Geometric standardized mean difference and its application to meta-analysis

TL;DR

This paper introduces a geometric approach to standardizing mean differences in meta-analysis, accommodating unequal variances, and proposes estimators with improved statistical properties validated through simulations and real data.

Contribution

It presents a novel geometric method for SMD calculation under unequal variances and develops estimators with superior bias and coverage properties.

Findings

Hedges-type estimator shows lower bias and MSE

Simulation confirms improved coverage probability

Real data analysis demonstrates practical utility

Abstract

The standardized mean difference (SMD) is a widely used measure of effect size, particularly common in psychology, clinical trials, and meta-analysis involving continuous outcomes. Traditionally, under the equal variance assumption, the SMD is defined as the mean difference divided by a common standard deviation. This approach is prevalent in meta-analysis but can be overly restrictive in clinical practice. To accommodate unequal variances, the conventional method averages the two variances arithmetically, which does not allow for an unbiased estimation of the SMD. Inspired by this, we propose a geometric approach to averaging the variances, resulting in a novel measure for standardizing the mean difference with unequal variances. We further propose the Cohen-type and Hedges-type estimators for the new SMD, and derive their statistical properties including the confidence intervals. Simulation results show that the Hedges-type estimator performs optimally across various scenarios, demonstrating lower bias, lower mean squared error, and improved coverage probability. A real-world meta-analysis also illustrates that our new SMD and its estimators provide valuable insights to the existing literature and can be highly recommended for practical use.