Ignorability and coarse data: some biomedical examples

Heitjan and Rubin (1991, Annals of Statistics 19, 2244-2253) define data to be "coarse" when one observes not the exact value of the data but only some set (a subset of the sample space) that contains the exact value. This definition covers a number of incomplete-data problems arising in biomedicine, including rounded, heaped, censored, and missing data. In analyzing coarse data, it is common to proceed as though the degree of coarseness is fixed in advance--in a word, to ignore the randomness in the coarsening mechanism. When coarsening is actually stochastic, however, inferences that ignore this randomness may be seriously misleading. Heitjan and Rubin (1991) have proposed a general model of data coarsening and established conditions under which it is appropriate to ignore the stochastic nature of the coarsening. The conditions are that the data be coarsened at random [a generalization of missing at random (Rubin, 1976, Biometrika 63, 581-592)] and that the parameters of the data and the coarsening process be distinct. This article presents detailed applications of the general model and the ignorability conditions to a variety of coarse-data problems arising in biomedical statistics. A reanalysis of the Stanford Heart Transplant Data (Crowley and Hu, 1977, Journal of the American Statistical Association 72, 27-36) reveals significant evidence that censoring of pretransplant survival times by transplantation was nonignorable, suggesting a greater benefit from cardiac transplantation than previous analyses had found.