Skip to main content
SpringerLink
Log in

Bayesian ridge regression for survival data based on a vine copula-based prior

  • Original Paper
  • Published:
AStA Advances in Statistical Analysis Aims and scope Submit manuscript

A Correction to this article was published on 14 February 2023

This article has been updated

Abstract

Ridge regression estimators can be interpreted as a Bayesian posterior mean (or mode) when the regression coefficients follow multivariate normal prior. However, the multivariate normal prior may not give efficient posterior estimates for regression coefficients, especially in the presence of interaction terms. In this paper, the vine copula-based priors are proposed for Bayesian ridge estimators under the Cox proportional hazards model. The semiparametric Cox models are built on the posterior density under two likelihoods: Cox’s partial likelihood and the full likelihood under the gamma process prior. The simulations show that the full likelihood is generally more efficient and stable for estimating regression coefficients than the partial likelihood. We also show via simulations and a data example that the Archimedean copula priors (the Clayton and Gumbel copula) are superior to the multivariate normal prior and the Gaussian copula prior.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Change history

References

  • Aas, K., Czado, C., Frigessi, A., Bakken, H.: Pair-copula constructions of multiple dependence. Insurance Math. Econ. 44, 182–198 (2009)

    Article  MathSciNet  Google Scholar 

  • Abonazel, M.R., Taha, I.M.: Beta ridge regression estimators: simulation and application. Commun. Stat. Simul. Comput. (2021). https://doi.org/10.1080/03610918.2021.1960373

    Article  Google Scholar 

  • Arashi, M., Roozbeh, M., Hamzah, N.A., Gasparini, M.: Ridge regression and its applications in genetic studies. PLoS ONE 16, e0245376 (2021)

    Article  Google Scholar 

  • Arbel, J., Lijoi, A., Nipoti, B.: Full Bayesian inference with hazard mixture models. Comput. Stat. Data Anal. 93, 359–372 (2016)

    Article  MathSciNet  Google Scholar 

  • Armagan, A., Zaretzki, R.L.: Model selection via adaptive shrinkage with t priors. Comput. Stat. 25, 441–461 (2010)

    Article  MathSciNet  Google Scholar 

  • Assaf, A.G., Tsionas, M., Tasiopoulos, A.: Diagnosing and correcting the effects of multicollinearity: Bayesian implications of ridge regression. Tour. Manage. 71, 1–8 (2019)

    Article  Google Scholar 

  • Avalos, B.R., Klein, J.L., Kapoor, N., Tutschka, P.J., Klein, J.P., Copelan, E.A.: Preparation for marrow transplantation in Hodgkin’s and non-Hodgkin’s lymphoma using Bu/CY. Bone Marrow Transpl. 12, 133–138 (1993)

    Google Scholar 

  • Box, G.E.P., Tiao, G.C.: Bayesian Inference in Statistical Analysis. Wiley (1992)

    Book  Google Scholar 

  • Burzykowski, T., Molenberghs, G., Buyse, M., Geys, H., Renard, D.: Validation of surrogate end points in multiple randomized clinical trials with failure time end points. J. R. Stat. Soc. Ser. C (appl. Stat.) 50, 405422 (2001)

    MathSciNet  Google Scholar 

  • Chang, B., Joe, H.: Prediction based on conditional distributions of vine copulas. Comput. Stat. Data Anal. 139, 45–63 (2019)

    Article  MathSciNet  Google Scholar 

  • Czado, C.: Analyzing Dependent Data with Vine Copulas. Lecture Notes in Statistics. Springer (2019)

    Book  Google Scholar 

  • Chang, Y.C., Mastrangelo, C.: Addressing multicollinearity in semiconductor manufacturing. Qual. Reliab. Eng. Int. 27, 843–854 (2011)

    Article  Google Scholar 

  • Chipman, H.: Bayesian variable selection with related predictors. Can. J. Stat. 24, 17–36 (1996)

    Article  MathSciNet  Google Scholar 

  • Corradin, R., Nieto-Barajas, L.E., Nipoti, B.: Optimal stratification of survival data via Bayesian nonparametric mixtures. Econom. Stat. 22, 17–38 (2022)

    MathSciNet  Google Scholar 

  • Cox, D.R.: Regression models and life-tables. J. R. Stat. Soc.: Ser. B (methodol.) 34, 187–202 (1972)

    MathSciNet  Google Scholar 

  • Durante, F., Sempi, C.: Principles of Copula Theory. CRC Press, Boca Raton (2016)

    Google Scholar 

  • Emura, T., Michimae, H., Matsui, S.: Dynamic risk prediction via a joint frailty-copula model and IPD meta-analysis: building web applications. Entropy 24, 589 (2022)

    Article  MathSciNet  Google Scholar 

  • Emura, T., Sofeu, C., Rondeau, V.: Conditional copula models for correlated survival endpoints: individual patient data meta-analysis of randomized controlled trials. Stat. Methods Med. Res. 30, 26342650 (2021)

    Article  MathSciNet  Google Scholar 

  • Flórez, A.J., Abad, A.A., Molenberghs, G., Van Der Elst, W.: Generating random correlation matrices with fixed values: an application to the evaluation of multivariate surrogate endpoints. Comput. Stat. Data Anal. 142, 106834 (2020)

    Article  MathSciNet  Google Scholar 

  • Flórez, A.J., Molenberghs, G., Van der Elst, W., Abad, A.A.: An efficient algorithm to assess multivariate surrogate endpoints in a causal inference framework. Comput. Stat. Data Anal. 172, 107494 (2022)

    Article  MathSciNet  Google Scholar 

  • García, C.B., García, J., López Martín, M.M., Salmerón, R.: Collinearity: revisiting the variance inflation factor in ridge regression. J. Appl. Stat. 42, 648–661 (2015)

    Article  MathSciNet  Google Scholar 

  • Griffin, J., Brown, P.: Hierarchical shrinkage priors for regression models. Bayesian Anal. 12, 135–159 (2017)

    Article  MathSciNet  Google Scholar 

  • Gruber, M.H.J.: Improving Efficiency by Shrinkage: The James-Stein and Ridge Regression Estimators. CRC Press (1998)

    Google Scholar 

  • Hoerl, A.E., Kennard, R.W.: Ridge regression: Biased estimation for nonorthogonal problems. Technometrics 12, 55–67 (1970)

    Article  Google Scholar 

  • Hoerl, R.W.: Ridge regression: a historical context. Technometrics 62, 420–425 (2020)

    Article  MathSciNet  Google Scholar 

  • Huard, D., Evin, G., Favre, A.C.: Bayesian copula selection. Comput. Stat. Data Anal. 51, 809–822 (2006)

    Article  MathSciNet  Google Scholar 

  • Ibrahim, J.G., Chen, M.H., Sinha, D.: Bayesian Survival Analysis. Springer, New York (2001)

    Book  Google Scholar 

  • Joe, H.: Generating random correlation matrices based on partial correlations. J. Multivar. Anal. 97, 2177–2189 (2006)

    Article  MathSciNet  Google Scholar 

  • Kalbfleisch, J.D.: Nonparametric Bayesian analysis of survival time data. J. R. Stat. Soc. B 40, 214–221 (1978)

    Google Scholar 

  • Killiches, M., Kraus, D., Czado, C.: Examination and visualisation of the simplifying assumption for vine copulas in three dimensions. Aust. N. Z. J. Stat. 59, 95–117 (2017)

    Article  MathSciNet  Google Scholar 

  • Klein, J.P., Moeschberger, M.L.: Survival Analysis: Techniques for Censored and Truncated Data, 2nd edn. Springer (2013)

    Google Scholar 

  • Klein, J.P., Van Houwelingen, H.C., Ibrahim, J.G., Scheike, T.H.: Handbook of Survival Analysis. CRC Press, Boca Raton (2014)

    Google Scholar 

  • Kurowicka, D., Cooke, R.: A parameterization of positive definite matrices in terms of partial correlation vines. Linear Algebra Appl. 372, 225–251 (2003)

    Article  MathSciNet  Google Scholar 

  • Kurowicka, D., Cooke, R.M.: Uncertainty Analysis with High Dimensional Dependence Modelling. Wiley, Hoboken (2006)

    Book  Google Scholar 

  • Kwon, S., Ha, I.D., Shih, J.H., Emura, T.: Flexible parametric copula modelling approaches for clustered survival data. Pharm. Stat. 21(1), 69–88 (2022)

    Article  Google Scholar 

  • Lewandowski, D., Kurowicka, D., Joe, H.: Generating random correlation matrices based on vines and extended onion method. J. Multivar. Anal. 100(9), 1989–2001 (2009)

    Article  MathSciNet  Google Scholar 

  • Lustbader, E.D.: Relative risk regression diagnosis. In: Moolgavkar, S.H., Prentice, R.L. (eds.) Modern Statistical Methods in Chronic Disease Epidemiology. SIAM, Philadelphia (1986)

    Google Scholar 

  • Loesgen, K.H.: A generalization and Bayesian interpretation of ridge-type estimators with good prior means. Stat. Pap. 31, 147 (1990)

    Article  MathSciNet  Google Scholar 

  • Michimae, H., Emura, T.: Bayesian ridge estimators based on copula-based joint prior distributions for regression parameters. Comput. Stat. 37, 2741–2769 (2022). https://doi.org/10.1007/s00180-022-01213-8

    Article  Google Scholar 

  • Michimae, H., Matsunami, M., Emura, T.: Robust ridge regression for estimating the effects of correlated gene expressions on phenotypic traits. Environ. Ecol. Stat. 27, 41–72 (2020)

    Article  Google Scholar 

  • Montgomery, D.C., Peck, E.A., Vining, G.G.: Introduction to Linear Regression Analysis, 5th edn. Wiley (2012)

    Google Scholar 

  • Nagler, T., Bumann, C., Czado, C.: Model selection in sparse high-dimensional vine copula models with an application to portfolio risk. J. Multivar. Anal. 172, 180–192 (2019)

    Article  MathSciNet  Google Scholar 

  • Nelsen, R.B.: An Introduction to Copulas. Springer (2006)

    Google Scholar 

  • Park, T., Casella, G.: The Bayesian lasso. J. Am. Stat. Assoc. 103, 681–686 (2008)

    Article  MathSciNet  Google Scholar 

  • Peng, M., Xiang, L., Wang, S.: Semiparametric regression analysis of clustered survival data with semi-competing risks. Comput. Stat. Data Anal. 124, 53–70 (2018)

    Article  MathSciNet  Google Scholar 

  • Pliskin, J.L.: A ridge-type estimator and good prior means. Commun. Stat.-Theory Methods 16, 3429–3437 (1987)

    Article  MathSciNet  Google Scholar 

  • Polson, N.G., Scott, J.G.: On the half-Cauchy prior for a global scale parameter. Bayesian Anal. 7, 887–902 (2012)

    Article  MathSciNet  Google Scholar 

  • Rubio, F.J., Yu, K.: Flexible objective Bayesian linear regression with applications in survival analysis. J. Appl. Stat. 44, 798–810 (2017)

    Article  MathSciNet  Google Scholar 

  • Salmerón, R., García, J., García, C., del Mar, L.M.: Transformation of variables and the condition number in ridge estimation. Comput. Stat. 33, 1497–1524 (2018)

    Article  MathSciNet  Google Scholar 

  • Sambasivan, R., Das, S., Sahu, S.K.: A Bayesian perspective of statistical machine learning for big data. Comput. Stat. 35, 893–930 (2020)

    Article  MathSciNet  Google Scholar 

  • Schepsmeier, U., Stöber, J.: Derivatives and Fisher information of bivariate copulas. Stat. Pap. 55, 525–542 (2014)

    Article  MathSciNet  Google Scholar 

  • Schafer, R.L., Roi, L.D., Wolfe, R.A.: A ridge logistic estimator. Commun. Stat.-Theory Methods 13, 99–113 (1984)

    Article  Google Scholar 

  • Scheipl, F., Kneib, T., Fahrmeir, L.: Penalized likelihood and Bayesian function selection in regression models. Adv. Stat. Anal. 97, 349–385 (2013)

    Article  MathSciNet  Google Scholar 

  • Segerstedt, B.: On ordinary ridge regression in generalized linear models. Commun. Stat.-Theory Methods 21, 2227–2246 (1992)

    Article  MathSciNet  Google Scholar 

  • Sinha, D., Ibrahim, J.G., Chen, M.H.: A Bayesian justification of Cox’s partial likelihood. Biometrika 90, 629–641 (2003)

    Article  MathSciNet  Google Scholar 

  • Spiegelhalter, D.J., Best, N.G., Carlin, B.P., van der Linde, A.: Bayesian measures of model complexity and fit. J. R. Stat. Soc. B 64, 583–640 (2002)

    Article  MathSciNet  Google Scholar 

  • Stan Development Team.: Stan Modeling Language Users Guide and Reference Manual, https://mc-stan.org (2017)

  • Stan Development Team.: RStan: The R interface to Stan. R package version 2.17.3: http://mc-stan.org (2018)

  • Stewart, G.W.: Collinearity and least squares regression. Stat. Sci. 2, 68–100 (1987)

    MathSciNet  Google Scholar 

  • Stöber, J., Joe, H., Czado, C.: Simplified pair copula constructions—limitations and extensions. J. Multivar. Anal. 119, 101–118 (2013)

    Article  MathSciNet  Google Scholar 

  • Van Wieringen, W.N.: Lecture Notes on Ridge Regression. Preprint. https://arxiv.org/pdf/1509.09169 (2020)

  • Van Wieringen, W.N., Kun, D., Hampel, R., Boulesteix, A.L.: Survival prediction using gene expression data: a review and comparison. Comput. Stat. Data Anal. 53, 1590–1603 (2009)

    Article  MathSciNet  Google Scholar 

  • Veerman, J.R., Leday, G.G., van de Wiel, M.A.: Estimation of variance components, heritability and the ridge penalty in high-dimensional generalized linear models. Commun. Stat. Simul. Comput. 51(1), 116–134 (2022)

    Article  MathSciNet  Google Scholar 

  • Verweij, P.J.M., van Houwelingen, H.C.: Penalized likelihood in Cox regression. Stat. Med. 13, 2427–2436 (1994)

    Article  Google Scholar 

  • Xue, X., Kim, M.Y., Shore, R.E.: Cox regression analysis in presence of collinearity: an application to assessment of health risks associated with occupational radiation exposure. Lifetime Data Anal. 13, 333–350 (2007)

    Article  MathSciNet  Google Scholar 

  • Yang, S.P., Emura, T.: A Bayesian approach with generalized ridge estimation for high-dimensional regression and testing. Commun. Stat.-Simul. Comput. 46, 6083–6105 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank Editor, Associate Editor, and two referees for their valuable suggestions that improved the paper. This work was supported by JSPS KAKENHI Grant Number JP21K12127.

Author information

Authors and Affiliations

  1. Department of Clinical Medicine (Biostatistics), School of Pharmacy, Kitasato University, Tokyo, 108-8641, Japan

    Hirofumi Michimae

  2. Biostatistics Center, Kurume University, 67 Asahi-Machi, Kurume, Japan

    Takeshi Emura

  3. Research Center for Medical and Health Data Science, The Institute of Statistical Mathematics, Tokyo, Japan

    Takeshi Emura

Authors
  1. Hirofumi Michimae
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takeshi Emura
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hirofumi Michimae.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The original online version of this article was revised: In Table 3 of this article, the first two values in the third column headed "n" were incorrect. The first value 200 should be 20, the second value 1.00 should be 200. Equation 12 has been corrected.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file1 (PDF 210 KB)

Supplementary file2 (ZIP 23 KB)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Michimae, H., Emura, T. Bayesian ridge regression for survival data based on a vine copula-based prior. AStA Adv Stat Anal 107, 755–784 (2023). https://doi.org/10.1007/s10182-022-00466-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10182-022-00466-4

Keywords

Search

Navigation