Quality-of-life outcomes collected during clinical trials often have considerable amounts of missing data, which, if not appropriately accounted for, may lead to bias in inferences. We introduce a method-of-moments (MM) estimating procedure for a model designed to handle nonignorable missingness arising in categorical data measured on independent populations. The missingness mechanism is assumed to be the same across the populations. We derive necessary and sufficient conditions for the identifiability of the model and fit the model to quality-of-life data collected as part of a breast cancer clinical trial. We compare the MM estimator to the maximum likelihood estimator in a simulation study.