{
  "abstract": "Background A common approach used to estimate per-protocol effects, or the effect of perfect adherence to a defined protocol on a specified outcome, is to artificially censor individuals when they deviate from the protocol. As adherence is not under control of the investigator, selection bias is a threat to the validity of per-protocol analyses. Methods Here, we describe nonparametric bounds for the per-protocol effect. Importantly, these bounds describe the range of all possible per-protocol effects concordant with the observed data under the sole additional assumption—beyond those of a standard intent-to-treat analysis—that adherence was accurately measured. Further, these bounds naturally follow from an intuitive understanding of the “best” and “worst” cases among trial participants. Finally, the per-protocol bounds are straightforward to compute with standard statistical analysis software. Results We describe and illustrate these bounds with two open-source data sets, one in the context of a binary outcome and the other with a survival outcome. These examples are accompanied by open-source code in several different software programs to aid implementation. Conclusions Per-protocol bounds offer a simple and intuitive assessment of the per-protocol effect under relatively weak assumptions and offer insight into the strength of the assumptions like no selection bias.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Epidemiology, Gillings School of Global Public Health, University of North Carolina at Chapel Hill",
    "Department of Biostatistics, Gillings School of Global Public Health, University of North Carolina at Chapel Hill",
    "Department of Population Health Science, Duke University"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 6,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Paul N. Zivich"
    },
    {
      "affs": [
        0
      ],
      "name": "Jessie K. Edwards"
    },
    {
      "affs": [
        0
      ],
      "name": "Valerie A. Lucas"
    },
    {
      "affs": [
        1
      ],
      "name": "Bonnie E. Shook-Sa"
    },
    {
      "affs": [
        2
      ],
      "name": "M. Alan Brookhart"
    },
    {
      "affs": [
        0
      ],
      "name": "Stephen R. Cole"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5tzgtm9bhezwf.bib",
  "cited_by": 0,
  "doi": "10.1186/s13063-026-09927-6",
  "fulltext_url": "https://doi.org/10.1186/s13063-026-09927-6",
  "id": "5tzgtm9bhezwf",
  "issns": [
    "1745-6215"
  ],
  "landing_url": "https://doi.org/10.1186/s13063-026-09927-6",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "lcc",
      "R5-920"
    ],
    [
      "openalex",
      "2613|Advanced Causal Inference Techniques"
    ]
  ],
  "openalex": "W7171563981",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-01",
  "publisher": "BMC",
  "ris": "https://research.draconitic.com/paper/5tzgtm9bhezwf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T13:46:21Z",
      "native_id": "4ff9c935d4174ea6901b650ac317910c",
      "source": "doaj",
      "url": "https://doi.org/10.1186/s13063-026-09927-6"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2613|Advanced Causal Inference Techniques",
      "subfield": "statistics"
    }
  ],
  "title": "Computing assumption-lean bounds for per-protocol effects",
  "title_text": "Computing assumption-lean bounds for per-protocol effects",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T13:46:21Z",
  "url": "https://research.draconitic.com/paper/5tzgtm9bhezwf",
  "venue": "Trials",
  "versions": []
}