Estimation in longitudinal studies with nonignorable dropout

A sampled subject with repeated measurements often drops out prior to the study end. Data observed from such a subject is longitudinal with monotone missing. If dropout at a time point t is only related to past observed data from the response variable, then it is ignorable and statistical methods are well developed. When dropout is related to the possibly missing response at t even after conditioning on all past observed data, it is nonignorable and statistical analysis is difficult. Without any further assumption, unknown parameters may not be identifiable when dropout is nonignorable. We develop a semiparametric pseudo likelihood method that produces consistent and asymptotically normal estimators under nonignorable dropout with the assumption that there exists a dropout instrument, a covariate related to the response variable but not related to the dropout conditioned on the response and other covariates. Although consistency and asymptotic normality for the proposed estimators can be established using a standard argument, their asymptotic covariance matrices are very complicated because the estimation at t uses estimators from all time prior to t. Our main effort is to derive easy-to-compute consistent estimators of the asymptotic covariance matrices for assessing variability or inference. For illustration, we present an example using the HIV-CD4 data and some simulation results.