Gaussianization-based Quasi-imputation and Expansion Strategies for Incomplete Correlated Binary Responses

New quasi-imputation and expansion strategies for correlated binary responses are proposed by borrowing ideas from random number generation. The core idea is to convert correlated binary outcomes to multivariate normal outcomes in a sensible way so that re-conversion to the binary scale, after performing multiple imputation, yields the original specified marginal expectations and correlations. This conversion process ensures that the correlations are transformed reasonably which in turn allows us to take advantage of well-developed imputation techniques for Gaussian outcomes. We use the phrase 'quasi' because the original observations are not guaranteed to be preserved. We argue that if the inferential goals are well-defined, it is not necessary to strictly adhere to the established definition of multiple imputation. Our expansion scheme employs a similar strategy where imputation is used as an intermediate step. It leads to proportionally inflated observed patterns, forcing the data set to a complete rectangular format. The plausibility of the proposed methodology is examined by applying it to a wide range of simulated data sets that reflect alternative assumptions on complete data populations and missing-data mechanisms. We also present an application using a data set from obesity research. We conclude that the proposed method is a promising tool for handling incomplete longitudinal or clustered binary outcomes under ignorable non-response mechanisms.