{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29685",
  "author_total": 1,
  "authors": [
    {
      "name": "Subir Hait"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4zv7yvpbxq6jj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29685",
  "fulltext_url": null,
  "id": "4zv7yvpbxq6jj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29685",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "openalex",
      "2613|Advanced Causal Inference Techniques"
    ]
  ],
  "openalex": "W7214320930",
  "pdf_url": "https://arxiv.org/pdf/2609.29685",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4zv7yvpbxq6jj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:21Z",
      "native_id": "2609.29685",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29685"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ME",
      "subfield": "statistics"
    }
  ],
  "title": "Aggregation Distortion in Multilevel Mediation: Estimand Geometry, Design Dependence, and Diagnostics",
  "title_text": "Aggregation Distortion in Multilevel Mediation: Estimand Geometry, Design Dependence, and Diagnostics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:21Z",
  "url": "https://research.draconitic.com/paper/4zv7yvpbxq6jj",
  "venue": "arXiv",
  "versions": []
}