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Robust Updating of a Risk Prediction Model by Integrating External Ranking Information

Nicholas C. Henderson

TL;DR

The approach introduces the ranking parameters associated with the regression coefficients of an internal risk model and estimates the internal risk model parameters by penalizing a ranking-based discrepancy measure between the ranking parameters and the rankings implied by the established prognostic model.

Abstract

Utilizing established risk factors and prognostic models can often improve the construction of a newer risk model that uses novel biomarkers in a smaller, internal study. However, directly borrowing information from an established prognostic model is often unsuitable due to differences in study populations, patient outcomes measured, and other specific features of the internal study design. To better enable the use of established prognostic information when constructing a novel risk model, we propose an estimation approach centered around the idea that the risk rankings rather than the risk scores from an established prognostic model are often more transportable to the internal study context. To leverage external ranking information, our approach introduces the ranking parameters associated with the regression coefficients of an internal risk model and estimates the internal risk model parameters by penalizing a ranking-based discrepancy measure between the ranking parameters and the rankings implied by the established prognostic model. Our method does not require the external prognostic model to have a specific form, but only requires one to compute risk score rankings from an external model. Simulation studies demonstrate that our method leads to competitive predictive performance and performs particularly well when the true internal and external prognostic models have high rank correlation but large discrepancies between their underlying risk scores. We demonstrate the use of our approach through the development of a prognostic model for advanced prostate cancer patients who were treated with an immune checkpoint inhibitor

Robust Updating of a Risk Prediction Model by Integrating External Ranking Information

TL;DR

The approach introduces the ranking parameters associated with the regression coefficients of an internal risk model and estimates the internal risk model parameters by penalizing a ranking-based discrepancy measure between the ranking parameters and the rankings implied by the established prognostic model.

Abstract

Utilizing established risk factors and prognostic models can often improve the construction of a newer risk model that uses novel biomarkers in a smaller, internal study. However, directly borrowing information from an established prognostic model is often unsuitable due to differences in study populations, patient outcomes measured, and other specific features of the internal study design. To better enable the use of established prognostic information when constructing a novel risk model, we propose an estimation approach centered around the idea that the risk rankings rather than the risk scores from an established prognostic model are often more transportable to the internal study context. To leverage external ranking information, our approach introduces the ranking parameters associated with the regression coefficients of an internal risk model and estimates the internal risk model parameters by penalizing a ranking-based discrepancy measure between the ranking parameters and the rankings implied by the established prognostic model. Our method does not require the external prognostic model to have a specific form, but only requires one to compute risk score rankings from an external model. Simulation studies demonstrate that our method leads to competitive predictive performance and performs particularly well when the true internal and external prognostic models have high rank correlation but large discrepancies between their underlying risk scores. We demonstrate the use of our approach through the development of a prognostic model for advanced prostate cancer patients who were treated with an immune checkpoint inhibitor
Paper Structure (18 sections, 1 theorem, 48 equations, 3 figures, 4 tables)

This paper contains 18 sections, 1 theorem, 48 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Internal Risk Models and Ranking Parameters
  3. Data Structure
  4. External Model Rankings and Internal Risk Model Ranking Parameters
  5. Association Measures between the Internal and External Risk Rankings.
  6. Estimating the Internal Risk Model
  7. Penalized Estimation
  8. Computation and Choice of $\nu$
  9. Hyperparameter Selection
  10. Simulation Studies
  11. Simulation Study 1: Linear External and Internal Risk Models
  12. Simulation Study 2: A Nonlinear External Risk Model
  13. Application: A prognostic model for prostate cancer patients receiving immunotherapy
  14. Discussion
  15. Further Details for Rank Association Measures
  16. ...and 3 more sections

Key Result

Theorem 1

When $\ell_{\lambda, \alpha}(\beta_{0}, \hbox{\boldmath $\beta$})$ has the form (eq:penalized_obj_form) and $D_{\bullet}^{\nu}(\cdot, \cdot)$ has the form (eq:general_disc_form), then for any $\hbox{\boldmath $\beta$} \in \mathbb{R}^{p}$ and $\hbox{\boldmath $\beta$}^{(t)} \in \mathbb{R}^{p}$, there where $\mathbf{W}_{t}$ and $\mathbf{V}_{t}$ are $n^{2} \times n^{2}$ diagonal matrices. The $k^{th}

Figures (3)

  • Figure 1: Schematic depicting which quantities are captured in the internal and external data sources and our transportability assumptions.
  • Figure 2: MSK-CHORD cohort: rank association between the fitted values from RASPER and the fitted values from the external model across different choices of $\lambda$. Rank association is measured by Kendall's $\tau$. For each fixed value of $\lambda$, the hyperparameter $\alpha$ is selected using leave-one-out cross-validation.
  • Figure 3: MSK-CHORD cohort: internal model risk rankings versus external rankings for the following methods: OLS, ridge regression, and RASPER using the Kendall association appraoch.

Theorems & Definitions (1)

  • Theorem 1