Quantifying the Effect Size of Exposure-Outcome Association Using δ-Score: Application to Environmental Chemical Mixture Studies

Vishal Midya, Jiangang Liao, Chris Gennings, Elena Colicino, Susan L. Teitelbaum, Robert O. Wright, Damaskini Valvi

Research output: Contribution to journalArticlepeer-review


Epidemiologists often study the associations between a set of exposures and multiple biologically relevant outcomes. However, the frequently used scale-and-context-dependent regression coefficients may not offer meaningful comparisons and could further complicate the interpretation if these outcomes do not have similar units. Additionally, when scaling up a hypothesis-driven study based on preliminary data, knowing how large to make the sample size is a major uncertainty for epidemiologists. Conventional p-value-based sample size calculations emphasize precision and might lead to a large sample size for small- to moderate-effect sizes. This asymmetry between precision and utility is costly and might lead to the detection of irrelevant effects. Here, we introduce the “ (Formula presented.) -score” concept, by modifying Cohen’s (Formula presented.). (Formula presented.) -score is scale independent and circumvents the challenges of regression coefficients. Further, under a new hypothesis testing framework, it quantifies the maximum Cohen’s (Formula presented.) with certain optimal properties. We also introduced “Sufficient sample size”, which is the minimum sample size required to attain a (Formula presented.) -score. Finally, we used data on adults from a 2017–2018 U.S. National Health and Nutrition Examination Survey to demonstrate how the (Formula presented.) -score and sufficient sample size reduced the asymmetry between precision and utility by finding associations between mixtures of per-and polyfluoroalkyl substances and metals with serum high-density and low-density lipoprotein cholesterol.

Original languageEnglish
Article number1962
Issue number10
StatePublished - Oct 2022


  • Cohen’s f
  • effect size
  • environmental health
  • sample size estimation
  • weighted quantile sum regression


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