Coarsening at random: characterizations, conjectures and counter-examples

The notion of coarsening at random CAR was introduced by Heitjan and Rubin to describe the most general form of randomly grouped censored or missing data for which the coarsening mechanism can be ignored when making likelihoodbased inference about the parameters of the distribution of the variable of interest The CAR assumption is popular and applications abound However the full implications of the assumption have not been realized Moreover a satisfactory theory of CAR for continuously distributed datawhich is needed in many applications particularly in survival analysishardly exists as yet This paper gives a detailed study of CAR We show that grouped data from a nite sample space always t a CAR model a nonparametric model for the variable of interest together with the assumption of an arbitrary CAR mechanism puts no restriction at all on the distribution of the observed data In a slogan CAR is everythingWe describe what would seem to be the most general way CAR data could occur in practice a sequential procedure called randomized monotone coarseningWe show that CAR mechanisms exist which are not of this type Such a coarsening mechanism uses information about the underlying data which is not revealed to the observer without this affecting the observers conclusions In a second slogan CAR is more than it seems This implies that if the analyst can argue from subjectmatter considerations that coarsened data is CAR he or she has knowledge about the structure of the coarsening mechanism which can be put to good use in nonlikelihoodbased inference procedures We argue that this is a valuable option in multivariate survival analysis We give a new denition of CAR in general sample spaces criticising earlier proposals and we establish parallel results to the discrete case The new denition focusses on the distribution rather than the density of the data It allows us to generalise the theory of CAR to the important situation where coarsening variables eg censoring times are partially observed as well as the variables of interes