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Tang, G., R. J A Little, and Trivellore Raghunathan. 2003. "Analysis of Multivariate Missing Data With Nonignorable Nonresponse." Biometrika, 90:747-764.
We consider multivariate regression analysis with missing data in the outcome variables, when the nonresponse mechanism depends on the underlying values of the responses and hence is nonignorable. Related problems include response-biased sampling where data are sampled with probability depending only on the univariate response. Our methods do not require specification of the form of the nonresponse mechanism. We show that, under certain regularity conditions, all the regression parameters can be identified from a conditional likelihood based on the complete cases, if the marginal distribution of the covariates is known. If the marginal distribution of the covariates is estimated from the data, then the regression parameters are identified from a pseudolikelihood resulting from substituting the estimated marginal distribution of the covariates in the above conditional likelihood. Simulation studies suggest that the pseudolikelihood method is approximately unbiased. In order to identify the model parameters, usually the dimension of the covariates and observed responses is required to be at least as large as the dimension of the missing responses. The method can also be modified to handle partial information about the missing-data mechanism. We also consider the special case where the missing data have a monotone pattern, where better use of the incomplete information can be made under certain assumptions.