Using fractional polynomials and restricted cubic splines to model non‐proportional hazards or time‐varying covariate effects in the Cox regression model
Why this work is in the frame
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Bibliographic record
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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.011 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it