Selection correction and sensitivity analysis for ordered treatment effect on count response

In estimating the effect of an ordered treatment τ on a count response y with an observational data where τ is self-selected (not randomized), observed variables x and unobserved variables ε can be unbalanced across the control group (τ = 0) and the treatment groups (τ = 1,..., J). While the imbalance in x causes 'overt bias' which can be removed by controlling for x, the imbalance in ε causes 'covert (hidden or selection) bias' which cannot be easily removed. This paper makes three contributions. First, a proper counter-factual causal framework for ordered treatment effect on count response is set up. Second, with no plausible instrument available for τ, a selection correction approach is proposed for the hidden bias. Third, a nonparametric sensitivity analysis is proposed where the treatment effect is nonparametrically estimated under no hidden bias first, and then a sensitivity analysis is conducted to see how sensitive the nonparametric estimate is to the assumption of no hidden bias. The analytic framework is applied to data from the Health and Retirement Study: the treatment is ordered exercise levels in five categories and the response is doctor office visits per year. The selection correction approach yields very large effects, which are however ruled out by the nonparametric sensitivity analysis. This finding suggests a good deal of caution in using selection correction approaches.