The Annals of Statistics

Ignorability and Coarse Data

Daniel F. Heitjan and Donald B. Rubin

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We present a general statistical model for data coarsening, which includes as special cases rounded, heaped, censored, partially categorized and missing data. Formally, with coarse data, observations are made not in the sample space of the random variable of interest, but rather in its power set. Grouping is a special case in which the degree of coarsening is known and nonstochastic. We establish simple conditions under which the possible stochastic nature of the coarsening mechanism can be ignored when drawing Bayesian and likelihood inferences and thus the data can be validly treated as grouped data. The conditions are that the data be coarsened at random, a generalization of the condition missing at random, and that the parameters of the data and the coarsening process be distinct. Applications of the general model and the ignorability condition are illustrated in a numerical example and described briefly in a variety of special cases.

Article information

Ann. Statist., Volume 19, Number 4 (1991), 2244-2253.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F99: None of the above, but in this section
Secondary: 62A10 62A15

Censord data coarsened at random grouped data heaped data interval-censored data missing at random missing data rounded data


Heitjan, Daniel F.; Rubin, Donald B. Ignorability and Coarse Data. Ann. Statist. 19 (1991), no. 4, 2244--2253. doi:10.1214/aos/1176348396.

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