary longitudinal pain data subject to informative dropout through remedication. J Am Stat Assoc 93(442):438–450
Albert SP (2000) A translational model for longitudinal binary data subject to nonignorbale missing data. Biometrics 56:602–608
Albert SP, Follmann DA, Wang SA, Suh BE (2002) A latent autoregressive model for longitudinal binary data subject to informative missingness. Biometrics 58:631–642
Albert SP, Follman DA (2007) Random effects and latent processes approaches for analyzing binary longitudinal data with missingness: a comparison of approaches using opiate clinical trial data. Stat Methods Med Res 16:417–439
Rizopoulus D, Verbeke G, Lesaffre E, Vanrenterghem Y (2008) A two-part joint model for the analysis of survival and longitudinal binary data with excess zeros. Biometrics 64:611
Su L (2012) A marginalized conditional linear model for longitudinal binary data when informative dropout occurs in continuous time. Biostatistics 13(2):355–368
Kyoung Y, Lee K (2015) Bayesian pattern mixture model for longitudinal binary data with nonignorable missingness. Commun Stat Appl Method 22(6):589–598
Chan JSK (2016) Bayesian informative dropout model for longitudinal binary data with random effects using conditional and joint modeling approaches. Biom J 58(3):549–569
Li Q, Su L (2018) Accommodating informative dropout and death: a joint modeling approach for longitudinal and semicompeting risks data. J R Stat Soc Ser C 67(1):145–163
Alfo M, Aitkin M (2000) Random coefficient models for binary longitudinal responses with attrition. Stat Comput 10:279–287
Daniels MJ, Hogan JW (2008) Missing data in longitudinal studies. CRC, New York
Molenberghs G, Kenward MG, Lesaffre E (1997) The analysis of longitudinal ordinal data with nonrandom dropout. Biometrika 84:33–34
Fletcher R (1987) Practical methods of optimization, 2nd edn. Willey, New York
Pinheiro JC, Bates DM (1995) Approximations to the log-likelihood function in the nonlinear mixed-effects model. J Comput Graph Stat 4:12–35
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This work was supported by the National Institutes of Health Grants HL077192 (AAJ).
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Jaffa, M.A., Jaffa, A.A. A Likelihood-Based Approach with Shared Latent Random Parameters for the Longitudinal Binary and Informative Censoring Processes. Stat Biosci 11, 597–613 (2019). https://doi.org/10.1007/s12561-019-09254-2
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DOI: https://doi.org/10.1007/s12561-019-09254-2