Log-linear models for a binary response with nonignorable nonresponse

We focus on the analysis of contingency tables that contain both completely and partially cross-classified data where one of the variables is a binary response variable subject to nonignorable non-response and the other variables are always observed. Log-linear models can be used to adjust for nonresponse in an augmented table where one index corresponds to whether or not the subject is a respondent. The maximum likelihood estimators (MLEs) of the expected cell frequencies in the augmented table can be obtained iteratively using the EM algorithm. ML estimation may yield boundary estimators for the expected frequencies of the nonrespondents' cells. As a result, the odds ratio which represents the nonresponse mechanism is estimated either as 0 or infinity and the estimate of the corresponding log-linear model parameter is infinite. Since such an extreme estimate is not anticipated, it is desirable to use a constraint which limits the range of the odds ratio. We propose a constrained maximum likelihood estimator which maximizes the likelihood function subject to a constraint on the odds ratio. Through a simulation study, the proposed estimators of the expected cell frequencies of the nonrespondents are shown to have smaller mean square errors than those of the MLEs. This estimation procedure is illustrated using polling data in Baker and Laird (1988). In addition, we provide some recommendations on how to choose constraints for practical applications.