Estimated conditional score function for missing mechanism model with nonignorable nonresponse

Missing data mechanism often depends on the values of the responses, which leads to nonignorable nonresponses. In such a situation, inference based on approaches that ignore the missing data mechanism could not be valid. A crucial step is to model the nature of missingness. We specify a parametric model for missingness mechanism, and then propose a conditional score function approach for estimation. This approach imputes the score function by taking the conditional expectation of the score function for the missing data given the available information. Inference procedure is then followed by replacing unknown terms with the related nonparametric estimators based on the observed data. The proposed score function does not suffer from the non-identifiability problem, and the proposed estimator is shown to be consistent and asymptotically normal. We also construct a confidence region for the parameter of interest using empirical likelihood method. Simulation studies demonstrate that the proposed inference procedure performs well in many settings. We apply the proposed method to a data set from research in a growth hormone and exercise intervention study.