Response probability estimation

This paper extends the ideas in Giommi (Proc. 45th Session of the Internat. Statistical Institute, Vol. 2 (1985) 577–578; Techniques d'enquête 13(2) (1987) 137–144) and, in Särndal and Swenson (Bull. Int. Statist. Inst. 15(2) (1985) 1–16; Int. Statist. Rev. 55(1987) 279–294). Given the parallel between a ‘three-phase sampling’ and a ‘sampling with subsequent unit and item nonresponse’, we apply results from three-phase sampling theory to nonresponse situation. To handle the practical problem of unknown distributions at the second and the third phases of selection (the response mechanisms) in the nonresponse case, we use two approaches of response probability estimation: response homogeneity groups (RHG) model (Särndal and Swenson, 1985, 1987) and the nonparametric estimation (Giommi, 1985, 1987). To motivate the three-phase selection, imputation procedures for item nonresponse are used with the RHG model for unit nonresponse. By means of a Monte Carlo study, we find that the regression-type estimators are the most precise of those studied under the two approaches of response probability estimation in terms of lower bias, mean square error and variance; variance estimator close to the true variance and achieved coverage rates closer to the nominal levels. The simulation study shows how poor the variance estimators are under the single imputation approach currently used to handle the problem of missing values.