Reference-State Bootstrap Correction and Kernel Characterization of Residual-Based Fit Indices in Structural Equation Models
Abstract
Sampling error affects the finite-sample values of residual-based fit indices. We show that the unweighted comparative fit index (CFIu) rests on a quadratic-kernel maximum mean discrepancy and that a Gaussian-kernel comparative fit index (KCFI) converges to CFIu as bandwidth increases. Because kernel discrepancies lack an analytic sampling correction, we extend expected-discrepancy bootstrap methods to a fitted-correlation reference state: the sample is transformed to reproduce the fitted correlation matrix, then resampled and refitted to estimate the exact-fit sampling component. Across 18 confirmatory factor analysis conditions, this correction reduced the root mean square error (RMSE) of both indices by 62%-87%. Under normality it matched an analytic residual-covariance estimator of CFIu within .0001, and in small samples it outperformed ordinary bootstrap correction in accuracy and availability. Corrected KCFI had lower RMSE than identically corrected CFIu in all 29 nonnormal conditions, even against a common target, but in six heavy-tailed small-sample conditions the analytic estimator was more accurate than both. The results validate the reference-state correction where it is needed and delimit where the analytic estimator is preferable.
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BibTeX
@misc{song2026referencestate,
title = {{Reference-State Bootstrap Correction and Kernel Characterization of Residual-Based Fit Indices in Structural Equation Models}},
author = {Eunjeong Song and Sehee Hong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13968},
eprint = {2610.13968},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13968},
}
RIS
TY - UNPB
TI - Reference-State Bootstrap Correction and Kernel Characterization of Residual-Based Fit Indices in Structural Equation Models
AU - Eunjeong Song
AU - Sehee Hong
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.13968
UR - https://arxiv.org/abs/2610.13968
AB - Sampling error affects the finite-sample values of residual-based fit indices. We show that the unweighted comparative fit index (CFI_u) rests on a quadratic-kernel maximum mean discrepancy and that a Gaussian-kernel comparative fit index (KCFI) converges to CFI_u as bandwidth increases. Because kernel discrepancies lack an analytic sampling correction, we extend expected-discrepancy bootstrap methods to a fitted-correlation reference state: the sample is transformed to reproduce the fitted correlation matrix, then resampled and refitted to estimate the exact-fit sampling component. Across 18 confirmatory factor analysis conditions, this correction reduced the root mean square error (RMSE) of both indices by 62%-87%. Under normality it matched an analytic residual-covariance estimator of CFI_u within .0001, and in small samples it outperformed ordinary bootstrap correction in accuracy and availability. Corrected KCFI had lower RMSE than identically corrected CFI_u in all 29 nonnormal conditions, even against a common target, but in six heavy-tailed small-sample conditions the analytic estimator was more accurate than both. The results validate the reference-state correction where it is needed and delimit where the analytic estimator is preferable.
ER -