Proxy Pattern-Mixture Analysis for a Binary Variable Subject to Nonresponse

Given increasing survey nonresponse, good measures of the potential impact of nonresponse on survey estimates are particularly important. Existing measures, such as the R-indicator, make the strong assumption that missingness is missing at random, meaning that it depends only on variables that are observed for respondents and nonrespondents. We consider assessment of the impact of nonresponse for a binary survey variable Y subject to nonresponse when missingness may be not at random, meaning that missingness may depend on Y itself. Our work is motivated by missing categorical income data in the 2015 Ohio Medicaid Assessment Survey (OMAS), where whether or not income is missing may be related to the income value itself, with low-income earners more reluctant to respond. We assume there is a set of covariates observed for nonrespondents and respondents, which for the item nonresponse (as in OMAS) is often a rich set of variables, but which may be potentially limited in cases of unit nonresponse. To reduce dimensionality and for simplicity we reduce these available covariates to a continuous proxy variable X, available for both respondents and nonrespondents, that has the highest correlation with Y, estimated from a probit regression analysis of respondent data. We extend the previously proposed proxy-pattern mixture (PPM) analysis for continuous outcomes to the binary outcome using a latent variable approach for modeling the joint distribution of Y and X. Our method does not assume data are missing at random but includes it as a special case, thus creating a convenient framework for sensitivity analyses. Maximum likelihood, Bayesian, and multiple imputation versions of PPM analysis are described, and robustness of these methods to model assumptions is discussed. Properties are demonstrated through simulation and with the 2015 OMAS data.