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The use of fractional polynomials to model continuous risk variables in epidemiology
Abstract
BACKGROUND: The traditional method of analysing continuous or ordinal risk factors by categorization or linear models may be improved. METHODS: We propose an approach based on transformation and fractional polynomials which yields simple regression models with interpretable curves. We suggest a way of presenting the results from such models which involves tabulating the risks estimated from the model at convenient values of the risk factor. We discuss how to incorporate several continuous risk and confounding variables within a single model. The approach is exemplified with data from the Whitehall I study of British Civil Servants. We discuss the approach in relation to categorization and non-parametric regression models. RESULTS: We show that non-linear risk models fit the data better than linear models. We discuss the difficulties introduced by categorization and the advantages of the new approach. CONCLUSIONS: Our approach based on fractional polynomials should be considered as an important alternative to the traditional approaches for the analysis of continuous variables in epidemiological studies.
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“NfL was entered as a linear predictor term after confirming linearity of the associations with fractional polynomial analyses (32).”
From A Method to Combine Neurofilament Light Measurements From Blood Serum and Plasma in Clinical and Population-Based Studies (2022) · cites this paper as [32]
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“Linearity in the logit was assessed with fractional polynomials (35).”
From Personality, Socioeconomic Status, and All-Cause Mortality in the United States (2010) · cites this paper as [35]
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Hippisley‐Cox, J., & Coupland, C. A. C. (2011) · British Journal of General Practice · doi:10.3399/bjgp12x616346 · Cited by 110
Dwyer, T., Hosmer, D. W., Hosmer, T. A., Venn, A., Blizzard, C. L., Granger, R. H., Cochrane, J., Blair, S. N., Shaw, J. E., Zimmet, P., et al. (2006) · International Journal of Obesity · doi:10.1038/sj.ijo.0803472 · Cited by 106
Ferrari, P., Licaj, I., Muller, D. C., Kragh Andersen, P., Johansson, M., Boeing, H., Weiderpass, E., Dossus, L., Dartois, L., Fagherazzi, G., et al. (2014) · BMJ Open · doi:10.1136/bmjopen-2014-005245 · Cited by 105
Peelen, L. M., de Keizer, N. F., Peek, N. B., Scheffer, G. J., van der Voort, P. H. J., & de Jonge, E. (2007) · Critical Care · doi:10.1186/cc5727 · Cited by 106
Stöhr, W., Back, D. J., Dunn, D. T., Sabin, C. A., Winston, A., Gilson, R. J. C., Pillay, D., Hill, T., Ainsworth, J., Pozniak, A., et al. (2008) · Antiviral Therapy · doi:10.1177/135965350801300507 · Cited by 104
Nair, M., Kurinczuk, J. J., Brocklehurst, P., Sellers, S. M., Lewis, G., & Knight, M. (2015) · BJOG An International Journal of Obstetrics & Gynaecology · doi:10.1111/1471-0528.13279 · Cited by 102
Clemens, T. L., Turner, S., & Dibben, C. J. L. (2017) · Environment International · doi:10.1016/j.envint.2017.07.018 · Cited by 102
Medici, M., Korevaar, T. I. M., Schalekamp–Timmermans, S., Gaillard, R., de Rijke, Y. B., Visser, W. E., Visser, W. E., Keizer‐Schrama, S. M. P. F. D. M., Albert Hofman, D., Hooijkaas, H., et al. (2014) · The Journal of Clinical Endocrinology & Metabolism · doi:10.1210/jc.2014-1505 · Cited by 103
Castelló, A., Martín, M., Ruı́z, A., Casas, A., Baena-Cañada, J. M., Lope, V., Antolín, S., Sánchez, P., Ramos, M., Antón, A., et al. (2015) · PLoS ONE · doi:10.1371/journal.pone.0126096 · Cited by 102
Is Obesity a Risk Factor for Mortality in Coronary Artery Bypass Surgery?
Jin, R., Grunkemeier, G. L., Furnary, A. P., & Handy, J. R. (2005) · Circulation · doi:10.1161/circulationaha.104.489880 · Cited by 100
Meat intake and risk of gastric cancer in the Stomach cancer Pooling (StoP) project
Ferro, A., Rosato, V., Rota, M., Costa, A. R., Morais, S., Pelucchi, C., Johnson, K. C., Hu, J., Palli, D., Ferraroni, M., et al. (2019) · International Journal of Cancer · doi:10.1002/ijc.32707 · Cited by 99
Wahrendorf, M., Reinhardt, J. D., & Siegrist, J. (2013) · PLoS ONE · doi:10.1371/journal.pone.0071893 · Cited by 104
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Dwyer, T., Ponsonby, A., Ukoumunne, O. C., Pezic, A., Venn, A., Dunstan, D. W., Barr, E., Blair, S. N., Cochrane, J., Zimmet, P., et al. (2011) · BMJ · doi:10.1136/bmj.c7249 · Cited by 97
Metadata
- Authors
- Royston, P., Ambler, G., & Sauerbrei, W.
- Affiliations
- Imperial College London (GB)
- Hammersmith Hospital (GB)
- University of Freiburg (DE)
- Venue
- International Journal of Epidemiology
- Publisher
- Oxford University Press (OUP)
- Published
- 1999-10-01
- Volume
- 28
- Issue
- 5
- Pages
- 964-974
- ISSN
- 0300-5771 (ISSN-L)
- 1464-3685
- OpenAlex
- W2105087430
- Cited by
- 1,066
- References
- 44
- Citations by year
- 2026: 15 · 2025: 26 · 2024: 30 · 2023: 31 · 2022: 36 · 2021: 51 · 2020: 51 · 2019: 41 · 2018: 67 · 2017: 80 · 2016: 46 · 2015: 66 · 2014: 63 · 2013: 67 · 2012: 56
- FWCI
- 8.54
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- International Journal of Epidemiology (publishedVersion)
- PubMed (publishedVersion)
- UCL Discovery (University College London) (submittedVersion)
- FreiDok plus (Universitätsbibliothek Freiburg) (submittedVersion)
- Funders
- Wellcome Trust
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods in Epidemiology
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
- Keywords
- fractional polynomials, risk transformation, risk curves, confounding variables, nonparametric regression, British civil servants
- MeSH terms
- Coronary Disease, Cross-Sectional Studies, Epidemiologic Methods, United Kingdom, Humans, Male, Models, Cardiovascular, Mortality, Neoplasms, Prospective Studies, Risk Factors, Sensitivity and Specificity, Linear Models, Odds Ratio, Risk Assessment, Stroke
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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