Contribution to Discussion of “Eelworms, Bullet Holes, and Geraldine Ferraro”

Abstract

Howard Wainer (1989) is to be thanked for his survey of some of the many problems awaiting those who attempt to model their ignorance. I liked his examples and generally agreed with his conclusions, which were on the whole cautionary. I did wonder, however, about the basis of his optimistic statements, such as thanks to the ingenious efforts of many statisticians.., .in many circumstances, there are enough tools available for us to do very well indeed and that augmenting traditional adjustment methods are some model-based schemes that can provide much-improved estimates. My comments relate to three points: the relations between different procedures for handling nonresponse, the paucity of examples of these procedures in action, and their dependence on uncheckable assumptions. In his section on modeling nonresponse, Wainer writes of two procedures enjoying popularity currently-selection modeling and mixture modeling-and presents a third, simplified selection modeling, which, he says, combines the best aspects of both procedures. I would like to draw attention to the fact that all three of these procedures are simply ways of describing the joint distribution p (y,r) of a variable Y of interest and the response indicator R (possibly conditional upon some concomitant variables X). Any mixture model has a corresponding selection model that is equivalent to it, and vice versa, with similar remarks applying to the simplified selection model. These models are distinguished only by the way in which the joint distribution p (y,r) is decomposed and parameterized. To repeat myself, given a mixture model, there is a selection model that produces the same joint distribution for the observables, so the two are distinguishable. In a discussion at the conference on which Wainer (1986) (and much of this paper) is based, Rubin conceded as much to Hartigan (p. 146), saying that selection and mixture modeling are not different in principle, but they are different in practice. This may well be true, but only because users of the two approaches are led to devise inequivalent models for p(y,r), because of anything to do with their preferred way of thinking about the model. There seem to be no studies where these models or methods have been