Application of potential outcomes to an intentional weight loss latent variable problem

Studies that explore the link between weight loss among obese individuals and mortality have met with mixed results. One possible explanation is that total weight loss may have contributions from weight loss that is intentional and weight loss that is unintentional. The latter may be due to some underlying condition that has a deleterious effect on subsequent mortality. Some studies have then focused on subjects who intend to lose weight. However, in a population there is no guarantee that weight loss among these individuals is due only to their intention. This paper extends the work of Coffey et al., (2005) who treated intentional weight loss as a latent variable. In particular, the problem is reformulated using potential outcomes. This formulation more clearly identifies a nonestimable correlation that arises because of the latent variable, and it allows for the incorporation of covariate information that can tighten estimable bounds for this correlation. We show in a data set from an experiment on mice that substantial tightening of bounds is possible with a covariate that is predictive of weight loss. These bounds can then, in turn, be used to estimate bounds on a causal parameter in a linear model.