Two-part models describe situations in which the ordered choice is part of a two-stage decision process. In a typical situation, an individual decides whether or not to participate in an activity, then, if so, decides how much. The first decision is a binary choice. The intensity outcome can be of several types – what interests us here is an ordered choice. In the example below, an individual decides whether or not to be a smoker. The intensity outcome is how much they smoke. The sample selection model is one in which the participation “decision” relates to whether the data on the outcome variable will be observed, rather than whether the activity is undertaken. This chapter will describe several types of two-part and sample selection models.

Inflation models

Harris and Zhao (2007) analyzed a sample of 28,813 Australian individuals' responses to the question “How often do you now smoke cigarettes, pipes or other tobacco products?” (Data are from the Australian National Drug Strategy Household Survey, NDSHS (2001).) Responses were “zero, low, moderate, high,” coded 0, 1, 2, 3. The sample frequencies of the four responses were 0.75, 0.04, 0.14, and 0.07. The spike at zero shows a considerable excess of zeros compared to what might be expected in an ordered choice model. The authors reason that there are numerous explanations for a zero response: “genuine nonsmokers, recent quitters, infrequent smokers who are not currently smoking and potential smokers who might smoke when, say, the price falls.” It is also possible that the zero response includes some individuals who prefer to identify themselves as nonsmokers. The question is ambiguously worded, but arguably, the group of interest is the genuine nonsmokers.