In longitudinal studies,
measurements are taken repeatedly over time on the same experimental unit.
These measurements are thus correlated. Missing data are very common in
longitudinal studies. A lot of research has been going on ways to appropriately
analyze such data set. Generalized Estimating Equations (GEE) is a popular
method for the analysis of non-Gaussian longitudinal data. In the presence of
missing data, GEE requires the strong assumption of missing completely at
random (MCAR). Multiple Imputation Generalized Estimating Equations (MIGEE),
Inverse Probability Weighted Generalized Estimating Equations (IPWGEE) and
Double Robust Generalized Estimating Equations (DRGEE) have been proposed as
elegant ways to ensure validity of the inference under missing at random (MAR).
In this study, the three extensions of GEE are compared under various dropout rates
and sample sizes through simulation studies. Under MAR and MCAR mechanism, the
simulation results revealed better performance of DRGEE compared to IPWGEE and
MIGEE. The optimum method was applied to real data set.
References
[1]
Barnard, J. and Meng, X.-L. (1999) Applications of Multiple Imputation in Medical Studies: From AIDS to NHANES. Statistical Methods in Medical Research, 8, 17-36.
https://doi.org/10.1177/096228029900800103
[2]
Little, R.J. and Rubin, D.B. (2002) Bayes and Multiple Imputation. Statistical Analysis with Missing Data, 200-220.
Durrant, G.B. (2009) Imputation Methods for Handling Item-Nonresponse in Practice: Methodological Issues and Recent Debates. International Journal of Social Research Methodology, 12, 293-304. https://doi.org/10.1080/13645570802394003
[5]
Rubin, D.B. (1978) Multiple Imputations in Sample Surveys—A Phenomenological Bayesian Approach to Nonresponse. Proceedings of the Survey Research Methods Section of the American Statistical Association, 1, 20-34.
[6]
Liang, K.-Y. and Zeger, S.L. (1986) Longitudinal Data Analysis Using Generalized Linear Models. Biometrika, 73, 13-22. https://doi.org/10.1093/biomet/73.1.13
[7]
Little, R.J. and Rubin, D.B. (2014) Statistical Analysis with Missing Data. John Wiley & Sons, Hoboken.
[8]
Robins, J.M., Rotnitzky, A. and 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.
https://doi.org/10.1080/01621459.1995.10476493
[9]
Tsiatis, A. (2007) Semiparametric Theory and Missing Data. Springer Science & Business Media, New York.
[10]
Aluko Omololu, S. and Mwambi, H. (2017) A Comparison of Three Different Enhancements of the Generalized Estimating Equations Method in Handling Incomplete Longitudinal Binary Outcome. Global Journal of Pure and Applied Mathematics, 13, 7669-7688.
[11]
Bang, H. and Robins, J.M. (2005) Doubly Robust Estimation in Missing Data and Causal Inference Models. Biometrics, 61, 962-973.
https://doi.org/10.1111/j.1541-0420.2005.00377.x
[12]
Chen, B. and Zhou, X.-H. (2011) Doubly Robust Estimates for Binary Longitudinal Data Analysis with Missing Response and Missing Covariates. Biometrics, 67, 830-842. https://doi.org/10.1111/j.1541-0420.2010.01541.x
[13]
da Silva, J.L.P., Colosimo, E.A. and Demarqui, F.N. (2015) Doubly Robust-Based Generalized Estimating Equations for the Analysis of Longitudinal Ordinal Missing Data. arXiv preprint arXiv:1506.04451.
[14]
Toledano, A.Y. and Gatsonis, C. (1999) Generalized Estimating Equations for Ordinal Categorical Data: Arbitrary Patterns of Missing Responses and Missingness in a Key Covariate. Biometrics, 55, 488-496.
https://doi.org/10.1111/j.0006-341X.1999.00488.x
[15]
Donneau, A.F., Mauer, M., Molenberghs, G. and Albert, A. (2015) A Simulation Study Comparing Multiple Imputation Methods for Incomplete Longitudinal Ordinal Data. Communications in Statistics-Simulation and Computation, 44, 1311-1338.
https://doi.org/10.1080/03610918.2013.818690
[16]
Kombo, A.Y., Mwambi, H. and Molenberghs, G. (2017) Multiple Imputation for Ordinal Longitudinal Data with Monotone Missing Data Patterns. Journal of Applied Statistics, 44, 270-287. https://doi.org/10.1080/02664763.2016.1168370
[17]
Agresti, A. (1989) Tutorial on Modeling Ordered Categorical Response Data. Psychological Bulletin, 105, 290. https://doi.org/10.1037/0033-2909.105.2.290
[18]
Lui, I. and Agresti, A. (2005) The Analysis of Ordered Categorical Data: An Overview and a Survey of Recent Developments. Test, 14, 1-73.
https://doi.org/10.1007/BF02595397
[19]
McCullagh, P. (1980) Regression Models for Ordinal Data. Journal of the Royal Statistical Society. Series B (Methodological), 42, 109-142.
[20]
Das, S. and Rahman, R.M. (2011) Application of Ordinal Logistic Regression Analysis in Determining Risk Factors of Child Malnutrition in Bangladesh. Nutrition Journal, 10, 124. https://doi.org/10.1186/1475-2891-10-124
[21]
Bender, R. and Grouven, U. (1998) Using Binary Logistic Regression Models for Ordinal Data with Non-Proportional Odds. Journal of Clinical Epidemiology, 51, 809-816. https://doi.org/10.1016/S0895-4356(98)00066-3
[22]
Wedderburn, R.W. (1974) Quasi-Likelihood Functions, Generalized Linear Models, and the Gauss-Newton Method. Biometrika, 61, 439-447.
[23]
McCullagh, P. and Nelder, J.A. (1989) Generalized Linear Models. No. 37 in Monograph on Statistics and Applied Probability, Chapman & Hall, London.
[24]
Lipsitz, S.R., Kim, K. and Zhao, L. (1994) Analysis of Repeated Categorical Data Using Generalized Estimating Equations. Statistics in Medicine, 13, 1149-1163.
https://doi.org/10.1002/sim.4780131106
[25]
Touloumis, A., Agresti, A. and Kateri, M. (2013) GEE for Multinomial Responses Using a Local Odds Ratios Parameterization. Biometrics, 69, 633-640.
https://doi.org/10.1111/biom.12054
[26]
Schafer, J.L. (2003) Multiple Imputation in Multivariate Problems When the Imputation and Analysis Models Differ. Statistica Neerlandica, 57, 19-35.
https://doi.org/10.1111/1467-9574.00218
[27]
Beunckens, C., Sotto, C. and Molenberghs, G. (2008) A Simulation Study Comparing Weighted Estimating Equations with Multiple Imputation Based Estimating Equations for Longitudinal Binary Data. Computational Statistics & Data Analysis, 52, 1533-1548. https://doi.org/10.1016/j.csda.2007.04.020
[28]
Satty, A., Mwambi, H. and Molenberghs, G. (2015) Different Methods for Handling Incomplete Longitudinal Binary Outcome Due to Missing at Random Dropout. Statistical Methodology, 24, 12-27. https://doi.org/10.1016/j.stamet.2014.10.002
[29]
Xie, F. and Paik, M.C. (1997) Multiple Imputation Methods for the Missing Covariates in Generalized Estimating Equation. Biometrics, 53, 1538-1546.
https://doi.org/10.2307/2533521
[30]
Moleberghs, G. and Verbeke, G. (2005) Models for Discrete Longitudinal Data. Springer, Berlin.
[31]
Scharfstein, D.O., Rotnitzky, A. and Robins, J.M. (1999) Adjusting for Nonignorable Drop-Out Using Semiparametric Nonresponse Models. Journal of the American Statistical Association, 94, 1096-1120.
https://doi.org/10.1080/01621459.1999.10473862
[32]
Touloumis, A. (2016) Simulating Correlated Binary and Multinomial Responses under Marginal Model Specification: The SimCorMultRes Package. The R Journal, 8, 79-91. https://journal.r-project.org/archive/2016/RJ-2016-034/index.html
[33]
R Core Team (2013) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. http://www.R-project.org/
[34]
Collins, L.M., Schafer, J.L. and Kam, C.-M. (2001) A Comparison of Inclusive and Restrictive Strategies in Modern Missing Data Procedures. Psychological Methods, 6, 330-351. https://doi.org/10.1037/1082-989X.6.4.330
[35]
Burton, A., Altman, D.G., Royston, P. and Holder, R.L. (2006) The Design of Simulation Studies in Medical Statistics. Statistics in Medicine, 25, 4279-4292.
https://doi.org/10.1002/sim.2673
[36]
Schafer, J.L. and Graham, J.W. (2002) Missing Data: Our View of the State of the Art. Psychological Methods, 7, 147-177. https://doi.org/10.1037/1082-989X.7.2.147
[37]
Van Buuren, S. (2007) Multiple Imputation of Discrete and Continuous Data by Fully Conditional Specification. Statistical Methods in Medical Research, 16, 219-242. https://doi.org/10.1177/0962280206074463
[38]
Pawitan, Y. (2001) In All Likelihood: Statistical Modelling and Inference Using Likelihood. Oxford University Press, Oxford.
[39]
Hogan, J.W., Roy, J. and Korkontzelou, C. (2004) Handling Drop-Out in Longitudinal Studies. Statistics in Medicine, 23, 1455-1497.
https://doi.org/10.1002/sim.1728
[40]
Dong, Y. and Peng, C.-Y.J. (2013) Principled Missing Data Methods for Researchers. SpringerPlus, 2, 222. https://doi.org/10.1186/2193-1801-2-222
[41]
Jolani, S., Van Buuren, S. and Frank, L.E. (2013) Combining the Complete-Data and Nonresponse Models for Drawing Imputations under MAR. Journal of Statistical Computation and Simulation, 83, 868-879.
https://doi.org/10.1080/00949655.2011.639773
[42]
Rotnitzky, A., Robins, J.M. and Scharfstein, D.O. (1998) Semiparametric Regression for Repeated Outcomes with Nonignorable Nonresponse. Journal of the American Statistical Association, 93, 1321-1339.
https://doi.org/10.1080/01621459.1998.10473795
[43]
Vansteelandt, S., Rotnitzky, A. and Robins, J. (2007) Estimation of Regression Models for the Mean of Repeated Outcomes under Nonignorable Nonmonotone Nonresponse. Biometrika, 94, 841-860. https://doi.org/10.1093/biomet/asm070
