On the testability of coarsening assumptions: A hypothesis test for subgroup independence

Since coarse(ned) data naturally induce set-valued estimators, analysts often assume coarsening at random (CAR) to force them to be single-valued. Focusing on a coarse categorical response variable and a precisely observed categorical covariate, we first re-illustrate the impossibility to test CAR and then contrast it to another type of coarsening called subgroup independence (SI). It turns out that - depending on the number of subgroups and categories of the response variable - SI can be point-identifying as CAR, but testable unlike CAR. A main goal of this paper is the construction of the likelihood-ratio test for SI. All issues are similarly investigated for the here proposed generalized versions, gCAR and gSI, thus allowing a more flexible application of this hypothesis test. The results are illustrated by the data of the German Panel Study “Labour Market and Social Security” (PASS).