Asymptotic Distribution of Robust Effect Size Index

TL;DR

This work develops a scalable analytic inference framework for the Robust Effect Size Index (RESI) by deriving its asymptotic distribution through estimating equations and the Delta method with a robust sandwich covariance. It introduces a truncated confidence interval to handle the non-negative RESI boundary and provides a comprehensive treatment of parametric and robust covariance cases. Through simulations in linear and logistic settings, RESI with robust covariance shows reduced bias and reliable coverage compared with traditional metrics, while offering substantial speedups over bootstrap inference. The method is demonstrated on ASD-related datasets (depression trajectories and SPARK CBCL data), revealing diagnostically large effects with modest sex-related differences and illustrating the practical impact for high-dimensional, heterogeneous biomedical data.

Abstract

The Robust Effect Size Index (RESI) is a recently proposed standardized effect size to quantify association strength across models. However, its confidence interval construction has relied on computationally intensive bootstrap procedures. We establish a general theorem for the asymptotic distribution of the RESI using a Taylor expansion that accommodates a broad class of models. Simulations under various linear and logistic regression settings show that RESI and its CI have smaller bias and more reliable coverage than commonly used effect sizes such as Cohen's d and f. Combining with robust covariance estimation yields valid inference under model misspecification. We use the methods to investigate associations of depression and behavioral problems with sex and diagnosis in Autism spectrum disorders, and demonstrate that the asymptotic approach achieves up to a 50-fold speedup over the bootstrap. Our work provides a scalable and reliable alternative to bootstrap inference, greatly enhancing the applicability of RESI to high-dimensional studies.

Figures (5)

• Figure 1: Bias (left) and coverage (right) performance of the unsigned (a) and signed (b) RESI estimators under different true effect sizes (columns) and error distributions (rows). "Normal" = homoskedastic, symmetric; "Gamma" = homoskedastic, asymmetric; "Hetero" = heteroskedastic. Curves show mean bias $\mathbb{E}[\widehat{S}_\beta - S_\beta]$ versus sample size; dashed line marks 0 for bias and 0.95 for coverage.
• Figure 2: Bias (left) and coverage (right) of signed and unsigned RESI estimator by true effect sizes (columns) and intercepts (rows). "Balanced": $\eta=0$; "Semi-balanced": $\eta = -1$; "Unbalanced": $\eta = -2$. Curves show mean bias $\mathbb{E}[\widehat{S}_\beta - S_\beta]$ versus sample size; dashed line marks 0 for bias and 0.95 for coverage.
• Figure 3: The model-estimated CDI Total score trajectories for ASD and TD across age and sex, with shaded bands indicating 95% confidence intervals. Lines represent the predicted mean scores at each age, while the ribbons reflect uncertainty around those estimates. This visualization illustrates how children’s depression severity changes with age and whether the developmental patterns differ between ASD/TD and males/females.
• Figure 4: The model-estimated dichotomized Total Problems T-score trajectories for males and females across age, with shaded bands indicating 95% confidence intervals. Lines represent the predicted mean scores at each age, while the ribbons reflect uncertainty around those estimates. This visualization illustrates how behavioral problem severity changes with age and whether the developmental patterns differ between males and females.
• Figure S1: The model-estimated continuous Total Problems T-score trajectories for males and females across age, with shaded bands indicating 95% confidence intervals. Lines represent the predicted mean scores at each age, while the ribbons reflect uncertainty around those estimates. This visualization illustrates how behavioral problem severity changes with age and whether the developmental patterns differ between males and females.

Theorems & Definitions (3)

• theorem 1: Asymptotic Distribution of $\widetilde{S}_\beta$
• corollary 1
• corollary 2