ArticleStatistics in medicine2022
Using fractional polynomials and restricted cubic splines to model non-proportional hazards or time-varying covariate effects in the Cox regression model.
Article in Statistics in medicine, 2022. The graph could read no effect estimate from its abstract, so it casts no vote on the map. Cited by 69 papers, 2 of them syntheses that pooled it.
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Who cites it
69 citing papers in PubMed, 2 syntheses or guidelines pooled it.
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- CT-based pulmonary vascular pruning and emphysema: clinical correlations and prognostic impacts in COPD.European radiology · 2026Article
- A Novel Approach for Analyzing Non-Proportional Hazards With Long-Term Survivors: The Defective Yang-Prentice Model.Biometrical journal. Biometrische Zeitschrift · 2026Article
- Estimating clinical trial hazard functions.Clinical trials (London, England) · 2026Article
- Using Polygenic Risk Scores to Refine Risk and Outcomes in High-grade Serous Ovarian Cancer in Australia.Cancer epidemiology, biomarkers & prevention : a publication of the American Association for Cancer Research, cosponsored by the American Society of Preventive Oncology · 2026Article
- Adiposity and Blood Pressure in Children and Adolescents: A National Cross-Sectional Study in China.Healthcare (Basel, Switzerland) · 2026Article
- Early discharge after clinical stabilization in acute decompensated heart failure: associations with short-term outcomes.Clinical research in cardiology : official journal of the German Cardiac Society · 2026Article
- Association Between Free Sugar Intake and Risk of Dyslipidemia in Chinese Adults-Evidence from the Chinese Food Consumption Survey and the China Health and Nutrition Survey.Foods (Basel, Switzerland) · 2026Article
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- The HFpEF-ABA score predicts adverse cardiac remodelling and incident heart failure: a UK biobank study.ESC heart failure · 2026Article
- The Burden of Cardiovascular Disease in Lymphoma Survivors: A Population-Based Matched Cohort Study.JACC. Advances · 2026Article
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- Behavioral Components of Life's Essential 8 Are More Strongly Associated with Cognitive Function than Clinical Factors in Chinese Older Adults.International journal of behavioral medicine · 2026Article
- Pre-Dialysis Trajectory of Brain Natriuretic Peptide Levels and Body Weight in Chronic Kidney Disease Patients: A Predictive Marker for Unplanned Dialysis Initiation.Kidney medicine · 2026Article
- Plasma proteomic signature of the human menstrual cycle.Nature medicine · 2026Article
- Development and validation of a prediction model for ischemic stroke recurrence risk in patients with ischemic stroke and atrial fibrillation.Neurological sciences : official journal of the Italian Neurological Society and of the Italian Society of Clinical Neurophysiology · 2026Article
9 more citing papers are in PubMed but not listed here.
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3 authors.
Funding
Abstract
The Cox proportional hazards model is used extensively in clinical and epidemiological research. A key assumption of this model is that of proportional hazards. A variable satisfies the proportional hazards assumption if the effect of that variable on the hazard function is constant over time. When the proportional hazards assumption is violated for a given variable, a common approach is to modify the model so that the regression coefficient associated with the given variable is assumed to be a linear function of time (or of log-time), rather than being constant or fixed. However, this is an unnecessarily restrictive assumption. We describe two different methods to allow a regression coefficient, and thus the hazard ratio, in a Cox model to vary as a flexible function of time. These methods use either fractional polynomials or restricted cubic splines to model the log-hazard ratio as a function of time. We illustrate the utility of these methods using data on 12 705 patients who presented to a hospital emergency department with a primary diagnosis of heart failure. We used a Cox model to assess the association between elevated cardiac troponin at presentation and the hazard of death after adjustment for an extensive set of covariates. SAS code for implementing the restricted cubic spline approach is provided, while an existing Stata function allows for the use of fractional polynomials.
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