Joint Generalized Estimating Equations for Multivariate Longitudinal Binary Outcomes with Missing Data: An Application to Acquired Immune Deficiency Syndrome Data

Abstract

In a large, prospective longitudinal study designed to monitor cardiac abnormalities in children born to HIV-infected women, instead of a single outcome variable, there are multiple binary outcomes (e.g., abnormal heart rate, abnormal blood pressure, abnormal heart wall thickness) considered as joint measures of heart function over time. In the presence of missing responses at some time points, longitudinal marginal models for these multiple outcomes can be estimated using generalized estimating equations (GEE) (Liang and Zeger, 1986), and consistent estimates can be obtained under the assumption of a missing completely at random (MCAR) mechanism. When the missing data mechanism is missing at random (MAR), that is the probability of missing a particular outcome at a time-point depends on observed values of that outcome and the remaining outcomes at other time points, we propose joint estimation of the marginal models using a single modified GEE based on an EM-type algorithm. The proposed method is motivated by the longitudinal study of cardiac abnormalities in children born to HIV-infected women and analyses of these data are presented to illustrate the application of the method. Further, in an asymptotic study of bias, we show that under an MAR mechanism in which missingness depends on all observed outcome variables, our joint estimation via the modified GEE produces almost unbiased estimates, provided the correlation model has been correctly specified, whereas estimates from standard GEE can lead to substantial bias.