# Sample-constrained partial identification with application to selection bias

## TL;DR

This paper develops a statistical inference method for partial identification problems involving optimization over estimated sets, with applications to selection bias and causal inference, providing more informative bounds and practical implementation.

## Contribution

It introduces a new asymptotic confidence interval for optimal values in partial identification, applicable to selection bias, and enhances sensitivity analysis with auxiliary data.

## Key findings

- The method produces valid confidence intervals for the optimal value.
- It yields more informative bounds using auxiliary population information.
- The approach is demonstrated with simulations and real data from UK Biobank.

## Abstract

Many partial identification problems can be characterized by the optimal value of a function over a set where both the function and set need to be estimated by empirical data. Despite some progress for convex problems, statistical inference in this general setting remains to be developed. To address this, we derive an asymptotically valid confidence interval for the optimal value through an appropriate relaxation of the estimated set. We then apply this general result to the problem of selection bias in population-based cohort studies. We show that existing sensitivity analyses, which are often conservative and difficult to implement, can be formulated in our framework and made significantly more informative via auxiliary information on the population. We conduct a simulation study to evaluate the finite sample performance of our inference procedure and conclude with a substantive motivating example on the causal effect of education on income in the highly-selected UK Biobank cohort. We demonstrate that our method can produce informative bounds using plausible population-level auxiliary constraints. We implement this method in the R package selectioninterval.