We study the problem of ignorability in likelihood-based inference from incomplete categorical data. Two versions of the coarsened at random assumption (car) are distinguished, their compatibility with the parameter distinctness assumption is investigated and several conditions for ignorability that do not require an extra parameter distinctness assumption are established.
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It is shown that car assumptions have quite different implications depending on whether the underlying complete-data model is saturated or parametric. In the latter case, car assumptions can become inconsistent with observed data.