We consider data-analysis settings where data are missing not at random. In these cases, the two basic modeling approaches are 1) pattern-mixture models, with separate distributions for missing data and observed data, and 2) selection models, with a distribution for the data preobservation and a missing-data mechanism that selects which data are observed. These two modeling approaches lead to distinct factorizations of the joint distribution of the observed-data and missing-data indicators. In this paper, we explore a third approach, apparently originally proposed by J. W. Tukey as a remark in a discussion between Rubin and Hartigan, and reported by Holland in a two-page note, which has been so far neglected.Data analyses typically rely upon assumptions about the missingness mechanisms that lead to observed versus missing data, assumptions that are typically unassessable. We explore an approach where the joint distribution of observed data and missing data are specified in a nonstandard way. In this formulation, which traces back to a representation of the joint distribution of the data and missingness mechanism, apparently first proposed by J. W. Tukey, the modeling assumptions about the distributions are either assessable or are designed to allow relatively easy incorporation of substantive knowledge about the problem at hand, thereby offering a possibly realistic portrayal of the data, both observed and missing. We develop Tukeytextquoterights representation for exponential-family models, propose a computationally tractable approach to inference in this class of models, and offer some general theoretical comments. We then illustrate the utility of this approach with an example in systems biology.All raw input and processed output data are available in Dryad (DOI: 10.5061/dryad.rg367 and DOI: 10.5061/dryad.d644f).