Selection and Stability of Parametric Bivariate Copula Models for Joint Modelling of Overall and Progression-Free Survival
Author(s)
Daniel J. Sharpe, PhD1, Ashley E. Tate, PhD2, Tuli De, PhD3, Jacqueline Vanderpuye-Orgle, MSc, PhD3;
1Parexel International Ltd, London, United Kingdom, 2Parexel, Amsterdam, Netherlands, 3Parexel International Ltd, Billerica, MA, USA
1Parexel International Ltd, London, United Kingdom, 2Parexel, Amsterdam, Netherlands, 3Parexel International Ltd, Billerica, MA, USA
Presentation Documents
OBJECTIVES: Bivariate copula functions represent the joint distribution of associated survival outcomes, such as progression-free (PFS) and overall survival (OS), by linking a pair of marginal distributions. Consideration of parametric distributions for both the marginal survival and copula functions leads to an excessive number of candidate models that is unfeasible to assess in practice. We explored trends in sensitivity of model estimates to guide feasible selection procedures.
METHODS: We investigated the stability of short-term survival extrapolations and patient-level correlation coefficients (Spearman’s rho) across bivariate copula models, employing one of seven standard parametric models for the choices of both marginal survival distributions and one of six copula functions (Clayton, Hougaard, Frank, Joe, Plackett, and Gaussian) to represent association. Goodness-of-fit was assessed by the Akaike information criterion. Models were applied to synthetic data simulating PFS-OS outcomes among patients with metastatic colorectal cancer treated with FOLFOXIRI plus bevacizumab in the phase III TRIBE2 study.
RESULTS: For all choices of marginal distributions, the Hougaard copula gave the best fit, followed by the Joe and Gaussian copulas. The single best-fitting bivariate model was based on log-logistic marginals, but the preferred marginal model varied between copulas. Although Spearman’s rho was sensitive to both the copula and marginal distributions, the estimates obtained from any given copula were generally consistent when considering only the four marginal distributions that gave reasonable fits to the PFS and OS endpoints (e.g., range for Hougaard copula: 0.40-0.61; restricted range: 0.40-0.46). Estimates for 5-year OS were generally stable to the choice of copula (e.g., range for log-logistic marginals: 22.3-25.3%).
CONCLUSIONS: The preferred copula function and estimates for Spearman’s rho were consistent across appropriate marginal distributions, and survival estimates were generally stable to the copula function. This behaviour supports a two-step model selection procedure for bivariate survival models, where the marginal distributions are chosen based on univariate fits.
METHODS: We investigated the stability of short-term survival extrapolations and patient-level correlation coefficients (Spearman’s rho) across bivariate copula models, employing one of seven standard parametric models for the choices of both marginal survival distributions and one of six copula functions (Clayton, Hougaard, Frank, Joe, Plackett, and Gaussian) to represent association. Goodness-of-fit was assessed by the Akaike information criterion. Models were applied to synthetic data simulating PFS-OS outcomes among patients with metastatic colorectal cancer treated with FOLFOXIRI plus bevacizumab in the phase III TRIBE2 study.
RESULTS: For all choices of marginal distributions, the Hougaard copula gave the best fit, followed by the Joe and Gaussian copulas. The single best-fitting bivariate model was based on log-logistic marginals, but the preferred marginal model varied between copulas. Although Spearman’s rho was sensitive to both the copula and marginal distributions, the estimates obtained from any given copula were generally consistent when considering only the four marginal distributions that gave reasonable fits to the PFS and OS endpoints (e.g., range for Hougaard copula: 0.40-0.61; restricted range: 0.40-0.46). Estimates for 5-year OS were generally stable to the choice of copula (e.g., range for log-logistic marginals: 22.3-25.3%).
CONCLUSIONS: The preferred copula function and estimates for Spearman’s rho were consistent across appropriate marginal distributions, and survival estimates were generally stable to the copula function. This behaviour supports a two-step model selection procedure for bivariate survival models, where the marginal distributions are chosen based on univariate fits.
Conference/Value in Health Info
2025-05, ISPOR 2025, Montréal, Quebec, CA
Value in Health, Volume 28, Issue S1
Code
PT7
Topic
Methodological & Statistical Research
Disease
SDC: Oncology