Inferences About a Quantile Shift Measure of Effect Size When There Is a Covariate

Rand R Wilcox

doi:10.5539/ijsp.v11n2p52

Abstract

When comparing two independent groups, a possible appeal of the quantile shift measure of effect size is that its magnitude takes into account situations where one or both distributions are skewed. Extant results indicate that a percentile bootstrap method performs reasonably well given the goal of making inferences about this measure of effect size. The goal here is to suggest a method for making inferences about this measure of effect size when there is a covariate. The method is illustrated with data dealing with the wellbeing of older adults.

Highlights

Consider two independent groups having unknown distributions
When comparing two independent groups, a possible appeal of the quantile shift measure of effect size is that its magnitude takes into account situations where one or both distributions are skewed
Extant results indicate that a reasonably accurate confidence interval for Q can be computed via a percentile bootstrap method (e.g., Wilcox, 2022b)

Summary

Introduction

Consider two independent groups having unknown distributions. Here, the first group is viewed as a control group and the other group is an experimental group. Q reflects the extent the median of the experimental group is unusual relative to the control group and is generally known as a quantile shift measure of effect size. To underscore some concerns when dealing with skewed distributions, it helps to first note that under normality, ∆ = 0.2 indicates that the mean of the experimental group corresponds to the 0.42 quantile of the control group. Given that ∆ = 0.2 is viewed as a small effect size when dealing with normal distributions, it follows that δq = 0.08 is considered small as well. It is readily verified that the reverse can happen where δq suggests a small effect size in contrast to ∆ This same concern occurs for any measure of effect size that implicitly assumes that the distributions are symmetric. A p-value does not indicate the probability of a correct decision

The Proposed Method

Simulation Results

Illustration

Concluding Remarks