A New Approach to Estimation of Response Probabilities when Missing Data are Not missing at Random

Often the probability of responding depends directly on the outcome value. This case can be treated by postulating a parametric model for the distribution of the outcomes before nonresponse and a model for the response mechanism. The two models define a parametric model for the joint distribution of the outcomes and response indicators, and therefore the parameters of these models can be estimated by maximization of the likelihood corresponding to this distribution. Modeling the distribution of the outcomes before nonresponse, however, can be problematic since no data is available from this distribution. We propose an alternative approach that allows estimation of the parameters of the response model by first estimating the outcomes distribution of the respondents, and then solving an estimating equation defined by the census likelihood of the response indicators.