Figure 1 depicts the sample frequency distributions of four health-related outcomes:

Panel (a): the number of days on which adult respondents reported engaging in vigorous or moderate exercise in the prior week (VM7; 2015 Health Survey of England (HSE))

Panel (b): the number of chronic conditions from a list of eight, reported by adult respondents (CC8; 2021 Behavioral Risk Factors Surveillance System (BRFSS))

Panel (c): the number of physically healthy days in the previous 30 days, reported by adult respondents (HD30; 2021 BRFSS)

Panel (d): the PHQ-2 depression screener score for adult respondents (PHQ2; 2020 Medical Expenditure Panel Survey (MEPS))

Although the health phenomena they describe are quite different, these four outcomes share two features that define this paper's main focus: they are measured as nonnegative integers and they are bounded from below and above. While many familiar health measures possess both measurement attributes it will turn out that some of these, like the PHQ-2 score in panel (d), have essential measurement properties that differentiate them from those shown in panels (a)-(c).

This paper assesses analytical strategies designed to respect the bounded count structures of outcomes like those whose distributions are shown in panels (a)-(c). As emphasized in what follows, a wide range of questions may be asked of such data. It will be shown that for some purposes bounded count outcomes (BCOs) pose few special analytical considerations while for others the measurement structure of BCOs raises distinct analytical issues.1

In what follows: section 2 describes the measurement and structure of BCO data and provides the paper's key assumptions, considers evaluation and treatment effects when the outcomes of interest are bounded counts, and addresses some particular data structures and estimation considerations; section 3 considers specification and estimation of models of BCO distributions' conditional moments; section 4 explores specification and estimation of BCO conditional probability models; section 5 explores in depth the nature of conditional moment and conditional probability partial effects, as these are often of primary interest in empirical work; and section 6 concludes. Most technical material is relegated to the online Supplementary Material. Strategies for implementing the paper's main ideas and corresponding Stata programs are also discussed and provided in the Supplementary Material.