## A Simple Interval Estimator for the Ratios of Two Medians

By Maxwell Cairns, La Trobe University

A simple Interval estimator for the ratios of two medians was given by Bonett and Price in a 2002 paper in the Journal of Statistical Computation and Simulation.

Their method for estimating the confidence intervals for ratio of medians from two independent populations provided a convenient way to compare medians. Their method is simple, effective and was also suitable for the limitations of the time. These limitations include the computational power of computers, less powerful programming packages and programs which did not have required functions. Notably these calculations could be done using Excel.

With the creation of this simple method we were able to create confidence intervals for the ratios of medians from two independent populations. This was good because at the time often point estimates were being reported and we prefer to report confidence intervals.

The Bonett-Price method was designed for ratios of median quantiles and we also wondered whether coverages could be improved with larger sample sizes. Therefore, it became necessary for a new method to be created to accurately calculate the confidence intervals for large sample sizes which we now have to deal with.

Therefore, we created a method which was designed to principally deal with these larger sample sizes. Furthermore we designed our method to deal with the case where the quantiles are not the same.

Our method relies on using quantile density functions to estimate the variance of the two quantiles using kernel density estimators. Having found this it is a simple matter to find the confidence intervals for the ratio of two quantiles.

Over the course of the six weeks numerous simulations were run both using our method and the Bonett-Price method. These included tests for ratios of medians. We further showed that our method provides good results for ratios of other quantiles. We found that our method is slightly more conservative with coverage rarely below nominal.

References:

Price, RM & Bonett, DG (2002) Distribution-Free Confidence Intervals for Difference and Ratio of Medians, Journal of Statistical Computation and Simulation, 72:2, 119-124, DOI: 10.1010/00949650212140

Maxwell Cairns was one of the recipients of a 2015/16 AMSI Vacation Research Scholarship.