Statistical Inference Under Nonignorable Sampling and Nonresponse—An Empirical Likelihood Approach

Statistical models are often based on sample surveys. When the sample selection probabilities and/or the response probabilities are related to a model outcome variable, even after conditioning on the model covariates, the model holding for the observed data is different from the model holding in the population, resulting in biased inference if not accounted for properly. Accounting for sample selection bias is relatively simple because the sample selection probabilities are usually known. Accounting for nonignorable nonresponse is much harder since the response probabilities are, in practice, unknown. In this article, we develop a new approach for modelling complex survey data, which accounts simultaneously for nonignorable sampling and nonresponse. Our proposed approach combines the nonparametric empirical likelihood with a parametric model for the response probabilities, which contains the outcome variable as one of the covariates. Combining the model holding for the responding units with the model for the response probabilities enables extracting the model holding for the missing data and imputing them. We propose ways of testing the underlying model holding for the respondents’ data. Simulation results illustrate the good performance of the approach in terms of parameter estimation and imputation. We conclude with an application to the household expenditure survey in Israel, carried out by Israel’s Central Bureau of Statistics. The survey collects information on the socio-demographic characteristics of each member of the sampled households (HH), as well as detailed information on the HH income and expenditure. The total sample size was n = 12,136 with 7,827 responding HHs. The target estimated parameter in this application is the population mean of the gross HH income.