A Law of Iterated Expectation Primer for Causal Inference

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ID: 332222
2026
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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.
Reference Key
openalex_W7165396636 Use this key to autocite in the manuscript while using SciMatic Manuscript Manager or Thesis Manager
Authors Ashley Isaac Naimi, Razieh Nabi, Lindsay Jane Collin, Zhaohua Zeng, Paul N. Zivich, Stephen R. Cole
Journal american journal of epidemiology
Year 2026
DOI
10.1093/aje/kwag258
URL
https://doi.org/10.1093/aje/kwag258
Keywords Keywords not found

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