{
  "abstract": "The g-formula is a foundational tool for identifying causal effects in observational data. This tool is based on the law of iterated expectation, a key mathematical identity in statistics. However, the notation with which the law of iterated expectation and the g-formula is expressed can be opaque to those with little background in statistics. We provide a primer introducing the law of iterated expectation, the integration notation used to express it, and its role for causal effect identification via the g-formula. Combined with the assumptions of causal consistency, positivity, and conditional exchangeability, the law of iterated expectation yields a causal standardization formula (the g-formula) in two nonparametrically equivalent forms: a non-iterative conditional expectation (NICE) form involving a single weighted average of conditional outcome means, and an iterative conditional expectation (ICE) form involving nested expectations. We illustrate both forms using three progressively complex numerical examples: a time-fixed example with a single binary confounder, a time-fixed example with discrete and continuous confounders, and a time-varying example with two timepoints. We provide clarity on what the law of iterated expectation is, how it is related to the g-formula, and how to gain intuition of its mathematical formulations in actual data examples that can be generalized to a range of settings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.20078",
  "author_total": 6,
  "authors": [
    {
      "name": "Ashley I. Naimi"
    },
    {
      "name": "Razieh Nabi"
    },
    {
      "name": "Lindsay J. Collin"
    },
    {
      "name": "Zhaohua Zeng"
    },
    {
      "name": "Paul N. Zivich"
    },
    {
      "name": "Stephen R. Cole"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/51p9gy55g5xqm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.20078",
  "fulltext_url": null,
  "id": "51p9gy55g5xqm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.20078",
  "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.OT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2606.20078",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/51p9gy55g5xqm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:14Z",
      "native_id": "2606.20078",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.20078"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.OT",
      "subfield": "statistics"
    }
  ],
  "title": "A Law of Iterated Expectation Primer for Causal Inference",
  "title_text": "A Law of Iterated Expectation Primer for Causal Inference",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:14Z",
  "url": "https://research.draconitic.com/paper/51p9gy55g5xqm",
  "venue": "arXiv",
  "versions": []
}