Sequential models for coarsening and missingness

Abstract

In a companion paper we described what intuitively would seem to be the most general possible way to generate Coarsening at Random mechanisms a sequential procedure called randomized monotone coarsening Counterexamples showed that CAR mechanisms exist which cannot be represented in this way Here we further develop these results in two directions Firstly we consider what happens when data is coarsened at random in two or more phases We show that the resulting coarsening mechanism is not CAR anymore but under suitable assumptions is identied and can provide interesting alternative analysis of data under a nonCAR model Secondly we look at sequential mechanisms for generating MAR data missing components of a multivariate random vector Randomised monotone missingness schemes in which one variable at a time is observed and depending on its value another variable is chosen or the procedure is terminated supply in our opinion the broadest class of physically interpretable MAR mechanisms We show that every randomised monotone missingness scheme can be represented by a Markov monotone missingness scheme in which the choice of which variable to observe next only depends on the set of previously observed variables and their values not on the sequence in which they were measured We also show that MAR mechanisms exist which cannot be represented sequentially