Exact Binomial Confidence Intervals for Randomized Response

Abstract

We consider the problem of finding an exact confidence interval for a proportion that is estimated using randomized response. For many randomized response schemes, this is equivalent to finding an exact confidence interval for a bounded binomial proportion. Such intervals can be obtained by truncating standard exact binomial confidence intervals, but the truncated intervals may be empty or misleadingly short. We address this problem by using exact confidence intervals obtained by inverting a likelihood ratio test that takes into account that the proportion is bounded. A simple adjustment is made to keep the intervals from being excessively conservative. An R function for computing the intervals is available as online supplementary material.

KEY WORDS:

• Binomial distribution
• Bounded binomial proportion
• Likelihood ratio
• Simple random sampling

SUPPLEMENTARY MATERIALS

R Code: Code for computing the new exact adjusted likelihood ratio confidence intervals. (Code.txt)

This work was done while A. Pérez was a Master’s student at Villanova University. The authors thank the editor, an associate editor, and the two referees for their comments and suggestions.