The derivation of a latent threshold instrumental variables model

Measuring the causal effect of a treatment from observational data is often difficult because the treatment status of a subject may be confounded with nonrandomized factors, such as those that affect a subject's choice of treatment. An approach to remedying this problem is through the use of instrumental variables. We extend the instrumental variables framework proposed by Angrist, Imbens and Rubin (1996) by introducing a latent "threshold to receive treatment" parameter for each unit in the study. Incorporation of latent thresholds in the model permits the inclusion of discrete or continuous instruments, covariate information, and flexible choices of distributions. We apply our methodology to examine the effect of cardiac catheterization on short-term survival of a cohort of elderly heart attack patients.