Doubly Robust Imputation of Incomplete Binary Longitudinal Data

Estimation in binary longitudinal data by using generalized estimating equation (GEE) becomes complicated in the presence of missing data because standard GEEs are only valid under the restrictive missing completely at random assumption. Weighted GEE has therefore been proposed to allow the validity of GEE's under the weaker missing at random assumption. Multiple imputation offers an attractive alternative, by which the incomplete data are pre-processed, and afterwards the standard GEE can be applied to the imputed data. Nevertheless, the imputation methodology requires correct specification of the imputation model. Dual imputation method provides a new way to increase the robustness of imputations with respect to model misspecification. The method involves integrating the so-called doubly robust ideas into the imputation model. Focusing on incomplete binary longitudinal data, we combine DIM and GEE (DIM-GEE) and study the relative performance of the new method in a case study of obesity among children, as well as a simulation study. Citation: Jolani S, van Buuren S (2014) Doubly Robust Imputation of Incomplete Binary Longitudinal Data. J Biomet Biostat 5: 194. doi:10.4172/21556180.1000194 J Biomet Biostat ISSN: 2155-6180 JBMBS, an open access journal Page 2 of 6 Volume 5 • Issue 3 • 1000194 This paper is organized as follows. In GEE for binary longitudinal data, an overview of GEE for analyzing longitudinal binary data is given. MI is briefly outlined in multiple imputation. The new imputation method is presented in dual imputation based GEE. In case study, DIM-GEE is used to analyze a case study of obesity among children and to compare with MI-GEE. A simulation study comparing DIM-GEE with MI-GEE was conducted and results are presented in simulation study. GEE for Binary Longitudinal Data Suppose the random variable Yij denotes a sequence of binary measurements at time j, j = 1,...,N for subject i, i = 1,...,n. The observed value yij is a realization of the binary response variable Yij, and we assume independence across subjects. The focus of this study is on the marginal models that describe the binary outcome vector, given a set of predictor variables. The association structure (correlation among the components) is captured by an assumed model. Let πij denote the marginal probability of observing a ‘success' for subject i at time j, i.e., πij = E(Yij) = P(Yij = 1), (1 ) ij ij ij ij ij Y π ε π π − = − be standardized deviation between the data and the model predictor for subject i at time j, and 1 2 1 2 1 2 3 1 2 3 ( ), ( ),... ij j ij ij ij j j ij ij ij E E ρ ε ε ρ ε ε ε = = be associations among responses. Following Bahadur [15], the model can be represented by 1 1 ( ) ( ) (1 ) , π π − = = ∏ − ij ij y y N i i j ij ij f c y y (1) where yi = (yi1,...,yiN) is a vector of measurements for subject i, and 1 2 1 2 3 1 2 1 2 3 1 2 1 2 3 1... 1 ( ) 1 ... ... , ij ij ij ij ij i ij j e e ij j j e e e i N i iN j j j j j c e e ρ ρ ρ < < < = + + + + ∑ ∑ y