MOVER-R and Penalized MOVER-R Confidence Intervals for the Ratio of Two Quantities

Abstract

Developing a confidence interval for the ratio of two quantities is an important task in statistics because of its omnipresence in real world applications. For such a problem, the MOVER-R (method of variance recovery for the ratio) technique, which is based on the recovery of variance estimates from confidence limits of the numerator and the denominator separately, was proposed as a useful and efficient approach. However, this method implicitly assumes that the confidence interval for the denominator never includes zero, which might be violated in practice. In this article, we first use a new framework to derive the MOVER-R confidence interval, which does not require the above assumption and covers the whole parameter space. We find that MOVER-R can produce an unbounded confidence interval, just like the well-known Fieller method. To overcome this issue, we further propose the penalized MOVER-R. We prove that the new method differs from MOVER-R only at the second order. It, however, always gives a bounded and analytic confidence interval. Through simulation studies and a real data application, we show that the penalized MOVER-R generally provides a better confidence interval than MOVER-R in terms of controlling the coverage probability and the median width.

KEYWORDS:

• Ratio of two parameters
• Second order approximation
• Skewed distribution
• Unbounded confidence interval
• Variance recovery

Supplementary Materials

Supplementary materials include the appendixes, supplementary tables, the data, as well as the R code to replicate the data example.

Disclosure Statement

The authors report there are no competing interests to declare.

Acknowledgments

The authors would like to thank the editor, the associate editors, and the anonymous referees for their constructive comments which greatly improved this article.

Additional information

Funding

This study is supported, in part, by the National Institutes of Health under award numbers R03DE024198 (NL), R03DE025646 (NL), K01HL140333 (ML), and R03HD092854 (ML), by the National Science Foundation under award number DMS 2002865 (XL and NL), the Natural Science Foundation of China under award number 82173628 (PY), and by the Fundamental Research Funds for School of Public Health, Tongji Medical College, Huazhong University of Science and Technology, under award number 2022gwzz03 (PW).