Estimates for small area compositions subjected to informative missing data

Estimation of small area (or domain) compositions may suffer from informative missing data, if the probability of missing varies across the categories of interest as well as the small areas. We develop a double mixed modeling approach that combines a random effects mixed model for the underlying complete data with a random effects mixed model of the differential missing-data mechanism. The effect of sampling design can be incorporated through a quasi-likelihood sampling model. The associated conditional mean squared error of prediction is approximated in terms of a three-part decomposition, corresponding to a naive prediction variance, a positive correction that accounts for the hypothetical parameter estimation uncertainty based on the latent complete data, and another positive correction for the extra variation due to the missing data. We illustrate our approach with an application to the estimation of Municipality household compositions based on the Norwegian register household data, which suffer from informative under-registration of the dwelling identity number.