On February 14, 2023, a prestigious statistics journal "The American Statistician" published online the latest research results of the research group of Prof. Ping Yin from Huazhong University of Science and Technology and Prof. Nianjun Liu from Indiana University Bloomington entitled "MOVER-R and penalized MOVER-R confidence intervals for the ratio of two quantities" (https://www.tandfonline.com/doi/full/10.1080/00031305.2023.2173294)[1]. Peng Wang, a junior faculty at Huazhong University of Science and Technology, is the first author, Yilei Ma, a doctoral student, is a co-author, and Prof. Ping Yin and Prof. Nianjun Liu are the co-corresponding authors. This study first extended the original MOVER-R to the case where the confidence interval (CI) for denominator contains zero, and further proposed the penalized MOVER-R, which is shown to have a narrower confidence width.

The ratio-type parameter is often encountered in medical studies, such as the median response dose, the incremental risk-benefit ratio, etc. For such ratios, in addition to presenting point estimates in data analysis, it is also necessary to provide CIs to evaluate the precision of point estimates. However, when the point estimates of either the numerator or denominator does not follow a symmetric distribution, the Fieller method, the delta method and the bootstrap method may underestimate the coverage probability. To overcome this problem, the MOVER-R manages the skewness of the sampling distribution of estimated quantities by allowing asymmetric CIs for the numerator and the denominator[2-3]. However, the original MOVER-R implicitly assumes that the CI for denominator never includes zero. In practice, due to the influence of effect size, sample size and significance level, there is no guarantee that the denominator is significantly deviated from 0. How to account for the case where the CI for denominator contains zero? And how to deal with the defect that the CI for the ratio will be unbounded when the CI for denominator contains zero? Focusing on the above issues, they conducted the research based on the penalized likelihood theory.

To extend the original MOVER-R to the case where the CI for denominator does not contain zero, this study had shown that the MOVER-R CI for the ratio  can be defined as the set of  for which the MOVER-D[4] CI for  includes zero. Compared to the original MOVER-R, even if the denominator CI includes zero, the new method can still give the valid CI for . In particular, when the CIs for the numerator and denominator are symmetrical, the MOVER-R CI gives the same confidence limits as the Fieller CI. In order to overcome the problem that the MOVER-R CI will be unbounded when the CI for denominator includes zero, this study further proposed the penalized MOVER-R CI. By adopting a penalized likelihood approach to estimate the denominator, the denominator tends to be far away from the singular point zero, and a suitable penalty parameter can even make the CI for the denominator never include zero, thus always producing a bounded CI[5]. This study gave a closed-form solution for penalized MOVER-R CI, and further proved that the new method differs from MOVER-R only at the second order.

Through theoretical derivation, numerical simulations as well as real data analysis, this study confirmed that the penalized MOVER-R has a better performance than MOVER-R in terms of coverage probability and median width. The new proposed method provides a more robust and effective choice for CI construction for ratios in medical research.

The research was supported by the National Natural Science Foundation of China and the Fundamental Research Funds for the School of Public Health, Tongji Medical College, Huazhong University of Science and Technology.

References：

[1] Wang, P., Ma, Y., Xu, S., Wang, Y. X., Zhang, Y., Lou, X., Li, M., Wu, B., Gao, G., Yin, P., and Liu, N. (2023). MOVER-R and penalized MOVER-R confidence intervals for the ratio of two quantities. The American Statistician. https://doi.org/10.1080/00031305.2023.2173294.

[2] Donner, A. and Zou, G. Y. (2012). Closed-form confidence intervals for functions of the normal mean and standard deviation. Statistical Methods in Medical Research 21(4), 347-359.

[3] Newcombe, R. G. (2016). MOVER-R confidence intervals for ratios and products of two independently estimated quantities. Statistical Methods in Medical Research 25(5), 1774-1778.

[4] Zou, G. Y. and Donner, A. (2008). Construction of confidence limits about effect measures: a general approach. Statistics in Medicine 27(10), 1693-1702.

[5] Wang, P., Xu, S., Wang, Y. X., Wu, B., Fung, W. K., Gao, G., Liang, Z., and Liu, N. (2021). Penalized Fieller’s confidence interval for the ratio of bivariate normal means. Biometrics 77(4), 1355-1368.