Imputation using response probability

Abstract

The authors propose a new ratio imputation method using response probability. Their estimator can be justified either under the response model or under the imputation mo del; it is thus doubly protected against the failure of either of these models. The authors also propose a variance estimator that can be justified under the two models. Their methodology is applicable whether the re sponse probabilities are estimated or known. A small simulation study illustrates their technique. Utilisation de probabilit ´ es de r´ eponse ` a des fins d'imputation R´´ e : Les auteurs proposent une nouvelle md'imputation par quotient bassur les probabilit ´ es Imputation is a commonly used method of compensating for item nonresponse in sample sur- veys. Reasons for conducting imputation are to facilitate analyses using complete data analysis methods, to ensure that the results obtained by different an alyses are consistent with one another, and to reduce nonresponse bias. Kalton (1983) and Groves, Dillman, Eltinge & Little (2002) provide a comprehensive overview of imputation methods in survey sampling. Many imputation methods such as ratio imputation or regression imputation use auxiliary information that is observed throughout the sample. Such imputation methods require assump- tions about the distribution of the study variable. The imputation model refers to the assumptions about the variables collected in the survey and the relation ship among these variables. Another model, called the response model, is also commonly adopted in the analysis of missing data. The response model refers to the assumptions about the probability of obtaining responses from the sample for the item. One of the commonly used response models is the uniform response model, where the responses are assumed to be independent and identically distributed within the impu- tation cell. Rao & Shao (1992), Rao & Sitter (1995) and Shao & Steel (1999) discuss inference using the imputed estimator under the uniform response model. However, for the other nonuni- form response models such as the logistic response model, imputation methods incorporating the response model are relatively underdeveloped, although analyses incorporating the response model are quite popular in the nonimputation context. Examples include Rosenbaum (1987), Robins, Rotnitzky & Zhao (1994), and Lipsitz, Ibrahim & Zhao (1999). In this article, we provide an imputation methodology that combines the imputation model and the response model. The proposed method can be justified u nder either one of the two approaches. That is, it is justified if either a response mode l or an imputation model can be correctly specified. Thus, the resulting estimator is doubl y protected against the failure of the as- sumed model. (Scharfstein, Rotnitsky & Robins 1999). The basic project is introduced under the ratio imputation model in Section 2. The proposed method is further discussed in Section 3. In Section 4, we propose a replication variance estimator that can be justified under the two models. In Section 5, we discuss the proposed imputation method when the response probabilities are