Aggregation Distortion in Multilevel Mediation: Estimand Geometry, Design Dependence, and Diagnostics
Abstract
Researchers often summarize mediation in clustered data with a single-level product-of-coefficients estimator. We show what this pooled analysis estimates and why the target depends on the design. The pooled a-path is weighted by the within-between variance composition of the treatment. The pooled b-path uses a different weight, based on the mediator variance that remains after residualizing on treatment and covariates. Because these weights generally differ, the pooled indirect effect can differ from a level-respecting combination of the within- and between-level indirect effects. We derive the pooled probability limit and split this difference into a nonlinearity term and a weight-mismatch term. Equal b-paths remove the distortion, but equal a-paths do not. Allowing the two weights to vary gives a bilinear surface with saddle geometry, sharp corner bounds, and a zero-distortion frontier. We also define a design-standardized indirect effect and practical diagnostics. Simulations verify the identities and show that a common decompose-first, cluster-weighted construction can overstate apparent distortion. In an ECLS-K:2011 illustration with 8,363 children in 753 baseline schools, the within-level weights were 0.723 and 0.896, a difference of -0.173 (95% bootstrap CI [-0.193, -0.151]). The conventional construction produced 1.76 times the absolute distortion of the identity-consistent construction. The identity-consistent AD interval included zero. Thus, the application shows unequal weighting and sensitivity to construction, not evidence of nonzero population distortion.
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BibTeX
@misc{hait2026aggregation,
title = {{Aggregation Distortion in Multilevel Mediation: Estimand Geometry, Design Dependence, and Diagnostics}},
author = {Subir Hait},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29685},
eprint = {2609.29685},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29685},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Aggregation Distortion in Multilevel Mediation: Estimand Geometry, Design Dependence, and Diagnostics
AU - Subir Hait
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29685
UR - https://arxiv.org/abs/2609.29685
AB - Researchers often summarize mediation in clustered data with a single-level product-of-coefficients estimator. We show what this pooled analysis estimates and why the target depends on the design. The pooled a-path is weighted by the within-between variance composition of the treatment. The pooled b-path uses a different weight, based on the mediator variance that remains after residualizing on treatment and covariates. Because these weights generally differ, the pooled indirect effect can differ from a level-respecting combination of the within- and between-level indirect effects. We derive the pooled probability limit and split this difference into a nonlinearity term and a weight-mismatch term. Equal b-paths remove the distortion, but equal a-paths do not. Allowing the two weights to vary gives a bilinear surface with saddle geometry, sharp corner bounds, and a zero-distortion frontier. We also define a design-standardized indirect effect and practical diagnostics. Simulations verify the identities and show that a common decompose-first, cluster-weighted construction can overstate apparent distortion. In an ECLS-K:2011 illustration with 8,363 children in 753 baseline schools, the within-level weights were 0.723 and 0.896, a difference of -0.173 (95% bootstrap CI [-0.193, -0.151]). The conventional construction produced 1.76 times the absolute distortion of the identity-consistent construction. The identity-consistent AD interval included zero. Thus, the application shows unequal weighting and sensitivity to construction, not evidence of nonzero population distortion.
ER -