Beyond the Mean: A Weighted k-Sample Omnibus Variance-Ratio Statistic for Covariate Balance Diagnostics
Abstract
Rubin's variance ratio (VR) complements the standardized mean difference by detecting covariate imbalance in spread, but no purpose-built omnibus extension exists for more than two groups. We introduce FVR, a size-weighted quadratic combination of pairwise log-variance-ratios that generalizes VR to k groups, together with geometric-mean and maximum pairwise variance-ratio comparators. FVR has an exact relationship to Rubin's VR at k = 2 and a mathematical structure directly parallel to Cohen's f. In Monte Carlo simulations spanning four variance-driven bias mechanisms, k = 3, 4, 6, and sample sizes from 200 to 10,000, FVR and the geometric-mean statistic generally tracked downstream estimation bias at least as well as the maximum statistic. FVR retained substantially more signal when many groups shared a very small sample, while absolute-bias thresholds were unstable at N = 200. Across mechanisms, FVR<0.10 was reasonably reassuring, values above approximately 0.30 generally indicated concern, and intermediate values were context-dependent. The statistic is implemented for arbitrary analysis weights in the Stata command varatio and illustrated using a multivalued-treatment application.